Oil, Equities, and the Zero Lower Bound

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Oil, Equities, and the Zero Lower Bound Deepa Datta, Benjamin K. Johannsen, Hannah Kwon, and Robert J. Vigfusson 2018-058 Please cite this paper as: Datta, Deepa, Benjamin K. Johannsen, Hannah Kwon, and Robert J. Vigfusson (2018). “Oil, Equities, and the Zero Lower Bound,” Finance and Economics Discussion Series 2018-058. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2018.058. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Oil, Equities, and the Zero Lower Bound Deepa Datta, Benjamin K. Johannsen, Hannah Kwon, and Robert J. Vigfusson∗ August 16, 2018 Abstract From late 2008 to 2017, oil and equity returns were more positively correlated than in other periods. Inaddition,weshowthatbothoilandequityreturnsbecamemoreresponsivetomacroeconomic news. We provide empirical evidence and theoretical justification that these changes resultedfromnominalinterestratesbeingconstrainedbythezerolowerbound(ZLB).Although theZLBalterstheeconomicenvironmentintheory,supportiveempiricalevidencehasbeenlacking. Our paper provides clear evidence of the ZLB altering the economic environment, with implicationsfortheeffectivenessoffiscalandmonetarypolicy. JELClassifications: F31,F41,E30,E01,C81 ∗The views in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System. We thank Martin Bodenstein, Craig Burnside, Franc¸ois Gourio, Luca Guerrieri, Lee Smith, Johannes Wieland,andJingCynthiaWu,forhelpfulcommentsanddiscussion. WethankAnastaciaDialynasforhercontributions totheinitialempiricalinvestigations. Commentsandsuggestionscanbedirectedtorobert.j.vigfusson@frb.gov. 1

1 Introduction Wedocumentthatthebehaviorofoilandequityreturnschangeddramaticallyfromlate2008to2017. Duringthisperiod,oilandequityreturnsbecamehighlycorrelated. Atothertimes,theyaretypically uncorrelated. Also in contrast to historical experience, from 2008 to 2017, oil and equity returns became responsive to macroeconomic news surprises such as unanticipated changes in nonfarm payrolls. We provide both empirical evidence and theoretical justification that these changes resulted fromnominalinterestratesbeingconstrainedbythezerolowerbound(ZLB).Althoughalargetheoretical literature has argued that the ZLB alters the economic environment, empirical support for this proposition has been lacking, especially for measures of economic activity. As such, our paper’s major contribution is to provide strong evidence of the ZLB altering the economic environment, which has implications for the effectiveness of both fiscal and monetary policy and for the desirability of a fastexitfromZLBepisodes. As can be seen in Panel (a) of Figure 1, the correlation between oil and equity returns increased sharplyin2008. Between1983and2008,thecorrelationfluctuatedaroundzero,onlyturningsharply negative in response to events such as the Gulf War in 1990. The correlation rose drastically in late 2008,reachingashighas0.65in2010andthenaveragingaround0.50throughlate2013. Thereafter, the correlation moved lower. We provide evidence that this correlation is broad based with equity returnsforadisparategroupofsectorsallshowinganincreasedcorrelationwithoil. Given that this observed increase in correlation coincides with the onset of the ZLB period in the U.S.economy,onemightwonderwhethertheZLBcausesthisincreasedcorrelation. Weprovideboth theoretical and empirical evidence in favor of this causal relationship. We present a formal analysis withaNewKeynesianmodelthatisaugmentedtoincludeoil.1 UsingourNewKeynesianmodel,we showthatoilandequityreturnsbecomemorecorrelatedattheZLB.Themechanismforthisincreased correlation arises from the monetary authority being constrained at the ZLB. When the ZLB binds, the nominal interest rate does not respond to changes in inflation. By contrast, away from the ZLB, changes in inflation lead to more than a one-for-one change in the nominal rate. Consequently, a 1OurmodelissimilarinstructuretoBodensteinetal.(2013),althoughtheydonotconsiderequityprices. 2

shock that causes oil prices and inflation to rise will increase real interest rates when the ZLB does not bind, but decrease real interest rates when it does bind. The differing dynamics of real interest ratesinducedbytheZLBchangethedynamicsofequitypricesand,asaresult,thecontemporaneous comovementofoilandequityreturns. Tocomplementourone-countrymodel,weconsideraninternationalextensionwithtwocountries. MotivatedbyJapan’sexperienceattheZLBsincethemid-1990sandMexico’sexperienceawayfrom theZLBsince2008,weconstructthemodelsothatthereisalargecountry(whichwethinkofasthe United States) and a small country. We argue that the predictions of the two-country version of the modelareroughlyconsistentwithdatafromtheUnitedStates,Japan,andMexico. Building on our model’s theoretical implications, we provide further empirical evidence of the role of the ZLB by studying how oil and equity returns respond to identified shocks. In particular, we report how much oil and equity returns change on the day of a surprise in U.S. macroeconomic announcements. We identify our shocks as the difference between actual economic announcements (such as nonfarm payrolls) and the average forecast from a week earlier. We show that, in contrast to historicalexperience,oilandequityreturnsbecameandremainedresponsivetomacroeconomicnews surprises,suchasunanticipatedchangesinnonfarmpayrolls,forseveralyears. Theseresultsbuildon and extend the existing literature. For example, consistent with our finding that the response changes with the onset of the ZLB, Kilian and Vega (2011) report that oil prices do not have statistically significantresponsestomacroeconomicnewssurprisesovertheperiodfrom1983to2008. Likewise, using data from 1957 to 2000, Boyd et al. (2005) claim that equities responded positively to bad news in expansions and negatively to bad news in recessions. Our results differ in that the increased responsivenessofequityreturnspost2008hasoutlastedtherecessionandinsteadseemstoberelated tothelowlevelofnominalinterestrates. Although we provide both a consistent theoretical model and supportive empirical evidence that theZLBcausedthischangingrelationshipbetweenoilandequityreturns,alternativeexplanationsare conceivable. Forexample,theincreasedfinancializationofcommoditiesorgreateruncertaintyrelated 3

to the financial crisis could potentially explain this increased correlation.2 As such, we statistically test the relative merits of explanations based on measures of the ZLB (either a Taylor-rule-implied interest rate or the shadow rate of Wu and Xia (2016)) and explanations based on other variables, including open interest in oil futures contracts, the VIX, and the uncertainty indexes of Jurado et al. (2015) and Baker et al. (2015). Overall, we find that the variation in sensitivity to macroeconomic newssurprisesisbestexplainedbymeasuresofmonetarypolicybeingconstrainedbytheZLB. Finally, we provide a structural vector autoregression (VAR) analysis of the same correlation, albeit at a monthly frequency. In our structural VAR work, the reason for the change in correlation wasnotbecauseaggregatedemandorsupplyshocksbecamemoreimportant. Instead,consistentwith ourZLB-drivenhypothesis,shocksthathaveanimmediateimpactonoilreturnsbutnotonaggregate demandoroilsupplywentfromcausinganegativecorrelationtoapositivecorrelation. To summarize, we present multiple pieces of empirical evidence that are consistent with the ZLB changing the correlation between oil and equity returns. These results should help focus the debate overwhicheconomicmodelsaremostappropriateforstudyingrecentconditions. 1.1 Relationship to literature TheincreasedcorrelationbetweenoilandequityreturnshasbeendiscussedbyLombardiandRavazzolo (2016) and Serletis and Xu (2016). Lombardi and Ravazzolo (2016) are concerned with the implications of time-varying correlation for portfolio allocation. Serletis and Xu (2016) include a time dummyvariableintheiranalysisstartinginlate2008,whichtheyassociatewiththeZLB.Relativeto these earlier studies, our paper provides three contributions. First, we offer a theoretical explanation fortheempiricalchangecorrelationinaDSGEmodel. Second,weempiricallytestpredictionsofthe model beyond the increased correlation of oil and equity prices at the ZLB. Third, we test alternative hypotheses why the correlation between oil and equity returns may have increased (e.g., increased 2Theliteratureonfinancializationofcommoditiesislargeandnotwhollyinagreement. Forexample,TangandXiong (2012) argue that financialization plays an important role in price movements. In contrast, Irwin and Sanders (2012), Fattouhetal.(2013),andHamiltonandWu(2015)findamorelimitedroleforfinancialization. 4

financialization or an increased prevelance of demand shocks) against the ZLB driving the change in correlation. Our empirical evidence is supportive of a large literature of models in which economic outcomes are different under the ZLB. A non-exhaustive list of representative papers includes the following. Eggertsson (2011), Christiano et al. (2011), Woodford (2011), and Erceg and Linde (2014) show that, in their models, fiscal multipliers were much larger under the ZLB. Likewise, Eggertsson et al. (2014) present a theoretical model, in which structural reform, which is normally expansionary, is contractionary when monetary policy is constrained. In the model of Caballero et al. (2015), the role ofcapitalflowschangesundertheZLB. Relative tothe theoreticalliterature on theZLB, theempirical literature testingfor ZLBeffects is lessextensive. DuporandLi(2015)presentsomeempiricalevidence,includingwhetherprofessional forecasters revised their inflation expectations commensurate with their output forecast revisions in response to government stimulus measures. Plante et al. (2016) study the relationship between uncertainty and GDP growth at the ZLB and find that they have been more negatively correlated during theZLBperiod,aspredictedbytheNewKeynesianmodel. Wieland(Forthcoming)exploreswhether reductions in oil supply are contractionary at the ZLB and fails to find strong evidence. Garin et al. (Forthcoming) study how the economy responds to TFP shocks. They, too, argue that their empirical results are at odds with the New Keynesian model at the ZLB. Although these papers would seem to cast doubt on the ZLB mechanism, they are limited by the small number of observations under which the ZLB is binding because they use monthly or quarterly data. Our paper complements these previousstudiesbyusinghigherfrequencydatabasedondailypricechanges. Inadditiontoproviding more observations, high frequency data offers the additional benefits that the timing assumptions are more plausible and that the shocks are more likely to be unanticipated than shocks that are identified at the monthly or quarterly frequency, as discussed in Ramey (2016) and Nakamura and Steinsson (2018). Swanson and Williams (2014) use high-frequency data to study the ZLB. They show that longerterm interest rates become less responsive to macroeconomic news surprises after 2008, which they 5

attributetotheZLB.Relativetothatpaper,acontributionofourworkisshowingthattheZLBaffects notonlyinterestratesbymakingthemlessresponsivetosurprises,butalsootherassetprices,including oil and equities, by making them more responsive. One important methodological contribution of our paper relative to Swanson and Williams is that, beyond reporting results for time-varying responsivenessaswasdonebySwansonandWilliams,weestimateandtestdirectlythehypothesisthat theresponsivenessvarieswithmonetarypolicyconditions,asmeasuredbyaninterestrateimpliedby a modified Taylor rule. Furthermore, we also test alternative hypotheses that attribute the change in responsivenesseithertothefinancializationofoilmarketsortoincreaseduncertaintyinthecrisisera, andshowthattheevidenceinfavoroftheZLBisstronger. 2 The increased correlation between oil and equities Thecorrelationbetweendailyoilandequityreturnsincreasedmarkedlyinlate2008(Panel(a)ofFigure 1). Our measure for the price of oil is the closing value, in dollars per barrel, of the front-month futures contract for West Texas Intermediate (WTI) crude oil for delivery in Cushing, Oklahoma obtained from NYMEX.3 For equities, we use the Fama–French value-weighted portfolio of all NYSE, AMEX,andNASDAQstocks.4 Table1presentssummarystatisticsforthesemeasuresoveroursampleperiod,whichcoversApril6,1983,throughDecember31,2017. Tocalculatereturns,wedropdayswithmissingvaluesforanyofourprimaryvariablesofinterest: WTIfuturesprices,metalsprices,interestrates,andtheleveloftheequitypriceindeximpliedbythe Fama–Frenchequityreturns(whichincludedividends). Then,wecalculate“daily”returnsasthe100 times the log difference of these consecutive closing prices, thereby ensuring that the daily returns 3The series reports the official daily closing prices from the New York Mercantile Exchange, posted daily at ftp: //ftp.cmegroup.com/pub/settle/nymex_future.csv. In contrast, Kilian and Vega (2011) use the daily spot price for WTI crude oil for delivery (freight on board) in Cushing, Oklahoma, as reported by the U.S. Energy Information Administration (EIA). Analyses using the EIA series, or the physical spot price for Brent Forties Oseberg crude oil, obtained from Bloomberg, generate similar results—Bloomberg Finance LP, Bloomberg Terminals (Open, Anywhere,andDisasterRecoveryLicenses). Ofthese,weprefertheWTInearbyfuturesprice,asitsmoreprecisetiming allows us to better relate it to the macroeconomic announcements. In supplementary analysis, we also use the WTI far futuresprice,whichwedefineasthepriceofthefurthestavailableDecembercontract. 4Fama–French data downloaded from http://mba.tuck.dartmouth.edu/pages/faculty/ken. french/data_library.html. 6

are calculated over the same period for each variable. Panel (a) of Figure 1 depicts the correlation of thesedailyreturnsforoilandequitiesusingarollingsampleofoneyear.5 Next, we use a Chow test for the simple regression of oil returns on equity returns to determine whether there is a break in the oil–equity relationship and find a statistically significant break date of September 22, 2008. Table 2 reports the estimated equity beta for three sample periods: the full sample, pre-break, and post-break. As shown in Table 2, the coefficient is slightly negative for the pre-break sample, but is large, positive, and significant for the post-break sample. The coefficient of 0.79 in the post-break sample implies that during this period, a daily return of 1 percent on the equityindexisassociatedwithanoilreturnofabout0.79percent. Wefindsimilarresultswhenusing alternative measures of oil prices, including the physical spot prices for WTI and Brent crude oil. Consistent with the lower variation in far futures prices as compared to nearby futures (reported in Table 1), we find that the results when using the WTI far futures series are qualitatively similar but quantitativelysmaller. To demonstrate that the break in the relationship extends beyond the oil market, we also use the metals spot index constructed by the Commodities Research Bureau. Applying the Chow test to the regression of metals on equity returns also implies a statistically significant break date of September 30,2008. Aswithoil,Table2showsthattheslopecoefficientonequityreturnsisessentiallyzerofor thepre-breaksample,butismuchlargerandstatisticallysignificantforthepost-breaksample. Using the standard Andrews supremum-Wald critical value based upon 15 percent trimming of the sample asinStockandWatson(2003),allofthesebreakdateswerefoundtobestatisticallysignificantatthe 1-percentlevel. Finally, to ensure that the increased correlation between oil and equity returns is not being driven byfluctuationsintheenergycomponentoftheequitymarket,weseparatelyregressoiloneachofthe 12Fama–Frenchindustryportfolios,determinedbySICcodes,aswellasonreturnsfortheS&P500 Ex-Energy index obtained from Bloomberg (Ticker: SPXXEGP).6 The results of the related Chow tests are presented in Panel B of Table 2. In the pre-break sample, returns in all of the non-energy 5InAppendixB,weshowthatthissustainedincreaseisalsovisiblewhenusingwindowsizesrangingfromonemonth tothreeyears,andwhenusingreturnscalculatedatthedaily,weekly,monthly,andquarterlyfrequencies. 6PanelAofAppendixTableB.1presentssummarystatisticsfortheindustryportfolios. 7

related sectors are negatively associated with oil prices. Only the energy sector shows a positive, statisticallysignificantrelationshipbeforethebreakin2008. Incontrast,post-break,allofthesectors display a positive and statistically significant relationship similar to that of the energy sector. These resultsconfirmthatourfindingofanincreasedcorrelationbetweenoilandequityreturnsisnotbeing driven exclusively by equity prices for energy producers. Instead, the increased correlation between oilandequityreturnsisbroadbased. 2.1 The desired policy rate and the ZLB Given that the relationship between oil and equity returns seems to be dependent on economic conditions, we now formally estimate how this relationship varies with the proximity of the stance of desired monetary policy to the ZLB. Because the federal funds rate may be censored or constrained neartheZLB,divergingfromthedesiredstanceofmonetarypolicy,weconstructanalternativemeasureofthestanceofmonetarypolicy. First,wedefinethenotionalrateasthepredictionforthefederal fundsrateusingthemodifiedTaylorruleasinBernanke(2015). AsseeninPanel(b)ofFigure1,the notional rate closely tracks the observed federal funds rate until the ZLB era. Next, we construct our desired policy rate as being equal to the observed federal funds rate when the notional rate is above zero, and being equal to the notional rate when the notional rate is below zero (and the federal funds ratemaybecensored). Thenotionalrateisdefinedas ˜ NR = π +y +0.5(π −2)+2, (1) t t t t where π is inflation and y is the output gap. To measure inflation, π , we use the deflator for core t t t personal consumption expenditures, which excludes food and energy prices. For the output gap, y , t we use published estimates prepared by Federal Reserve staff for FOMC meetings through 2009 and then use estimates produced and published by the Congressional Budget Office through 2017.7 We 7Published estiates prepared by Federal Reserve staff for FOMC meetings through 2009 can be obtained from Bernanke (2015) at https://www.brookings.edu/wp-content/uploads/2015/04/ Taylor-Rule-Data.xlsx or from the Philadelphai Fed’s Real-Time Data Research Center at https: //www.philadelphiafed.org/research-and-data/real-time-center/greenbook-data/ gap-and-financial-data-set. After 2009, potential output is measured using the CBO’s estimate of potential output, available on FRED and https://alfred.stlouisfed.org/series/downloaddata? seid=GDPPOT, and output is measured using the Philadelphia Fed’s real-time GDP series, available at https: 8

˜ use real-time data when available. Panel (b) of Figure 1 depicts the desired monetary policy rate, R , t showinghowitisacombinationoftheobservedfederalfundsrateandthenotionalrate. Weestimatetheequitybetaforoilasafunctionofthedesiredpolicyrateusingthemodel ˜ ˜ Oil = α(R )+β(R )Equity +ε . (2) t t t t t Theestimatesofα andβ solvethekernelregressionproblem (cid:32) (cid:33) (cid:110) (cid:111) (cid:88) R ˜ −R ˜ αˆ(R ˜ ),β ˆ (R ˜ ) = argmin φ t k (Oil −α−βEquity )2. (3) k k t t α,β h t ˜ ˜ ˆ ˜ For each R , αˆ(R ) and β(R ) minimize the weighted regression of oil returns on equity returns, k k k ˜ ˜ using all available observations. These weights are based on the difference R −R , normalized by t k a constant, h, evaluated using φ, the standard normal density function. For this regression, we set ˜ h equal to one, and results are robust to other values. For each R , the weights on the observations k ˜ ˜ declineasthedistancebetweenR andR increases. Theintuitionforthissetupisthateachestimated t k ˆ ˜ ˜ ˜ β(R )placesmoreweightontheobserved(Oil ,Equity )whenR isclosetoR . k t t t k ˜ Figure 2 plots our estimate of β(R ) and provides further evidence that oil and equities have k ˜ stronger co-movement (i.e., β(R ) is larger) when interest rates are low, and in particular, when the k notional rate is negative. That is, when the Taylor rule would imply nominal interest rates that are lowerthantheZLB,wefindthatthecorrelationbetweenoilandequityreturnsishigh. Having established an empirical linkage between oil and equity returns that seems to depend on the ZLB, the next section provides a theoretical background for our work and motivates additional empiricalexercises. 3 A DSGE model with oil To study the theoretical effect of the ZLB on the relationship between oil and equity returns, we use a medium-scale, New Keynesian model augmented with oil, similar to the model in Bodenstein et al. (2013). Asintheprevioussection,wefindthattheZLBdramaticallychangesthebehaviorofoiland //www.philadelphiafed.org/research-and-data/real-time-center/real-time-data/ data-files/routput. 9

equity returns. Most of our analysis is conducted in a one-country model. In Section 3.7, we extend theanalysistoamulticountrysetting. 3.1 Households Therepresentativehouseholdhasanexpectedutilityfunctiongivenby (cid:34) (cid:35) (cid:88) ∞ (C −hC ¯ )1−σ L1+ϕ (cid:18) B (cid:19) E βj t+j t+j−1 −χ t+j +η V t+j . (4) t t+j 1−σ 1+ϕ P C,t+j j=0 ¯ Here, 0 < β < 1, C denotes consumption, C is average aggregate consumption, 0 ≤ h < 1 controls t t consumption habit, L denotes hours worked, P is the price of consumption, and B /P are real t C,t t C,t bond holdings. We include real bond holdings in the utility function as in Fisher (2015) to capture changes in the spread between risky and risk-free assets. We couple the bonds in the utility function with the preference shifter, η , to allow the spread to change over time. In our model, η plays an t t analogous role to the spread shock in Smets and Wouters (2007).8 Consumption is an aggregate of non-oilgoods,Y ,andoil,O ,where C,t C,t C = (cid:18) ω1−ρCYρC +(1−ω )1−ρC (cid:18) O C,t (cid:19)ρC (cid:19) ρ 1 C . (5) t C C,t C µ C,t AsinBodensteinetal.(2013),µ isaprocessthataffectspreferencesforoilconsumption. C,t Thehouseholdfacesaper-periodbudgetconstraintgivenby 1 B t +P C,t C t +P Y,t I t = (1+R t−1 )4 B t−1 +R K,t K t +W t L t +T t , (6) whereP isthepriceofnon-oiloutput,I isinvestment,K arecapitalholdings,R isthenominal Y,t t t K,t rental rate of capital, W is the nominal wage rate, R is the annualized net nominal interest rate, and t t T arelump-sumprofits,taxes,andtransfers. Capitalevolvesaccordingto t (cid:32) (cid:33) φ (cid:18) I (cid:19)2 K t K = (1−δ)K +I 1− −1 . (7) t+1 t t 2 I t−1 8We normalize real bonds to be in zero net supply and assume that V is increasing, concave, and has some positive andnegativesupport. 10

The parameter φ controls adjustment costs to changes in investment, as in Christiano et al. (2005). K Toconnectourmodeltothedata,itisusefultonotethatthepriceofcapitalisgivenby Λ P t+1 C,t P = β [P (1−δ)+R ]. (8) K,t K,t+1 K,t+1 Λ P t C,t+1 The variable Λ is the marginal utility of consumption in period t. The ex-post nominal returns on t capitalandoil(inlogs)aregivenby (cid:18) (cid:19) P (1−δ)+R K,t K,t Equity = log andEquity = log(P /P ), (9) t t O,t O,t−1 P K,t−1 whereP isthenominalpriceofaunitofoil. O,t 3.2 Firms Perfectlycompetitivefirmsproducefinaloutput,Y ,usingintermediateinputs,X (i). Theproduction t t technologyanddemandcurves(derivedfromperfectcompetition)aregivenby (cid:18)(cid:90) 1 (cid:19) ν− ν 1 (cid:18) P (i) (cid:19)−ν ν−1 X,t Y t = X t (i) ν di andX t (i) = Y t , (10) P 0 Y,t whereν > 1andP (i)isthepriceofX (i). Aunitmeasureofmonopolistsproducewith X,t t X (i) = (cid:18) ω1−ρX(V (i))ρX +(1−ω )1−ρX (cid:18) O X,t (i) (cid:19)ρX (cid:19) ρ 1 X . (11) t X t X µ X,t The variable O (i) is oil used in production and, as in Bodenstein et al. (2013), µ is an ex- X,t X,t ogenous and stochastic process that shifts the usefulness of oil in production. We assume V (i) = t A K (i)αL (i)1−α, where A is the economy-wide level of technology. Monopolists take demand t t t t curvesasgivenandmaximizeexpecteddiscountedprofits (cid:88) ∞ Λ (cid:20) (1+τ )P (i) (cid:21)(cid:18) P (i) (cid:19)−ν E βj t+j X X,t+j −MC X,t+j Y . (12) t t+j t+j Λ P P t C,t+j Y,t+j j=0 Here,MC isrealmarginalcostandτ isasubsidytooffsetsteady-statedistortionsduetomonopoly t X power. FirmsaresubjecttoCalvopricingfrictions,asinChristianoetal.(2005). Ineachperiod,firm i has probability 1−ξ that it can update its price optimally. Otherwise, the firm updates its price by theinflationratefornon-oiloutputinthepreviousperiod. 11

3.3 Oil market In each period oil supply, O , is exogenously determined. Households purchase oil using non-oil t output. Our assumption is akin to assuming that oil must be purchased from abroad using non-oil output.9 Marketclearingintheoilmarketandtheresourceconstraintimply (cid:90) 1 P O,t O = O (i)di+O andY = Y +G +I + O , (13) t X,t C,t t C,t t t t P 0 Y,t whereG aregovernmentpurchases. t 3.4 Government policy ThefiscalauthoritypurchasesG . Lumpsumtaxesaresettosatisfythegovernmentbudgetconstraint, t (cid:110) (cid:111) ˜ period-by-period,withB = 0. ThemonetaryauthoritysetsR = max 0,R ,where t t t (cid:16) 1+R ˜ t (cid:17) 4 1 = (cid:18) (cid:104) 1+R ˜ t−1 (cid:105)1 4 (cid:21)γ (cid:32) [1+R] 1 4 (cid:16)π π Y ∗ ,t (cid:17)θπ (cid:18) Y Y t ∗ (cid:19)θY (cid:33)1−γ (14) t ˜ The variable R is the notional interest rate that the monetary authority would set if it were not cont strainedbytheZLB.10 IntheTaylor-typeruleforR ˜ ,Risthesteady-stateannualizednetnominalint terestrate,γ controlstheamountofinterestratesmoothingintheTaylor-typerule,π ≡ P /P Y,t Y,t Y,t−1 denotes the inflation rate of prices for non-oil output, π∗ is the monetary authority’s target rate of inflation,andθ > 1tosatisfytheTaylorprinciple. π We specified the monetary policy rule in terms of π so that movements in oil prices would not Y,t feed through as quickly to movements in the notional interest rate.11 The Taylor rule also includes the deviation of output from its potential level, Y∗, defined as the level of output that would prevail t with no nominal rigidities. We include an interest rate smoothing term to make the rule similar to those considered in the DSGE literature. Removing the interest rate smoothing term would make the effects of the ZLB even more dramatic in two ways. First, the nominal interest rate would respond evenmoretochangesininflationandtheoutputgapduringnormaltimesascomparedtoattheZLB. 9Ourmainresultsarelittlechangedifweassumethattheoilsupplyisownedbythehousehold. 10In the model, there is no distinction between the notional rate and the desired policy rate. In Section 2.1, the two differbecauseweusetheobservedfederalfundsratewhentheTaylor-ruleimpliednotionalrateisabovezero. 11Ourmainresultsaresimilarifthemonetaryauthorityrespondstoconsumption-goodspriceinflation. 12

˜ Second, with interest rate smoothing, negative values of R imply low future interest rates. Without t ˜ interestratesmoothing,negativelevelsofR wouldhavenoeffectonfutureinterestrates. t In our simulations, we will also consider an alternative version of our model in which there is no ˜ ZLBconstraintsothat,forallt,R = R . WeanalyzetheeffectsoftheZLBbycomparingtheresults t t fromourbenchmarkmodelwiththisalternativeversion. 3.5 Calibration and solution strategy For the parameters that are specific to the oil market, we draw on the DSGE literature that has incorporated oil supply. Following Bodenstein et al. (2013), we set ρ = ρ = −1.5 so that the elasticity C X of substitution for oil is 0.4. We set ω = 0.03 and ω = 0.027. In steady state, these parameters C X implythatoilusedinproductionisabout1.8timesoilusedforfinalconsumptionandthattheoverall oilshareoftheeconomyisalittleover4percent,consistentwithevidenceinBodensteinetal.(2013). For the parameters of our model not related to oil, we use parameter values commonly found in the DSGE literature. We set the parameter governing consumption habit, h, to 0.7, in line with Boldrin et al. (2001). We set δ = 0.025, as in Christiano et al. (2005), who draw on Christiano and Eichenbaum (1992). The parameter α is set to 0.33 so that the steady-state labor share of payments to labor and capital is roughly 0.67. We set φ = 3, in line with Bodenstein et al. (2013). The K value 1−ξ governs how often firms can update their prices optimally. We set ξ = 0.75. This value is slightly higher than the value implied by evidence in Nakamura and Steinsson (2008) but slightly lower than the value implied by estimates in Gust et al. (2017). As in Christiano et al. (2005), we set σ = 1, ϕ = 1, and we normalize steady-state labor supply to be 1. We set β = 0.9975 to imply a steady-state risk-free real interest rate of 1 percent. The parameter ν governs substitution between different monopolists’ output. We set ν = 7, which is within the range of values considered in Altig etal.(2011),andimpliessteady-statemarkupsof15percent. Forgovernmentpolicy,wecalibratesteady-stategovernmentpurchasestobe20percentofsteadystate output. We set the monetary authority’s inflation objective to 2 percent annual inflation. We set θ = 1.5tosatisfytheTaylorprinciple,θ = 0.25,andγ = 0.75. π Y 13

Our model has six exogenous processes, G , A , µ , µ , η , and O . We assume that each t t X,t C,t t t of these processes is an AR(1) in log deviations from steady state, except for η which is specified t as an AR(1) in levels because it has a zero steady state. As in Bodenstein et al. (2013), we set µ = µ andrefertothisshockasanoildemandshock. Wecalibrateµ tohavepersistence0.95 C,t X,t X,t and shock volatility to 0.01. This calibration yields similar unconditional autocorrelation properties for oil demand as those in Bodenstein et al. (2011). Similar to Bodenstein et al. (2011), we specify A to have persistence 0.89 and shock volatility 0.015 and O to have persistence 0.99 and shock t t volatility0.018. WecalibrateG tohavepersistence0.85andshockvolatility0.01. Werefertoη asa t t spreadshock,andspecifyittobeanAR(1)processinlevelswithpersistence0.95andshockvolatility 0.0005. We normalize V(cid:48)(0) to be equal to Λ in steady state so that a shock to η plays the same role t as the spread shock in Smets and Wouters (2007). Gust et al. (2017) consider a similar shock to η , t and use persistence 0.85. However, they report that the data seem to prefer a more persistent process for η , so we use 0.95. We set the shock volatility for η so that the ZLB binds roughly 10 percent of t t thetime. WesolvethemodelusingthemethodologyofGuerrieriandIacoviello(2015). Theirsolutionstrategyinvolvesafirst-orderperturbationtothemodel,whichisappliedpiecewisesoastoaccommodate the ZLB. Guerrieri and Iacoviello (2015) show that their solution methodology performs well, even when compared to fully non-linear numerical solutions. The main advantage of using the methodology of Guerrieri and Iacoviello (2015) is that it is able to accommodate the number of state variables impliedbymedium-scaleDSGEmodels. 3.6 The effects of the ZLB on oil and equity returns We simulate one million periods from our model to generate data that we can use to analyze how the correlation between oil and equity returns change as the notional interest rate changes. We report localcorrelations,constructedusingthelocalmean,variance,andcovarianceofoilandequityreturns, whicharecomputedusingmethodologyanalogoustoEquation3. Wesimulatetwodifferentversions 14

ofourmodel. Inoneversion,weincludetheZLBconstraint. Intheother,thereisnoZLBconstraint, ˜ andR equalsR forallt. t t Figure 3 shows the model-implied value of the local correlation between oil and equity returns ˜ as a function of R . This correlation can be related to the rolling correlations between oil and equity t returns, as shown in Panel (a) of Figure 1. When the interest rate is above the ZLB, the correlation is negative. As reported by the black line, for the model that includes the ZLB constraint, when the interest rate is at the ZLB (and the notional interest rate is less than the ZLB) the correlation is positive. Figure3alsoshowsthecorrelationbetweenoilandequityreturnsforthemodelwithoutthe ZLB constraint. In this version, as when the interest rate is above zero, the correlation between oil ˜ andequityreturnsisnegativeforanyvalueofR . t To understand why the correlation between oil and equity returns changes at the ZLB, we run a kernel regression similar to Equation 3, using either oil or equity returns as the dependent variable and the structural shocks in our model as explanatory variables. We scale our structural shocks by theirstandarddeviation. ˆ ˜ Figure4showsthecoefficientβ(R)estimatedusingdatafromourmodelwhenequityreturnsare used as the dependent variable. Each panel of the figure shows the coefficient estimates for the two differentdatasets,generatedbyourtwomodelversions(withandwithouttheZLBconstraint). Panel (a) shows that, with the ZLB constraint, positive technology shocks cause equity returns to rise away from the ZLB, but fall when the notional rate is negative. When interest rates are positive, the rental rate of capital rises with the improvement in its marginal product. At the ZLB, the positive technologyshockcausesinflationtofallbecausemarginalcostfalls. Thenominalinterestratecannot fall in response to the decline in inflation, causing real interest rates to rise. The rise in real interest rates offsets the direct effects of the technology shock and causes equity returns to fall. Panel (b) showsthatequityreturnsalwaysfallinresponsetopositivespreadshocks. Equityreturnsfallbecause households have an increased desire to hold bonds rather than capital. Away from the ZLB, real interestratesfallinresponsetoapositivespreadshockbecauseoutputandinflationfall. AttheZLB, 15

the nominal interest rate cannot fall in the same way, causing real interest rates to rise and equity returnstorespondmorestrongly. Panel (c) shows that government spending shocks have little effect on equity returns away from the ZLB. The increase in demand due to increased government consumption is offset by an increase in real interest rates. At the ZLB, real interest rates do not rise in the same way, and equity returns rise. As showninPanel(d),wefindthat theresponseofequityreturnstooil demand shockschanges signattheZLB.AwayfromtheZLB,oildemandshocksincreasethepriceofoilusedforproduction, causing marginal cost, inflation, and real interest rates to rise. The rise in real interest rates leads to thelowequityreturn. AttheZLB,therealinterestratefallsinresponsetotheriseininflation,causing equityprices,andthusequityreturns,torise. Panel (e) of Figure 4 shows the response of equity prices to a positive oil supply shock. Away from the ZLB, oil supply shocks cause equity prices (and also returns) to rise, because an increase in oil supply reduces marginal cost, lowering inflation and the real interest rate. At the ZLB, equity pricesincreaselessbecausethenominalinterestratedoesnotrespond. Unlikefortechnologyshocks, thesignoftheresponseofequityreturnstooilsupplyshocksdoesnotchangeattheZLBbecause,in our model, oil supply shocks are assumed to have very persistent effects that are expected to outlast a particular ZLB episode. After the ZLB no longer binds, the persistent increase in oil supply causes inflation and real interest rates to be low, increasing consumption. These longer-run effects offset the effects of the low short-term real interest rates. If the oil supply process is modeled instead with a much lower persistence of 0.10 instead of 0.99, then the response of equity prices is negative at the ZLB(shownasthedashed-dottedanddottedlines).12 ˆ ˜ Figure 5 shows the coefficient β(R) estimated using data from our model when oil returns are used as the dependent variable. Each panel of the figure shows the coefficient estimates for the two different data sets, generated by our two model versions (with and without the ZLB constraint). Panel (a) shows that positive technology shocks always cause oil prices to fall. The reason is that less oil is needed in production, so oil demand declines. At the ZLB, the oil price decline is larger 12Oil supply could be modeled as in Leduc et al. (2016) as having very persistent and transitory components. For parsimony and consistency with Bodenstein et al. (2013), we elected to show separately the two cases of high and low persistence. Ourcalculatedlocalcorrelationsaresimilarifweusepersistence0.10fortheoilsupplyprocess. 16

because the decline in inflation that comes with a positive technology shock is not accompanied by a decline in the nominal interest rate, which causes demand to be relatively low. Panel (b) shows that positive spread shocks also cause oil prices to fall. The reason is that households would prefer to save in bonds than purchase oil consumption. As in the case of positive technology shock, at the ZLB the nominal interest rate does not fall, magnifying the effects of the shock. Panel (c) shows that apositivegovernmentspendingshockcausesoilpricestorisebecauseoutputdemandrisesandoilis used in production. At the ZLB, the nominal interest rate does not respond to the increase in output and inflation, and the effects are larger than away from the ZLB. Panels (d) and (e) show the effects of positive oil demand and supply shocks. Oil demand shocks cause oil prices to rise and oil supply shockscausepricestofall. ThereislittleeffectfromtheZLBbecause,inourmodel,monetarypolicy respondstoameasureofinflationthatdoesnotincludeoilconsumption. The change in correlation between oil and equity prices can now be understood by considering theeffectsofshocksonoilandequityreturnsjointly. TheZLBdoesnotchangethesignoftheeffect of each structural shock on oil prices, and in some cases magnifies the effects. The ZLB changes the sign of the response of equity returns to technology shocks and oil demand shocks. In both cases, oil and equity returns move in the same direction in response to a shock at the ZLB, whereas away from the ZLB they move in opposite directions. The effects of spread shocks and government spending shocks on equity prices are magnified at the ZLB and oil and equity returns move in the same at the ZLB. The effects of oil supply shocks are muted at the ZLB, while away from the ZLB, oil supply shocks move oil and equity returns in opposite directions. Overall, the ZLB causes greater positive co-movement between oil and equity returns, which increases the local correlation for low values of ˜ R . t Consistent with our empirical findings, our DSGE model shows that if monetary policy is constrained by the ZLB, then the correlation between oil and equity returns rises. Moreover, at the ZLB, thesignoftheresponseofequityreturnstocertainstructuralshockschanges,andtheeffectsofsome shocks are magnified. In Section 4, we show that these model predictions hold in the data. But first, wepresentsomeinternationalevidence. 17

3.7 International considerations Having shown that the ZLB is theoretically consistent with the observed increase in the correlation between oil and equity returns in the United States, we now turn to international considerations. We aremotivatedbyJapan’sexperienceatthelowerboundsincethe1990saswellasMexico’sexperience awayfromthelowerboundsince2008(seeFigure6). Our model is, in most respects, the two-country analogue of our one-country model. We assume that non-oil output is an aggregate of home and foreign goods. Additionally, we assume that consumerspreferthegoodfromthehomecountryrelativelymorethanthegoodfromtheforeigncountry (homebias).13 WeincorporatenominalrigiditiesusingCalvo-stylestickyprices,asinourone-country model, and assume that firms set prices in the currency in which the good is sold (so-called “localcurrency pricing”). We assume that households experience preference shocks for bonds, as in our one-country model. So as to accommodate this setup, we assume either that only risk-free nominal bonds are traded in international asset markets or that there is financial autarky (we present results for both cases). In our model, the world is composed of a large country (with size 0.9), which we think of as the United States, and a small country (with size 0.1). Similar to our one-country model, we assume oil supply is exogenous and owned by neither country so that changes in oil prices do not transferwealthacrosscountries. AdetaileddescriptionofthemodelisgiveninAppendixD. To isolate the effects of the ZLB in either the small or the large country, we only ever impose the ZLB in one country. In the other country, we assume that the nominal interest rate is unconstrained. Panel (a) of Figure 7 shows that in the large country, the ZLB changes the correlation between oil and equity returns in similar ways to our one-country model, regardless of our assumption about internationalassetmarkets. Intuitively,becausethelargecountrycomprisesmostoftheworld,adding a small country to the model has little effect. As shown in Panel (b) of Figure 7, in the large country, thecorrelationbetweenoilandequityreturnsisunchangediftheZLBisbindinginthesmallcountry. Intuitively,thesmallcountryhaslittleeffectonthelargecountryingeneral,sotheZLBbindinginthe small country also has little effect. Thus, our model predicts little change in the correlation between 13Theelasticityofsubstitutionbetweenhomeandforeigngoodsiscalibratedtobe1.5. 18

oil and equity returns in the large country when the small country is at the ZLB, which is consistent withourfindingthattheU.S.correlationchangedonlyafter2008. Figure 8 displays the correlation between oil and small country equity returns when the small country is constrained by the ZLB. In Panel (a), we see that the correlation between oil and small country equity returns changes in the small country’s currency, regardless of our assumptions about internationalfinancialmarkets,buttoasomewhatsmallerextentunderfinancialautarky. InPanel(b), weseethatinthelargecountry’scurrency,thecorrelationdoesnotchangeiftherearetradednominal bonds. The reason is that the ZLB is not binding in the large country, so nominal returns in the large country respond to shocks in similar ways, regardless of the ZLB in the small country. Under financial autarky, the correlation changes because the household in the large country is unable to purchasebondsinthesmallcountrytoarbitragenominalreturns. Figure 9 shows rolling correlations between oil and equity returns in Japan.14 The correlations are computed in yen in Panel (a) and dollars in Panel (b). We have daily data on exchange rates, oil prices, and equity prices, and these data do not neatly match up because of the time difference between the United States and Japan. We compute the rolling correlations for monthly returns under theassumptionthatthetimezonedifferenceswillhavelittleeffectatamonthlyhorizon.15 Consistent with our model, the correlation in yen rose during the period in which the Japan was at the ZLB. There is a somewhat smaller increased correlation in dollars during that period, which is consistent with our model with no financial integration. In both currencies, there is a decline in the correlation between oil and equity returns in Japan in the late 2000s. Consistent with our model, this is around thetimethattheBankofJapantemporarilyraisedthediscountrate(Figure6). Figure 10 displays the correlation between oil and small country equity returns when the large country is constrained by the ZLB. In the small country, when the large country is at the ZLB, the correlation changes in the large country’s currency, but not in the small country’s currency. In the smallcountry,becausetheZLBisnotbinding,thenominalinterestraterespondstoshocksinsimilar ways, regardless of the desired policy rate in the big country. As a result, the real rate channel that 14EquityreturnsforJapancomefromBloomberg—BloombergFinanceLP,BloombergTerminals(Open, Anywhere, andDisasterRecoveryLicenses). 15FigureB.2inAppendixBshowsthatthecorrelationslooksimilarforweeklyreturnsandvariouswindowlengths. 19

causes the change in correlation in the one-country model is not present, and the correlation in the smallcountry’scurrencyisunchangedwhenthebigcountryisattheZLB.However,theexchangerate is heavily influenced by shocks in the big country, and its properties change when the large country is at the ZLB. This change translates into an increase in the correlation between the small country’s equityreturnsandoilreturnsmeasuredinthebigcountry’scurrency. Figure11showsrollingcorrelationsbetweenoilandequityreturnsinMexico.16 Thecorrelations are computed in dollarsin Panel (a) and in pesos in Panel (b). As for Japan, we show here the rolling correlations for monthly returns.17 Consistent with our model, the correlation in dollars rose during the period in which the United States was at the ZLB. Although our model predicts little change in the correlation when measured in the small country’s currency, the observed correlation in pesos did increase during the first part of the ZLB period, though not by as much as correlation in dollars. One possible explanation for the increase in the oil and equity return correlation measured in pesos is that the discount rate in Mexico was held constant from 2009 to 2013 (Figure 6). Additionally, in our model the large country being at the ZLB does not constrain the small country’s monetary policy. In realitythisassumptionmaynothold. 4 Estimating the response to macroeconomic news Having presented theoretical results and also international evidence, we now turn our attention to further empirical evidence for the United States. Our empirical evidence relates to the theoretical predictionofourDSGEmodelthatattheZLB,thesignoftheresponseofequityreturnstostructural shockschangesandtheeffectsofsomeshocksaremagnified. 16Equity returns for Mexico come from Haver Analytics, Haver Analytics, http://www.haver.com/our_ data.html. 17FigureB.3inAppendixBshowsthedailyandweeklyreturns. 20

4.1 Macroeconomic news surprises To test the New Keynesian model developed in the previous section, we need to identify shocks. One challenge in the existing literature is that using quarterly data limits the number of observations. Another challenge is the ongoing debate about the plausibility of identifying assumptions. We avoid these issues by looking at the response at the daily frequency to macroeconomic news, which is definedasthedifferencebetweentheannouncedvalueofamacroeconomicstatisticanditspreviously expectedvaluefromasurvey. Itisimportanttonotethatnewsaboutmacroeconomicannouncements is not what macroeconomists would call a news shock. A Beaudry/Portier-style news shock, as in Barskyetal.(2014),isinformationaboutthefuturestateoftheworld. Incontrast,ourmacroeconomic newsannouncementsprovideinformationaboutthecurrent stateoftheworld. We measure macroeconomic news using the same approach that has been well established in the empirical literature such as Beechey and Wright (2009) and Kilian and Vega (2011). We use survey results from Action Economics as the expected U.S. macroeconomic fundamentals.18 Macroeconomics news is defined as the difference between the announced realization of the macroeconomic aggregatesandthesurveyexpectations. WefocusonthevariablesthatSwansonandWilliams(2014) use in their analysis of interest rate movements during the ZLB period: capacity utilization, consumer confidence, core CPI, GDP (advance), initial claims, ISM manufacturing, leading indicators, newhomesales,nonfarmpayrolls,corePPI,retailsalesexcludingautos,andtheunemploymentrate. FollowingSwansonandWilliams(2014),ourregressionsamplebeginsinJanuary1990,whenallbut two of the surprises are available. Our sample ends in December 2017, five years later than that of SwansonandWilliams(2014). Since the units of measurement differ across the news indicators, we follow the common practice inthisliteratureandnormalizethesurprisecomponentofeachnewsannouncementbyitsfullsample standard deviation. This normalization allows the responses to be comparable across all announce- 18ActionEconomics,LLC,ActionEconomicsWeeklySurvey,http://www.actioneconomics.com/index. php. 21

ments. Therefore,foreachindicatorj attimet,thesurprisecomponents is jt (A −E ) jt jt s = , (15) jt σ j where A denotes the announced value of indicator j and E refers to the market’s expectation of jt jt indicator j prior to the announcement. To calculate σ , which is the standard deviation of the surj prise component (A −E ), we use the entire sample period available for each surprise. Following jt jt Beechey and Wright (2009), we flip the sign for unemployment and initial jobless claims announcements, so that all positive surprises represent stronger-than-expected growth. Summary statistics for the surprise component of each announcement, (A −E ), can be found in Panel B of Appendix jt jt TableB.1. As discussed in Beechey and Wright (2009), the response of asset prices to news events occurs very rapidly, often completely adjusting within 15 minutes of the announcement. However, as was also noted in Beechey and Wright, although intra-daily regressions provide more efficient estimates of the reactions to news announcements, the daily estimators also are consistent. It would seem reasonable to expect a similar result for oil prices. In addition, by using daily data, our results are most comparable to those reported in Kilian and Vega (2011).19 Using high-frequency data, Rosa (2014) reports statistically significant results for the responses of oil prices to macroeconomic news over the 1999 to 2011 sample. However, he does not consider the role of time variation, which we emphasizehere,andwhichmayexplainthedifferencebetweentheresultsreportedinRosaandthose inKilianandVega. 4.2 Sensitivity during the ZLB period WenowtestwhetherthesensitivitytomacroeconomicnewssurpriseschangesduringtheZLBperiod, as would be predicted by our model. Oil and equity returns are calculated as in Section 2 as 100 times the log difference in daily prices, with an adjustment for dividend payments. For interest rates, consistent with Swanson and Williams (2014), our dependent variable is the daily change in basis 19StudiesusinghigherfrequencypricesincludeHalova(2012),whichlooksathowoilandnaturalgasrespondtonews aboutoilandnaturalgasinventories. 22

points for the market yield on U.S. Treasury securities at a constant maturity of 2 years. We also includemarketyieldsat1-yearand10-yearconstantmaturityforcomparison. Our estimation procedure is similar to those found in earlier papers, such as Kilian and Vega (2011). Weestimatetheeffectofnewssurprisesusingthemodel Y = α+βs +ε . (16) t t t In this model, s = {s ,...,s } and β = {β ,...,β }. Each s refers to the standardized macroet 1t 12t 1 12 jt conomic news surprise for announcement j on day t. Each β measures the response of the variable j Y to a one-standard-deviation surprise for the corresponding announcement s . This regression is j estimatedseparatelyforeachassetthatweareinterestedin,soY ∈ {Oil ,Equity ,InterestRate }. t t t t By estimating the response on an announcement day, we attempt to isolate the immediate reaction of asset prices to the news announcement as much as possible. As discussed earlier, this strategy has alreadybeenappliedsuccessfullytonumerousfinancialassetsintheliterature,includinginAndersen etal.(2003)andKilianandVega(2011). Theregressionmodelisestimatedusingdataforonlythose daysonwhichatleastonenewsannouncementwasmade.20 To get a baseline estimate for responsiveness to surprises, we first report the estimates over the pre-ZLB era, which covers January 1990 through March 2009. In the pre-ZLB columns of Table 3, the generally small coefficient estimates and lack of statistical significance indicate that both oil and equity returns are not responsive to macroeconomic news. In contrast, the larger coefficients and t-statistics for interest rates indicate their responsiveness to surprises over this period, which is consistentwiththeresultsinSwansonandWilliams(2014). These pre-ZLB era estimates can be compared with estimated betas for the ZLB period, which wereestimatedbyrestrictingthesampletotheperiodwhentheZLBisbinding(i.e.,whenthenotional rateimpliedbytheTaylorruleisnegative,April2009toDecember2014andJuly2015toDecember 2015). As reported in the ZLB era columns of Table 3 and in contrast to the lack of response during 20The regression sample includes all days with at least one announcement and with available data for our dependent variablesofinterest. Foreachdayinourregressionsample,wesets = 0forthosevariableswithoutanannouncement jt onthatday. Inordertopreventthese0’sfrombiasingthecoefficients,thes aredemeanedusingthemeanofthes in jt jt theregressionsample. Wealsoconsideredresultswithallnon-announcementdaysincluded. Makingthechangedidnot alterourresults. 23

the pre-ZLB era, oil and equities respond strongly to surprises during the ZLB period. During the ZLB era and in comparison to the pre-ZLB era, interest rates respond less to news. This decline in the interest rate response is consistent with the results in Swanson and Williams (2014). Figure 12 summarizestheseresults. Our next set of regressions estimates the average response to all the news announcements made during a particular era. Estimating the average response provides two benefits. First, the average response can summarize the information contained in 12 individual responses. Because the news announcements have already been standardized, the average response is a sensible statistic. Second, theaverageresponsecanbeestimatedforthepost-ZLBeraofJanuary2016toDecember2017. With only two years of data in the post-ZLB era, the individual β would be estimated using relatively few j observations. Incontrast,poolingacrosstheobservationsprovidesamorereliableestimate. Forthese reasons,weestimateanaverageresponseusingapooledmodel Y = α+βS +ε . (17) t t t In the pooled model, we pool the news surprises to generate S = (cid:80)12 s , and then we estimate t j=1 jt β, the average response to a one-standard-deviation surprise. By pooling the data, we increase the numberofobservationsusedtoestimateβ. Table 4 reports results for the pre-ZLB era, the ZLB era, and the post-ZLB era. For both oil and equities, the average responsiveness to news surprises is low before the ZLB, jumps up during the ZLB period, and then declines thereafter. As such, these results are supportive of the conjecture that theZLBplayedanimportantroleindeterminingtheresponsivenessofoilandequityreturns. Likewise,asinSwansonandWilliams(2014),short-terminterestratesarelesssensitivetomacroeconomic news during the ZLB era than during the pre-ZLB era. In the post-ZLB era, oil and equity returnsonceagainbecomelessresponsivetonews,andshort-terminterestratesbecomeslightlymore responsiveinthepost-ZLBperiod. Althoughthepost-ZLBresponsesforoilandequitiesaresimilarto thepre-ZLBresponses,theestimatedresponseofinterestratesinthepost-ZLBperiodremainssomewhat attenuated (see Figure 12). This attenuation may reflect the mechanisms discussed in Swanson andWilliams(2014)regardinglowinterestrateenvironments. 24

4.3 Kernel regression using the desired policy rate Having shown that the change in the response of macroeconomic news seems to be coincident with the ZLB period in the United States, we now test this hypothesis more directly. We use a kernel regressionsetupsimilartotheoneinEquation3toestimatecoefficientsonpooledsurprisesthatvary withotherunderlying,orcontrollingvariables,Z , t Y = α(Z )+β(Z )S +ε . (18) t t t t t Theestimatesofα andβ solvethekernelregressionproblem (cid:18) (cid:19) (cid:110) (cid:111) (cid:88) Z −Z αˆ(Z ),β ˆ (Z ) = argmin φ t k (Y −α−βS )2. (19) k k t t α,β h t In particular, we can estimate how the responsiveness to surprises changes based on our estimate of the desired monetary policy rate from Section 2.1. When using the desired policy rate as the kernel ˜ ˆ ˜ variable, Z = R , the coefficients β(R ) are estimated by placing more weight on the observed rek k k ˜ ˜ sponsestosurprisesondayswhenR isclosetoR . Figure13plotstheresultofthepooledsurprises t k estimation. Theresultsprovidedirectevidenceofthehighersensitivityofoilandequitiestomacroeconomic news announcements during periods with lower desired rates, and the higher sensitivity of interestratestomacroeconomicnewsannouncementsduringperiodswithhigherdesiredrates. Next, to test for statistical significance of these results, we construct a test statistic F(Z) that comparesthesumofsquaredresidualsforthekernelregressionmodeltothesumofsquaredresiduals forarestrictedmodel,inwhichαandβ donotvaryacrosstimeorwithanyothercontrollingvariables SSR−SSR(Z) F(Z) = , (20) SSR(Z) where (cid:88)(cid:16) (cid:17)2 (cid:88)(cid:16) (cid:17)2 ˆ ˆ SSR(Z) = Y −αˆ(Z )−β(Z )S andSSR = Y −αˆ −βS . (21) t t t t t t t t To determine the associated p-value, we compare this test statistic F(Z) to a distribution of Fsim generated using a wild bootstrap procedure. To generate the simulated distribution, we run 1000 ˆ simulations. Foreachsimulationi,weusetherestrictedmodelestimatesforαˆ,β,andεˆ togenerate: t Ysim = αˆ−β ˆ S +ν ∗εˆ. NotethattheYsimpreservesanyexistingserialcorrelationintheexplanatory it t it t it 25

variables by leaving the S variable fixed and preserves heteroscedasticity by scaling the residuals εˆ t t by ν = 1 − 2B , where B is a Bernoulli random variable, B ∼ B(1,0.5).21 Using these Ysim, it it it it it weestimateboththerestrictedandunrestrictedmodels,andgeneratetheresultingdistributionoftest statistics. We use this distribution to determine how frequently one would observe in this simulated distributiontheempiricalteststatisticcomputedusingtheactualdata. Using this test statistic and simulated distribution, we test the null hypothesis that the restricted model, in which the coefficients α and β do not vary, is equivalent to the unrestricted model, in ˜ which the coefficients are allowed to vary with the desired policy rate, R . We find statistically k significant improvement in model fit for all three of our dependent variables. As reported in the first row of Table 5, the p-values for oil, equities, and interest rates are 0.07, 0.02, and less than 0.001, respectively. 4.4 The shadow rate The previous section provides strong evidence that oil and equities are more sensitive to macroeconomic news surprises when the desired policy rate is negative. We now turn to testing alternative hypotheses for these findings. In this subsection, we study the Wu and Xia (2016) shadow rate rather thanourdesiredpolicyrate. The Wu–Xia shadow rate is a market-implied driver of the short-term rate that is allowed to be negative during the ZLB period. The shadow rate is estimated using a dynamic term structure model and thus incorporates information from observed longer-term rates during the ZLB era with the historicalrelationshipbetweenshort-andlonger-termrates. AsseeninPanel(a)ofFigure14,incontrast toourTaylor-rule-impliednotionalrate,theWu–Xiameasureispositivein2009andmostnegativein 2014. The factors affecting this rate include the monetary policy rate, the expected time at the ZLB, and various risk premia. In particular, unconventional monetary policy (UMP) can lower the shadow ratewhereasitwouldnothavethesamedirecteffectontheTaylor-rule-impliednotionalrate. 21Thischoiceofscalingvariableν isbasedonDavidsonandFlachaire(2008). Ifweweretouseascalingvariable it withastandardnormaldistributioninstead,thefourthmomentofthesimulatedresidualswouldbeartificiallyexaggerated. 26

One potential benefit of using the shadow rate in our kernel regressions—as opposed to our notional rate implied by the Taylor rule—is that it might better capture the effect of UMP through observed longer-term rates. In theory, the use of the shadow rate in the kernel regression should help usexaminetheextenttowhichUMPisasubstituteforinterestratepolicy. IfUMPisafullyeffective substitute, it would be equivalent to the model in Section 3 in which the ZLB is not binding, and we should not see heightened sensitivity to shocks in the ZLB period.22 However, if UMP is only somewhateffective,wewouldseesomeadditionalsensitivitytoshocks. Wetestthemodelwiththeshadowrateagainstthealternativeinwhichthesensitivitytonewssurprises does not vary and find statistically significant improvement in model fit for equity and interest rate responsiveness to surprises, though not for oil. As reported in the second row of Panel (a) in Table5,thep-valuesforoil,equities,andinterestratesare0.01,0.02,andlessthan0.001,respectively. Finally, we estimate a model that allows the coefficients on the surprises to vary with both the desired policy rate and the shadow rate.23 Using this model, we test the null hypotheses that a model includingthetworatesisequivalenttoamodelincludingjustthedesiredpolicyrateorjusttheshadow rate. Wefindthatingeneral,oncethemodelcoefficientsareallowedtovarywithoneofthetworates, the inclusion of the second rate does not result in a statistically significant improvement in model fit (Panel(b),Table5). Assuch,thetworatesappeartobesubstitutesforeachotherinourregressions. 4.5 Uncertainty and financialization Although we have presented a theoretical justification for why the ZLB could induce changes in the responsiveness to macroeconomic shocks, one might speculate about alternative explanations. In particular, the common folk wisdom that all correlations go to one in a crisis suggests that increased uncertainty could be an alternative driver of the elevated oil–equity correlation and the increased responsiveness to macroeconomic news surprises during the ZLB period. To test this conjecture, we use three different measures of uncertainty. First, we use the 90-day moving average of the daily 22WuandXia(2016)provideadiscussionofhowUMPundoestheconstraintsimpliedbytheZLB. 23When using two controlling variables, the coefficients in the model Y = α(Z ,Z ) + β(Z ,Z )S + ε are t 1t 2t 1t 2t t t (cid:110) (cid:111) (cid:16) (cid:17) (cid:16) (cid:17) estimatedbysolving αˆ(Z 1k ,Z 2k ),βˆ(Z 1k ,Z 2k ) =argmin α,β (cid:80) t φ Z1t h − 1 Z1k φ Z2t h − 2 Z2k (Y t −α−βS t )2. 27

series for economic policy uncertainty from Baker et al. (2015). Second, we use the 90-day horizon measureoffinancialuncertaintyfromJuradoetal.(2015). Finally,weusethe90-daymovingaverages oftheVIX,whichisameasureofoptions-impliedstockmarketvolatility. Accordingtothismeasure, market uncertainty began rising in 2007, spiked sharply in 2008 at the height of the financial crisis, andremainedelevatedforafewyearsafterthat. AllthreeofthesemeasuresaredepictedinPanel(b) ofFigure14,andsummarystatisticsarereportedinPanel(d)ofTable1. A second alternative hypothesis is that with increased financialization of oil markets, the greater overlap between oil market and other financial market participants resulted in greater sensitivity of oil to general market conditions. According to this theory, the oil market would react much more strongly to events that earlier would have moved only equity markets. We proxy for financialization using the 90-day rolling average of the open interest across all futures contracts for WTI crude oil on NYMEX,asdepictedinPanel(c)ofFigure14. To test these alternative hypotheses, we re-estimate the kernel regression of our three dependent variables on macroeconomic news surprises in Equation 18 using each of our alternative controlling variablesinturn. Wetesteachofthesemodelsagainstthealternativeinwhichthesensitivitytonews surprises does not vary. We also estimate the models using pairs of controlling variables. We test the null hypothesis that a model including the desired policy rate along with one of the alternative controlling variables is equivalent to a model including just the desired policy rate or just the alternative controllingvariable. Table 6 summarizes the hypotheses being tested and the p-values that result from the wild bootstrap procedure for each test. In Panel (a), we generally find that the inclusion of the alternative variables tends to not improve the fit of the models for oil or equities in a statistically significant way against the restricted alternative in which the coefficients are non-varying. The exception is for the inclusion of open interest, which improves model fit for equities and interest rates—but, counter to the financialization hypothesis, not for oil. In Panel (b), we find that when allowing the coefficients to vary with the desired policy rate, the addition of a second controlling variable sometimes provides a statistically significant improvement in model fit for the interest rate, but again not for oil or for 28

equities. Lastly, we find in Panel (c) of Table 5 that even after including the alternative kernel variables, the addition of the desired policy rate to the kernel often results in a statistically significant improvement in model fit. In conclusion, we find that the variation in sensitivity to macroeconomic news surprises for oil, equities, and interest rates is better explained by our measure of constrained monetarypolicythanbythealternativemeasuresofuncertaintyandoilmarketfinancialization. 5 Additional evidence: A structural VAR Althoughwehaveshownthatoilandequityreturnshavebecomemoreresponsivetomacroeconomic data announcements, we have not yet documented the role of structural shocks, such as oil supply shocks or aggregate demand shocks, in changing the correlation. As discussed in Kilian and Park (2009),thesourceoftheshockcandeterminewhetheroilpriceincreasesareassociatedwithincreases or decreases in equity prices. For example, an increase in the size or frequency of aggregate demand shockscouldalsohavecontributedtotheincreaseintheoil-equitycorrelationduringtheZLBperiod. In this section, we consider the role of structural shocks by estimating a monthly structural VAR. WethenusetheVAR’simpliedstructuralmovingaveragerepresentationtodecomposethecorrelation into its contributions from the various structural shocks. This exercise is analogous to a variance decomposition but is done for a correlation between two variables rather than for the variance of a singlevariable. Our estimated structural VAR is similar to that of Kilian and Park (2009), featuring four monthly variables: thelogdifferenceofglobaloilproduction∆O,thedetrendedlevelofglobalrealeconomic activity A, the log difference in the real oil price ∆P, and real U.S. equity returns ∆E.24 Given our interest in the correlation between oil and equity returns, we estimate the VAR using the first difference of real oil prices and 12 lags, whereas Kilian and Park (2009) estimate their VAR using 24GlobaloilproductiondataisobtainedfromtheU.S.EnergyInformationAdministration’sMonthlyEnergyReview (Table 11.1b). The detrended global real economic activity index, as in Kilian and Park (2009) and Kilian (2009), was downloadedonMarch27,2018fromhttp://www-personal.umich.edu/˜lkilian/reaupdate.txt. The nominal oil price is the refiner acquisition cost of imported crude oil, obtained from the U.S. Energy Information Administration’s Petroleum Marketing Monthly (Table 1). Nominal U.S. equity returns reflect the market return on the Fama–Frenchvalue-weightedportfolioofallNYSE,AMEX,andNASDAQstocks. Realoilpricesandrealequityreturns areconstructedusingtheU.S.CPIforallurbanconsumers,obtainedfromtheBureauofLaborStatistics. 29

the level of real oil prices and 24 lags.25 We identify shocks to the structural VAR using a standard recursivenessassumption,withthefollowingidentifyingassumptions. IdentifyingAssumptionsintheStructuralVAR Shock ImmediateResponse DelayedResponse OilSupplyShock ∆E,∆P,A,∆O AggregateDemandShock ∆E,∆P,A ∆O OilResidualShock ∆E,∆P A,∆O EquityResidualShock ∆E ∆P,A,∆O We remain agnostic about the final two shocks, labeling them as generic residual shocks to the relevant variables. The estimated VAR implies the following moving average representation for the h-stepaheadforecasterrors, h−1 (cid:88) y −y = Θ w , (22) t+h t+h|t i t+h−i i=0 where Θ is a 4-by-4 matrix of moving average coefficients implied by the estimated VAR and struci turalfactorizationofthevariance-covariancematrixofthereducedformresiduals. Ratherthandiscussingimpulseresponsestothestructuralshocks,weinsteadfocusonourpaper’s central question of what causes changes in the correlation between oil and equity returns. To do so we decompose the correlation into its contribution from the various structural shocks. Although a covariancedecompositionisseldomfeaturedintheliterature,tocalculateacovariancedecomposition is just a straightforward application of textbook calculations for a variance decomposition (Hamilton (1994)). Thecovariancecalculationrequirestwokindsofmatrices. ThefirstisMSPE(h)thevalueofthe h-stepaheadforecastvariance-covariancematrixconditionalonallshocks h−1 MSPE(h) = E (cid:16) (cid:0) y −y (cid:1)(cid:0) y −y (cid:1)(cid:48) (cid:17) = (cid:88) Θ IΘ(cid:48), (23) t+h t+h|t t+h t+h|t i i i=0 25Asshownintheappendix,estimatingtheVARusingthelevelofrealoilpricesand/or24lagsdoesnotsubstantively changeouranalysisaboutthecorrelationbetweenoilandequityreturns.Ourtotalsampleperiodis1974to2017,whereas KilianandPark(2009)useasampleperiodof1974to2006. 30

where I is the identity matrix. The second is MSPE (h), the h-step ahead forecast variancej covariancematrixconditionalononlythej-shock h−1 (cid:88) MSPE (h) = Θ E Θ(cid:48), (24) j i j i i=0 whereallelementsofE areequaltozeroexceptthej-th,j-thelement,whichequalsone. j Thecorrelationbetweenoilandequityreturnscanbedefinedusingtermsfromthesetwomatrices. In particular, define σ (h) as the square root of the 3,3 element of MSPE(h), σ (h) as the square p e root of the 4,4 element of MSPE(h), and σ (h) as the 3,4 element of MSPE(h). Furthermore, pe defineσ (h),thecovarianceconditionaljustonshockj,asthe3,4elementofMSPE (h). Having pe,j j defined these terms, we can then write the correlation between oil and equity returns ρ (h) as the pe followingequation 4 σ (h) (cid:88) σ (h) pe pe,j ρ (h) = = . (25) pe σ (h)σ (h) σ (h)σ (h) p e p e j=1 For any shock, a larger σ (h) indicates a larger contribution of the j-th shock to the overall correpe,j lation. WeestimatetheVARseparatelyforanearlysampleofJanuary1974toMarch2009andforalate sample of April 2009 to December 2017. Table 7 reports our results for h = 1000, which is large enoughthatρ (h)approximateswellthecorrelationbetweenoilandequityreturns. Consistentwith pe ourresultsfordailydata,theoverallcorrelationρ (h)formonthlydataincreasesfromnegative0.10 pe in the early sample to positive 0.33 in the late sample. As reported in Table 7, the correlation change was not driven by oil supply or aggregate demand shocks. Instead, during the ZLB period, shocks that have an immediate impact on oil prices but not oil demand and supply went from causing a negative correlation to a positive correlation. This change in the correlation is almost entirely due to the oil residual shock, which contributes negative 0.11 to the oil–equity correlation in the early sample and positive 0.24 in the late sample. The negative effect of these shocks in the pre-ZLB sample is consistent with the empirical results reported in Kilian and Park (2009), and the positive effect in the late sample is consistent with how the ZLB alters the response in our DSGE model discussed in Section3. Thesemonthlyoilresidualshocksare,bythemselves,notverycleanlyidentified,highlighting 31

the benefits of our work in Section 4, which uses higher frequency data. In high frequency data the timingassumptionsaremoreplausibleandtheshocksaremorelikelytobeunanticipatedthanshocks thatareidentifiedatthemonthlyorquarterlyfrequency(Ramey,2016). 6 Conclusion Startinginlate2008,thecorrelationbetweenoilandequityreturns,whichpreviouslyhadbeeneither smallornegative,increaseddramaticallyandremainedelevatedthereafter. Ourmainargumentisthat this correlation change is evidence that the ZLB alters the dynamic behavior of the economy. Our argument is supported by the following observations. First and most obviously, the ZLB becomes binding and the correlation increases at the same time. Second, this jump in correlation is consistent with a standard New Keynesian model in which the ZLB is binding and is not consistent with the same model that ignores the ZLB. A multicountry version of the New Keynesian model is also consistent with international experience at the ZLB. Third, consistent with theory, we show that oil and equity returns became more responsive to macroeconomic news announcements when the ZLB binds. Fourth,weconsideralternativehypothesesthatcouldaltertheresponsivenessofoilandequity returnsrelatedtofinancializationanduncertainty. Ourempiricalevidenceshowsthatthesealternative explanations do not improve the fit relative to conditioning on the ZLB. Finally, we use a structural VARonmonthlydataandagainfindevidencesupportiveofourZLB-drivenhypothesis. As such, our results complement and extend the findings of Swanson and Williams (2014) for interest rates by showing that activity measures such as oil and equity prices are also affected by the ZLB. Our findings are more supportive than some previous empirical work that the ZLB alters the economic environment. We would argue that our results should be preferred given our use of daily datawithclearlyidentifiedshocks. A large literature now exists that shows that the ZLB theoretically changes the dynamic behavior of the economy. However, empirical macroeconomic evidence that the ZLB actually causes these changeshasbeenscarce. ThescarcityofevidencefortheUnitedStateslikelyreflectsthatinterestrates were well above zero until 2008, and consequently the number of monthly or quarterly observations 32

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A Figures and tables Figure1: Oil-EquityCorrelationandPolicyRates (a)Oil-EquityRollingCorrelation (b)PolicyRates Panel(a): Rollingcorrelationbetweendailyoilandequityreturns. Thedateaxismarkstheendoftheone-yearrolling windowoverwhichthecorrelationiscalculated. Panel(b): Thefederalfundsrateandthenotionalinterestrateimplied bythemodifiedTaylorruleinEquation1((Bernanke,2015)). Thenotionalinterestrateisintendedtocapturethetarget federal funds rate as implied by the current state of the economy, without censoring due to the zero lower bound. The thickerlinesrepresenttheseriesforthedesiredpolicyrate,R˜ ,definedasthenotionalratewhenitisnegative,andthe k observedfederalfundsrateotherwise. 36

Figure2: EquityBetaforOil 0.8 0.4 0 -0.4 -0.8 -5 0 5 10 Desired Policy Rate ateB ytiuqE Note:Thedotsshowtheestimatesofβ(R˜ )usingthekernelregressioninEquation2,Oil =α(R˜ )+β(R˜ )Equity +ε . k t t t t t The shaded region represents a 90 percent confidence interval for the estimated β(R˜ ), based on the wild bootstrap as k describedinSection4.3. Figure3: LocalCorrelationforOilandEquityReturns 0.8 ZLB 0.6 No ZLB 0.4 0.2 0 -0.2 -4 -2 0 2 4 6 8 Desired Policy Rate Note: ThesolidblacklinesshowthelocalregressionslopeforthemodelthatincludestheZLBconstraint. Thedashed redlinesshowthelocalregressionslopeforthemodelthatdoesnotincludetheZLBconstraint. Wecomputethelocal correlationsusingonemillionsimulatedperiodsfromourmodel. 37

Figure4: EquityReturnKernelRegression (a)TechnologyShock 0.5 0 -0.5 ZLB No ZLB -1 -4 -2 0 2 4 6 8 Desired Policy Rate (b)SpreadShock (c)GovernmentSpendingShock 0 0.2 -0.5 0.1 -1 0 -1.5 -0.1 -4 -2 0 2 4 6 8 -4 -2 0 2 4 6 8 Desired Policy Rate Desired Policy Rate (d)OilDemandShock (e)OilSupplyShock 0.1 0.1 0.05 0 0 ZLB, low persistence -0.05 -0.1 No ZLB, low persistence -0.1 -4 -2 0 2 4 6 8 -4 -2 0 2 4 6 8 Desired Policy Rate Desired Policy Rate Note: ThesolidblacklinesshowthelocalregressionslopeforthemodelthatincludestheZLBconstraint. Thedashed redlinesshowthelocalregressionslopeforthemodelthatdoesnotincludetheZLBconstraint. Wecomputethelocal regression slopes using one million simulated periods from our model. In Panel (e), the dashed-dotted blue line is analogoustotheblacksolidline, butforthemodelinwhichthepersistenceofoilsupplyisreducedfrom0.99to0.10. Thedottedgreenlineissimilarlyanalogoustothedashedredline. 38

Figure5: OilReturnKernelRegression (a)TechnologyShock 0 -1 -2 ZLB -3 No ZLB -4 -2 0 2 4 6 8 Desired Policy Rate (b)SpreadShock (c)GovernmentSpendingShock 0 0.8 0.6 -1 0.4 0.2 0 -2 -0.2 -4 -2 0 2 4 6 8 -4 -2 0 2 4 6 8 Desired Policy Rate Desired Policy Rate (d)OilDemandShock (e)OilSupplyShock 0 2 1.5 -2 1 0.5 -4 0 -6 -4 -2 0 2 4 6 8 -4 -2 0 2 4 6 8 Desired Policy Rate Desired Policy Rate Note: ThesolidblacklinesshowthelocalregressionslopeforthemodelthatincludestheZLBconstraint. Thedashed redlinesshowthelocalregressionslopeforthemodelthatdoesnotincludetheZLBconstraint. Wecomputethelocal regressionslopesusingonemillionsimulatedperiodsfromourmodel. 39

Figure6: PolicyRatesintheUnitedStates,Japan,andMexico 9 8 7 6 5 4 3 2 1 0 1995 2000 2005 2010 2015 Date tnecreP dezilaunnA ,etaR yciloP United States Mexico Japan Note: The solid black line shows the midpoint of the target range for the federal funds rate. The blue dash-dotted line shows the discount rate in Japan. The red dashed line shows the discount rate in Mexico. Data for the United States and Japan come from FRED (https://fred.stlouisfed.org). The series for the United States is constructed fromtheFREDseriesnamednamedDFEDTAR,DFEDTARL,andDFEDTARU.TheseriesforJapanistheFREDseries named INTDSRJPM193N. The series from Mexico comes from Banco de Me´xico (http://www.banxico.org. mx/estadisticas/statistics.html). Figure7: Oil-EquityCorrelationinLargeCountry,InternationalModel (a)ZLBinLargeCountry (b)ZLBinSmallCountry 0.8 0.8 Traded Bonds Traded Bonds 0.6 0.6 Financial Autarky Financial Autarky 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 -4 -2 0 2 4 6 8 -4 -2 0 2 4 6 8 Desired Policy Rate, Large Country Desired Policy Rate, Small Country Note: The figures show the local correlation between oil returns and large country equity returns, with both series expressedinthelargecountrycurrency. Weassumetheworldsizeis1,withthesizeofthelargecountryequalto0.9and thesizeofthesmallcountryequalto0.1. InPanel(a),theZLBisimposedforthelargecountryandthelocalcorrelation is calculated over the range of large country desired policy rates, while in Panel (b), the ZLB is imposed for the small countryandthelocalcorrelationiscalculatedovertherangeofsmallcountrydesiredpolicyrates. 40

Figure8: Oil-EquityCorrelationinSmallCountry,withSmallCountryZLB (a)SmallCountryCurrency (b)LargeCountryCurrency 0.8 0.8 Traded Bonds Traded Bonds 0.6 0.6 Financial Autarky Financial Autarky 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 -4 -2 0 2 4 6 8 -4 -2 0 2 4 6 8 Desired Policy Rate, Small Country Desired Policy Rate, Small Country Note: Thefiguresshowthelocalcorrelationbetweenoilreturnsandsmallcountryequityreturnswhenthesmallcountry isconstrainedbytheZLB.Weassumetheworldsizeis1,withthesizeofthelargecountryequalto0.9andthesizeof thesmallcountryequalto0.1. InPanel(a),thereturnsareexpressedinthesmallcountry’scurrency,whileinPanel(b), thereturnsareexpressedinthelargecountry’scurrency. Figure9: Oil-EquityRollingCorrelation,Japan (a)Yen (b)Dollars 1 1 0.5 0.5 0 0 1 yr 1 yr -0.5 -0.5 2 yr 2 yr 3 yr 3 yr -1 -1 1985 1990 1995 2000 2005 2010 2015 1985 1990 1995 2000 2005 2010 2015 Note: Monthly returns computed as log differences in the TOPIX equity index on last trading day of each period. Oil returnscomputedaslogdifferencesinoilpriceonlasttradingdayofeachperiod. Legendlabelscorrespondtolengthof rolling window. Correlations dated at end of rolling window. Currency conversion done using exchange rates from the H.10releasefromtheFederalReserve. 41

Figure10: Oil-EquityCorrelationinSmallCountry,withLargeCountryZLB (a)LargeCountryCurrency (b)SmallCountryCurrency 0.8 0.8 Traded Bonds Traded Bonds 0.6 0.6 Financial Autarky Financial Autarky 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 -4 -2 0 2 4 6 8 -4 -2 0 2 4 6 8 Desired Policy Rate, Large Country Desired Policy Rate, Large Country Note: Thefiguresshowthelocalcorrelationbetweenoilreturnsandsmallcountryequityreturnswhenthelargecountry isconstrainedbytheZLB.Weassumetheworldsizeis1,withthesizeofthelargecountryequalto0.9andthesizeof thesmallcountryequalto0.1. InPanel(a),thereturnsareexpressedinthelargecountry’scurrency,whileinPanel(b), thereturnsareexpressedinthesmallcountry’scurrency. Figure11: Oil-EquityRollingCorrelation,Mexico (a)Dollars (b)Pesos 1 1 0.5 0.5 0 0 1 yr 1 yr -0.5 -0.5 2 yr 2 yr 3 yr 3 yr -1 -1 1985 1990 1995 2000 2005 2010 2015 1985 1990 1995 2000 2005 2010 2015 Note: Monthly returns computed as log differences in the BOLSA equity index on last trading day of each period. Oil returnscomputedaslogdifferencesinoilpriceonlasttradingdayofeachperiod. Legendlabelscorrespondtolengthof rolling window. Correlations dated at end of rolling window. Currency conversion done using exchange rates obtained fromtheBankofMexico. 42

Figure12: ResponsivenesstoSurprisesintheZLBPeriod (a)Oil 1 0.5 0 -0.5 ateB esirpruS cu con cpi gdp icl ism ind nhs nfp ppi rtl ur pooled Pooled Pooled Pre-ZLB ZLB Pooled ZLB Pre-ZLB Post-ZLB (b)Equity 1 0.5 0 -0.5 ateB esirpruS cu con cpi gdp icl ism ind nhs nfp ppi rtl ur pooled (c)InterestRate 6 5 4 3 2 1 0 -1 ateB esirpruS cu con cpi gdp icl ism ind nhs nfp ppi rtl ur pooled Note: The bars represent the estimated coefficients from the pre-ZLB, and ZLB, and post-ZLB eras for the regression Y =α+βs +ε ,wheres isthevectorofstandardizedanddemeanednewsondayt. Thefinalthreebarsineachpanel t t t t representtheβ’sfromtheregressionusingpooledsurprises, Y = α+βS +ε , whereS = (cid:80)12 s . Thepre-ZLB t t t t j=1 jt eraregressionscoverJanuary1990toMarch2009;theZLBeraregressionscoverApril2009toDecember2014andJuly 2015 to December 2015; and the post-ZLB era regressions cover January 2016 to December 2017. Following Beechey andWright(2009),weflipthesignforunemploymentandinitialjoblessclaimsannouncements,sothatpositivesurprises representastronger-than-expectedeconomy. SeeSection4.2formoredetailandTable3fortheassociatedannouncement codesandregressionstatistics. 43

Figure13: SurpriseBetasandtheDesiredPolicyRate (a)Oil 0.4 0.2 0 -0.2 -0.4 -5 0 5 10 Desired Policy Rate ateB esirpruS (b)Equity 0.2 0.1 0 -0.1 -0.2 -5 0 5 10 Desired Policy Rate ateB esirpruS (c)InterestRate 3 2 1 0 -5 0 5 10 Desired Policy Rate ateB esirpruS Note: For each of our dependent variables, Y ∈ {Oil ,Equity ,InterestRate }, we estimate the regression Y = t t t t t α(R˜ )+β(R˜ )S +ε ,usingthedesiredpolicyrateasthecontrollingvariableinthekernelregression. Theshadedregion t t t t representsa90percentconfidenceintervalfortheestimatedβ(R˜ )basedonthewildbootstrap. t 44

Figure14: AlternativeTheories (a)PolicyRates (b)Uncertainty (c)Financialization Panel (a): Policy rate measures include the Wu and Xia (2016) shadow rate, the notional interest rate implied by the modified Taylor rule in Equation 1 ((Bernanke, 2015)), and the federal funds rate. Panel (b): Uncertainty measures includethe90-daymovingaverageofthedailyseriesforeconomicpolicyuncertaintyfromBakeretal.(2015),the90dayhorizonmeasureoffinancialuncertaintyfromJuradoetal.(2015),andthe90-daymovingaverageoftheVIX.The economic policy uncertainty measure is multiplied by 100 in this panel for ease of comparison to the other two series. Panel(c):Thefinancializationmeasureisthe90-daymovingaverageofopeninterestacrossallmaturitiesofWTIfutures contracts,expressedinmillionsofcontracts. 45

Table1: SummaryStatistics Variable Obs. StartDate Mean St. Dev. Min. Max. PanelA:PrimaryVariablesofInterest Oilreturns(WTInearbyfutures) 8584 1983-Apr-06 0.01 2.40 -40.05 22.80 Equityreturns 8584 1983-Apr-06 0.04 1.09 -19.13 9.89 ∆Interestrate(2year) 8584 1983-Apr-06 -0.09 6.15 -84.00 38.00 PanelB:AlternativeMeasures WTIphysicalspotreturns 7910 1986-Jan-03 0.01 2.56 -40.64 21.70 Brentphysicalspotreturns 8388 1983-May-17 0.01 2.32 -40.71 27.82 WTIfarfuturesreturns 8584 1983-Apr-06 0.01 1.37 -10.35 10.80 Metalsreturns 8584 1983-Apr-06 0.02 0.86 -10.29 9.40 ∆Interestrate(1year) 8584 1983-Apr-06 -0.09 5.45 -83.00 52.00 ∆Interestrate(10year) 8584 1983-Apr-06 -0.09 6.39 -75.00 39.00 S&P500excl. energy 4946 1998-Jan-02 0.02 1.22 -9.11 10.10 PanelC:InternationalEquityReturns TOPIX(indollars) 7920 1983-Apr-06 0.02 1.40 -17.26 10.53 TOPIX(inyen) 7926 1983-Apr-06 0.01 1.29 -15.81 12.86 BOLSA(indollars) 8100 1983-Apr-06 0.07 2.10 -31.23 23.29 BOLSA(inpesos) 8106 1983-Apr-06 0.13 1.75 -20.24 23.58 PanelD:ControllingVariables ˜ Desiredpolicyrate(R) 8187 1985-Dec-19 3.10 3.45 -4.60 9.81 Wu–Xiashadowrate(SR) 8250 1985-Dec-19 3.24 3.18 -2.99 9.81 Economicpolicyuncert. (EPU) 9914 1980-Jan-01 0.94 0.14 0.72 1.43 Financialuncertainty(FNU) 12051 1985-Jan-01 101.50 38.50 40.95 232.72 VIX 7304 1990-Jan-02 19.45 6.97 10.37 55.03 Openinterest(OI) 8338 1986-Jan-15 0.81 0.59 0.07 2.44 Notes: The price of oil is the closing value, in dollars per barrel, of the front-month futures contract for West Texas Intermediate (WTI) crude oil for delivery in Cushing, Oklahoma obtained from NYMEX. Equity returns are obtained fromtheFama-Frenchvalue-weighteddailyreturnonallNYSE,AMEX,andNASDAQstocks(whichincludedividends), andareconvertedtopricelevels. The WTI physical spot price is for WTI crude oil for delivery (freight on board) in Cushing, Oklahoma, as reported by the U.S. Energy Information Administration. The Brent physical spot price is for Brent Forties Oseberg crude oil, obtained from Bloomberg. The WTI far futures price is the price of the furthest available December contract for WTI crudeoil, andisobtainedfromNYMEX.ThemetalspriceistheCommoditiesResearchBureauMetalsIndexobtained fromBloomberg. TheS&P500Ex-EnergyandTOPIXarealsoobtainedfromBloomberg,whiletheBOLSAisobtained fromHaverAnalytics. TheyenexchangerateisobtainedfromtheFederalReserveH.10. release,andthepesoexchange rateisobtainedfromtheBankofMexico. Thedesiredpolicyrateisdefinedasthenotionalratewhenitisnegative,andtheobservedfederalfundsrateotherwise. See Equation 1 for details on construction of the notional interest rate. For the remaining controlling variables, we use the shadow rate constructed in Wu and Xia (2016), the 90-day moving average of the daily series for economic policy uncertaintyfromBakeretal.(2015), the90-dayhorizonmeasureoffinancialuncertaintyfromJuradoetal.(2015), the 90-daymovingaveragesoftheVIXobtainedfromBloomberg,andthe90-daymovingaverageofopeninterestmeasured inmillionsofcontractsobtainedfromtheCFTC. 46

stluseRtseTwohC :2elbaT kaerB-tsoP kaerB-erP elpmaSlluF tats-t β tats-t β tats-t β etaDkaerB .sbO snruteRytiuqEfoxednIlluFehtnosnoissergeR:AlenaP )43.12( 97.0 )33.5-( 61.0- )51.8( 91.0 22-peS-8002 4858 snruterserutufybraenITW )35.12( 08.0 )27.4-( 51.0- )91.8( 12.0 22-peS-8002 0197 snrutertopslacisyhpITW )94.62( 58.0 )86.3-( 11.0- )78.01( 52.0 22-peS-8002 8838 snrutertopslacisyhptnerB )26.02( 73.0 )12.3-( 60.0- )55.7( 01.0 22-peS-8002 4858 snruterserutufrafITW )41.51( 42.0 )41.2( 20.0 )19.11( 01.0 03-peS-8002 4858 snruterslateM sexednIdnasrotceSytiuqEsuoiraVnoliOfosnoissergeR:BlenaP )17.61( 58.0 )13.9-( 92.0- )09.0( 20.0 22-peS-8002 4858 selbarudnonremusnoC )92.91( 15.0 )05.7-( 71.0- )83.6( 11.0 22-peS-8002 4858 selbarudremusnoC )77.22( 07.0 )79.4-( 41.0- )42.01( 22.0 22-peS-8002 4858 gnirutcafunaM )88.73( 79.0 )99.22( 74.0 )83.73( 26.0 12-voN-8002 4858 ygrenE )12.12( 48.0 )94.7-( 12.0- )33.5( 31.0 22-peS-8002 4858 slacimehC )62.81( 66.0 )19.3-( 70.0- )75.4( 80.0 22-peS-8002 4858 tnempiuqessenisuB )15.81( 86.0 )66.6-( 71.0- )35.4( 01.0 22-peS-8002 4858 snoitacinummoceleT )40.81( 57.0 )79.0-( 30.0- )63.01( 82.0 22-peS-8002 4858 seitilitU )41.51( 76.0 )45.01-( 72.0- )33.1-( 30.0- 22-peS-8002 4858 spohS )97.31( 16.0 )79.7-( 02.0- )22.0( 00.0 22-peS-8002 4858 erachtlaeH )76.61( 24.0 )06.8-( 12.0- )34.5( 01.0 22-peS-8002 4858 ecnaniF )40.02( 86.0 )05.6-( 81.0- )58.6( 51.0 22-peS-8002 4858 rehtO )99.81( 47.0 )85.4-( 81.0- )72.9( 62.0 22-peS-8002 6494 ygrene .lcxe005P&S )99.3( 80.0 )32.4-( 91.0- )20.1( 20.0 12-guA-1991 0297 xednIytiuqEXIPOT )93.22( 46.0 )26.1( 30.0 )74.31( 22.0 92-guA-8002 5565 xednIytiuqEASLOB + rotceSytiuqEβ+α= liOnoissergerehtmorfsatebrotcesytiuqestroperBlenaP. ε+ ytiuqEβ+α= YnoissergerehtmorfsatebytiuqestroperAlenaP:setoN t t t t t ottsetwohCdradnatsehtgniylpparetfadetamitseerastluserkaerb-tsopdna,kaerb-erp,etadkaerbehT .detropererascitsitats-tdna,β,snoitavresboelpmaslluF . ε t otdnuoferewsetadkaerbesehtfollA .selpmaskaerb-tsopdna-erpehtnonursnoissergerrofsrorrederauqsfomusehtseziminimhcihwetadkaerbehtenimreted saelpmasehtfognimmirt%51nopudesabeulavlacitircdlaW-mumerpusswerdnAdradnatsehtgnisunehw)8.7= F(level%1ehttatnacfiingisyllacitsitatseb tirc .)3002(nostaWdnakcotSni 47

setamitsEateBesirpruSetaRtseretnIdna,ytiuqE,liO :3elbaT etartseretnI ytiuqE liO areBLZ areBLZ-erp areBLZ areBLZ-erp areBLZ areBLZ-erp tats-t β tats-t β tats-t β tats-t β tats-t β tats-t β )07.0( 42.0 )94.4( 29.1 )11.0-( 10.0- )30.1( 80.0 )91.0( 40.0 )46.1( 52.0 )uc(noitazilituyticapaC )10.2( 66.0 )34.3( 75.1 )09.0( 11.0 )95.1-( 31.0- )14.0( 90.0 )41.1( 91.0 )noc(ecnedfinocremusnoC )67.0-( 43.0- )49.4( 80.2 )44.0-( 70.0- )60.3-( 32.0- )43.0-( 01.0- )35.1( 32.0 )ipc(IPCeroC )14.0( 13.0 )31.2( 15.1 )55.0-( 51.0- )82.0( 40.0 )72.0( 31.0 )94.0-( 21.0- )pdg(ecnavdaPDG )94.3( 66.0 )46.5( 81.1 )99.0( 70.0 )42.1( 50.0 )37.2( 33.0 )26.0( 50.0 )mlc(smialclaitinI )38.2( 60.1 )13.8( 25.3 )80.4( 55.0 )76.0( 50.0 )11.3( 47.0 )91.0-( 30.0- )msi(gnirutcafunamMSI )52.0-( 01.0- )83.0( 72.0 )76.1( 42.0 )49.2-( 83.0- )18.0-( 12.0- )69.0( 52.0 )dni(srotacidnignidaeL )03.0( 61.0 )54.2( 59.0 )02.0( 40.0 )05.0-( 30.0- )15.0( 71.0 )02.0( 30.0 )shn(selasemohweN )75.9( 82.5 )41.21( 01.5 )13.3( 66.0 )16.1-( 21.0- )15.1( 35.0 )13.0( 50.0 )pfn(slloryapmrafnoN )16.1( 27.0 )24.1( 75.0 )86.1( 72.0 )47.1-( 31.0- )15.1( 34.0 )11.0-( 20.0- )ipp(IPPeroC )58.2( 76.1 )74.5( 06.2 )47.1( 73.0 )04.1( 21.0 )51.1( 34.0 )08.1-( 13.0- )ltr(sotua .xeselasliateR )19.0-( 63.0- )99.3( 89.1 )17.0( 01.0 )16.0( 50.0 )77.0( 91.0 )83.1-( 52.0- )ru(etartnemyolpmenU 51.0 51.0 50.0 10.0 30.0 10.0 derauqs-R denaemeddnadezidradnatsforotcevehtsi serehw, ε+ sβ+α = Ynoissergerehtmorf,βrotcevehtfo βstnemelelaudividniehtstroperelbatehT :setoN t t t t j ehtnidedulcnierasnoitavresbo0332 .esirprusswennoitaiveddradnatsenoaotelbairavtnednepedhcaefosesnopserehterusaem βsretemarapehT .tyadnoswen j 5102yluJdna4102rebmeceDot9002lirpA(snoissergerareBLZehtnidedulcnierasnoitavresbo836dna,)9002hcraMot0991yraunaJ(snoissergerareBLZ-erp ehtpiflew,)9002(thgirWdnayehceeBgniwolloF .evitagensietarlanoitonelurrolyaTehthcihwgniruddoirepehtsadenfiedsiareBLZehT .)5102rebmeceDot eromrof2.4noitceSeeS .ymonocedetcepxe-naht-regnortsatneserpersesirprusevitisoptahtos ,stnemecnuonnasmialcsselbojlaitinidnatnemyolpmenurofngis .liated 48

Table4: EstimatedAverageResponsetoMacroeconomicNews DependentVariableY Pre-ZLB ZLB Post-ZLB t Oil β 0.04 0.24 0.06 t-stat 0.87 3.59 0.32 Equity β -0.03 0.15 0.07 t-stat -1.23 3.85 1.24 InterestRate(2years) β 1.89 0.65 0.75 t-stat 16.46 5.75 2.96 InterestRate(1year) β 1.47 0.21 0.37 t-stat 15.52 4.03 2.28 InterestRate(10years) β 1.48 1.20 0.74 t-stat 13.19 5.78 2.42 Observations 2330 763 230 Notes: The table reports the value of β from the regression using pooled surprises, Y = α+βS +ε , where S is t t t t theaverageofstandardizedanddemeanednewsondayt. β measurestheresponseofeachdependentvariabletoaone standarddeviationnewssurprise. TheZLBeraisdefinedastheperiodduringwhichtheTaylor-ruleimpliednotionalrate isnegative. SeeSection4.2,andFigure12,whichplotsthesevaluesforOil,Equity,andInterestRate. Table5: DotheZLBmeasuresimprovemodelfit? DependentVariable(Y ) t NullHypothesis Oil Equity Int. Rate ˜ a. H : β = β(R ) 0.07 0.02 0.00 0 k H : β = β(SR ) 0.01 0.02 0.00 0 k ˜ b. H : β(SR ) = β(SR ,R ) 0.90 0.47 0.23 0 k k k ˜ ˜ H : β(R ) = β(R ,SR ) 0.57 0.77 0.60 0 k k k ˜ c. H : β(FNU ) = β(FNU ,R ) 0.22 0.13 0.00 0 k k k ˜ H : β(EPU ) = β(EPU ,R ) 0.16 0.07 0.00 0 k k k ˜ H : β(VIX ) = β(VIX ,R ) 0.11 0.18 0.00 0 k k k ˜ H : β(OI ) = β(OI ,R ) 0.09 0.04 0.00 0 k k k Notes: The table reports for each dependent variable Y ∈ {Oil ,Equity ,InterestRate } the p-values for the test of t t t t each null hypothesis listed. Rejection of the null hypothesis provides evidence that the variable being tested is able to improvethefitofthekernelregressionY = α(.)+β(.)S +ε ,relativetothealternativelisted. Panel(a)testsmodels t t t inwhichthesensitivitytonewssurprisesdoesnotvaryagainstmodelsinwhichthesensitivityvarieswithasinglepolicy rate. Panel(b)teststhenullhypothesisthatamodelincludingthetworatesisequivalenttoamodelincludingjustoneof thetworates. Panel(c)teststhenullhypothesisthatamodelincludingthedesiredpolicyrateandoneofthealternative variablesisequivalenttoamodelincludingjustthealternativecontrollingvariable. SeeTableB.1forcontrollingvariable summarystatistics,andSections4.3through4.5formoredetail. 49

Table6: Dothealternativecontrollingvariablesimprovemodelfit? DependentVariable(Y ) t NullHypothesis Oil Equity Int. Rate a. H : β = β(FNU ) 0.11 0.43 0.18 0 k H : β = β(EPU ) 0.41 0.36 0.00 0 k H : β = β(VIX ) 0.89 0.87 0.00 0 k H : β = β(OI ) 0.13 0.01 0.00 0 k ˜ ˜ b. H : β(R ) = β(R ,FNU ) 0.29 0.45 0.04 0 k k k ˜ ˜ H : β(R ) = β(R ,EPU ) 0.41 0.31 0.04 0 k k k ˜ ˜ H : β(R ) = β(R ,VIX ) 0.81 0.83 0.00 0 k k k ˜ ˜ H : β(R ) = β(R ,OI ) 0.54 0.40 0.24 0 k k k Notes: ThetablereportsforeachdependentvariableY ∈ {Oil ,Equity ,InterestRate }thep-valuesassociatedwith t t t t the test of each null hypothesis listed. Rejection of the null hypothesis provides evidence that the controlling variable beingtestedisabletoimprovethefitofthekernelregressionY = α(.)+β(.)S +ε ,relativetothealternativelisted. t t t Panel (a) tests models in which the sensitivity to news surprises does not vary against models in which the sensitivity varieswithoneofthealternativecontrollingvariables. Panel(b)testswhetheramodelincludingjustthedesiredpolicy rateisequivalenttoamodelincludingoneofthealternativecontrollingvariablesalongwiththedesiredpolicyrate. See TableB.1forcontrollingvariabledefinitionsandsummarystatistics,andSections4.3through4.5formoredetail. Table7: StructuralVARDecompositionoftheCorrelationbetweenOilandEquityReturns ———————–Contributionof——————— Correlation OilSupply Agg. Demand OilResid. EquityResid. ρ (h) σpe,1(h) σpe,2(h) σpe,3(h) σpe,4(h) pe σp(h)σe(h) σp(h)σe(h) σp(h)σe(h) σp(h)σe(h) Jan. 1974–Mar. 2009 -0.100 0.000 0.020 -0.114 -0.006 Apr. 2009–Dec. 2017 0.327 0.043 0.013 0.240 0.031 Notes: The table reports the structural decomposition of the correlation between monthly oil and equity returns based on the vector autoregression (VAR) described in Section 5. The decomposition is based on Equation 25. The VAR is estimatedindependentlyforeachreportedsample,andthevalueofh=1000. 50

For Online Publication Oil, Equities, and the Zero Lower Bound Deepa Datta, Benjamin K. Johannsen, Hannah Kwon, and Robert J. Vigfusson* August 16, 2018 Abstract The following appendixes are supplementary material for our paper “Oil, Equities, and the ZeroLowerBound.” AppendixBcontainsadditionaltablesandfiguresfromourempiricalwork. Appendix C contains details about our benchmark New Keynesian model with oil. Appendix D containsdetailsaboutourtwo-countryNewKeynesianmodelwithoil. ∗The views in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System. We thank Martin Bodenstein, Craig Burnside, Franc¸ois Gourio, Luca Guerrieri, Lee Smith, Johannes Wieland,andJingCynthiaWu,forhelpfulcommentsanddiscussion. WethankAnastaciaDialynasforhercontributions totheinitialempiricalinvestigations. Commentsandsuggestionscanbedirectedtorobert.j.vigfusson@frb.gov. 1

B Additional figures and tables FigureB.1: OilandEquityCorrelation-Robustness The rolling window correlations between oil and equity returns are presented here. The four panels illustratethecorrelationsforreturnscalculatedoverdaily,weekly,monthly,andquarterlyfrequencies. Thelinesineachpanelshowtherollingwindowsofvariouslengths(1monthupto3years). (a)DailyReturns (b)WeeklyReturns 1 1 0.5 0.5 0 0 1 mo 3 mo 3 mo -0.5 6 mo -0.5 6 mo 2 yr 2 yr 3 yr 3 yr 1 yr 1 yr -1 -1 1985 1990 1995 2000 2005 2010 2015 1985 1990 1995 2000 2005 2010 2015 (c)MonthlyReturns (d)QuarterlyReturns 1 1 0.5 0.5 0 0 -0.5 6 mo -0.5 2 yr 2 yr 3 yr 3 yr 1 yr 1 yr -1 -1 1985 1990 1995 2000 2005 2010 2015 1985 1990 1995 2000 2005 2010 2015 2

FigureB.2: Oil-EquityRollingCorrelation,Japan (a)Daily,Dollars (b)Daily,Yen 1 1 0.5 0.5 0 0 1 yr 1 yr -0.5 -0.5 2 yr 2 yr 3 yr 3 yr -1 -1 1985 1990 1995 2000 2005 2010 2015 1985 1990 1995 2000 2005 2010 2015 (c)Weekly,Dollars (d)Weekly,Yen 1 1 0.5 0.5 0 0 1 yr 1 yr -0.5 -0.5 2 yr 2 yr 3 yr 3 yr -1 -1 1985 1990 1995 2000 2005 2010 2015 1985 1990 1995 2000 2005 2010 2015 (e)Monthly,Dollars (f)Monthly,Yen 1 1 0.5 0.5 0 0 1 yr 1 yr -0.5 -0.5 2 yr 2 yr 3 yr 3 yr -1 -1 1985 1990 1995 2000 2005 2010 2015 1985 1990 1995 2000 2005 2010 2015 Note: Legend labels correspond to length of rolling window. Correlations dated at end of rolling window. Currency conversion done using exchange rates from the H.10 release from the Federal Reserve. Period returns computed as log changes in equity index on last trading day of each period. Oilreturnscomputedaslogchangesinoilpriceonlasttradingdayofeachperiod. 3

FigureB.3: Oil-EquityRollingCorrelation,Mexico (a)Daily,Dollars (b)Daily,Pesos 1 1 0.5 0.5 0 0 1 yr 1 yr -0.5 -0.5 2 yr 2 yr 3 yr 3 yr -1 -1 1985 1990 1995 2000 2005 2010 2015 1985 1990 1995 2000 2005 2010 2015 (c)Weekly,Dollars (d)Weekly,Pesos 1 1 0.5 0.5 0 0 1 yr 1 yr -0.5 -0.5 2 yr 2 yr 3 yr 3 yr -1 -1 1985 1990 1995 2000 2005 2010 2015 1985 1990 1995 2000 2005 2010 2015 (e)Monthly,Dollars (f)Monthly,Pesos 1 1 0.5 0.5 0 0 1 yr 1 yr -0.5 -0.5 2 yr 2 yr 3 yr 3 yr -1 -1 1985 1990 1995 2000 2005 2010 2015 1985 1990 1995 2000 2005 2010 2015 Note: Legend labels correspond to length of rolling window. Correlations dated at end of rolling window. Currency conversion done using exchange rates from the Bank of Mexico. Period returns computed as log changes in equity index on last trading day of each period. Oil returns computed as logchangesinoilpriceonlasttradingdayofeachperiod. 4

TableB.1: SummaryStatistics: EquitySectorReturnsandMacroeconomicNewsSurprises Variable Obs. StartDate Mean St. Dev. Min. Max. PanelA:EquitySectorReturns Consumernondurables 8584 1983-Apr-06 0.05 0.95 -18.67 8.83 Consumerdurables 8584 1983-Apr-06 0.03 1.46 -20.27 9.12 Manufacturing 8584 1983-Apr-06 0.05 1.21 -22.61 9.55 Energy 8584 1983-Apr-06 0.04 1.46 -21.60 17.24 Chemicals 8584 1983-Apr-06 0.05 1.10 -21.33 9.40 Businessequipment 8584 1983-Apr-06 0.04 1.54 -22.43 14.95 Telecommunications 8584 1983-Apr-06 0.04 1.23 -18.26 13.21 Utilities 8584 1983-Apr-06 0.04 0.97 -13.77 12.67 Shops 8584 1983-Apr-06 0.05 1.15 -18.32 10.43 Healthcare 8584 1983-Apr-06 0.05 1.15 -19.71 10.29 Finance 8584 1983-Apr-06 0.04 1.42 -16.08 15.62 Other 8584 1983-Apr-06 0.03 1.18 -18.13 9.43 PanelB:MacroeconomicNewsSurprises Capacityutilization(cu) 357 1988-Apr-18 -0.01 0.35 -1.57 1.40 Consumerconfidence(con) 316 1991-Jul-30 0.25 5.12 -14.00 13.30 CoreCPI(cpi) 341 1989-Aug-18 -0.01 0.11 -0.34 0.40 GDPadvance(gdp) 123 1987-Apr-23 0.08 0.74 -1.68 1.80 Initialclaims(clm) 1303 1991-Jul-18 0.05 18.08 -85.00 94.00 ISMmanufacturing(ism) 333 1990-Feb-01 0.03 1.97 -6.30 7.40 Leadingindicators(ind) 455 1980-Feb-29 0.02 0.31 -1.80 2.00 Newhomesales(nhs) 353 1988-Mar-29 5.43 56.77 -166.00 249.00 Nonfarmpayrolls(nfp) 395 1985-Feb-01 -8.29 100.29 -328.00 408.50 CorePPI(ppi) 337 1989-Aug-11 -0.02 0.24 -1.20 1.07 Retailsalesex. autos(rtl) 454 1980-Feb-13 -0.03 0.66 -2.40 5.13 Unemploymentrate(ur) 453 1980-Feb-07 0.04 0.16 -0.60 0.60 Notes: InPanel(a),the12industry-specificequityreturnsseriesareobtainedfromtheFama-Frenchdatalibrary,andare convertedtolevels. Tocalculatereturns,wedropdayswithmissingvaluesforoil,metals,interestrates,orequities,and thencalculate“daily”returnsasthe100timesthelogdifferenceoftheseconsecutiveclosingprices. ForPanel(b)only, newssurprisesaredefinedasthedifferencebetweentheannouncedrealizationofthemacroeconomicaggregatesandthe survey expectations. Prior to use in regression analysis, each surprise is divided by the full sample standard deviation reported above. Following Beechey and Wright (2009), we flip the sign for unemployment and initial jobless claims announcementsthroughoutthepaper,sothatpositivesurprisesrepresentastronger-than-expectedeconomy. 5

TableB.2: StructuralVARDecompositionoftheCorrelationbetweenOilandEquityReturns ———————–Contributionof——————— Corr. OilSupply Agg. Demand OilResid. EquityResid. Lags ρ (h) σpe,1(h) σpe,2(h) σpe,3(h) σpe,4(h) pe σp(h)σe(h) σp(h)σe(h) σp(h)σe(h) σp(h)σe(h) OilPriceinDifferences Jan. 1974–Mar. 2009 12 -0.100 0.000 0.020 -0.114 -0.006 Jan. 1974–Mar. 2009 24 -0.125 0.001 0.011 -0.129 -0.008 Jan. 1974–Dec. 2006 12 -0.172 -0.003 0.005 -0.165 -0.010 Jan. 1974–Dec. 2006 24 -0.176 0.001 0.008 -0.178 -0.008 Apr. 2009–Dec. 2017 12 0.327 0.043 0.013 0.240 0.031 OilPriceinLevels Jan. 1974–Mar. 2009 12 -0.096 0.004 0.014 -0.111 -0.003 Jan. 1974–Mar. 2009 24 -0.138 -0.000 0.001 -0.132 -0.007 Jan. 1974–Dec. 2006 12 -0.172 -0.000 0.002 -0.166 -0.008 Jan. 1974–Dec. 2006 24 -0.183 -0.001 0.003 -0.179 -0.006 Apr. 2009–Dec. 2017 12 0.326 0.046 0.013 0.242 0.025 Notes: The table reports the structural decomposition of the correlation between monthly oil and equity returns based on the monthly VAR described in Section 5. The decomposition is based on Equation 25. The VAR is estimated independently for each reported sample. When the VAR is estimated using the log-level of the oil price (instead of the log difference),wecalculatethecorrelationanddecompositionsforoilandequityreturnsusingtheimpliedmovingaverage representationforoilreturns. Thevalueofh=1000. Boldedrowsdenoteourbenchmarkresults,asreportedinthemain text. 6

C Benchmark New Keynesian model In this appendix, we describe our benchmark New Keynesian model. We use a medium-scale New Keynesianmodelandaddendogenousoildemandalongwithexogenousoilsupplyalongthelinesof Bodensteinetal.(2013). C.1 Household Therepresentativehouseholdmaximizes E (cid:88) ∞ βj (cid:32)(cid:0) C t+j −hC ¯ t+j−1 (cid:1)1−σ − χ L1+φ +log(η )V (cid:18) B t (cid:19) (cid:33) (C.1) t 1−σ 1+φ t+j t P C,t j=0 ¯ where C is consumption, C is average aggregate consumption, L is hours worked, B is nominal t t t t bondholdings,andP isthepriceoftheconsumptiongood. Thestochasticvariableη isapreference C,t t shifterthancapturesincreaseddesiretoholdsafenominalassets. Thebudgetconstraintis B +P C +P I = (1+R )1/4B +R K +W L +T (C.2) t C,t t Y,t t t−1 t−1 K,t t t t t where P is the price of non-oil output, R is the net annual nominal interest rate, W is the wage Y,t t t rate,R istherentalrateoncapital,K ,I isinvestment,andT arelump-sumprofitsandtaxes. The K,t t t t capitalaccumulationequationis (cid:32) (cid:33) φ (cid:18) I (cid:19)2 K t K = I 1− −1 +(1−δ)K . (C.3) t+1 t t 2 I t−1 Thedefinitionofconsumptionis C = (cid:18) ω1−ρC (Y )ρC +(1−ω )1−ρC (cid:18) O C,t (cid:19)ρC (cid:19) ρ 1 C . (C.4) t C C,t C µ C,t 7

The household creates the consumption good to minimize the cost of consumption. That is, the householdsolves min P Y +P O (C.5) Y,t C,t O,t C,t YC,t,OC,t subjecttotheconstraintthat (cid:18) ω1−ρC (Y )ρC +(1−ω )1−ρC (cid:18) O C,t (cid:19)ρC (cid:19) ρ 1 C ≥ C . (C.6) C C,t C µ t C,t Here Y is non-oil output used for consumption and O is oil that is consumed by the household. C,t C,t Thenthefirst-orderconditionsare (cid:18) (cid:19) 1 P Y,t ρC−1 Y = ω C , (C.7) C,t C t P C,t O = (cid:18) P O,t (cid:19) ρC 1 −1 C (1−ω )µρC ρC −1. (C.8) C,t P t C C,t C,t Theidealpriceindexforfinalconsumptionisgivenby P C,t = (cid:16) ω C (P Y,t )ρC ρC −1 +(1−ω C )(P O,t µ C,t )ρC ρC −1 (cid:17)ρC ρC −1 . (C.9) Thefirst-orderconditionsofthehouseholdare (cid:0) ¯ (cid:1)−σ C −hC = Λ , (C.10) t t−1 t Λ W /P = χLφ, (C.11) t t C,t t (cid:18) (cid:19) B Λ Λ = log(η )V(cid:48) t +β(1+R )1/4E t+1 , (C.12) t t t t P π C,t C,t+1 8

(cid:34)(cid:32) (cid:33) (cid:35) P φ (cid:18) I (cid:19)2 I (cid:18) I (cid:19) Y,t K t t t Λ =Q 1− −1 − φ −1 t t K P 2 I I I C,t t−1 t−1 t−1 (cid:18) I (cid:19) I2 +βE Q φ t+1 −1 t+1, (C.13) t t+1 K I I2 t t (cid:20) (cid:21) R K,t+1 Q = βE Q (1−δ)+Λ . (C.14) t t t+1 t+1 P C,t+1 C.2 Goods aggregators Perfectly competitive firms aggregate intermediate inputs into non-oil output, Y . Non-oil output is a t compositeofgoodspurchasedfrommonopolists. Wedenotethequantitypurchasedformmonopolist ibyX (i). Theintermediateinputsareaggregatedaccordingto t (cid:18)(cid:90) 1 ν−1 (cid:19) ν− ν 1 Y t = X t (i) ν di . (C.15) 0 Demandcurvesarethenoftheform (cid:18) P (i) (cid:19)−ν X,t X (i) = Y . (C.16) t t P Y,t Here, P (i) is the price of X (i). Perfect competition implies the ideal price index for Y is given X,t t t by (cid:18)(cid:90) 1 (cid:19) 1− 1 ν P = P (i)1−ν di . (C.17) Y,t X,t 0 C.3 Monopolists We introduce price stickiness as a Calvo-style price-setting friction. Monopolists are only able to optimize their price with probability ξ in each period. If monopolist i can update its price, it chooses ˜ P (i)tomaximize X,t ∞ (cid:32) ˜ (cid:33)(cid:32) ˜ (cid:33)−ν (cid:88) P (i) P (i) X,t ˜ X,t ˜ E Λ X (1+τ )−MC X Y (C.18) t t+j t,j X t+j t,j t+j P P C,t+j Y,t+j j=0 9

where    1 j = 1 ˜ X = . (C.19) t,j   π Y,t ×π Y,t+1 ×···×π Y,t+j−1 else ˜ ˜ Here,X capturesindexationtopastpricechanges. TheFOCwithrespecttoP (i)is X,t (cid:34) (cid:88) ∞ P ˜ P (cid:18) P (cid:19)−ν E (βξ)jΛ X,t C,t X ˜ Y,t X ˜ Y t t+j t,j t,j t+j P P P C,t C,t+j Y,t+j j=0 (cid:35) 1 ν (cid:18) P (cid:19)−ν Y,t ˜ − MC X Y = 0. (C.20) t+j t,j Y,t+j 1+τ ν −1 P X Y,t+j ˜ ˜ Here we set P (i) = P for all firms that can update their price because they all face the same X,t X,t problem. Thenwehave F p˜ = F (C.21) 1,t X,t 2,t ˜ wherep˜ ≡ P /P andF andF aredefinedas X,t X,t C,t 1,t 2,t (cid:88) ∞ P (cid:18) P (cid:19)−ν F ≡ E (βξ)jΛ C,t X ˜1−ν Y,t Y (C.22) 1,t t t+j P t,j P t+j C,t+j Y,t+j j=0 (cid:88) ∞ 1 ν (cid:18) P (cid:19)−ν F ≡ E (βξ)jΛ MC X ˜−ν Y,t Y . (C.23) 2,t t t+j 1+τ ν −1 t+j t,j P t+j X Y,t+j j=0 ThevariablesF andF canbeexpressedas 1,t 2,t F = Λ Y +βξE π1−νπ−1 πν F , (C.24) 1,t t t t Y,t C,t+1 Y,t+1 1,t+1 1 ν F = Λ MC Y +βξE π−νπν F , (C.25) 2,t t 1+τ ν −1 t t t Y,t Y,t+1 2,t+1 X where π ≡ P /P (C.26) Y,t Y,t Y,t−1 and π ≡ P /P . (C.27) C,t C,t C,t−1 10

Theidealpriceindexforretailgoodsevolvesaccordingto (cid:16) (cid:17) 1 P = (1−ξ)P ˜1−ν +ξπ1−ν P1−ν 1−ν (C.28) Y,t X,t Y,t−1 Y,t−1 sothat (cid:32) (cid:33) 1 p1−ν 1−ν p = (1−ξ)p˜1−ν +ξπ1−ν Y,t−1 . (C.29) Y,t X,t Y,t−1 π1−ν C,t C.4 Marginal cost In this subsection we drop the i index from firm-specific quantities. The firm solves the following costminimizationproblem min W L +R K +P O (C.30) t t K,t t O,t t Kt,Lt subjecttotheconstraintthat (cid:18) (ω )1−ρV VρV +(1−ω )1−ρV (cid:18) O X,t (cid:19)ρV (cid:19) ρ 1 V ≥ X . (C.31) V t V µ t X,t Here V = A KαL1−α (C.32) t t t t andA isastochasticprocessthatrepresentsaggregatetechnology. Thefirst-orderconditionsare t R (cid:18) L (cid:19)1−α Kt = MC (X )1−ρV (ω )1−ρV VρV−1αA t , (C.33) P t t V t t K Ct t W (cid:18) L (cid:19)−α t = MC (X )1−ρV (ω )1−ρV VρV−1(1−α)A t , (C.34) P t t V t t K C,t t P (cid:18) O (cid:19)ρV−1 1 O,t = MC (X )1−ρV (1−ω )1−ρV X,t . (C.35) t t V P µ µ C,t X,t X,t 11

C.5 Oil Market Thereisanexogenoussupplyofoil,O . Oil-marketclearingimplies t O +O = O . (C.36) C,t X,t t C.6 Goods market clearing Weassumethatoilispaidforusingnon-oiloutput. So,goodsmarketclearingimplies P O,t Y +G +I +(O +O ) = Y . (C.37) C,t t t C,t X,t t P Y,t C.7 Aggregation Aggregatingacrossfirmsyields (cid:90) 1(cid:18) P X,t (i) (cid:19)−ν Y di = (cid:90) 1(cid:18) (ω )1−ρV V (i)ρV +(1−ω )1−ρV (cid:18) O X,t (i) (cid:19)ρV (cid:19) ρ 1 V di (C.38) t V t V P µ 0 Y,t 0 X,t = (cid:90) 1(cid:18) (ω )1−ρV +(1−ω )1−ρV (cid:18) O X,t (i) (cid:19)ρV (cid:19) ρ 1 V V (i)di (C.39) V V t V (i)µ 0 t X,t = (cid:90) 1(cid:18) (ω )1−ρV +(1−ω )1−ρV (cid:18) O X,t (i) (cid:19)ρV (cid:19) ρ 1 V A (cid:18) K t (i) (cid:19)α L (i)di. (C.40) V V t t V (i)µ L (i) 0 t X,t t Fromcostminimization,theratios OX,t(i) and Kt(i) arecommonacrossfirms. Then Vt(i)µX,t Lt(i) (cid:90) 1(cid:18) P X,t (i) (cid:19)−ν Y di = (cid:18) (ω )1−ρV VρV +(1−ω )1−ρV (cid:18) O X,t (cid:19)ρV (cid:19) ρ 1 V (C.41) P t V t V µ 0 Y,t X,t sothat d−1Y = (cid:18) (ω )1−ρV VρV +(1−ω )1−ρV (cid:18) O X,t (cid:19)ρV (cid:19) ρ 1 V (C.42) t t V t V µ X,t wherethelastequationfollowswithoutthe(i)’sbecauseallfirmschoosethesamecapital-to-laborratiofromtheconstant-returns-to-scaleproductiontechnology. Herethedispersionterm,d ,represents t 12

theresourcecostsofpricedispersionandcanbewrittenrecursivelyas d = (1−ξ)(p˜ )−ν +ξπ−ν πν d . (C.43) t X,t Y,t−1 Y,t t−1 C.8 Government ˜ ThemonetaryauthorityfollowsatruncatedTaylorrule. Thedesiredpolicyrate,R evolvesaccording t to (cid:104) 1+R ˜ (cid:105)1/4 = (cid:18) (cid:104) 1+R ˜ (cid:105)1/4 (cid:19)γ (cid:32) (cid:16) [1+R]1/4 (cid:17)(cid:16)π Y,t (cid:17)θπ (cid:18) Y t (cid:19)θY (cid:33)1−γ whereθ > 1. (C.44) t t−1 π YN π t Here, R is the steady-state annualized net nominal interest rate, π is the target rate of inflation. The natural rate of output, YN is defined as the level of output that would prevail under flexible prices, t given the entire history of shocks. The fiscal authority balances its budget with lump sum taxes so thatB = 0. Governmentpurchases,G ,followsanAR(1). Toincorporatethezerolowerbound, t t (cid:110) (cid:111) ˜ R = max 0,R . (C.45) t t C.9 Equilibrium A rational expectations equilibrium is a sequence of prices and quantities that have the property that the household and firm optimality conditions are satisfied, the goods market, labor market, and oil markets clear, and the nominal interest rate and government purchases evolve as specified. To solve for a rational expectations equilibrium, we solve for the following 24 endogenous objects: C , Λ , t t L , w ≡ Wt , Y , R , MC , π , K , I , Q , r ≡ RK,t, p ≡ PY,t, p˜ , F , F , d , π , t t PC,t t t t C,t t t t K,t PC,t Y,t PC,t X,t 1,t 2,t t Y,t V , O , Y , O , p ≡ PO,t, R ˜ . To determine these variables, we require that the following t X,t C,t C,t O,t PC,t t 24 equations hold: (C.3), (C.7), (C.8), (C.9), (C.10), (C.11), (C.12), (C.13), (C.14), (C.37), (C.21), (C.24), (C.25), (C.26), (C.29), (C.32), (C.33), (C.34), (C.35), (C.36), (C.42), (C.43), (C.44), (C.45). The budget constraint of the household clears by Walras’ law. We linearize the model around non- 13

stochastic steady state. We incorporate the zero lower bound using the methodology of Guerrieri and Iacoviello(2015). WeutilizetheOccBinsolverfromGuerrieriandIacoviello(2015),whichinteracts withDynare(seeAdjemianetal.(2011)). C.10 Steady state To determine steady state, we assume that target inflation is π. So, π = π = π. The intertemporal C Y Eulerequationdetermines(1+R ˜ )1/4 = (1+R)1/4 = πβ−1. WenormalizedL = 1. Firmoptimality and symmetry of the equilibrium imply p˜ = 1. Because of our indexation assumption, there is no X pricedispersioninsteadystate,sod = 1. Wewillnormalizethepriceofoiltobep = 1(wehaveto O findO instead). Asaresult,p = 1,meaningQ = Λ. Marginalcostisgivenby Y ν −1 MC = (1+τ ) = 1 (C.46) X ν Frompricingoptimality F = F = (1−βξ)−1ΛY (C.47) 1 2 Therentalrateofcapitalis 1−β(1−δ) r = (C.48) K β Fromournormalizationofp O O = (1−ω )C (C.49) C C and Y = ω C (C.50) C C Themarginalutilityofconsumptiongives ([1−h]C)−σ = Λ (C.51) 14

Notethat I = δK (C.52) and Y = I +Y +G+O +O (C.53) C C X FromthedefinitionofV wehave V = Kα (C.54) Thismeans δK +C +G+O X = (cid:0) (ω V )1−ρV (Kα)ρV +(1−ω V )1−ρV (O X )ρV (cid:1) ρ 1 V (C.55) Weknowthatcostminimizationimplies r K = MC (cid:0) (ω V )1−ρV VρV +(1−ω V )1−ρV (O X )ρV (cid:1)1− ρV ρV (ω)1−ρV VρV−1α (cid:18) 1 (cid:19)1−α (C.56) K w = MC (cid:0) (ω V )1−ρV VρV +(1−ω V )1−ρV (O X )ρV (cid:1)1− ρV ρV (cid:16)ω V (cid:17)1−ρV (1−α) (cid:18) 1 (cid:19)−α (C.57) V K 1 = MC (cid:0) (ω V )1−ρV VρV +(1−ω Y )1−ρV (O X )ρV (cid:1)1− ρV ρV (1−ω V )1−ρV (O X )ρV−1 (C.58) Meaning (cid:18) ω (cid:19)1−ρV (cid:18) Kα(cid:19)ρV−1 (cid:18) 1 (cid:19)1−α V r = α (C.59) K 1−ω O K V X and O X = r K − ρV 1 −1 (cid:18) 1− ω V ω (cid:19)−1 (Kα)αρV 1 −1 (cid:18) K 1 (cid:19) ρ 1 V − − α 1 (C.60) V So, r K = (cid:32) (ω V )1−ρV (Kα)ρV +(1−ω V )1−ρV (cid:32) r K − ρV 1 −1 (cid:18) 1− ω ω V (cid:19)−1 (Kα)αρV 1 −1 (cid:18) K 1 (cid:19) ρ 1 V − − α 1 (cid:33)ρV (cid:33)1− ρV ρV V (cid:18) 1 (cid:19)1−α ×MC(1−ω )1−ρV (Kα)ρV−1α (C.61) V K 15

with r known and MC known, we can solve for K. With K we get O , V, and then w. With K X the intratemporal Euler equation, we get χ. We have Y from production technology, C from market clearing,O fromO = (1−ω )C. WithbothO andO wehaveO. Therestfollowseasily. C C C X C 16

D Two-country model Here we extend our one-country model to a two-country environment. We assume that there are two countries,homeandforeign. Thehomecountryissizenandtheforeigncountryissize1−n. Weare onlygoingtoallownon-state-contingenthomeandforeignnominalbondstobetradedinternationally. Our model features Calvo-style sticky prices and so-called “local-currency pricing.” Again we add endogenousoildemandalongwithexogenousoilsupplyalongthelinesofBodensteinetal.(2013). D.1 Household Therepresentativehouseholdinthehomecountrymaximizes E (cid:88) ∞ βj (cid:32)(cid:0) C t+j −hC ¯ t+j−1 (cid:1)1−σ − χ L1+φ t 1−σ 1+φ t+j j=0 (cid:33) (cid:18) (cid:19) (cid:18) (cid:19) B B NER +log(η )V H,t +log(η∗)V F,t t . (D.1) t P t P C,t C,t ¯ Here C is per-capita consumption, C is average aggregate per-capita consumption, L is per-capita t t t hours worked, B is per-capita nominal home bond holdings, B is per-capita nominal foreign H,t F,t bondholdings,P isthepriceofthehomeconsumptiongoodinthehomecurrencyunit,andNER C,t t isthenominalexchangeratequotedasthepriceoftheforeigncurrencyunit. Thestochasticvariables η and η∗ are preference shifters than capture the desire to hold safe nominal assets in the home and t t foreigncurrency. Thebudgetconstraintis φ (cid:18) B NER (cid:19)2 B +B +P C +P I + b F,t t P = (1+R )1/4B (D.2) H,t F,t C,t t Y,t t Y,t t−1 H,t−1 2 P C,t + (cid:0) 1+R∗ (cid:1)1/4 B NER +R K +W L +T t−1 F,t−1 t K,t t t t t whereP isthepriceofnon-oiloutputinthehomecountry,R istheannualizednetnominalinterest Y,t t rateonthehomebond,R∗istheannualizednetnominalinterestrateonthehomebond,W isthewage t t rate, R is the rental rate on capital, K is per-capita capital holdings, I is per-capita investment, K,t t t 17

(cid:16) (cid:17)2 and T are per-capita lump-sum profits and taxes. The term φ b BF,tNERt P is a carrying cost of t 2 PC,t Y,t holdingthehome-countrybond. Fromapracticalperspective,φ issettoasmallnumberandthisterm b ensures stationarity in the model. See Schmitt-Grohe´ and Uribe (2003). The capital accumulation equationis (cid:32) (cid:33) φ (cid:18) I (cid:19)2 K t K = I 1− −1 +(1−δ)K . (D.3) t+1 t t 2 I t−1 Thedefinitionofconsumptionis C = (cid:18) ω1−ρC (Y )ρC +(1−ω )1−ρC (cid:18) O C,t (cid:19)ρC (cid:19) ρ 1 C . (D.4) t C C,t C µ OC,t Thehouseholdcreatestheconsumptiongoodtominimizecosts min P Y +P O (D.5) Y,t C,t O,t C,t YC,t,OC,t subjecttotheconstraintthat (cid:18) ω1−ρC (Y )ρC +(1−ω )1−ρC (cid:18) O C,t (cid:19)ρC (cid:19) ρ 1 C ≥ C (D.6) C C,t C µ t OC,t where Y is non-oil output used for consumption and O is oil that is consumed by the household. C,t C,t Thenthefirst-orderconditionsare (cid:18) (cid:19) 1 P Y,t ρC−1 Y = ω C (D.7) C,t C t P C,t O = (cid:18) P O,t (cid:19) ρC 1 −1 C (1−ω )µρC ρC −1 (D.8) C,t P t C OC,t C,t Theidealpriceindexforfinalconsumptionisgivenby P C,t = (cid:16) ω C (P Y,t )ρC ρC −1 +(1−ω C ) (cid:0) P O,t µ OC,t (cid:1) ρC ρC −1 (cid:17)ρC ρC −1 . (D.9) 18

Thehousehold-widefirst-orderconditionsare (cid:0) ¯ (cid:1)−σ C −hC = Λ (D.10) t t−1 t Λ W /P = χLφ (D.11) t t C,t t (cid:18) (cid:19) B Λ Λ = log(η )V(cid:48) H,t +β(1+R )1/4E t+1 (D.12) t t t t P π C,t C,t+1 (cid:18) (cid:19) B P B Λ +φ F,t NER Y,t =log(η∗)V(cid:48) F,t NER t B P t P t P t C,t C,t C,t Λ NER +β(1+R∗)1/4E t+1 t+1 (D.13) t t π NER C,t+1 t (cid:34)(cid:32) (cid:33) (cid:35) P φ (cid:18) I (cid:19)2 I (cid:18) I (cid:19) Y,t K t t t Λ =Q 1− −1 − φ −1 t t K P 2 I I I C,t t−1 t−1 t−1 (cid:18) I (cid:19) I2 +βE Q φ t+1 −1 t+1 (D.14) t t+1 K I I2 t t (cid:20) (cid:21) R K,t+1 Q = βE Q (1−δ)+Λ . (D.15) t t t+1 t+1 P C,t+1 Therepresentativeforeignhouseholdmaximizes E (cid:88) ∞ βj (cid:32)(cid:0) C t ∗ +j −hC ¯ t ∗ +j−1 (cid:1)1−σ − χ (cid:0) L∗ (cid:1)1+φ t 1−σ 1+φ t+j j=0 (cid:32) (cid:33) (cid:32) (cid:33)(cid:33) B∗ B∗ +log(η )V H,t +log(η∗)V F,t (D.16) t P∗ NER t P∗ C,t t C,t whereC∗ isper-capitaconsumption,C ¯∗ isaverageaggregateper-capitaconsumption,L∗ isper-capita t t t hours worked, B∗ is per-capita home nominal bond holdings, B∗ is per-capita foreign nominal H,t F,t bonds, and P∗ is the price of the foreign consumption good in the foreign currency unit. Note that C,t η and η∗ are the same preference shifters as for the home household. In this way, we capture global t t 19

demandforthedesiretoholdsafenominalassetsinonecurrencyoranother. Thebudgetconstraintis φ (cid:18) B∗ (cid:19)2 B∗ +P∗ C∗ +P∗ I∗ +B∗ NER−1 + b H,t P∗ = F,t C,t t Y,t t H,t t 2 P∗ NER Y,t Ct t (cid:0) 1+R∗ (cid:1)1/4 B∗ +(1+R )1/4B∗ NER−1 +R∗ K∗+W∗L∗ +T∗ (D.17) t−1 F,t−1 t−1 H,t−1 t K,t t t t t The term φ b (cid:16) B H ∗ ,t (cid:17)2 P∗ is a carrying cost of holding the home-country bond. The capital accu- 2 P t ∗NERt C,t mulationequationis (cid:32) (cid:33) φ (cid:18) I∗ (cid:19)2 K∗ = I∗ 1− K t −1 +(1−δ)K∗. (D.18) t+1 t 2 I∗ t t−1 Thenthefirst-orderconditionsforthecompositeconsumptiongoodare (cid:32) (cid:33) 1 P∗ ρC−1 Y∗ = Y,t ω C∗, (D.19) C,t P∗ C t C,t (cid:32) (cid:33) 1 O C ∗ ,t = P P O ∗ ∗ ,t ρC−1 C t ∗(1−ω C ) (cid:0) µ∗ OC,t (cid:1) ρC ρC −1 . (D.20) C,t Theidealpriceindexforfinalconsumptionisgivenby P C ∗ ,t = (cid:18) ω C (cid:0) P Y ∗ ,t (cid:1) ρC ρC −1 +(1−ω C ) (cid:16) P O ∗ ,t µ∗ OC,t (cid:17) ρC ρC −1 (cid:19)ρC ρC −1 . (D.21) Thehousehold-widefirst-orderconditionsare (cid:0) C∗ −hC ¯∗ (cid:1)−σ = Λ∗ (D.22) t t−1 t Λ∗W∗/P∗ = χ(L∗)φ (D.23) t t C,t t (cid:32) (cid:33) B∗ P∗ B∗ Λ∗ NER Λ∗ +φ H,t Y,t = log(η )V(cid:48) H,t NER−1 +β(1+R )1/4E t+1 t (D.24) t b NER P∗ P∗ t P∗ t t t π∗ NER t C,t C,t C,t C,t+1 t+1 20

(cid:32) (cid:33) B∗ Λ∗ Λ∗ = log(η∗)V(cid:48) F,t +β(1+R∗)1/4E t+1 (D.25) t t P∗ t t π∗ C,t C,t+1 (cid:34)(cid:32) (cid:33) (cid:35) P∗ φ (cid:18) I∗ (cid:19)2 I∗ (cid:18) I∗ (cid:19) Y,t Λ∗ =Q∗ 1− K t −1 − t φ t −1 P∗ t t 2 I∗ I∗ K I∗ C,t t−1 t−1 t−1 (cid:18) I∗ (cid:19)(cid:18) I∗ (cid:19)2 +βE Q φ t+1 −1 t+1 (D.26) t t+1 K I∗ I∗ t t (cid:34) (cid:35) R∗ Q∗ = βE Q∗ (1−δ)+Λ∗ K,t+1 . (D.27) t t t+1 t+1P∗ C,t+1 Notethatwedefinetherealexchangerate,RER ,sothat t NER P∗ t C,t RER = . (D.28) t P C,t D.2 Goods aggregators In each country, perfectly competitive firms aggregate country-specific intermediate inputs into Y , H,t Y ,Y∗ ,andY∗ . ThevaluesY andY arecompositesofgoodspurchasedfrommonopolistsby F,t H,t F,t H,t F,t perfectlycompetitivefirmswhoproduceusing (cid:18) 1 (cid:19) ν 1 (cid:18)(cid:90) n ν−1 (cid:19) ν− ν 1 Y H,t = X H,t (i) ν di (D.29) n 0 (cid:18) 1 (cid:19) ν 1 (cid:18)(cid:90) 1−n ν−1 (cid:19) ν− ν 1 Y F,t = X F,t (i) ν di (D.30) 1−n 0 Demandcurvesarethenoftheform 1 (cid:18) P (i) (cid:19)−ν H,t X (i) = Y (D.31) H,t H,t n P H,t and 1 (cid:18) P (i) (cid:19)−ν F,t X (i) = Y . (D.32) F,t F,t 1−n P F,t 21

Thezeroprofitcondition,alongwiththesedemandcurves,impliestheidealpriceindexisgiveby (cid:18) 1 (cid:90) n (cid:19) 1− 1 ν P = P (i)1−ν di . (D.33) H,t H,t n 0 Similarly, (cid:18) 1 (cid:90) 1−n (cid:19) 1− 1 ν P = P (i)1−ν di . (D.34) F,t F,t 1−n 0 Theforeigncountryissymmetric. Demandcurvesareoftheform (cid:32) (cid:33)−ν 1 P∗ (i) X∗ (i) = H,t Y∗ (D.35) H,t n P∗ H,t H,t and (cid:32) (cid:33)−ν 1 P∗ (i) X∗ (i) = F,t Y∗ . (D.36) F,t 1−n P∗ F,t F,t Thezeroprofitconditions,alongwiththesedemandcurves,implyidealpriceindexes (cid:18) 1 (cid:90) n (cid:19) 1− 1 ν P∗ = P∗ (i)1−ν di (D.37) H,t n H,t 0 and (cid:18) 1 (cid:90) 1−n (cid:19) 1− 1 ν P∗ = P∗ (i)1−ν di . (D.38) F,t 1−n F,t 0 D.3 Retailers Non-oiloutput,Y ,iscreatedbycombininggoodsfromcountriesHandF(Y andY )using t H,t F,t Y = (cid:0) ω1−ρ(Y )ρ +(1−ω)1−ρ(Y )ρ(cid:1) ρ 1 (D.39) t H,t F,t where ω ≡ 1 − (1−n)Ω. The value 0 < Ω ≤ 1 captures home bias if it is less than one (see Faia andMonacelli(2008)). Profitsaregivenby P (cid:0) ω1−ρ(Y )ρ +(1−ω)1−ρ(Y )ρ(cid:1) ρ 1 −P Y −P Y (D.40) Y,t H,t F,t H,t H,t F,t F,t 22

whereP isthenominalpriceofY ,P isthenominalpriceofY . Demandcurvesarethen H,t H,t F,t F,t (cid:18) (cid:19) 1 P ρ−1 H,t Y = ωY (D.41) H,t t P Y,t and (cid:18) (cid:19) 1 P ρ−1 F,t Y = (1−ω)Y . (D.42) F,t t P Y,t There is free entry for retailers, so profits are zero. Substituting demand curves into the profits expressionyieldstheidealpriceindex (cid:16) ρ ρ (cid:17)ρ− ρ 1 P Y,t = ωP H ρ− ,t 1 +(1−ω)(P F,t )ρ−1 (D.43) Non-oil output in the foreign country, Y∗, are created by combining goods for countries H and F t (Y∗ andY∗ )using H,t F,t Y∗ = (cid:0) (ω∗)1−ρ(cid:0) Y∗ (cid:1)ρ +(1−ω∗)1−ρ(cid:0) Y∗ (cid:1)ρ(cid:1) ρ 1 (D.44) t F,t H,t whereω∗ ≡ 1−nΩ∗. Thevalue0 < Ω∗ ≤ 1captureshomebiasifitislessthatone. Profitsaregiven by P∗ (cid:0) (ω∗)1−ρ(cid:0) Y∗ (cid:1)ρ +(1−ω∗)1−ρ(cid:0) Y∗ (cid:1)ρ(cid:1) ρ 1 −P∗ Y∗ −P∗ Y∗ (D.45) Y,t F,t H,t F,t F,t H,t H,t whereP∗ isthenominalpriceofY∗ ,P∗ isthenominalpriceofY∗ . Demandcurvesaregivenby H,t H,t F,t F,t (cid:32) (cid:33) 1 P∗ ρ−1 Y∗ = H,t (1−ω∗)Y∗ (D.46) H,t P∗ t Y,t and (cid:32) (cid:33) 1 P∗ ρ−1 Y∗ = F,t ω∗Y∗. (D.47) F,t P∗ t Y,t TheidealpriceindexforY∗ isgivenby t P∗ = (cid:16) ω∗ (cid:0) P∗ (cid:1) ρ− ρ 1 +(1−ω∗) (cid:0) P∗ (cid:1) ρ− ρ 1 (cid:17)ρ− ρ 1 . (D.48) Y,t F,t H,t 23

Wedefine, π ≡ P /P (D.49) Y,t Y,t Y,t−1 and π∗ ≡ P∗ /P∗ . (D.50) Y,t Y,t Y,t−1 D.4 Monopolists WeintroducepricestickinessasaCalvo-styleprice-settingfriction. Monopolistssettheirpriceinthe currency where their goods are sold (so-called “local-currency pricing”). Monopolists are only able tooptimallyupdatetheirpricewithprobabilityξ ineachperiod. IfmonopolistiinthecountryHcan optimallyupdateitsprice,itchoosesP ˜ (i)andP ˜∗ (i)tomaximize H,t H,t ∞ (cid:40)(cid:32) ˜ (cid:33)(cid:32) ˜ (cid:33)−ν (cid:88) P (i) P (i) H,t ˜ H,t ˜ E Λ X (1+τ )−MC X Y t t+j H,t,j X t+j H,t,j H,t+j P P C,t+j H,t+j j=0 (cid:32) NER P ˜∗ (i) (cid:33)(cid:32) P ˜∗ (i) (cid:33)−ν (cid:41) + t+j H,t X ˜∗ (1+τ )−MC H,t X ˜∗ Y∗ (D.51) P H,t,j X t+j P∗ H,t,j H,t+j C,t+j H,t+j where    1 j = 1 ˜ X = , (D.52) H,t,j   π H,t ×π H,t+1 ×···×π H,t+j−1 else and    1 j = 1 X ˜∗ = . (D.53) H,t,j   π∗ ×π∗ ×···×π∗ else H,t H,t+1 H,t+j−1 Here, π ≡ P /P (D.54) H,t H,t H,t−1 and π∗ ≡ P∗ /P∗ . (D.55) H,t H,t H,t−1 24

The variables X ˜ and X ˜∗ capture indexation to past price changes. The first-order condition H,t,j H,t,j ˜ withrespecttoP (i)is H,t (cid:34) ∞ ˜ (cid:88) P P E (βξ)jΛ H,t C,t X ˜ t t+j H,t,j P P C,t C,t+j j=0 (cid:35) 1 ν (cid:18) P (cid:19)−ν H,t ˜ − MC X Y = 0 (D.56) t+j H,t,j H,t+j 1+τ ν −1 P X H,t+j Thenwehave F p˜ = K (D.57) H,t H,t H,t ˜ wherep˜ ≡ P /P andF andK aregivenby H,t H,t C,t H,t H,t (cid:88) ∞ P (cid:18) P (cid:19)−ν F ≡ E (βξ)jΛ C,t X ˜ H,t X ˜ Y (D.58) H,t t t+j H,t,j H,t,j H,t+j P P C,t+j H,t+j j=0 and (cid:88) ∞ 1 ν (cid:18) P (cid:19)−ν K ≡ E (βξ)jΛ MC H,t X ˜ Y . (D.59) H,t t t+j t+j H,t,j H,t+j 1+τ ν −1 P X H,t+j j=0 Thesecanbewrittenas F = Λ Y +βξE π1−νπ−1 πν F (D.60) H,t t H,t t H,t C,t+1 H,t+1 H,t+1 and 1 ν K = Λ MC Y +βξE π−νπν K (D.61) H,t t 1+τ ν −1 t H,t t H,t H,t+1 H,t+1 X Theidealpriceindexforhomegoodsinthehomemarketisgivenby (cid:16) (cid:17) 1 P = (1−ξ)P ˜1−ν +ξπ1−ν P1−ν 1−ν . (D.62) H,t H,t H,t−1 H,t−1 Then (cid:32) (cid:33) 1 p1−ν 1−ν p = (1−ξ)p˜1−ν +ξπ1−ν H,t−1 . (D.63) H,t H,t H,t−1 π1−ν C,t 25

Thefirst-orderconditionwithrespecttoP ˜∗ (i)is H,t (cid:34) E (cid:88) ∞ (βξ)jΛ NER t+j NER t P C ∗ ,t p˜∗ P C,t X ˜∗ t t+j NER P H,tP H,t,j t C,t C,t+j j=0 (cid:35)(cid:32) (cid:33)−ν MC ν P∗ − t+j H,t X ˜∗ Y∗ = 0 (D.64) 1+τ ν −1 P∗ H,t,j H,t+j X H,t+j wherep˜∗ ≡ P ˜∗ /P∗ . Thenwehave H,t H,t C,t F∗ RER p˜∗ = K∗ (D.65) H,t t H,t H,t where (cid:88) ∞ NER P (cid:32) P∗ (cid:33)−ν F∗ ≡ E (βξ)jΛ t+j C,t X ˜∗ H,t X ˜∗ Y∗ (D.66) H,t t t+j NER P H,t,j P∗ H,t,j H,t+j j=0 t C,t+j H,t+j (cid:88) ∞ 1 ν (cid:32) P∗ (cid:33)−ν K∗ ≡ E (βξ)jΛ MC H,t X ˜∗ Y∗ . (D.67) H,t t t+j 1+τ ν −1 t+j P∗ H,t,j H,t+j j=0 X H,t+j Thesevariablescanbewrittenas NER F∗ = Λ Y∗ +βξE t+1 (cid:0) π∗ (cid:1)1−ν π−1 (cid:0) π∗ (cid:1)ν F∗ (D.68) H,t t H,t t NER H,t C,t+1 H,t+1 H,t+1 t 1 ν K∗ = Λ MC Y∗ +βξE (cid:0) π∗ (cid:1)−ν (cid:0) π∗ (cid:1)ν K∗ (D.69) H,t t 1+τ ν −1 t H,t t H,t H,t+1 H,t+1 X Theidealpriceindexforhomegoodsintheforeignmarketisgivenby (cid:18) (cid:19) 1 P∗ = (1−ξ) (cid:16) P ˜∗ (cid:17)1−ν +ξ (cid:0) π∗ P∗ (cid:1)1−ν 1−ν (D.70) H,t H,t H,t−1 H,t−1 sothat (cid:32) (cid:0) p∗ (cid:1)1−ν(cid:33) 1− 1 ν p∗ = (1−ξ) (cid:0) p˜∗ (cid:1)1−ν +ξ (cid:0) π∗ (cid:1)1−ν H,t−1 . (D.71) H,t H,t H,t−1 (cid:0) π∗ (cid:1)1−ν C,t 26

The foreign firms are symmetric. If monopolist i can update its price, it chooses P ˜∗ (i) and F,t ˜ P (i)tomaximize F,t (cid:88) ∞ (cid:40)(cid:32) P ˜∗ (i) (cid:33)(cid:32) P ˜∗ (i) (cid:33)−ν E Λ∗ F,t X ˜∗ (1+τ )−MC∗ F,t X ˜∗ Y∗ t t+j P∗ F,t,j X t+j P∗ F,t,j F,t+j j=0 t+j F,t+j (cid:32) (cid:33)(cid:32) (cid:33)−ν (cid:41) ˜ ˜ P (i) P (i) + F,t X ˜ (1+τ )−MC∗ F,t X ˜ Y (D.72) NER P∗ F,t,j X t+j P F,t,j F,t+j t+j t+j F,t+j where    1 j = 1 ˜ X = , (D.73) F,t,j   π F,t ×π F,t+1 ×···×π F,t+j−1 else and    1 j = 1 X ˜∗ = . (D.74) F,t,j   π∗ ×π∗ ×···×π∗ else F,t F,t+1 F,t+j−1 where π∗ ≡ P∗ /P∗ (D.75) F,t F,t F,t−1 and π ≡ P /P . (D.76) F,t F,t F,t−1 Thefirst-orderconditionwithrespecttoP ˜∗ (i)is F,t (cid:34) (cid:88) ∞ P ˜∗ P∗ E (βξ)jΛ∗ F,t C,t X ˜∗ t t+j P∗ P∗ F,t,j j=0 C,t C,t+j (cid:35)(cid:32) (cid:33)−ν 1 ν P∗ − MC∗ F,t X ˜∗ Y∗ = 0. (D.77) 1+τ ν −1 t+j P∗ F,t,j F,t+j X F,t+j Wewritethisas F∗ p˜∗ = K∗ (D.78) F,t F,t F,t 27

wherep˜∗ ≡ P ˜∗ /P∗ , F,t F,t C,t F∗ = Λ∗Y∗ +βξE (cid:0) π∗ (cid:1)1−ν (cid:0) π∗ (cid:1)−1(cid:0) π∗ (cid:1)ν F∗ (D.79) F,t t F,t t F,t C,t+1 F,t+1 F,t+1 and 1 ν K∗ = Λ∗ MC∗Y∗ +βξE (cid:0) π∗ (cid:1)−ν (cid:0) π∗ (cid:1)ν K∗ . (D.80) F,t t1+τ ν −1 t F,t t F,t F,t+1 F,t+1 X Theidealpriceindeximplies (cid:32) (cid:0) p∗ (cid:1)1−ν(cid:33) 1− 1 ν p∗ = (1−ξ) (cid:0) p˜∗ (cid:1)1−ν +ξ (cid:0) π∗ (cid:1)1−ν F,t−1 . (D.81) F,t F,t F,t−1 (cid:0) π∗ (cid:1)1−ν C,t wherep∗ ≡ P∗ /P∗ . Thefirst-orderconditionwithrespecttoP ˜ (i)is F,t F,t C,t F,t (cid:34) (cid:88) ∞ NER P P∗ E (βξ)jΛ∗ t C,t p˜ C,t X ˜ t t+j NER NER P∗ F,t P∗ F,t,j j=0 t+j t C,t C,t+j (cid:35) MC∗ ν (cid:18) P (cid:19)−ν t+j F,t ˜ − X Y = 0. (D.82) F,t,j F,t+j 1+τ ν −1 P X F,t+j Wecanwritethisas p˜ F,t F = K (D.83) F,t F,t RER t wherep ≡ P /P , F,t F,t C,t NER F = Λ∗Y +βξE t π1−ν (cid:0) π∗ (cid:1)−1 (π )ν F (D.84) F,t t F,t t NER F,t C,t+1 F,t+1 F,t+1 t+1 and 1 ν K = Λ∗ MC∗Y +βξE π−ν (π )ν K . (D.85) F,t t1+τ ν −1 t F,t t F,t F,t+1 F,t+1 X Thepriceindeximpliesthat (cid:32) (cid:33) 1 (p )1−ν 1−ν p = (1−ξ)(p˜ )1−ν +ξ(π )1−ν F,t−1 . (D.86) F,t F,t F,t−1 (π )1−ν C,t 28

D.5 Marginal cost In this subsection we drop the i index because it should be understood that all quantities are the quantitypurchasedbyfirmi. Thefirmsolvesthefollowingcostminimizationproblem min W L +R K +P O (D.87) t t Kt t O,t t Kt,Lt subjecttotheconstraintthat (cid:18) (ω )1−ρV VρV +(1−ω )1−ρV (cid:18) V O,t (cid:19)ρV (cid:19) ρ 1 V ≥ X (D.88) V t V µ t VO,t where V = A KαL1−α (D.89) t t t t andA andA∗ arestochasticprocesses. Thefirst-orderconditionsforthehomefirmsare t t R (cid:18) L (cid:19)1−α Kt = MC (X )1−ρV (ω )1−ρV VρV−1αA t (D.90) P t t V t t K Ct t W (cid:18) L (cid:19)−α t = MC (X )1−ρV (ω )1−ρV VρV−1(1−α)A t (D.91) P t t V t t K C,t t P (cid:18) V (cid:19)ρV−1 1 O,t = MC (X )1−ρV (1−ω )1−ρV O,t . (D.92) t t V P µ µ C,t VO,t VO,t Foreignfirmsminimize min W∗L∗ +R∗ K∗ +P∗ O∗ (D.93) t t Kt t O,t t K∗,L∗ t t subjecttotheconstraintthat (cid:32) (cid:32) V∗ (cid:33)ρV (cid:33) ρ 1 V (ω )1−ρV (V∗)ρV +(1−ω )1−ρV O,t ≥ X∗ (D.94) V t V µ∗ t VO,t where V∗ = A∗(K∗)α(L∗)1−α. (D.95) t t t t 29

Thefirst-orderconditionsfortheforeignfirmsare R∗ (cid:18) L∗ (cid:19)1−α Kt = MC∗(X∗)1−ρV (ω )1−ρV (V∗)ρV−1αA∗ t (D.96) P∗ t t V t t K∗ Ct t W∗ (cid:18) L∗ (cid:19)−α t = MC∗(X∗)1−ρV (ω )1−ρV (V∗)ρV−1(1−α)A∗ t (D.97) P∗ t t V t t K∗ C,t t P∗ (cid:32) V∗ (cid:33)ρV−1 1 O,t = MC∗(X∗)1−ρV (1−ω )1−ρV O,t . (D.98) P∗ t t V µ∗ µ∗ C,t VO,t VO,t D.6 Oil Market Thereisanexogenoussupplyofoil,O . Oil-marketclearingimplies t (cid:0) (cid:1) n(V +O )+(1−n) V∗ +O∗ = O . (D.99) O,t C,t O,t C,t t Thepriceissetflexiblysothatthemarketclearsand P = NER P∗ . (D.100) O,t t O,t D.7 Goods market clearing Weassumethatoilispaidforusingnon-oiloutput. So,goodsmarketclearingimplies P φ (cid:18) B NER (cid:19)2 O,t b F,t t Y +G +I +(O +V ) + = Y (D.101) C,t t t C,t O,t t P 2 P Y,t C,t and (cid:32) (cid:33)2 P∗ φ B∗ Y∗ +G∗ +I∗ + (cid:0) O∗ +V∗ (cid:1) O,t + b H,t = Y∗. (D.102) C,t t t C,t O,t P∗ 2 P∗ NER t Y,t C,t t The quadratic costs of bond holdings show up in the resource constraint because we assume that non-oiloutputisusedtopaythosecosts. 30

D.8 Bond market clearing We assume that only the home bond can be traded internationally and that both home and foreign bondsareinzeronetsupply. So, nb +(1−n)b∗ = 0 (D.103) H,t H,t and nb +(1−n)b∗ = 0. (D.104) F,t F,t D.9 Aggregation Aggregatingacrosshomefirmsyields (cid:90) n 1 (cid:18) P (i) (cid:19)−ν (cid:90) n 1 (cid:32) P∗ (i) (cid:33)−ν n H,t Y di+(1−n) H,t Y∗ di n P H,t n P∗ H,t 0 H,t 0 H,t = (cid:90) n(cid:18) (ω )1−ρV V (i)ρV +(1−ω )1−ρV (cid:18) V O,t (i) (cid:19)ρV (cid:19) ρ 1 V di (D.105) V t V µ 0 VO,t = (cid:90) n(cid:18) (ω )1−ρV +(1−ω )1−ρV (cid:18) V O,t (i) (cid:19)ρV (cid:19) ρ 1 V V (i)di (D.106) V V t V (i)µ 0 t VO,t = (cid:90) n(cid:18) (ω )1−ρV +(1−ω )1−ρV (cid:18) V O,t (i) (cid:19)ρV (cid:19) ρ 1 V A (cid:18) K t (i) (cid:19)α L (i)di. (D.107) V V t t V (i)µ L (i) 0 t VO,t t Duetoconstant-returns-to-scale,theratios VO,t(i) and Kt(i) arecommonacrossfirms. Then Vt(i)µVO,t Lt(i) (cid:90) n 1 (cid:18) P (i) (cid:19)−ν (cid:90) n 1 (cid:32) P∗ (i) (cid:33)−ν n H,t Y di+(1−n) H,t Y∗ di n P H,t n P∗ H,t 0 H,t 0 H,t = n (cid:18) (ω )1−ρV VρV +(1−ω )1−ρV (cid:18) V O,t (cid:19)ρV (cid:19) ρ 1 V (D.108) V t V µ VO,t sothat d Y +d∗ 1−n Y∗ = (cid:18) (ω )1−ρV VρV +(1−ω )1−ρV (cid:18) V O,t (cid:19)ρV (cid:19) ρ 1 V (D.109) H,t H,t H,t n H,t V t V µ VO,t 31

whered andd∗ areappropriatelydefined. Similarly, H,t H,t n (cid:32) (cid:32) V∗ (cid:33)ρV (cid:33) ρ 1 V d Y +d∗ Y∗ = (ω )1−ρV (V∗)ρV +(1−ω )1−ρV O,t . (D.110) F,t 1−n F,t F,t F,t V t V µ∗ VO,t Herethedispersiontermscanbewrittenrecursivelyas d = (1−ξ)pν (p˜ )−ν +ξπ−ν πν d , (D.111) H,t H,t H,t H,t−1 H,t H,t−1 d∗ = (1−ξ) (cid:0) p∗ (cid:1)ν (cid:0) p˜∗ (cid:1)−ν +ξ (cid:0) π∗ (cid:1)−ν (cid:0) π∗ (cid:1)ν d∗ , (D.112) H,t H,t H,t H,t−1 H,t H,t−1 d∗ = (1−ξ) (cid:0) p∗ (cid:1)ν (cid:0) p˜∗ (cid:1)−ν +ξ (cid:0) π∗ (cid:1)−ν (cid:0) π∗ (cid:1)ν d∗ , (D.113) F,t F,t F,t F,t−1 F,t F,t−1 d = (1−ξ)(p )ν (p˜ )−ν +ξπ−ν (π )ν d . (D.114) F,t F,t F,t F,t−1 F,t F,t−1 D.10 Government ˜ In each country, the monetary authority follows a truncated Taylor rule. The desired policy rates, R t andR ˜∗ evolvesaccordingto t (cid:16) 1+R ˜ (cid:17)1/4 = (cid:18) (cid:16) 1+R ˜ (cid:17)1/4 (cid:19)γ (cid:32) (1+R)1/4 (cid:16)π Y,t (cid:17)θπ (cid:18) Y t (cid:19)θY (cid:33)1−γ (D.115) t t−1 π YN t where θ > 1. Here, R is the steady-state annualized net nominal interest rate, π is the target rate of π inflation. Intheforeigncountry (cid:16) 1+R ˜ ∗ (cid:17)1/4 = (cid:18) (cid:16) 1+R ˜ ∗ (cid:17)1/4 (cid:19)γ∗ (cid:32) (1+R∗)1/4 (cid:18)π Y ∗ ,t (cid:19)θ π ∗ (cid:18) Y t ∗ (cid:19)θ Y ∗ (cid:33)1−γ∗ (D.116) t t π∗ YN∗ t where θ > 1. Here, R∗ is the steady-state annualized net nominal interest rate, π∗ is the target rate π of inflation. The natural rate of output, YN∗ is defined as the level of output that would prevail under t flexibleprices,giventheentirehistoryofshocks. Thefiscalauthoritiesbalancesitsbudgetwithlump sum taxes so bonds are in zero net supply. Government purchases, G and G∗, follow independent t t 32

AR(1)processes. Toincorporatethezerolowerbound, (cid:110) (cid:111) ˜ R = max 0,R . (D.117) t t Fortheforeigncountry,weignorethezerolowerbound,sothat R∗ = R ˜∗. (D.118) t t Weignorethezerolowerboundfortheforeigncountrybecausewewanttostudyhowabindinglower boundinthehomecountryaffectstheforeigncountry. D.11 Equilibrium A rational expectations equilibrium is a sequence of prices and quantities that have the property that the household and firm optimality conditions are satisfied, the goods market, labor market, and oil markets clear, and the nominal interest rate and government purchases evolve as specified. To solve forarationalexpectationsequilibrium,wesolveforthefollowing35endogenousobjects: C ,Λ ,L , t t t w ≡ Wt, Y , Y , R , MC , π , K , I , Q , r ≡ RK,t, Y , p ≡ PF,t, p ≡ PH,t, p˜ , F , t Pt H,t F,t t t C,t t t t K,t PC,t t F,t PC,t H,t PC,t H,t H,t K , d , π , p˜ , F , K , d , π , b , b , p ≡ PY,t, V , V , Y , O , p ≡ PO,t, π , H,t H,t H,t F,t F,t F,t F,t F,t H,t F,t Y,t PC,t t O,t C,t C,t O,t PC,t Y,t the35starversions,aswellas∆NER ≡ NERt andRER . t NERt−1 t We linearize the model around non-stochastic steady state. Given parameter values, we study the uniqueboundedrationalexpectationsequilibriumfromthelinearizedmodel. Todeterminethesevariables, we require that the linearized versions following 72 equations hold: (D.3), (D.7), (D.8), (D.9), (D.10),(D.11),(D.12),(D.13),(D.14),(D.15),(D.18),(D.19),(D.20),(D.21),(D.22),(D.23),(D.24), (D.25), (D.26), (D.27), (D.28), (D.41), (D.42), (D.43), (D.46), (D.47), (D.48), (D.101), (D.102), (D.103), (D.104), (D.57), (D.60), (D.61), (D.54), (D.63), (D.65), (D.68), (D.69), (D.55), (D.71), (D.78),(D.79),(D.80),(D.75),(D.81),(D.83),(D.84),(D.85),(D.76),(D.86),(D.89),(D.90),(D.91), (D.92),(D.95),(D.96),(D.97),(D.98),(D.99),(D.100),(D.109),(D.110),(D.111),(D.112),(D.113), (D.114),(D.115),(D.116),(D.49),(D.50),(D.17). Thehomehouseholdbudgetconstraint(D.2)clears 33

by Walras’ law. Note that to solve for the natural rate of output, we find the equilibrium of a similar economywhereξ = 0. Welinearizethemodelaroundnon-stochasticsteadystate. Weincorporatethe zero lower bound using the methodology of Guerrieri and Iacoviello (2015). We utilize the OccBin solverfromGuerrieriandIacoviello(2015),whichinteractswithDynare(seeAdjemianetal.(2011)). D.12 Steady State We assume that government policy is symmetric between the home and foreign county and that the target inflation rate is π. So, π = π∗ = π = π∗ = π = π∗ = π = π∗ = π. The intertemporal C C H H F F Y Y Euler equations determine (1+R)1/4 = (1+R∗)1/4 = πβ−1. We normalized L = L∗ = 1 (we will have to find χ instead of L). From the definition of steady state, with symmetric inflation targets ∆NER = 1. We define initial conditions so that RER = 1. In our steady state, there are no net homebondholdingsintheforeigncountrybecauseofthequadraticcostsofholdingthem. Similarly, there are no net foreign bond holdings in the home country. From firm optimality and symmetry of the equilibrium, p = p∗ = p = p∗ = 1. This also gives us that p˜ = p˜∗ = p˜ = p˜∗ = 1. H H F F H H F F Because of our inflation indexation assumption, there is no price dispersion in steady state, so d = H d = d∗ = d∗ = 1. We will normalize the price of oil to be p = p∗ = 1 (we have to find O F H F O O instead). Asaresult,p = p∗ = 1,meaningQ = ΛandQ∗ = Λ∗. Marginalcostisgivenby Y Y ν −1 MC = MC∗ = (1+τ ). (D.119) X ν Frompricingoptimality F = K = (1−βξ)−1Λ∗Y (D.120) F F F F∗ = K∗ = (1−βξ)−1Λ∗Y∗ (D.121) F F F F = K = (1−βξ)−1ΛY (D.122) H H H F∗ = K∗ = (1−βξ)−1ΛY∗. (D.123) H H H 34

Therentalrateofcapitalis 1−β(1−δ) r = r∗ = . (D.124) K K β Fromournormalizationofp andp∗, O O O = (1−ω )C (D.125) C C O∗ = (1−ω )C∗ (D.126) C C and Y = ω C (D.127) C C Y∗ = ω C∗. (D.128) C C Themarginalutilityofconsumptionimplies (C[1−h])−σ = Λ (D.129) (C∗[1−h])−σ = Λ∗ (D.130) Notethat Y = (1−(1−n)Ω)Y (D.131) H Y = (1−n)ΩY (D.132) F Y∗ = nΩ∗Y∗ (D.133) H Y∗ = (1−nΩ∗)Y∗ (D.134) F and I = δK (D.135) I∗ = δK∗ (D.136) 35

Y = I +Y +G+O +V (D.137) C C O Y∗ = I∗ +Y∗ +G∗ +O∗ +V∗. (D.138) C C O Our aggregate variables are expressed in per-capita terms, and we are going to consider a symmetric steadystatewhereY = Y∗. From d H Y H +d∗ H 1− n n Y H ∗ = (cid:0) (ω V )1−ρV VρV +(1−ω V )1−ρV (V O )ρV (cid:1) ρ 1 V (D.139) weget (1−(1−n)Ω)Y +(1−n)Ω∗Y∗ = Y = Y∗ (D.140) where Y = (cid:0) (ω V )1−ρV VρV +(1−ω V )1−ρV (V O )ρV (cid:1) ρ 1 V . (D.141) Wecanseethisfrom d F 1− n n Y F +d∗ F Y F ∗ = (cid:0) (ω V )1−ρV (V∗)ρV +(1−ω V )1−ρV (V O ∗)ρV (cid:1) ρ 1 V (D.142) whichyields nΩY +(1−nΩ∗)Y∗ = (cid:0) (ω V )1−ρV (V∗)ρV +(1−ω V )1−ρV (V O ∗)ρV (cid:1) ρ 1 V (D.143) whichmeanstheequalitiesabovehold. Thismeans δK +ω C C +G = (cid:0) (ω V )1−ρV (Kα)ρV +(1−ω V )1−ρV (V O )ρV (cid:1) ρ 1 V . (D.144) FromthedefinitionofV wehave V = Kα (D.145) and V∗ = (K∗)α. (D.146) 36

Define X ≡ (cid:0) (ω V )1−ρV VρV +(1−ω V )1−ρV (V O )ρV (cid:1) ρ 1 V . (D.147) Fromcostminimization,weknowthat (cid:18) 1 (cid:19)1−α r = MC(X)1−ρV (ω )1−ρV VρV−1α (D.148) K V K (cid:18) 1 (cid:19)−α w = MC(X)1−ρV (ω )1−ρV VρV−1(1−α) (D.149) V K 1 = MC(X)1−ρV (1−ω )1−ρV (V )ρV−1. (D.150) V O Define X∗ ≡ (cid:0) (ω V )1−ρV (V∗)ρV +(1−ω V )1−ρV (V O ∗)ρV (cid:1) ρ 1 V . (D.151) Fromcostminimization,weknowthat (cid:18) 1 (cid:19)1−α r∗ = MC∗(X∗)1−ρV (ω )1−ρV (V∗)ρV−1α (D.152) K V K∗ (cid:18) 1 (cid:19)−α w∗ = MC∗(X∗)1−ρV (ω )1−ρV (V∗)ρV−1(1−α) (D.153) t V K∗ 1 = MC∗(X∗)1−ρV (1−ω )1−ρV (V∗)ρV−1. (D.154) t V O Then (cid:18) ω (cid:19)1−ρV (cid:18) Kα(cid:19)ρV−1 (cid:18) 1 (cid:19)1−α V r = α . (D.155) K 1−ω V K V O Then V O = r K − ρV 1 −1 (cid:18) 1− ω V ω (cid:19)−1 (Kα)αρV 1 −1 (cid:18) K 1 (cid:19) ρ 1 V − − α 1 . (D.156) V 37

So r K = (cid:32) (ω V )1−ρV (Kα)ρV +(1−ω V )1−ρV (cid:32) r K − ρV 1 −1 (cid:18) 1− ω V ω (cid:19)−1 (Kα)αρV 1 −1 (cid:18) K 1 (cid:19) ρ 1 V − − α 1 (cid:33)ρV (cid:33)1− ρV ρV V (cid:18) 1 (cid:19)1−α ×MC(ω )1−ρV (Kα)ρV−1α (D.157) V K with r known and MC known, we can solve for K. With K we get V and then w. With the K householdintratemporalEulerequation,wegetχ. WehaveY from Y = (cid:0) (ω V )1−ρV VρV +(1−ω V )1−ρV (V O )ρV (cid:1) ρ 1 V (D.158) Weknow Y +G+I +(O +V ) = Y (D.159) C C O andY = ω C andO = (1−ω )C meaning C C C C C +G+I +V = Y (D.160) O With C, we get Y and O . Combined with V (and the star versions), we get O. The rest follows C C O easily. D.13 Calibration and solution strategy For parameters that are common with our one-country model, we use the same values as in our onecountry model, which are specified in Section 3.5. We set n = 0.9 so that the large country has size 0.9 and the small country has size 0.1. We assume that monetary policy is symmetric across the two countriesandthatthetargetlevelofinflationis2percentatanannualizedrate. WesetΩ = Ω∗ = 0.4 to incorporate home bias. This value implies that in steady state 96 percent of non-oil expenditure in the big country is on goods from the big country. In the small country, in steady state 64 percent of 38

non-oilexpenditureisongoodsfromthesmallcountry. Wesetρ = 1/3sothattheelasticitybetween domesticandforeigngoodsis1.5. Wecomputethenaturalrateofoutputasthelevelofoutputunderflexiblepricesinbothcountries. As in our one-country model, we solve the mode using the methodology of Guerrieri and Iacoviello (2015). Their solution strategy involves a first-order perturbation to the model, which is applied piecewise so as to accommodate the ZLB. We only ever impose the ZLB in one country or the other. The main advantage of using the methodology of Guerrieri and Iacoviello (2015) is that it is able to accommodatethenumberofstatevariablesimpliedbymedium-scaleDSGEmodels. Inourcase,the numberofstatevariablesisevenlargerbecauseofthesecondcountry. 39