The Taylor Rule and Interval Forecast For Exchange Rates

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 963 January 2009 Revised March 2009 The Taylor Rule and Interval Forecast For Exchange Rates Jian Wang and Jason J. Wu NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from Social Science Research Network electronic library at http://www.ssrn.com/.

The Taylor Rule and Interval Forecast For Exchange Rates Jian Wang and Jason J. Wu* Abstract: This paper attacks the Meese-Rogoff (exchange rate disconnect) puzzle from a different perspective: out-of-sample interval forecasting. Most studies in the literature focus on point forecasts. In this paper, we apply Robust Semi-parametric (RS) interval forecasting to a group of Taylor rule models. Forecast intervals for twelve OECD exchange rates are generated and modified tests of Giacomini and White (2006) are conducted to compare the performance of Taylor rule models and the random walk. Our contribution is twofold. First, we find that in general, Taylor rule models generate tighter forecast intervals than the random walk, given that their intervals cover out-of-sample exchange rate realizations equally well. This result is more pronounced at longer horizons. Our results suggest a connection between exchange rates and economic fundamentals: economic variables contain information useful in forecasting the distributions of exchange rates. The benchmark Taylor rule model is also found to perform better than the monetary and PPP models. Second, the inference framework proposed in this paper for forecast-interval evaluation can be applied in a broader context, such as inflation forecasting, not just to the models and interval forecasting methods used in this paper. Keywords: the Exchange Rate Disconnect Puzzle; Exchange Rate Forecast; Interval Forecasting. JEL Codes: C14, C53, and F31 * Author notes: Wang is a senior economist in the Research Department of the Federal Reserve Bank of Dallas. He can be reached at Jian.Wang@dal.frb.org. Wu is an economist in the Division of Banking Supervision and Regulation of the Federal Reserve Board. He can be reached at Jason.J.Wu@frb.gov. We wish to thank Menzie Chinn, Charles Engel, Bruce Hansen, Jesper Linde, Enrique Martinez-Garcia, Tanya Moldtsova, David Papell, Mark Wynne, and Ken West for invaluable discussions. 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.

1 Introduction Recentstudiesexploretheroleofmonetarypolicyrules,suchasTaylorrules,inexchangeratedetermination. They find empirical support in these models for the linkage between exchange rates and economic fundamentals. Our paper extends this literature from a different perspective: interval forecasting. We find that the Taylor rule models can outperform the random walk, especially at long horizons, in forecasting twelve OECD exchange rates based on relevant out-of-sample interval forecasting criteria. The benchmark Taylor rule model is also found to perform relatively better than the standard monetary model and the purchasing power parity (PPP) model. Inaseminalpaper,MeeseandRogoff(1983)findthateconomicfundamentals-suchasthemoneysupply, trade balance and national income - are of little use in forecasting exchange rates. They show that existing models cannot forecast exchange rates better than the random walk in terms of out-of-sample forecasting accuracy. This finding suggests that exchange rates may be determined by something purely random rather than economic fundamentals. Meese and Rogoff’s (1983) finding has been named the Meese-Rogoff puzzle in the literature. In defending fundamental-based exchange rate models, various combinations of economic variables and econometricmethodshavebeenusedinattemptstooverturnMeeseandRogoff’sfinding. Forinstance,Mark (1995)findsgreaterexchangeratepredictabilityatlongerhorizons.1 Groen(2000)andMarkandSul(2001) detect exchange rate predictability by using panel data. Kilian and Taylor (2003) find that exchange rates can be predicted from economic models at horizons of 2 to 3 years, after taking into account the possibility of nonlinear exchange rate dynamics. Faust, Rogers, and Wright (2003) find that the economic models consistently perform better using real-time data than revised data, though they do not perform better than the random walk. Recently,thereisagrowingstrandofliteraturethatusesTaylorrulestomodelexchangeratedetermination. EngelandWest(2005)derivetheexchangerateasapresent-valueassetpricefromaTaylorrulemodel. They also find a positive correlation between the model-based exchange rate and the actual real exchange rate between the US dollar and the Deutschmark (Engel and West, 2006). Mark (2007) examines the role of Taylor-rule fundamentals for exchange rate determination in a model with learning. In his model, agents use least-square learning rules to acquire information about the numerical values of the model’s coefficients. He finds that the model is able to capture six major swings of the real Deutschmark-Dollar exchange rate from 1973 to 2005. Molodtsova and Papell (2009) find significant short-term out-of-sample predictability 1ChinnandMeese(1995)andMacDonaldandTaylor(1994)findsimilarresults. However, thelong-horizonexchangerate predictabilityinMark(1995)hasbeenchallengedbyKilian(1999)andBerkowitzandGiorgianni(2001)insubsequentstudies.

of exchange rates with Taylor-rule fundamentals for 11 out of 12 currencies vis-´a-vis the U.S. dollar over the post-Bretton Woods period. Molodtsova, Nikolsko-Rzhevskyy, and Papell (2008a, 2008b) find evidence of out-of-sample predictability for the dollar/mark nominal exchange rate with forecasts based on Taylor rule fundamentals using real-time data, but not revised data. Chinn (forthcoming) also finds that Taylor rule fundamentals do better than other models at the one year horizon. With a present-value asset pricing model as discussed in Engel and West (2005), Chen and Tsang (2009) find that information contained in the cross-country yield curves are useful in predicting exchange rates. OurpaperjoinstheaboveliteratureofTaylor-ruleexchangeratemodels. However,weaddresstheMeese and Rogoff puzzle from a different perspective: interval forecasting. A forecast interval captures a range in which the exchange rate may lie with a certain probability, given a set of predictors available at the time of forecast. Our contribution to the literature is twofold. First, we find that for twelve OECD exchange rates, the Taylor rule models in general generate tighter forecast intervals than the random walk, given that their intervals cover the realized exchange rates (statistically) equally well. This finding suggests an intuitive connection between exchange rates and economic fundamentals beyond point forecasting: the use of economic variables as predictors helps narrow down the range in which future exchange rates may lie, compared to random walk forecast intervals. Second, we propose an inference framework for cross-model comparison of out-of-sample forecast intervals. The proposed framework can be used for forecast-interval evaluation in a broader context, not just for the models and methods used in this paper. For instance, the framework can also be used to evaluate out-of-sample inflation forecasting. As we will discuss later, we in fact derive forecast intervals from estimates of the distribution of changes in the exchange rate. Hence, in principle, evaluations across models can be done based on distributions instead of forecast intervals. However, focusing on interval forecasting performance allows us to compare models in two dimensions that are more relevant to practitioners: empirical coverage and length. While the literature on interval forecasting for exchange rates is sparse, several authors have studied out-of-sample exchange rate density (distribution) forecasts, from which interval forecasts can be derived. Diebold, Hahn and Tay (1999) use the RiskMetrics model of JP Morgan (1996) to compute half-houraheaddensityforecastsforDeutschmark/DollarandYen/Dollarreturns. ChristoffersenandMazzotta(2005) provide option-implied density and interval forecasts for four major exchange rates. Boero and Marrocu (2004)obtainone-day-aheaddensityforecastsfortheEuronominaleffectiveexchangerateusingself-exciting threshold autoregressive (SETAR) models. Sarno and Valente (2005) evaluate the exchange rate density forecasting performance of the Markov-switching vector equilibrium correction model that is developed by Clarida, Sarno, Taylor and Valente (2003). They find that information from the term structure of forward 2

premia help the model to outperform the random walk in forecasting out-of-sample densities of the spot exchange rate. More recently, Hong, Li, and Zhao (2007) construct half-hour-ahead density forecasts for Euro/Dollar and Yen/Dollar exchange rates using a comprehensive set of univariate time series models that capture fat tails, time-varying volatility and regime switches. There are several common features across the studies listed above, which make them different from our paper. First, the focus of the above studies is not to make connections between the exchange rate and economicfundamentals. Thesestudiesusehighfrequencydata,whicharenotavailableformostconventional economic fundamentals. For instance, Diebold, Hahn, and Tay (1999) and Hong, Li, and Zhao (2007) use intra-daydata. WiththeexceptionofSarnoandValente(2005),allthestudiesfocusonlyonunivariatetime series models. Second, these studies do not consider multi-horizon-ahead forecasts, perhaps due to the fact that their models are often highly nonlinear. Iterating nonlinear density models multiple horizons ahead is analytically difficult, if not infeasible. Lastly, the above studies assume that the densities are analytically definedforagivenmodel. Thesemiparametricmethodusedinthispaperdoesnotimposesuchrestrictions. Our choice of the semiparametric method is motivated by the difficulty of using macroeconomic models in exchange rate interval forecasting: these models typically do not describe the future distributions of exchange rates. For instance, the Taylor rule models considered in this paper do not describe any features of the data beyond the conditional means of future exchange rates. We address this difficulty by applying Robust Semiparametric forecast intervals (hereon RS forecast intervals) of Wu (2009).2 This method is useful since it does not require the model be correctly specified, or contain parametric assumptions about the future distribution of exchange rates. We apply RS forecast intervals to a set of Taylor rule models that differ in terms of the assumptions on policy and interest rate smoothing rules. Following Molodtsova and Papell (2009), we include twelve OECD exchange rates (relative to the US dollar) over the post-Bretton Woods period in our dataset. For these twelve exchange rates, the out-of-sample RS forecast intervals at different forecast horizons are generated fromtheTaylorrulemodelsandthencomparedwiththoseoftherandomwalk. Theempiricalcoveragesand lengths of forecast intervals are used as the evaluation criteria. Our empirical coverage and length tests are modified from Giacomini and White’s (2006) predictive accuracy tests in the case of rolling, but fixed-size, estimation samples. For a given nominal coverage (probability), the empirical coverage of forecast intervals derived from a forecasting model is the probability that the out-of-sample realizations (exchange rates) lie in the intervals. The length of the intervals is a measure of its tightness: the distance between its upper and lower bound. 2Forbrevity,weomitRSandsimplysayforecastintervalswhenwebelievethatitcausesnoconfusion. 3

In general, the empirical coverage is not the same as its nominal coverage. Significantly missing the nominal coverage indicates poor quality of the model and intervals. One certainly wants the forecast intervals to contain out-of-sample realizations as close as possible to the probability they target. Most evaluation methods in the literature focus on comparing empirical coverages across models, following the seminal work of Christoffersen (1998). Following this literature, we first test whether forecast intervals of the Taylor rule models and the random walk have equally accurate empirical coverages. The model with more accurate coverages is considered the better model. In the cases where equal coverage accuracy cannot be rejected, we further test whether the lengths of forecast intervals are the same. The model with tighter forecast intervals provides more useful information about future values of the data, and hence is considered as a more useful forecasting model. It is also important to establish what this paper is not attempting. First, the inference procedure does not carry the purpose of finding the correct model specification. Rather, inference is on how useful models areingeneratingforecastintervals,measuredintermsofempiricalcoveragesandlengths. Second,thispaper does not consider the possibility that there might be alternatives to RS forecast intervals for the exchange ratemodelsweconsider. Somemodelsmightperformbetterifparametricdistributionassumptions(e.g. the forecast errors are conditionally heteroskedastic and t−distributed) or other assumptions (e.g. the forecast errors are independent of the predictors) are added. One could presumably estimate the forecast intervals differently based on the same models, and then compare those with the RS forecast intervals, but this is out of the scope of this paper. As we described, we choose the RS method for the robustness and flexibility achieved by the semiparametric approach. Our benchmark Taylor rule model is from Engel and West (2005) and Engel, Wang, and Wu (2008). For the purpose of comparison, several alternative Taylor rule models are also considered. These setups have been studied by Molodtsova and Papell (2009) and Engel, Mark, and West (2007). In general, we find that the Taylor rule models perform better than the random walk model, especially at long horizons: the models either have more accurate empirical coverages than the random walk, or in cases of equal coverage accuracy, the models have tighter forecast intervals than the random walk. The evidence of exchange rate predictabilityismuchweakerincoverageteststhaninlengthtests. Inmostcases,theTaylorrulemodelsand therandomwalkhavestatisticallyequallyaccurateempiricalcoverages. So,undertheconventionalcoverage test,therandomwalkmodelandtheTaylorrulemodelsperformequallywell. However,theresultsoflength testssuggestthatTaylorrulefundamentalsareusefulingeneratingtighterforecastsintervalswithoutlosing accuracy in empirical coverages. We also consider two other popular models in the literature: the monetary model and the model of 4

purchasing power parity (PPP). Based on the same criteria, both models are found to perform better than the random walk in interval forecasting. As with the Taylor rule models, most evidence of exchange rate predictability comes from the length test: economic models have tighter forecast intervals than the random walk given statistically equivalent coverage accuracy. The PPP model performs worse than the benchmark Taylor rule model and the monetary model at short horizons. The benchmark Taylor rule model performs slightly better than the monetary model at most horizons. Our findings suggest that exchange rate movements are linked to economic fundamentals. However, we acknowledge that the Meese-Rogoff puzzle remains difficult to understand. Although Taylor rule models offerstatisticallysignificantlengthreductionsovertherandomwalk,thereductionoflengthisquantitatively small, especially at short horizons. Forecasting exchange rates remains a difficult task in practice. There are some impressive advances in the literature, but most empirical findings remain fragile. As mentioned in Cheung,Chinn,andPascual(2005),forecastsfromeconomicfundamentalsmayworkwellforsomecurrencies duringcertainsampleperiodsbutnotforothercurrenciesorsampleperiods. Engel, Mark, andWest(2007) recently show that a relatively robust finding is that exchange rates are more predictable at longer horizons, especially when using panel data. We find greater predictability at longer horizons in our exercise. It would be of interest to investigate connections between our findings and theirs. Several recent studies have attacked the puzzle from a different angle: there are reasons that economic fundamentals cannot forecast the exchange rate, even if the exchange rate is determined by these fundamentals. Engel and West (2005) show that existing exchange rate models can be written in a present-value asset-pricing format. In these models, exchange rates are determined not only by current fundamentals but alsobyexpectationsofwhatthefundamentalswillbeinthefuture. Whenthediscountfactorislarge(close to one), current fundamentals receive very little weight in determining the exchange rate. Not surprisingly, the fundamentals are not very useful in forecasting. Nason and Rogers (2008) generalize the Engel-West theorem to a class of open-economy dynamic stochastic general equilibrium (DSGE) models. Other factors such as parameter instability and mis-specification (for instance, Rossi 2005) may also play important roles in understanding the puzzle. It is interesting to investigate conditions under which we can reconcile our findings with these studies. Theremainderofthispaperisorganizedasfollows. Sectiontwodescribestheforecastingmodelsweuse, aswellasthedata. Insectionthree, weillustratehowtheRSforecastintervalsareconstructedfromagiven model. We also propose loss criteria to evaluate the quality of the forecast intervals and test statistics that arebasedonGiacominiandWhite(2006). Sectionfourpresentsresultsofout-of-sampleforecastevaluation. Finally, section five contains concluding remarks. 5

2 Models and Data Seven models are considered in this paper. Let m = 1,2,...,7 be the index of these models and the first model be the benchmark model. A general setup of the models takes the form of: s −s =α +β0 X +ε , (1) t+h t m,h m,h m,t m,t+h wheres −s ish-periodchangesofthe(log)exchangerate,andX containseconomicvariablesthatare t+h t m,t usedinmodelm. Followingtheliteratureoflong-horizonregressions, bothshort-andlong-horizonforecasts are considered. Models differ in economic variables that are included in matrix X . In the benchmark m,t model, (cid:20) (cid:21) X 1,t ≡ π t −π t ∗ y t gap−y t gap∗ q t , where π (π∗) is the inflation rate, and ygap (ygap∗) is the output gap in the home (foreign) country. The t t t t real exchange rate q is defined as q ≡ s +p∗−p , where p (p∗) is the (log) consumer price index in the t t t t t t t home (foreign) country. This setup is motivated by the Taylor rule model in Engel and West (2005) and Engel, Wang, and Wu (2008). The next subsection describes this benchmark Taylor rule model in detail. We also consider the following models that have been studied in the literature: (cid:20) (cid:21) • Model 2: X 2,t ≡ π t −π t ∗ y t gap−y t gap∗ (cid:20) (cid:21) • Model 3: X 3,t ≡ π t −π t ∗ y t gap−y t gap∗ i t−1 −i∗ t−1 , where i t (i∗ t ) is the short-term interest rate in the home (foreign) country. (cid:20) (cid:21) • Model 4: X 4,t ≡ π t −π t ∗ y t gap−y t gap∗ q t i t−1 −i∗ t−1 • Model 5: X ≡q 5,t t (cid:20) (cid:21) • Model 6: X 6,t ≡ s t −[(m t −m∗ t )−(y t −y t ∗)] , where m t (m∗ t ) is the money supply and y t (y t ∗) is total output in the home (foreign) country. • Model 7: X ≡0 7,t Models 2-4 are the Taylor rule models studied in Molodtsova and Papell (2009). Model 2 can be considered as the constrained benchmark model in which PPP always holds. Molodtsova and Papell (2009) include interestratelagsinmodels3and4totakeintoaccountpotentialinterestratesmoothingrulesofthecentral 6

bank.3 Model 5 is the purchasing power parity (PPP) model and model 6 is the monetary model. Both models have been widely used in the literature. See Molodtsova and Papell (2009) for the PPP model and Mark (1995) for the monetary model. Model 7 is the driftless random walk model (α ≡ 0).4 Given a 7,h dateτ andhorizonh, theobjectiveistoestimatetheforecastdistributionofs −s conditionalonX , τ+h τ m,τ and subsequently build forecast intervals from the estimated forecast distribution. Before moving to the econometric method, we first describe the Taylor rule model that motivates the setup of our benchmark model. 2.1 Benchmark Taylor Rule Model Our benchmark model is the Taylor rule model that is derived in Engel and West (2005) and Engel, Wang, and Wu (2008). Following Molodtsova and Papell (2009), we focus on models that depend only on current levels of inflation and the output gap.5 The Taylor rule in the home country takes the form of: ¯i =¯i+δ (π −π¯)+δ ygap+u , (2) t π t y t t where¯i is the central bank’s target for the short-term interest rate at time t,¯i is the equilibrium long-term t rate, π is the inflation rate, π¯ is the target inflation rate, and ygap is the output gap. The foreign country t t is assumed to follow a similar Taylor rule. In addition, we follow Engel and West (2005) to assume that the foreign country targets the exchange rate in its Taylor rule: ¯i∗ =¯i+δ (π∗−π¯)+δ ygap∗+δ (s −s¯)+u∗, (3) t π t y t s t t t wheres¯ isthetargetedexchangerate. AssumethattheforeigncountrytargetsthePPPleveloftheexchange t rate: s¯ =p −p∗, where p and p∗ are logarithms of home and foreign aggregate prices. In equation (3), we t t t t t assume that the policy parameters take the same values in the home and foreign countries. Molodtsova and Papell (2009) denote this case as “homogeneous Taylor rules”. Our empirical results also hold in the case of heterogenous Taylor rules. To simplify our presentation, we assume that the home and foreign countries have the same long-term inflation and interest rates. Such restrictions have been relaxed in our econometric model after we include a constant term in estimations. 3The coefficients on lagged interest rates in the home and foreign countries can take different values in Molodtsova and Papell(2009). 4Wealsotriedtherandomwalkwithadrift. Itdoesnotchangeourresults. 5Clarida,Gali,andGertler(1998)findempiricalsupportforforward-lookingTaylorrules. Forward-lookingTaylorrulesare ruledoutbecausetheyrequireforecastsofpredictors,whichcreatesadditionalcomplicationsinout-of-sampleforecasting. 7

We do not consider interest rate smoothing in our benchmark model. That is, the actual interest rate (i ) is identical to the target rate in the benchmark model: t i =¯i . (4) t t Molodtsova and Papell (2009) consider the following interest rate smoothing rule: i =(1−ρ)¯i +ρi +ν , (5) t t t−1 t where ρ is the interest rate smoothing parameter. We include these setups in models 3 and 4. Note that our estimation methods do not require the monetary policy shock u and the interest rate smoothing shock ν t t to satisfy any assumptions, aside from smoothness of their distributions when conditioned on predictors. Substituting the difference of equations (2) and (3) into Uncovered Interest-rate Parity (UIP), we have:   s =E  (1−b) X ∞ bj(p −p∗ )−b X ∞ bj(cid:2) δ (ygap−ygap∗)+δ (π −π∗ ) (cid:3)  , (6) t t t+j t+j y t+j t+j π t+j t+j   j=0 j=0 wherethediscountfactorb= 1 . Undersomeconditions,thepresentvalueassetpricingformatinequation 1+δs (6) can be written into an error-correction form:6 s −s =α +β z +ε , (7) t+h t h h t t+h where the deviation of the exchange rate from its equilibrium level is defined as: z =s −p +p∗+ b (cid:2) δ (ygap−ygap∗)+δ (π −π∗) (cid:3) . (8) t t t t 1−b y t t π t t We use equation (7) as our benchmark setup in calculating h-horizon-ahead out-of-sample forecasting intervals. Accordingtoequation(8),thematrixX inequation(1)includeseconomicvariablesq ≡s +p∗−p , 1,t t t t t ygap−ygap∗, and π −π∗. t t t t 6Seeappendixformoredetail. Whilethelong-horizonregressionformatofthebenchmarkTaylormodelisderiveddirectly from the underlying Taylor rule model, this is not the case for the models with interest rate smoothing (models 3 and 4). Molodtsova and Papell (2009) only consider the short-horizon regression for the Taylor rule models. We include long-horizon regressionsofthesemodelsonlyforthepurposeofcomparison. 8

2.2 Data Theforecastingmodelsandthecorrespondingforecastintervalsareestimatedusingmonthlydatafortwelve OECD countries. The United States is treated as the foreign country in all cases. For each country we synchronize the beginning and end dates of the data across all models estimated. The twelve countries and periods considered are: Australia (73:03-06:6), Canada (75:01-06:6), Denmark (73:03-06:6), France (77:12- 98:12), Germany (73:03-98:12), Italy (74:12-98:12), Japan (73:03-06:6), Netherlands (73:03-98:12), Portugal (83:01-98:12), Sweden (73:03-04:11), Switzerland (75:09-06:6), and the United Kingdom (73:03-06:4). The data is taken from Molodtsova and Papell (2009).7 With the exception of interest rates, the data is transformed by taking natural logs and then multiplying by 100. The nominal exchange rates are end-ofmonth rates taken from the Federal Reserve Bank of St. Louis database. Output data (y ) are proxied by t Industrial Production (IP) from the International Financial Statistics (IFS) database. IP data for Australia and Switzerland are only available at a quarterly frequency, and hence are transformed from quarterly to monthly observations using the quadratic-match average option in Eviews 4.0 by Molodtsova and Papell (2009). Following Engel and West (2006), the output gap (ygap) is calculated by quadratically de-trending t the industrial production for each country. Prices data (p ) are proxied by Consumer Price Index (CPI) from the IFS database. Again, CPI for t Australia is only available at a quarterly frequency and the quadratic-match average is used to impute monthly observations. Inflation rates are calculated by taking the first differences of the logs of CPIs. The money market rate from IFS (or “call money rate”) is used as a measure of the short-term interest rate set by the central bank. Finally, M1 is used to measure the money supply for most countries. M0 for the UK and M2 for Italy and Netherlands is used due to the unavailability of M1 data. 3 Econometric Method Foragivenmodelm, theobjectiveistoestimatefromequation(1)thedistributionofs −s conditional τ+h τ ondataX thatisobserveduptotimeτ. Thisistheh-horizon-aheadforecast distribution oftheexchange m,τ rate, from which the corresponding forecast interval can be derived. For a given α, the forecast interval of coverage α∈(0,1) is an interval in which s −s is supposed to lie with a probability of α. τ+h τ Models m = 1,...,7 in equation (1) provide only point forecasts of s −s . In order to construct τ+h τ forecast intervals for a given model, we apply robust semiparametric (RS) forecast intervals to all models. 7We thank the authors for the data, which we downloaded from David Papell’s website. For the exact line numbers and sourcesofthedata,seethedataappendixofMolodtsovaandPapell(2009). 9

The nominal α-coverage forecast interval of s −s conditional on X can be obtained by the following τ+h τ m,τ three-step procedure: Step 1. Estimate model m by OLS and obtain residuals ε bm,t+h ≡ s t+h −s t −α bm,h +βb m 0 ,h X m,t , for t = 1,...,τ −h. Step 2. For a range of values of ε (sorted residuals {ε }τ−h), estimate the conditional distribution of bm,t+h t=1 ε |X by: m,τ+h m,τ Pτ−h1(ε ≤ε)K (X −X ) Pb(ε m,τ+h ≤ε|X m,τ )≡ t=1 P b τ m − , h t+ K h (X b −X m,t ) m,τ , (9) t=1 b m,t m,τ where K (X −X ) ≡ b−dK((X −X )/b), K(·) is a multivariate Gaussian kernel with a b m,t m,τ m,t m,τ dimension the same as that of X , and b is the smoothing parameter or bandwidth.8 m,t Step 3. Find the (1−α)/2 and (1+α)/2 quantiles of the estimated distribution, which are denoted by ε(1−α)/2 and ε(1+α)/2, repectively. The estimate of the α-coverage forecast interval for s − s bm,h bm,h τ+h τ conditional on X is: m,τ Ib m α ,τ+h ≡(βb m 0 ,h X m,τ +ε bm (1 , − h α)/2,βb m 0 ,h X m,τ +ε b ( m 1 , + h α)/2) (10) For each model m, the above method uses the forecast models in equation (1) to estimate the location of the forecast distribution, while the nonparametric kernel distribution estimate is used to estimate the shape. As a result, the interval obtained from this method is semiparametric. Wu (2009) shows that under some weak regularity conditions, this method consistently estimates the forecast distribution,9 and hence the forecast intervals of s −s conditional on X , regardless of the quality of model m. That is, the τ+h τ m,τ forecast intervals are robust. Stationarity of economic variables is one of those regularity conditions. In our models, exchange rate differences, interest rates and inflation rates are well-known to be stationary, while empirical tests for real exchange rates and output gaps generate mixed results. These results may be driven by the difficulty of distinguishing a stationary, but persistent, variable from a non-stationary one. In this paper, we take the stationarity of these variables as given. Model 7 is the random walk model. The estimator in equation (9) becomes the Empirical Distribution Function (EDF) of the exchange rate innovations. Under regularity conditions, equation (9) consistently estimates the unconditional distribution of s −s , and can be used to form forecast intervals for s . τ+h τ τ+h 8WechoosebusingthemethodofHall,Wolff,andYao(1999). 9Itisconsistentinthesenseofconvergenceinprobabilityastheestimationsamplesizegoestoinfinity. 10

Theforecastintervalsofeconomicmodelsandtherandomwalkarecompared. Ourinterestistotestwhether RS forecast intervals based on economic models are more accurate than those based on the random walk model. We focus on the empirical coverage and the length of forecast intervals in our tests. Following Christoffersen (1998) and related work, the first standard we use is the empirical coverage. The empirical coverage should be as close as possible to the nominal coverage (α). Significantly missing the nominal coverage indicates the inadequacy of the model and predictors for the given sample size. For instance, if 90% forecast intervals calculated from a model contain only 50% of out-of-sample observations, the model can hardly be identified as useful for interval forecasting. This case is called under-coverage. In contrast, over-coverage implies that the intervals could be reduced in length (or improved in tightness), but the forecast interval method and model are unable to do that for the given sample size. An economic model is said to outperform the random walk if its empirical coverage is more accurate than that of the random walk. On the other hand, the empirical coverage of an economic model may be equally accurate as that of the random walk model, but the economic model has tighter forecast intervals than the random walk. We argue that the lengths of forecast intervals signify the informativeness of the intervals given that these intervals haveequallyaccurateempiricalcoverages. Inthiscase,theeconomicmodelisalsoconsideredtooutperform the random walk in forecasting exchange rates. The empirical coverage and length tests are conducted at both short and long horizons for the six economic models relative to the random walk for each of the twelve OECD exchange rates. We use tests that are applications of the unconditional predictive accuracy inference framework of Giacomini and White (2006). Unlike the tests of Diebold and Mariano (1995) and West (1996), our forecast evaluation tests do not focus on the asymptotic features of the forecasts. Rather, in the spirit of Giacomini and White (2006), we are comparing the population features of forecasts generated by rolling samples of fixed sample size. This contrasts to the traditional forecast evaluation methods in that although it uses asymptotic approximations to do the testing, the inference is not on the asymptotic properties of forecasts, but on their population finite sample properties. We acknowledge that the philosophy of this inference framework remains a point of contention, but it does tackle three important evaluation difficulties in this paper. First, it allows for evaluation of forecast intervals that are not parametrically derived. The density evaluation methods developed in well-known studies such as Diebold, Gunther, Tay (1998), Corradi and Swanson (2006a) and references within Corradi and Swanson (2006b) require that the forecast distributions be parametrically specified. Giacomini and White’s (2006) method overcomes this challenge by allowing comparisons among parametric, semiparametric and nonparametric forecasts. As a result, in the cases of 11

semiparametricandnonparametricforecasts, italsoallowscomparisonofmodelswithpredictorsofdifferent dimensions, as evident in our exercise. Second, by comparing the finite sample properties of RS forecast intervalsderivedfromdifferentmodels, weavoidrejectingmodelsthataremis-specified,10 butarenonetheless good approximations useful for forecasting. Finally, we can individually (though not jointly) test whether theforecastintervalsdifferintermsofempiricalcoveragesandlengths, forthegivenestimationsample, and are not confined to focus only on empirical coverages or holistic properties of forecast distribution, such as probability integral transform. 3.1 Test of Equal Empirical Coverages Suppose the sample size available to the researcher is T and all data are collected in a vector W . Our t inference procedure is based on a rolling estimation scheme, with the size of the rolling window fixed while T → ∞. Let T = R+N and R be the size of the rolling window. For each horizon h and model m, a sequence of N(h) = N +1−h α-coverage forecast intervals are generated using rolling data: {W }R for t t=1 forecast for date R+h, {W }R+1 for forecast for date R+h+1, and so on, until forecast for date T is t t=2 generated using {W }R+N(h)−1. t t=N(h) Under this fixed-sample-size rolling scheme, for each finite h we have N(h) observations to compare the empirical coverages and lengths across m models (m = 1,2,...,7). By fixing R, we allow the finite sample propertiesoftheforecastintervalstobepreservedasT →∞. Thus,theforecastintervalsandtheassociated forecast losses are simply functions of a finite and fixed number of random variables. We are interested in approximating the population moments of these objects by taking N(h) → ∞. A loose analogy would be finding the finite-sample properties of a certain parameter estimator when the sample size is fixed at R, by a bootstrap with an arbitrarily large number of bootstrap replications. We conduct individual tests for the empirical coverages and lengths. In each test, we define a correspondingforecastloss, proposeateststatisticandderiveitsasymptoticdistribution. Asdefinedinequation (10), let Ibα be the h−horizon ahead RS forecast interval of model m with a nominal coverage of α. For m,τ+h out-of-sample forecast evaluation, we require Ibα to be constructed using data from t = τ −R+1 to m,τ+h t=τ. The coverage accuracy loss is defined as: h i2 CLα m,h = P(Y τ+h ∈Ib m α ,τ+h )−α . (11) Foreconomicmodels(m=1,...,6),thegoalistocomparethecoverageaccuracylossofRSforecastintervals 10WhileRSintervalsremedymis-specificationsasymptotically,itdoesnotguaranteesuchcorrectionsinagivenfinitesample. 12

of model m with that of the random walk (m=7). The null and alternative hypotheses are: H : ∆CLα ≡CLα −CLα =0 0 m,h 7,h m,h H : ∆CLα 6=0. A m,h Define the sample analog of the coverage accuracy loss in equation (11): T−h !2 α X CdL m,h = N(h)−1 1(Y τ+h ∈Ib m α ,τ+h )−α , τ=R where1(Y τ+h ∈Ib m α ,τ+h )isanindexfunctionthatequalsonewhenY τ+h ∈Ib m α ,τ+h ,andequalszerootherwise. Applying the asymptotic test of Giacomini and White (2006) to the sequence {1(Y τ+h ∈ Ib m α ,τ+h )}T τ= − R h and applying the Delta method, we can show that p N(h)(∆CdL α m,h −∆CLα m,h )→ d N(0,Γ 0 m,h Ω m,h Γ m,h ), (12) d where → denotes convergence indistribution, andΩ is the long-run covariance matrixbetween 1(Y ∈ m,h τ+h Ib m α ,τ+h ) and 1(Y τ+h ∈Ib 7 α ,τ+h ). The matrix Γ m,h is defined as: (cid:20) (cid:21)0 (cid:16) (cid:16) (cid:17) (cid:17) (cid:16) (cid:16) (cid:17) (cid:17) Γ m,h ≡ 2 P Y τ+h ∈Ib m α ,τ+h −α 2 P Y τ+h ∈Ib 7 α ,τ+h −α . Γ m,h can be estimated consistently by its sample analog Γbm,h , while Ω m,h can be estimated by some HAC estimator Ωbm,h , such as Newey and West (1987).11 The test statistic for coverage test is defined as: p α Ctα ≡ N(h)∆CdL m,h → d N(0,1) (13) m,h q Γb0 m,h Ωbm,h Γbm,h 3.2 Test of Equal Empirical Lengths Define the length loss as: h (cid:16) (cid:17)i LLα ≡E leb Ib α , (14) m,h m,τ+h 11WeuseNeweyandWest(1987)forourempiricalwork,withawindowwidthof12. 13

where leb(·) is the Lesbesgue measure. To compare the length loss of RS forecast intervals of economic models m=1,2,...,6 with that of the random walk (m=7), the null and alternative hypotheses are: H : ∆LLα ≡LLα −LLα =0 0 m,h 7,h m,h H : ∆LLα 6=0. A m,h The sample analog of the length loss for model m is defined as: T−h α X LcL =N(h)−1 leb(Ib α ). m,h m,τ+h τ=R Directly applying the test of Giacomini and White (2006), we have p N(h)(∆LcL α m,h −∆LLα m,h )→ d N(0,Σ m,h ), (15) (cid:16) (cid:17) (cid:16) (cid:17) whereΣ m,h isthelong-runvarianceofleb Ib 7 α ,τ+h −leb Ib m α ,τ+h . LetΣbm,h betheHACestimatorofΣ m,h . The test statistic for empirical length is defined as: p α Ltα ≡ N(h)∆LcL m,h → d N(0,1). (16) m,h q Σbm,h 3.3 Discussion The coverage accuracy loss function is symmetric in our paper. In practice, an asymmetric loss function may be better when looking for an exchange rate forecast model to help make policy or business decisions. Under-coverage is arguably a more severe problem than over-coverage in practical situations. However, the focus of this paper is the disconnect between economic fundamentals and the exchange rate. Our goal is to investigate which model comes closer to the data: the random walk or fundamental-based models. It is not critical in this case whether coverage inaccuracy comes from the under- or over-coverage. We acknowledge thattheuseofsymmetriccoveragelossremainsacaveat,especiallysinceweareusingthecoverageaccuracy test as a pre-test for the tests of length. Clearly, there is a tradeoff between the empirical coverage and the lengthofforecastintervals. Giventhesamecenter,12 intervalswithunder-coveragehaveshorterlengthsthan intervalswithover-coverage. Inthiscase,thelengthtestisinfavorofmodelsthatsystematicallyunder-cover the targeted nominal coverage when compared to a model that systematically over-covers. This problem 12Centerheremeansthehalfwaypointbetweentheupperandlowerboundforagiveninterval. 14

cannot be detected by the coverage accuracy test with symmetric loss function because over- and undercoverage are treated equally. However, our results in section 4 show that there is no evidence of systematic under-coveragefortheeconomicmodelsconsideredinthispaper. Forinstance,inTable1,one-month-ahead (h = 1) forecast intervals over-cover the nominal coverage (90%) for eight out of twelve exchange rates.13 Notethatunder-coveragedoesnotguaranteeshorterintervalseitherinourpaper, becauseforecastintervals of different models usually have different centers.14 In addition, we also compare the coverage of economic models and the random walk directly in an exercise not reported in this paper. There is no evidence that the coverage of economic models is systematically smaller than that of the random walk.15 As we have mentioned, comparisons across models can also be done at the distribution level. We choose interval forecasts for two reasons. First, interval forecasts have been widely used and reported by the practitioners. For instance, the Bank of England calculates forecast intervals of inflation in its inflation reports. Second, compared to evaluation metrics for density forecasts, the empirical coverage and length loss functions of interval forecasts, and the subsequent interpretations of test rejection/acceptance are more intuitive. 4 Results WeapplyRSforecastintervalsforeachmodelforagivennominalcoverageofα=0.9. Thereisnoparticular reason why we chose 0.9 as the nominal coverage. Some auxiliary results show that our qualitative findings donotdependonthechoiceofα. Duetodifferentsamplesizesacrosscountries,wechoosedifferentsizesfor therollingwindow(R)fordifferentcountries. Ourruleisverysimple: forcountrieswithT ≥300,wechoose R = 200, otherwise we set R = 150.16 Again, from our experience, tampering with R does not change the qualitative results, unless R is chosen to be unusually big or small. Fortimehorizonsh=1,3,6,9,12andmodelsm=1,...,7,weconstructasequenceofN(h)90%forecast intervals {Ib m 0. , 9 τ+h }T τ= − R h for the h-horizon change of the exchange rate s t+h −s t . Then we compare economic models and the random walk by computing empirical coverages, lengths and test statistics Ct0.9 and Lt0.9 m,h m,h asdescribedinsection3. Wefirstreporttheresultsofourbenchmarkmodel. Afterthat,resultsofalternative models are reported and discussed. 13ThesenineexchangeratesaretheDanishKroner,theFrenchFranc,theDeutschmark,theJapaneseYen,theDutchGuilder, thePortugueseEscudo,theSwissFranc,andtheBritishpound. Similarresultsholdatotherhorizons. 14WhencomparingtheintervalsforS τ+h −Sτ,therandomwalkmodelbuildstheforecastintervalaround0,whileeconomic modelmbuildsitaroundβb m 0 ,h Xm,τ. 15Resultsareavailableuponrequest. 16TheonlyexceptionisPortugal,whereonly192datapointswereavailable. Inthiscase,wechooseR=120. 15

4.1 Results of Benchmark Model Table 1 shows results of the benchmark Taylor rule model. For each time horizon h and exchange rate, the first column (Cov.) reports the empirical coverage for the given nominal coverage of 90%. The second column(Leng.) reportsthelengthofforecastintervals(thedistancebetweenupperandlowerbounds). The length is multiplied by 100 and therefore expressed in terms of the percentage change of the exchange rate. For instance, the empirical coverage and length of the one-month-ahead forecast interval for the Australian dollar are 0.895 and 7.114, respectively. It means that on average, with a chance of 89.5%, the one-monthaheadchange ofAUD/USDlies inan interval withlength7.114%. We use superscripts a, b, andc to denote that the null hypothesis of equal empirical coverage accuracy/length is rejected in favor of the Taylor rule model at a confidence level of 10%, 5%, and 1% respectively. Superscripts x, y, and z are used for rejections in favor of the random walk analogously. We summarize our findings in three panels. In the first panel ((1) Coverage Test), the row of “Model Better” reports the number of exchange rates that the Taylor rule model has more accurate empirical coverages than the random walk. The row of “RW Better” reports the number of exchange rates for which the random walk outperforms the Taylor rule model under the same criterion. In the second panel ((2) Length Test Given Equal Coverage Accuracy), a better model is the one with tighter forecast intervals given equal coverage accuracy. In the last panel ((1)+(2)), a better model is the one with either more accurate coverages, or tighter forecast intervals given equal coverage accuracy. For most exchange rates and time horizons, the Taylor rule model and the random walk model have statistically equally accurate empirical coverages. The null hypothesis of equal coverage accuracy is rejected in only six out of sixty tests (two rejections each at horizons 6, 9, and 12). Five out of six rejections are in favor of the Taylor rule model. That is, the empirical coverage of the Taylor rule model is closer to the nominal coverage than those of the random walk. However, based on the number of rejections (5) in a total of sixty tests, there is no strong evidence that the Taylor rule model can generate more accurate empirical coverages than the random walk. In cases where the Taylor rule model and the random walk have equally accurate empirical coverages, theTaylorrulemodelgenerallyhasequalorsignificantlytighterforecastintervalsthantherandomwalk. In forty-two out of fifty-four cases, the null hypothesis of equally tight forecast intervals is rejected in favor of the Taylor rule model. In contrast, the null hypothesis is rejected in only three cases in favor of the random walk. Theevidenceofexchangeratepredictabilityismorepronouncedatlongerhorizons. Athorizontwelve (h = 12), for all cases where empirical coverage accuracies between the random walk and the Taylor rule 16

model are statistically equivalent, the Taylor rule model has significantly tighter forecast intervals than the random walk. As for each individual exchange rate, the benchmark Taylor rule model works best for the French Franc, the Deutschmark, the Dutch Guilder, the Swedish Krona, and the British Pound: for all time horizons, the model has tighter forecast intervals than the random walk, while their empirical coverages are statistically equally accurate. The Taylor rule model performs better than the random walk in most horizons for the remaining exchange rates except the Portuguese Escudo, for which the Taylor rule model outperforms the random walk only at long horizons. 4.2 Results of Alternative Models Five alternative economic models are also compared with the random walk: three alternative Taylor rule models that are studied in Molodtsova and Papell (2009), the PPP model, and the monetary model. Tables 2-6 report results of these alternative models. In general, results of coverage tests do not show strong evidence that economic models can generate more accurate coverages than the random walk at either short or long horizons. However, after considering lengthtests,wefindthateconomicmodelsperformbetterthantherandomwalk,especiallyatlonghorizons. Taylorrulemodel4(thebenchmarkmodelwithinterestratesmoothingTable4)andthePPPmodel(Table 5) perform the best among alternative models. Results of these two models are very similar to that of the benchmark Taylor rule model. At horizon twelve, both models outperform the random walk for most exchange rates under our out-of-sample forecast interval evaluation criteria. The performance of Taylor rule model 2 (Table 2) and 3 (Table 3) is relatively less impressive than other models, but for more than half of the exchange rates, the economic models outperform the random walk at several horizons in out-of-sample interval forecasts. ComparingthebenchmarkTaylorrulemodel, thePPPmodelandthemonetarymodel,theperformance of the PPP model (Table 5) is worse than the other two models at short horizons. Compared to the Taylor rule and PPP models, the monetary model outperforms the random walk for a smaller number of exchange ratesathorizons6,9,and12. Overall,thebenchmarkTaylorrulemodelseemstoperformslightlybetterthan the monetary and PPP models. Molodtsova and Papell (2009) find similar results in their point forecasts. Table 7 shows results with heterogeneous Taylor rules.17 In this model, we relaxed the assumption that theTaylorrulecoefficientsarethesameinthehomeandforeigncountries. Wereplaceπ −π∗andygap−ygap∗ t t t t in matrix X of the benchmark model with δˆ π −δˆ∗π∗ and δˆ ygap −δˆ∗ygap∗, where δˆ , δˆ∗, δˆ , and δˆ∗ 1,t π t π t y t y t π π y y 17SeeAppendixA.3fordetails. 17

are Taylor rule coefficients estimated from the data of home and foreign countries. The main findings in the benchmark model also hold in Table 7. 4.3 Discussion After Mark (1995) first documents exchange rate predictability at long horizons, long-horizon exchange rate predictability has become a very active area in the literature. With panel data, Engel, Mark, and West (2007) recently show that the long-horizon predictability of the exchange rate is relatively robust in the exchange rate forecasting literature. We find similar results in our interval forecasts. The evidence of longhorizonpredictabilityseemsrobustacrossdifferentmodelsandcurrencieswhenbothempiricalcoverageand length tests are used. At horizon twelve, all economic models outperform the random walk for six exchange rates: the Australian Dollar, French Franc, Italian Lira, Japanese Yen, Swedish Krona, and the British Pound in the sense that interval lengths of economic models are smaller than those of the random walk, given equivalent coverage accuracy. This is true only for the Danish Kroner and Swiss Franc at horizon one. We also notice that there is no clear evidence of long-horizon predictability based on the tests of empirical coverage accuracy only. Molodtsova and Papell (2009) find strong out-of-sample exchange rate predictability for Taylor rule models even at the short horizon. In our paper, the evidence for exchange rate predictability at short horizons is not very strong. This finding may be a result of some assumptions we have used to simplify our computation. For instance, an α-coverage forecast interval in our paper is always constructed using the (1−α)/2 and (1+α)/2 quantiles. Alternatively, we can choose quantiles that minimize the length of intervals,giventhenominalcoverage.18 Inaddition,thedevelopmentofmorepowerfultestingmethodsmay alsobehelpful. TheevidenceofexchangeratepredictabilityinMolodtsovaandPapell(2009)ispartlydriven by the testing method recently developed by Clark and West (2006, 2007). We acknowledge that whether or not short-horizon results can be improved remains an interesting question, but do not pursue this in the currentpaper. Thepurposeofthispaperistoshowtheconnectionbetweentheexchangerateandeconomic fundamentals from an interval forecasting perspective. Predictability either at short- or long-horizons will serve this purpose. Though we find that economic fundamentals are helpful for forecasting exchange rates, we acknowledge that exchange rate forecasting in practice is still a difficult task. The forecast intervals from economic models are statistically tighter than those of the random walk, but they remain fairly wide. For instance, the distance between the upper and lower bound of three-month-ahead forecast intervals is usually a 20% 18SeeWu(2009)formorediscussion. 18

change of the exchange rates. Figures 1-3 show the length of forecast intervals generated by the benchmark Taylor rule model and the random walk for the British Pound, the Deutschmark, and the Japanese Yen at different horizons.19 At the horizon of 12 months, the length of forecast intervals in the Taylor rule model is usually smaller than that in the random walk. However, at shorter horizons, such as 1 month, the difference is quantitatively small. 5 Conclusion There is a growing strand of literature that uses Taylor rules to model exchange rate movements. Our paper contributes to the literature by showing that Taylor rule fundamentals are useful in forecasting the distribution of exchange rates. We apply Robust Semiparametric forecast intervals of Wu (2009) to a group of Taylor rule models for twelve OECD exchange rates. The forecast intervals generated by the Taylor rule models are in general tighter than those of the random walk, given that these intervals cover the realized exchange rates equally well. The evidence of exchange rate predictability is more pronounced at longer horizons, a result that echoes previous long-horizon studies such as Mark (1995). The benchmark Taylor rule model is also found to perform better than the monetary and PPP models based on out-of-sample interval forecasts. Though we find some empirical support for the connection between the exchange rate and economic fundamentals, we acknowledge that the detected connection is weak. The reductions of the lengths of forecast intervals are quantitatively small, though they are statistically significant. Forecasting exchange ratesremainsadifficulttaskinpractice. EngelandWest(2005)arguethatasthediscountfactorgetscloser toone,presentvalueassetpricingmodelsplacegreaterweightonfuturefundamentals. Consequently,current fundamentalshaveveryweakforecastingpowerandexchangeratesappeartofollowapproximatelyarandom walk. Under standard assumptions in Engel and West (2005), the Engel-West theorem does not imply that exchangerates are morepredictable atlongerhorizonsorthateconomic modelscanoutperformthe random walk in forecasting exchange rates based on out-of-sample interval forecasts. However, modifications to these assumptions may be able to reconcile the Engel-West explanation with empirical findings of exchange rate predictability. For instance, Engel, Wang, and Wu (2008) find that when there exist stationary, but persistent,unobservablefundamentals,forexampleriskpremium,theEngel-Westexplanationpredictslonghorizon exchange rate predictability in point forecasts, though the exchange rate still approximately follows a random walk at short horizons. It would also be of interest to study conditions under which our findings 19Figuresinothercountriesshowsimilarpatterns. Resultsareavailableuponrequest. 19

in interval forecasts can be reconciled with the Engel-West theorem. We believe other issues, such as parameter instability (Rossi, 2005), nonlinearity (Kilian and Taylor, 2003),realtimedata(Faust,Rogers,andWright,2003,Molodtsova,Nikolsko-Rzhevskyy,andPapell,2008a, 2008b),andmodelselection(SarnoandValente,forthcoming)areallcontributingtotheMeese-Rogoffpuzzle. Panel data are also found helpful in detecting exchange rate predictability, especially at long horizons. For instance, see Mark and Sul (2001), Engel, Mark, and West (2007), and Rogoff and Stavrakeva (2008). It would be interesting to incorporate these studies into interval forecasting. We leave these extensions for future research. 20

Table 1: Results of Benchmark Taylor Rule Model h=1 h=3 h=6 h=9 h=12 Cov. Leng. Cov. Leng. Cov. Leng. Cov. Leng. Cov. Leng. Australian Dollar 0.895 7.114 0.888 14.209c 0.959 21.140 0.942 26.613 0.963 29.175c Canadian Dollar 0.814 3.480 0.794 6.440c 0.738 8.483c 0.675 8.669c 0.596x 9.707c Danish Kroner 0.920 8.676c 0.939 17.415c 0.954 26.198 0.922 28.712c 0.968 37.123c French Franc 0.912 8.921c 0.860 15.728c 0.928c 26.007c 0.957 29.924c 0.934 36.883c Deutschmark 0.927 8.327c 0.879 18.634c 0.894 27.923c 0.960a 33.734c 0.969 39.618c Italian Lira 0.899 8.291c 0.875 18.305 0.910 26.788c 0.846 32.545c 0.874 37.151c Japanese Yen 0.915 9.633z 0.909 19.762 0.892 28.451c 0.932 33.793c 0.883 37.728c Dutch Guilder 0.917 8.726c 0.907 18.615c 0.933 27.458c 0.941b 30.902c 0.959a 40.177c Portuguese Escudo 0.901 8.580z 0.928 18.758z 0.894c 23.552b 0.825 27.086 0.867 32.092c Swedish Krona 0.839 7.360c 0.860 15.413c 0.874 23.930c 0.820 28.090c 0.834 37.432c Swiss Franc 0.947 9.358c 0.916 19.655 0.963 26.553c 0.963 30.780c 0.899 35.008c British Pound 0.919 8.413a 0.923 16.592c 0.912 23.317c 0.900 26.942c 0.855 25.905c (1) Coverage Test† Model Better 0 0 2 2 1 RW Better 0 0 0 0 1 (2) Length Test Given Equal Coverage Accuracy‡ Model Better 8 8 8 8 10 RW Better 2 1 0 0 0 (1)+(2)§ Model Better 8 8 10 10 11 RW Better 2 1 0 0 1 Note: –hdenotesforecasthorizonsformonthlydata. –For each horizon (h), the first column (Cov.) reports empirical coverages given a nominal coverage of 90%. The second column (Leng.) reportsthelengthofforecastintervalsintermsofpercentagechangeoftheexchangerate. Empiricalcoveragesandlengths areaveragesacrossN(h)out-of-sampletrials. –Superscripts a,b,c in the column of Cov. (Leng.) denote rejections of equal coverage accuracy (equal length) in favor of the economic model at a 10%, 5% and 1% confidence level respectively. Superscripts x,y,z are defined analogously for rejections in favoroftherandomwalk. †–Inthispanel,abettermodelistheonewithmoreaccurateempiricalcoverages. RWistheabbreviationofRandomWalk. ‡–Inthispanel,abettermodelistheonewithtighterforecastintervalsgivenequalcoverageaccuracy. §–In this panel, a better model is the one with either more accurate coverages or tighter forecast intervals given equal coverage accuracy. 21

Table 2: Results of Taylor Rule Model Two h=1 h=3 h=6 h=9 h=12 Cov. Leng. Cov. Leng. Cov. Leng. Cov. Leng. Cov. Leng. Australian Dollar 0.884 7.146y 0.899 15.086c 0.928 21.327 0.901 27.329 0.872 30.815b Canadian Dollar 0.825 3.442c 0.783 6.321c 0.814 8.490c 0.858 10.034c 0.825 11.921c Danish Kroner 0.915 8.753a 0.939 17.764c 0.954 27.479z 0.953 33.426y 0.942 40.717 French Franc 0.951 9.042c 0.930 18.783 0.949c 29.161c 0.936 34.994c 0.868 42.081c Deutschmark 0.917 9.090 0.869 19.217 0.952 29.746 0.941a 39.093z 0.980 44.571z Italian Lira 0.928 9.196 0.875 18.322 0.895 26.926c 0.869 35.883c 0.898a 41.235z Japanese Yen 0.915 9.568x 0.914 19.734 0.912 29.344c 0.937 36.834 0.942 44.385c Dutch Guilder 0.908 8.586c 0.888 18.782c 0.962 29.777 0.990 39.507z 0.990 47.514z Portuguese Escudo 0.916 8.005 0.957 17.924z 0.909c 24.270b 0.889 28.533z 0.883a 35.338c Swedish Krona 0.867 7.624 0.860 16.132 0.857 24.500c 0.837 32.825 0.811 37.772c Swiss Franc 0.941 9.953b 0.928 20.105 0.982 29.758c 0.994 38.267z 0.962 45.965z British Pound 0.919 8.627z 0.933 17.334c 0.922 26.227c 0.937 31.044c 0.957 36.397c (1) Coverage Test† Model Better 0 0 2 1 2 RW Better 0 0 0 0 0 (2) Length Test Given Equal Coverage Accuracy‡ Model Better 5 5 6 4 6 RW Better 3 1 1 4 3 (1)+(2)§ Model Better 5 5 8 5 8 RW Better 3 1 1 4 3 Note: –hdenotesforecasthorizonsformonthlydata. –For each horizon (h), the first column (Cov.) reports empirical coverages given a nominal coverage of 90%. The second column (Leng.) reportsthelengthofforecastintervalsintermsofpercentagechangeoftheexchangerate. Empiricalcoveragesandlengths areaveragesacrossN(h)out-of-sampletrials. –Superscriptsa,b,cinthecolumnofCov. (Leng.) denoterejectionsofequalcoverageaccuracy(equallength)infavoroftheeconomic model at a 10%, 5% and 1% confidence level respectively. Superscripts x,y,z are defined analogously for rejections in favor of the randomwalk. †–Inthispanel,abettermodelistheonewithmoreaccurateempiricalcoverages. RWistheabbreviationofRandomWalk. ‡–Inthispanel,abettermodelistheonewithtighterforecastintervalsgivenequalcoverageaccuracy. §–In this panel, a better model is the one with either more accurate coverages or tighter forecast intervals given equal coverage accuracy. 22

Table 3: Results of Taylor Rule Model Three h=1 h=3 h=6 h=9 h=12 Cov. Leng. Cov. Leng. Cov. Leng. Cov. Leng. Cov. Leng. Australian Dollar 0.884 7.229z 0.899 15.055c 0.881 21.055 0.885 26.359c 0.867 30.234c Canadian Dollar 0.831 3.453b 0.789 6.408c 0.814 8.629c 0.864 10.220c 0.819 11.971c Danish Kroner 0.920 8.753b 0.934 17.649c 0.949 27.523z 0.948 33.307 0.936 40.070 French Franc 0.951 9.171 0.740 14.488c 0.722 20.313c 0.915 35.562a 0.813 41.350c Deutschmark 0.908 9.020 0.897 19.303 0.914 29.676 0.901c 37.291a 0.878 44.761z Italian Lira 0.928 8.900a 0.875 17.206c 0.872 26.674c 0.839 34.819c 0.787 39.569c Japanese Yen 0.905 9.179c 0.878 18.907c 0.892 25.883c 0.927 31.259c 0.894 37.049c Dutch Guilder 0.927 8.910 0.907 19.204a 0.952 29.426a 0.951a 36.896c 0.959 46.321z Portuguese Escudo 0.930 7.961 0.942 16.883c 0.955a 23.786 0.905 26.620c 0.850 33.745c Swedish Krona 0.867 7.316c 0.848 15.017c 0.840 23.241c 0.791 29.265c 0.757 33.751c Swiss Franc 0.929 9.761b 0.922 19.517c 0.939 28.437b 0.926c 37.519 0.911 45.619 British Pound 0.929 8.239a 0.939 16.213c 0.927 23.951c 0.905 28.720c 0.952 34.900c (1) Coverage Test† Model Better 0 0 1 3 0 RW Better 0 0 0 0 0 (2) Length Test Given Equal Coverage Accuracy‡ Model Better 7 11 8 8 8 RW Better 1 0 1 0 2 (1)+(2)§ Model Better 7 11 9 11 8 RW Better 1 0 1 0 2 Note: –hdenotesforecasthorizonsformonthlydata. –For each horizon (h), the first column (Cov.) reports empirical coverages given a nominal coverage of 90%. The second column (Leng.) reportsthelengthofforecastintervalsintermsofpercentagechangeoftheexchangerate. Empiricalcoveragesandlengths areaveragesacrossN(h)out-of-sampletrials. –Superscripts a,b,c in the column of Cov. (Leng.) denote rejections of equal coverage accuracy (equal length) in favor of the economic model at a 10%, 5% and 1% confidence level respectively. Superscripts x,y,z are defined analogously for rejections in favoroftherandomwalk. †–Inthispanel,abettermodelistheonewithmoreaccurateempiricalcoverages. RWistheabbreviationofRandomWalk. ‡–Inthispanel,abettermodelistheonewithtighterforecastintervalsgivenequalcoverageaccuracy. §–In this panel, a better model is the one with either more accurate coverages or tighter forecast intervals given equal coverage accuracy. 23

Table 4: Results of Taylor Rule Model Four h=1 h=3 h=6 h=9 h=12 Cov. Leng. Cov. Leng. Cov. Leng. Cov. Leng. Cov. Leng. Australian Dollar 0.895 7.119 0.888 14.424c 0.928 20.966 0.927 25.304c 0.872 27.492c Canadian Dollar 0.814 3.425c 0.771 6.366c 0.698 8.019c 0.651 8.631c 0.494y 7.693c Danish Kroner 0.920 8.703c 0.929 17.536c 0.964 26.025 0.984x 30.891c 0.963 36.545c French Franc 0.892 8.361c 0.870 16.079c 0.938c 25.950c 0.883 30.016c 0.791 35.755c Deutschmark 0.927 8.314c 0.879 18.652c 0.894 26.803c 0.931c 33.350c 0.969 36.393c Italian Lira 0.891 8.663c 0.838 17.575c 0.865 26.387c 0.746 32.270c 0.724 36.422c Japanese Yen 0.905 9.157c 0.863 18.708c 0.866 24.417c 0.869 28.730c 0.851 31.470c Dutch Guilder 0.936 8.815 0.897 18.368c 0.914 26.700c 0.931c 30.036c 0.796 29.462c Portuguese Escudo 0.901 8.525z 0.913a 17.110 0.939c 23.461c 0.889 27.096a 0.917 28.778c Swedish Krona 0.861 7.289c 0.860 15.321c 0.869 23.340c 0.773 27.198c 0.728 31.843c Swiss Franc 0.947 9.149c 0.940 19.782a 0.811 22.796c 0.808 26.148c 0.671 26.683c British Pound 0.919 8.113a 0.913 15.765c 0.875 21.679c 0.825 27.312c 0.839 29.081c (1) Coverage Test† Model Better 0 1 2 2 0 RW Better 0 0 0 1 1 (2) Length Test Given Equal Coverage Accuracy‡ Model Better 9 11 8 9 11 RW Better 1 0 0 0 0 (1)+(2)§ Model Better 9 12 10 11 11 RW Better 1 0 0 1 1 Note: –hdenotesforecasthorizonsformonthlydata. –For each horizon (h), the first column (Cov.) reports empirical coverages given a nominal coverage of 90%. The second column (Leng.) reportsthelengthofforecastintervalsintermsofpercentagechangeoftheexchangerate. Empiricalcoveragesandlengths areaveragesacrossN(h)out-of-sampletrials. –Superscriptsa,b,cinthecolumnofCov. (Leng.) denoterejectionsofequalcoverageaccuracy(equallength)infavoroftheeconomic model at a 10%, 5% and 1% confidence level respectively. Superscripts x,y,z are defined analogously for rejections in favor of the randomwalk. †–Inthispanel,abettermodelistheonewithmoreaccurateempiricalcoverages. RWistheabbreviationofRandomWalk. ‡–Inthispanel,abettermodelistheonewithtighterforecastintervalsgivenequalcoverageaccuracy. §–In this panel, a better model is the one with either more accurate coverages or tighter forecast intervals given equal coverage accuracy. 24

Table 5: Results of Purchasing Power Parity Model h=1 h=3 h=6 h=9 h=12 Cov. Leng. Cov. Leng. Cov. Leng. Cov. Leng. Cov. Leng. Australian Dollar 0.895 7.114z 0.883 15.558 0.912 21.311 0.880 26.120c 0.856 30.316c Canadian Dollar 0.819 3.570z 0.806 6.872 0.767 9.615 0.728 11.078c 0.615 12.306c Danish Kroner 0.925 8.697c 0.939 18.333 0.938 25.887c 0.937 31.673c 0.957 37.447c French Franc 0.922 8.904c 0.940 18.029c 0.918c 25.786c 0.904 29.789c 0.802 34.209c Deutschmark 0.936 9.079 0.935 18.797c 0.942 27.677c 1.000 33.585c 0.990 40.570c Italian Lira 0.913 8.780c 0.868 17.767c 0.827 25.044c 0.769 30.190c 0.772 34.806c Japanese Yen 0.920 9.662z 0.899 19.903 0.912 28.689c 0.932 33.973c 0.899 38.568c Dutch Guilder 0.936 8.862y 0.935 18.904c 0.952 27.928c 1.000 33.468c 0.990 41.812c Portuguese Escudo 0.916 8.421y 0.928 19.027y 0.924c 23.918 0.857 27.450 0.867 32.467c Swedish Krona 0.861 7.541c 0.876 16.089 0.886 24.345c 0.855 31.744b 0.799 37.943c Swiss Franc 0.941 9.708c 0.946 19.694b 0.976 27.197c 0.950b 31.725c 0.880 36.235c British Pound 0.934 8.571y 0.933 16.954c 0.932 24.064c 0.947 28.761c 0.925a 31.372c (1) Coverage Test† Model Better 0 0 2 1 1 RW Better 0 0 0 0 0 (2) Length Test Given Equal Coverage Accuracy‡ Model Better 5 6 8 10 11 RW Better 6 1 0 0 0 (1)+(2)§ Model Better 5 6 10 11 12 RW Better 6 1 0 0 0 Note: –hdenotesforecasthorizonsformonthlydata. –For each horizon (h), the first column (Cov.) reports empirical coverages given a nominal coverage of 90%. The second column (Leng.) reportsthelengthofforecastintervalsintermsofpercentagechangeoftheexchangerate. Empiricalcoveragesandlengths areaveragesacrossN(h)out-of-sampletrials. –Superscripts a,b,c in the column of Cov. (Leng.) denote rejections of equal coverage accuracy (equal length) in favor of the economic model at a 10%, 5% and 1% confidence level respectively. Superscripts x,y,z are defined analogously for rejections in favoroftherandomwalk. †–Inthispanel,abettermodelistheonewithmoreaccurateempiricalcoverages. RWistheabbreviationofRandomWalk. ‡–Inthispanel,abettermodelistheonewithtighterforecastintervalsgivenequalcoverageaccuracy. §–In this panel, a better model is the one with either more accurate coverages or tighter forecast intervals given equal coverage accuracy. 25

Table 6: Results of Monetary Model h=1 h=3 h=6 h=9 h=12 Cov. Leng. Cov. Leng. Cov. Leng. Cov. Leng. Cov. Leng. Australian Dollar 0.879 7.108 0.848 15.151 0.830 20.090 0.770 24.642c 0.745 30.099b Canadian Dollar 0.842 4.027x 0.829 7.492 0.744 10.518 0.645 10.689b 0.675 12.993c Danish Kroner 0.905 8.770b 0.893 17.943 0.897 25.017c 0.853 28.581c 0.809 32.504c French Franc 0.922 8.791c 0.910 18.237c 0.949b 26.322c 0.957 31.032c 0.956 35.971c Deutschmark 0.908 8.595 0.841 17.436c 0.808 24.622c 0.772 28.052c 0.704 31.364c Italian Lira 0.913 8.858c 0.882 18.439b 0.925 26.585c 0.931 34.857c 0.913 40.885c Japanese Yen 0.930 9.556 0.919 19.374c 0.887 28.614c 0.864 33.401c 0.809 36.520c Dutch Guilder 0.917 8.753a 0.916 19.408 0.962 29.149b 0.970 38.173 0.898c 41.716c Portuguese Escudo 0.901 8.086 0.986 18.484 0.985 24.744 0.984 27.230 1.000 34.222 Swedish Krona 0.850 7.504a 0.848 17.097x 0.811 23.878c 0.826 31.287 0.805 34.710c Swiss Franc 0.905 9.078c 0.820 17.020c 0.732 21.212c 0.609 22.741c 0.513x 23.225c British Pound 0.909 7.811c 0.882 14.945c 0.787 20.788c 0.677 24.311c 0.656 26.374c (1) Coverage Test† Model Better 0 0 1 0 1 RW Better 0 0 0 0 1 (2) Length Test Given Equal Coverage Accuracy‡ Model Better 7 6 8 9 9 RW Better 1 1 0 0 0 (1)+(2)§ Model Better 7 6 9 9 10 RW Better 1 1 0 0 1 Note: –hdenotesforecasthorizonsformonthlydata. –For each horizon (h), the first column (Cov.) reports empirical coverages given a nominal coverage of 90%. The second column (Leng.) reportsthelengthofforecastintervalsintermsofpercentagechangeoftheexchangerate. Empiricalcoveragesandlengths areaveragesacrossN(h)out-of-sampletrials. –Superscripts a,b,c in the column of Cov. (Leng.) denote rejections of equal coverage accuracy (equal length) in favor of the economic model at a 10%, 5% and 1% confidence level respectively. Superscripts x,y,z are defined analogously for rejections in favoroftherandomwalk. †–Inthispanel,abettermodelistheonewithmoreaccurateempiricalcoverages. RWistheabbreviationofRandomWalk. ‡–Inthispanel,abettermodelistheonewithtighterforecastintervalsgivenequalcoverageaccuracy. §–In this panel, a better model is the one with either more accurate coverages or tighter forecast intervals given equal coverage accuracy. 26

Table 7: Results of Heterogenous Taylor Rules h=1 h=3 h=6 h=9 h=12 Cov. Leng. Cov. Leng. Cov. Leng. Cov. Leng. Cov. Leng. Australian Dollar 0.915 7.155 0.909 14.690c 0.959 20.547 0.963y 26.364 0.947 29.583c Canadian Dollar 0.825 3.526 0.794 6.525c 0.797 9.040c 0.787 10.370c 0.693 11.352c Danish Kroner 0.915 8.548c 0.929 17.930c 0.938 25.328c 0.890 30.504c 0.904 36.624c French Franc 0.912 8.864c 0.880 15.970c 0.845 20.693c 0.968 30.436c 0.714 22.789c Deutschmark 0.917 8.605c 0.907 18.356c 0.894 28.121c 0.911c 31.378c 0.939 33.779c Italian Lira 0.913 8.659c 0.890 18.664 0.887 25.840c 0.831 32.037c 0.693 32.236c Japanese Yen 0.920 9.637z 0.888 19.352b 0.871 28.018c 0.932 33.388c 0.878 36.859c Dutch Guilder 0.936 8.851 0.916 18.822c 0.942 27.259c 0.970 31.410c 0.990 39.882c Portuguese Escudo 0.916 8.881z 0.870 17.651 0.758 18.730c 0.746 23.852c 0.600 20.593c Swedish Krona 0.828 7.428c 0.854 15.658c 0.903 24.315c 0.861 29.866c 0.876 36.235c Swiss Franc 0.935 9.731c 0.940 19.797 0.970 27.227c 0.969 32.179c 0.937 36.567c British Pound 0.919 8.350 0.908 16.774c 0.828 20.811c 0.783 23.105c 0.720 23.286c (1) Coverage Test† Model Better 0 0 0 1 0 RW Better 0 0 0 1 0 (2) Length Test Given Equal Coverage Accuracy‡ Model Better 6 9 11 10 12 RW Better 2 0 0 0 0 (1)+(2)§ Model Better 6 9 11 11 12 RW Better 2 0 0 1 0 Note: –hdenotesforecasthorizonsformonthlydata. –Foreachhorizon(h),thefirstcolumn(Cov.) reportsempiricalcoveragesgivenanominalcoverageof90%. Thesecondcolumn (Leng.) reports the length of forecast intervals in terms of percentage change of the exchange rate. Empirical coverages and lengthsareaveragesacrossN(h)out-of-sampletrials. –Superscripts a,b,c in the column of Cov. (Leng.) denote rejections of equal coverage accuracy (equal length) in favor of the economicmodelata10%, 5%and1%confidencelevelrespectively. Superscriptsx,y,z aredefinedanalogouslyforrejectionsin favoroftherandomwalk. †–Inthispanel,abettermodelistheonewithmoreaccurateempiricalcoverages. RWistheabbreviationofRandomWalk. ‡–Inthispanel,abettermodelistheonewithtighterforecastintervalsgivenequalcoverageaccuracy. §–In this panel, a better model is the one with either more accurate coverages or tighter forecast intervals given equal coverage accuracy. 27

Figure1: LengthofForecastIntervalsforBenchmarkTaylorRuleandRandomWalkModels(BritishPound) 1.4 1.2 1 0.8 0.6 0.4 0.2 0 21M9891 7M0991 2M1991 9M1991 4M2991 11M2991 6M3991 1M4991 8M4991 3M5991 01M5991 5M6991 21M6991 7M7991 2M8991 9M8991 4M9991 11M9991 6M0002 1M1002 8M1002 3M2002 01M2002 5M3002 21M3002 7M4002 2M5002 Taylor Rule Model Random Walk Model (a) 1-month-aheadforecast 1.4 1.2 1 0.8 0.6 0.4 0.2 0 5M0991 21M0991 7M1991 2M2991 9M2991 4M3991 11M3991 6M4991 1M5991 8M5991 3M6991 01M6991 5M7991 21M7991 7M8991 2M9991 9M9991 4M0002 11M0002 6M1002 1M2002 8M2002 3M3002 01M3002 5M4002 21M4002 7M5002 Taylor Rule Model Random Walk Model (b) 6-month-aheadforecast 1.4 1.2 1 0.8 0.6 0.4 0.2 0 11M0991 6M1991 1M2991 8M2991 3M3991 01M3991 5M4991 21M4991 7M5991 2M6991 9M6991 4M7991 11M7991 6M8991 1M9991 8M9991 3M0002 01M0002 5M1002 21M1002 7M2002 2M3002 9M3002 4M4002 11M4002 6M5002 1M6002 Taylor Rule Model Random Walk Model (c) 12-month-aheadforecast Note: Ineachchart,thelengthofforecastintervalsisnormalizedbythefirstobservationofthebenchmarkTaylorrulemodel. 28

Figure2: LengthofForecastIntervalsforBenchmarkTaylorRuleandRandomWalkModels(Deutschmark) 1.2 1 0.8 0.6 0.4 0.2 0 21M9891 4M0991 8M0991 21M0991 4M1991 8M1991 21M1991 4M2991 8M2991 21M2991 4M3991 8M3991 21M3991 4M4991 8M4991 21M4991 4M5991 8M5991 21M5991 4M6991 8M6991 21M6991 4M7991 8M7991 21M7991 4M8991 8M8991 21M8991 Taylor Rule Model Random Walk Model (a) 1-month-aheadforecast 1.2 1 0.8 0.6 0.4 0.2 0 5M0991 9M0991 1M1991 5M1991 9M1991 1M2991 5M2991 9M2991 1M3991 5M3991 9M3991 1M4991 5M4991 9M4991 1M5991 5M5991 9M5991 1M6991 5M6991 9M6991 1M7991 5M7991 9M7991 1M8991 5M8991 9M8991 Taylor Rule Model Random Walk Model (b) 6-month-aheadforecast 1.2 1 0.8 0.6 00.44 0.2 0 11M0991 3M1991 7M1991 11M1991 3M2991 7M2991 11M2991 3M3991 7M3991 11M3991 3M4991 7M4991 11M4991 3M5991 7M5991 11M5991 3M6991 7M6991 11M6991 3M7991 7M7991 11M7991 3M8991 7M8991 11M8991 Taylor Rule Model Random Walk Model (c) 12-month-aheadforecast Note: Ineachchart,thelengthofforecastintervalsisnormalizedbythefirstobservationofthebenchmarkTaylorrulemodel. 29

Figure 3: Length of Forecast Intervals for Benchmark Taylor Rule and Random Walk Models (Japanese Yen) 1.2 1 0.8 0.6 00.44 0.2 0 21M9891 8M0991 4M1991 21M1991 8M2991 4M3991 21M3991 8M4991 4M5991 21M5991 8M6991 4M7991 21M7991 8M8991 4M9991 21M9991 8M0002 4M1002 21M1002 8M2002 4M3002 21M3002 8M4002 4M5002 21M5002 Taylor Rule Model Random Walk Model (a) 1-month-aheadforecast 1.4 1.2 1 0.8 0.6 0.4 0.2 0 5M0991 21M0991 7M1991 2M2991 9M2991 4M3991 11M3991 6M4991 1M5991 8M5991 3M6991 01M6991 5M7991 21M7991 7M8991 2M9991 9M9991 4M0002 11M0002 6M1002 1M2002 8M2002 3M3002 01M3002 5M4002 21M4002 7M5002 2M6002 Taylor Rule Model Random Walk Model (b) 6-month-aheadforecast 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 11M0991 6M1991 1M2991 8M2991 3M3991 01M3991 5M4991 21M4991 7M5991 2M6991 9M6991 4M7991 11M7991 6M8991 1M9991 8M9991 3M0002 01M0002 5M1002 21M1002 7M2002 2M3002 9M3002 4M4002 11M4002 6M5002 1M6002 Taylor Rule Model Random Walk Model (c) 12-month-aheadforecast Note: Ineachchart,thelengthofforecastintervalsisnormalizedbythefirstobservationofthebenchmarkTaylorrulemodel. 30

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APPENDIX A.1 Monetary and Taylor Rule Models In this section, we describe the monetary and Taylor rule models used in the paper. A.1.1 Monetary Model Assume the money market clearing condition in the home country is: m =p +γy −αi +v , t t t t t where m is the log of money supply, p is the log of aggregate price, i is the nominal interest rate, y is the t t t t log of output, and v is the money demand shock. A symmetric condition holds in the foreign country and t we use an asterisk in superscript to denote variables in the foreign country. Subtracting the foreign money market clearing condition from the home, we have: 1 i −i∗ = [−(m −m∗)+(p −p∗)+γ(y −y∗)+(v −v∗)]. (A.1.1) t t α t t t t t t t t The nominal exchange rate is equal to its purchasing power value plus the real exchange rate: s =p −p∗+q . (A.1.2) t t t t The uncovered interest rate parity in financial market takes the form: E s −s =i −i∗+ρ , (A.1.3) t t+1 t t t t whereρ istheuncoveredinterestrateparityshock. Substitutingequations(A.1.1)and(A.1.2)into(A.1.3), t we have s =(1−b)[m −m∗−γ(y −y∗)+q −(v −v∗)]−bρ +bE s , (A.1.4) t t t t t t t t t t t+1 35

where b=α/(1+α). Solving s recursively and applying the “no-bubbles” condition, we have: t   s =E  (1−b) X ∞ bj(cid:2) m −m∗ −γ(y −y∗ )+q −(v −v∗ ) (cid:3) −b X ∞ bjρ  . (A.1.5) t t t+j t+1 t+j t+1 t+1 t+j t+j t+j   j=0 j=1 In the standard monetary model, such as Mark (1995), purchasing power parity (q =0) and uncovered t interest rate parity hold (ρ = 0). Furthermore, it is assumed that the money demand shock is zero (v = t t v∗ =0) and γ =1. Equation (A.1.5) reduces to: t   s =E  (1−b) X ∞ bj(cid:0) m −m∗ −(y −y∗ ) (cid:1)  . t t t+j t+j t+j t+j   j=0 A.1.2 Taylor Rule Model WefollowEngelandWest(2005)toassumethatbothcountriesfollowtheTaylorruleandtheforeigncountry targets the exchange rate in its Taylor rule. The interest rate differential is: i −i∗ =δ (s −s¯∗)+δ (ygap−ygap∗)+δ (π −π∗)+v −v∗, (A.1.6) t t s t t y t t π t t t t where s¯∗ is the targeted exchange rate. Assume that monetary authorities target the PPP level of the t exchange rate: s¯∗ = p −p∗. Substituting this condition and the interest rate differential into the UIP t t t condition, we have: s =(1−b)(p −p∗)−b (cid:2) δ (ygap−ygap∗)+δ (π −π∗)+v −v∗(cid:3) −bρ +bE s , (A.1.7) t t t y t t π t t t t t t t+1 where b= 1 . Assuming that uncovered interest rate parity holds (ρ =0) and monetary shocks are zero, 1+δs t equation (A.1.7) reduces to the benchmark Taylor rule model in our paper:    X ∞ X ∞  s =E (1−b) bj(p −p∗ )−b bj(δ (ygap−ygap∗)+δ (π −π∗ )) . t t t+j t+j y t+j t+j π t+j t+j   j=0 j=0 A.2 Long-horizon Regressions In this section, we derive long-horizon regressions for the monetary model and the benchmark Taylor rule model. 36

A.2.1 Monetary Model In the monetary model:   s =E  (1−b) X ∞ bj(cid:0) m −m∗ −(y −y∗ ) (cid:1)  , t t t+j t+j t+j t+j   j=0 wherem andy arelogarithmsofdomesticmoneystockandoutput,respectively. Thesuperscript∗denotes t t the foreign country. Money supplies (m and m∗) and total outputs (y and y∗) are usually I(1) variables. t t t t The general form considered in Engel, Wang, and Wu(2008) is: ∞ s = (1−b) X bjE α 0 D t t t j=0 (I −Φ(L))∆D = ε (A.2.1) n t t E(ε |ε ,ε ,...) ≡ E (ε )=0,∀j ≥1, t+j t t−1 t t+j where n is the dimension of D and I is an n×n identity matrix. L is the lag operator and Φ(L) = t n φ L+φ L2+...+φ Lp. AssumeΦ(1)isnon-diagonalandthecovariancematrixofε isgivenbyΩ=E [ε ε0]. 1 2 p t t t t We assume that the change of fundamentals follows a VAR(p) process in our setup. From proposition 1 of Engel, Wang, Wu (2008), we know that for a fixed discount factor b and p≥2, 0 0 s −s =β z +δ ∆D +...+δ ∆D +ζ t+h t h t 0,h t p−2,h t−p+2 t+h is a correctly specified regression where the regressors and errors do not correlate. In the case of p=1, the long-horizon regressions reduces to s −s =β z +ζ . t+h t h t t+h Followingtheliterature,forinstanceMark(1995),wedonotinclude∆D anditslagsinourlong-horizon t regressions. The monetary model can be written in the form of (A.2.1) by setting D =[m m∗ y y∗]0, t t t t t α = [1 −1 −1 1]0. By definition, z = s −(m −m∗)+(y −y∗). This corresponds to β = 1, t t t t t t m,h X =s −(m −m∗)+(y −y∗) in equation (1) of section 3. m,t t t t t t 37

A.2.2 Taylor Rule Model In the Taylor rule model,    X ∞ X ∞  s =E (1−b) bj(p −p∗ )−b bj(δ (ygap−ygap∗)+δ (π −π∗ )) , t t t+j t+j y t+j t+j π t+j t+j   j=0 j=0 where p , ygap and π are domestic aggregate price, output gap and inflation rate, respectively. δ and δ t t t y π are coefficients of the Taylor rule model. The aggregate prices p and p∗ are usually I(1) variables. Inflation t t and output gap are more likely to be I(0). Engel, Wang, and Wu (2008) consider a setup which includes both stationary and non-stationary variables: ∞ ∞ X X s =(1−b) bjE [f ]+b bjE [f +u ] t t 1t+j t 2t+j 2t+j j=0 j=0 f =α0D ∼I(1) 1t 1 t f =α0∆D ∼I(0) 2t 2 t u =α0∆D ∼I(0) 2t 3 t (I −Φ(L))∆X =ε , (A.2.2) n t t where f and f (u )are observable (unobservable)fundamentals. ∆D is the firstdifference ofD , which 1t 2t 2t t t contains I(1) economic variables.20 Fromproposition2ofEngel,Wang,andWu(2008),weknowthatforafixeddiscountfactorbandh≥2, p−1 s −s =β˜ z + X δ˜0 ∆D +ζ˜ (A.2.3) t+h t h t k,h t−k t+h k=0 is a correctly specified regression, where the regressors and errors do not correlate. In the case of p=1, the long-horizon regressions reduces to: s −s =β˜ z +ζ˜ . t+h t h t t+h 20To incorporate I(0) economic variables, Dt contains the levels of I(1) variables and the summation of I(0) variables from negativeinfinitytotimet. 38

The Taylor rule model can be written into the form of (A.2.2) by setting " t t t t #0 X X X X D = p p∗ ygap ygap∗, π π∗ . t t t s s s s s=−∞ s=−∞ s=−∞ s=−∞ By definition, z = s − p + p∗ + b (δ (ygap − ygap∗) + δ (π − π∗)). This corresponds to β = t t t t 1−b y t t π t t m,h [1 b δ b δ ]andX =[q ygap−ygap∗ π −π∗], whereq =s −p +p∗ istherealexchangerate. 1−b y 1−b π m,t t t t t t t t t t β and X can be defined differently. For instance, β = 1 and X = s −p +p∗+ b (δ (ygap− m,h m,t m,h m,t t t t 1−b y t ygap∗)+δ (π −π∗)). Our results do not change qualitatively under this alternative setup. t π t t A.3 Model with Heterogeneous Taylor Rules In the benchmark model, we assume that the Taylor rule coefficients are the same in the home and foreign countries. In this appendix, we relax the assumption of homogeneous Taylor rules in the benchmark model. It is straightforward to show in this case that the benchmark model changes to: s −s =α +β z +ε , (A.3.1) t+h t h h t t+h where the deviation of the exchange rate from its equilibrium level is defined as: z =s −p +p∗+ b (cid:2) δ ygap−δ∗ygap∗+δ π −δ∗π∗(cid:3) . (A.3.2) t t t t 1−b y t y t π t π t According to equation (8), the matrix X in equation (1) includes economic variables q ≡ s +p∗ −p , 1,t t t t t δ ygap−δ∗ygap∗, and δ π −δ∗π∗.21 y t y t π t π t We first estimate the Taylor rules in the home and foreign countries according to equations (2) and (3). Thenq ≡s +p∗−p ,δˆ ygap−δˆ∗ygap∗,andδˆ π −δˆ∗π∗ areusedinthelong-horizonregressionsandinterval t t t t y t y t π t π t forecasts. The results are very similar to the benchmark model and reported in Table 7. 21AnotheroptiontoincorporateheterogenousTaylorrulesistoincludeqt,y t gap,y t gap∗,πt,andπ t ∗ inX1,t. Forinstance,see Moldtsova and Papell (forthcoming). However, increasing the number of regressors may cause the “curse of dimensionality” problemforoursemiparametricmethod. Tobecomparabletoourbenchmarkmodel,wedefineX1,theresuchthatthenumber ofregressorsisthesameasinthebenchmarkmodel. 39