A Nowcasting Model for Canada: Do U.S. Variables Matter?

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. A Nowcasting Model for Canada: Do U.S. Variables Matter? Daniela Bragoli and Michele Modugno 2016-036 Please cite this paper as: Bragoli,DanielaandMicheleModugno(2016). “ANowcastingModelforCanada: DoU.S. VariablesMatter?,”FinanceandEconomicsDiscussionSeries2016-036. Washington: Board of Governors of the Federal Reserve System, http://dx.doi.org/10.17016/FEDS.2016.036. 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.

A N M C : D U.S. OWCASTING ODEL FOR ANADA O V M ?∗ ARIABLES ATTER Daniela Bragolia and Michele Modugnob April 2016 Abstract We propose a dynamic factor model for nowcasting the growth rate of quarterly real Canadian gross domestic product. We show that the proposed model produces more accurate nowcasts than those produced by institutional forecasters, like the Bank of Canada, the The Organisation for Economic Co-operation and Development (OECD), and the surveycollectedbyBloomberg,whichreflectsthemedianforecastofmarketparticipants. We showthatincludingU.S.datainanowcastingmodelforCanadadramaticallyimprovesits predictive accuracy, mainly because of the absence of timely production data for Canada. Moreover,StatisticsCanadaproducesamonthlyrealGDPmeasurealongwiththequarterly one, and we show how to modify the state space representation of our model to properly linkthemonthlyGDPwithitsquarterlycounterpart. JELClassification: C33,C53,E37. Keywords: Nowcasting,Updating,DynamicFactorModel. ∗We would like to thank Now-Casting Economics for advice, feedback, and access to data. The opinions in thispaperarethoseoftheauthorsanddonotnecessarilyreflecttheviewoftheBoardofGovernorsoftheFederal ReserveSystem. aUniversita` Cattolica,viaNecchi9,29100Milano(Italy),e-mail: daniela.bragoli@unicatt.it bFederalReserveBoard,WashingtonDC,20551(US),e-mail: michele.modugno@frb.gov

1 Introduction Policymakers and market participants track the state of the economy on a daily basis, the former to make decisions about the conduct of (conventional and unconventional) monetary policies, macroprudential policies, and fiscal policies, and the latter to make decisions about theirinvestmentstrategies. Real grossdomestic product (GDP) isone of the mostmonitored indicators, but it isgenerallyreleasedonaquarterlybasisandwithadelaythatrangesfromfourweeks(asintheUnited KingdomandintheUnitedStates)toeightweeks(asinCanada). Given real GDP’s lack of timeliness, policymakers and market participants try to infer current economic conditions by monitoring other indicators that are linked to GDP growth but are releasedatahigherfrequencyandinamoretimelymanner. Statisticalofficesandcentralbank releasedataalmosteverydaythatcanhelpinferthelevelofrealGDP. ThefactthatvariablesotherthanGDParehighlyregardedbypractitionersmaybeinferred from global information services, such as Bloomberg and Forex Factory: They not only report a calendar of data releases but also a measure of importance for each, which reflects usage of the data by practitioners. In addition, Bloomberg conducts a survey and collects forecasts from analysts and economists on each release it reports and publishes it the day before the release is disseminated. The aim of this paper is to propose an econometric model that can give users the ability to inferthestateoftheeconomyintermsofrealGDPandthatcanbeeasilyupdatedwhenevernew information becomes available. The econometric framework that we propose is based on the seminalpaperbyGiannoneetal.(2006),whoshowhowtoproduceanaccuratenowcastofU.S. (quarterly) real GDP using monthly indicators by taking into account their different releasing times and therefore the ragged-edge shape of the data set by using of a dynamic factor model (DFM).Inparticular,wefollowBan´buraandModugno(2014),whoshowhowtoestimateDFM withdatareleasedatdifferentfrequenciesandwithdifferenttime-spancoverage. DFMs have been successfully applied to various economies, both developed and develop- 2

ing1. Althoughthestatisticalframeworkissimilartotheonealreadyappliedtoothereconomies, each country has its own peculiarities in terms of the relevance that each input series has in tracking real GDP; the timeliness of the releases; and the importance that market participants attribute to production, demand or trade. The aim of country-specific nowcasting models is also to learn about the characteristics of the data, how they interact with real GDP, and, more broadly,howtheeconomyofthatspecificcountryworks. To our knowledge, this is the first article in the nowcasting literature to consider the Canadian economy. Canada was the 11th largest economy as of 2015, with a nominal GDP of approximately U.S.$ 1.79 trillion. It is a member of the OECD and the Group of Eight, and is one of the world’s top 10 trading nations, with a highly globalized economy. Furthermore, the TorontoStockExchangeisthe7thlargeststockexchangeintheworldbymarketcapitalization, listing over 1,500 companies with a combined market capitalization of over U.S.$ 2 trillion as of 2015.2 Tracking the state of the Canadian economy is of interest not only for national policymakersandmarketparticipants,butfortheirinternationalcounterpartsaswell. Canadiandatahaveseveralpeculiaritiesthatdistinguishitfromothereconomies. Forexample, quarterly GDP is released with eight weeks of delay from the end of the reference period, whichislesstimelythantheUnitedStates(fourweeks)andtheeuroarea(sixweeks). However, StatisticsCanada,differentlyfromothercountries,alsopublishesamonthlyfiguretogetherwith quarterly GDP growth eight weeks after the end of the reference period (the monthly real GDP relative to January is available only in March; the quarterly real GDP relative to Q1 is available only in May). Moreover, compared with other countries, Canada lacks some important serieslinkedtoproduction-namelyindustrialproductionandcapacityutilization-thathavebeen proved to be important in tracking real GDP growth for other economies given their timeliness 1The United States (Lahiri and Monokroussos, 2013), the euro area (Angelini et al., 2010; Angelini et al., 2011; and Camacho and Perez-Quiros, 2010), France (Barhoumi et al., 2010), Ireland (D’Agostino et al., 2008; and Liebermann, 2012), the Netherlands (De Winter, 2011), the Czech Republic (Arnostova et al., 2011; and Rusna´k, 2013), New Zealand (Matheson, 2010), Norway (Aastveit et al., 2012; and Luciani and Ricci, 2014), Switzerland (Siliverstovs, 2012), China (Yiu and Chow, 2010), Turkey (Modugno et al., 2016), Brazil (Bragoli et al., 2015), Mexico (Caruso, 2015), and Indonesia (Luciani et al., 2015). Moreover, the same framework has been applied to nowcast other variables than real GDP; see, among others D’Agostino et al. (2015) for the euro areatradevariablesandModugno(2013)forU.S.inflation. 2seeFund(2015)andhttp://www.tmx.com/resource/en/117 3

andtheircorrelationwiththetargetvariable. Inordertoaddressthesepeculiarities,weproposeanewmodellingstrategythatcoherently takesintoaccounttherelationbetweenquarterlyrealGDPanditsmonthlycounterpart,andwe deal with the lack of timely production data by including some of the most important series of the US economy. The United States is Canada’s the first trade partner, with a total trade share ofabout70percentin2015.3 U.S.productionseries,giventhestronglinkagesbetweenthetwo economies,arecrucialtogettinganaccurateassessmentofwheretheCanadianeconomystands. Moreover,giventheinterconnectednessoftheU.S.andCanadianeconomies,weaddtwomore U.S. series that are particularly relevant for market participants, are among the most timely U.S. series, and have been shown to be crucial to nowcasting the U.S. real GDP: purchasing managersindex(PMI)andtotalnon-farmpayrolls.4. Results show that our model can deliver nowcasts that are at least as accurate as those produced by institutional forecasters such as the Bank of Canada, the OECD, and the Bloomberg survey. This result suggests that model-based forecasts that can quickly incorporate a potentially large set of new information can be a complementary tool to judgmental forecasts for participants assessing the state of the Canadian economy. Moreover, it confirms what has been found in other studies, namely that a linear time-invariant model does a good job, and hence thateventualnonlinearities,timevariations,andsoftinformation(suchasweatherconditionsor government decisions) that could be incorporated by judgment, do not provide important new information. According to this last result, the often-cited superiority of professional forecasts (seeAngetal.,2007;ClementsandHendry,2004;andJansenetal.,2012)turnsouttobeweak in our sample, confirming findings in Giannone et al. (2006) and Liebermann (2011). Moreover,weshowthatU.S.dataarecrucialtoobtaininganaccuraterealGDPnowcastforCanada: The lack of timely production data for the Canadian economy can be compensated for using U.S. data, confirming that the interconnectedness between the two economies can be exploited toproducemoreprecisenowcastsofCanadianrealGDP. The rest of the paper is structured as follows. Section 2 describes the structure of the data 3Seehttp://www.statcan.gc.ca/tables−tableaux/sum−som/l01/cst01/gblec02a−eng.htm. 4SeeBan´buraetal.(2012)andBan´buraetal.(2013). 4

releases in Canada. Section 3 introduces the model and estimation technique. Section 4 describes the Bank of Canada surveys and other benchmarks. Section 5 introduces the empirical analysisandcommentsontheresults. Section6concludes. 2 The data set Canadian real GDP growth is published by the statistical office two months (60 days) after theendofthequarter,whichmeansrealGDPgrowthin,forexample,thefirstquarter(January toMarch)isdisclosedonlyinMay. ThedelayofrealGDPgrowthdatapublicationinCanadais greaterthaninmostdevelopedcountries-thatis-fourweeksinthecaseoftheUnitedKingdom and the United States and six weeks for European countries and Japan. Nevertheless, Canada also releases GDP data on a monthly basis about 60 days after month end. The quarterly GDP figureisapproximatelyasummationofthemonthlydata.5 The aim of the statistical model we propose in this paper is to predict GDP before the official figures are published by taking advantage of the flow of other economic data releases that precede GDP publication and that allow us to update our prediction with each successive datarelease. Weincludeinourmodelvariableswhoseheadlinenumberisreportedbythemainstatistical sources. In addition, we consider indicators monitored by financial markets and the press. We choose the transformations that guarantee stationarity of the variables (see table 1), which are the same as those reported by the media and Bloomberg, making the comparison easier. We consideronlyrealdataandsurveys. Wedisregardprices,financialvariables,nominalvariables, and sector-specific series. This choice reflects the results of previous research in which the inclusionofthesevariablesdoesnotimprovethemodel’sforecastingperformance(seeBan´bura andModugno,2010;andBan´buraandModugno,2014). InthenowcastingmodelforCanada,wedecidetoalsoincludesomeoftheU.S.variables;in particular, we focus on the four most timely releases: namely U.S. manufacturing PMI, change 5BrazilalsoreleasesanominalmonthlyGDPseries. 5

in nonfarm employment, industrial production and capacity utilization. One of the aims of this article is to test whether the inclusion of U.S. variables significantly improves the nowcasting performanceofourmodelforCanada. The target variable is quarterly real GDP growth reported as a quarter-on-quarter (QoQ) transformation, the headline series on which market participants focus their attention. The input series, which are monthly, are reported as month-on-month (MoM) transformations with the exception of PMI manufacturing surveys (both for Canada and the United States), which are in levels but behave like a MoM series because of how they are constructed. U.S. capacity utilization and both U.S. and Canadian employment are reported as a monthly change, and motor vehicle sales, which are not seasonally adjusted, are reported as a year-on-year (YoY) transformation.6 Linking the QoQ target variable with MoM input series is standard in this literature (see, among others, Mariano and Murasawa, 2003, Camacho and Perez-Quiros, 2010 andBan´buraandModugno,2014). Table 1 reports some details on the selected series, including the timing of the release and the importance that the financial markets attach to the series according to the Bloomberg and ForexFactoryindexes. TheBloombergmeasure,whichisshownasapercentage,reflectsusage by Bloomberg subscribers; the Forex Factory measure, which is shown as low/medium/high, reflectsForexFactorysubscribers’judgment. One peculiarity of the Canadian data set is the fact that it includes the monthly real GDP publishedbyStatisticsCanadabutdoesnothaveanindexofindustrialproduction,whichisone ofthemostimportantpieceofharddataforothereconomiesgivenitstimeliness(U.S.industrial productionindex,forexample,hasapublicationlagofonly15days). The rest of the variables can be divided into four categories: surveys, labor, domestic demand, and trade indicators. Among surveys, we consider the PMI which is released at the beginning of the following month and rated highly relevant to the market by Forex but relatively unimportant by Bloomberg (15.4 percent). For labor, we include employment, released 6ToconstructthePMIindex,respondentsareaskedwhetherbusinessconditionsforanumberofvariableshave improved,deteriorated,orstayedthesamecomparedwiththepreviousmonth. Therefore,theindexiscomparable toaMoMtransformation. 6

one week after month end and rated important by Forex. For domestic demand, we track manufacturing shipments (or sales), manufacturing orders, retail sales and wholesale trade, motor vehicle sales, building permits and dwelling starts. Building permits and retail sales are considered relevant by both Bloomberg and Forex. The trade category includes exports and imports, which have a publication lag of one month and are considered highly important by Forex. Dwelling starts, which are released one week after month end, are considered relevant formarketparticipantsbyBloomberg(82.0percent) To conclude, the most timely data in Canada are the PMI, employment and dwelling starts, which are released with a publication lag of one week. Building permits and trade variables (exports and imports) are released five weeks after the end of the reference month. The rest of the hard data (sales and orders) are released with a six - to seven- week lag from the end of the referencemonth. If we consider only the Canadian data we immediately realize that there are few timely variable. Inthefollowingsections,wewillshowthattheintroductionoffourofthemosttimely U.S. variables, which are released at most 15 days after the end of the reference month and are related to the Canadian business cycle, will improve the nowcasting performance of our model forCanada. 7

Table1: Seriesusedinthemodel Name Timing Publishinglag Frequency Source Available Transf. Relevance Relevance from Forex Bloomberg USManufacturingPMI firstdays 5days M ISM Jan-48 Levels H 94.6 USChangeinEmployment firstdays 5days M BLS Feb-39 Levels H 99.1 USIndustrialProduction middle 15days M FRB Jan-21 MoM M 86.6 USCapacityUtilization middle 15days M FRB Jan-67 Monthlychange M 60.6 CanadaIveyPMI firstdays 5days M IveyUWO Jan-01 Levels H 15.4 CanadaBuildingPermits firstweek 35days M StatCan Jan-60 MoM H 71.8 CanadaEmployment firstweek 8days M StatCan Jan-66 Monthlychange H 30.8 CanadaDwellingStarts firstweek 9days M CMHC Jan-90 MoM L 82 CanadaImportsofGoods firstdays 35days M StatCan Jan-88 MoM H - CanadaExportsofGoods firstdays 35days M StatCan Jan-88 MoM H - CanadaManufacturingShipments middle 45days M StatCan Jan-92 MoM H - CanadaMotorVehicleSales middle 45days M StatCan Jan-46 YoY - 0 CanadaManufacturingOrders middle 45days M StatCan Jan-92 MoM - - CanadaWholesaleTrade 20thmonth 50days M StatCan Jan-93 MoM M 51.3 CanadaRetailSales 20thmonth 50days M StatCan Jan-91 MoM M 74.4 CanadaMonthlyGDP lastdays 60days M StatCan Jan-97 MoM H 84.6 CanadaRealGDP lastdays 2months Q StatCan Q1-81 QoQ - 92.3 Notes. Timing is approximately the number of days from the end of the reference period. Frequency indicates whether the series is monthly (M) or quarterly (Q). Available from indicates the starting date of the series. Sources are ISM (Institute for Supply Management), BLS (Bureau of Labor Statistics), FRB (Federal Reserve Board), IveyUWO (Richard Ivey School of Business, Univ of W. Ontario), StatCan (Statistics Canada), and CMHC (Canada Mortgage and Housing Corporation). Bloomberg reports the market relevance of each variable according to Bloomberg’s relevance index, whichrangesfrom0to100. ForexreportsthemarketrelevanceofeachvariableaccordingtoForexrelevanceindex,i.e.,L=Lowrelevance,M=medium relevance,andH=Highrelevance. 8

3 The nowcasting problem and the econometric framework Nowcasting GDP involves inferring its value after the reference quarters begins but before the official release of GDP data by exploiting information from other, higher-frequency variables7. Moreformally,thenowcastofGDP(yQ)canbedefinedasthelinearprojectionofyQ onthe t t availableinformationsetΩ ,whichcontainsmixed-frequencyvariables(x ,j = 1,...,J ) v kj,tj v+1 and is characterized by a “ragged-edge” structure because the time of the last available informationvariesfromseriestoseries. Each time new information arrives, a new nowcast is produced. This nowcast can be decomposedasfollows: P[yQ|Ω ] = P[yQ|Ω ]+P[yQ|I ]. (1) t v+1 t v t v+1 The new nowcast, P[yQ|Ω ], is just the sum of the old nowcast, P[yQ|Ω ], and the revision, t v+1 t v P[yQ|I ],where t v+1 I = x −P[x |Ω ], (2) v+1,j kj,tj kj,tj v with j = 1,...,J . This revision (I ) is the linear projection of our target variable on the v+1 v+1,j difference between the actual release of any variable (x ∈ Ω ) and what our model was kj,tj v+1 predictingforthatrelease(P[x |Ω ]): Thisrevisionistheonlyelementthatleadstoachange kj,tj v inthenowcastbecauseitisthe“unexpected”(withrespecttothemodel)partofthedatarelease andwecallit“news.” AsshownbyBan´buraandModugno(2010),therevisioncanbedecomposedasaweighted average of the news in the latest release. We can find a vector, B = [b ,...,b ] v+1 v+1,1 v+1,Jv+1 7Inthissection,wecloselyfollowBan´buraandModugno(2010) 9

suchthatthefollowingholds: Jv+1 (cid:88) P[yQ|Ω ]−P[yQ|Ω ] = B I = b (x −P[x |Ω ]). (3) t v+1 t v v+1 v+1 v+1,j kj,tj kj,tj v j=1 Themagnitudeofthenowcastrevisiondependsonboththesizeofthenewsandonitsrelevance forthetargetvariable,asrepresentedbytheassociatedweight,b .8 v+1,j Throughthismechanism,itispossibletotracethecontributionofeachseriestotherevision of the nowcast, in particular by linking the revision of the target variable nowcast with the unexpecteddevelopmentsoftheinputvariables. The model we use to compute the nowcast and the news is a dynamic factor model (DFM). This model produces a good representation of the data and, at the same time, guarantees parsimony. It exploits the fact that there is a large amount of co-movement among macroeconomic data series and, hence, that relatively few factors can explain the dynamics of many variables (seeSargentandSims,1977;Giannoneetal.,2005;andStockandWatson,2011). The general representation of a DFM can be written as a system with two types of equations: a measurement equation (equation 4) linking the observed series (that is, GDP and all theindicatorslistedintable1)toalatentstateprocess,andthetransitionequation(equation5), which describes the state process dynamics. Equations 4 and 5, written in a state space form, allow the use of the Kalman filter to obtain an optimal projection for both the observed and the state variables. The Kalman filter generates projections for all the variables in the model (GDP andalltheotherdatareleases). TheDFMisdescribedbythefollowingequations: y = Λf +e , (4) t t t f = A f +A f +...+A f +u u ∼ i.i.d.N(0,Q), (5) t 1 t−1 2 t−2 p t−p t t 8Equation3canbeconsideredageneralizationoftheKalmanfilterupdateequationtothecaseinwhichnew dataarriveinanon-synchronousmanner. SeeBan´buraandModugno(2010). 10

e = ρ e +v v ∼ i.i.d.N(0,σ2), (6) i,t i i,t−1 i,t i,t i where y = [y ;y ;...;y ](cid:48) denotes a set of standardized stationary monthly variables, f is t 1,t 2,t n,t t a vector of r unobserved common factors with zero mean and unit variance, Λ is a matrix of coefficients collecting the factor loadings for the monthly variables, ande = [e ;e ;...;e ](cid:48) t 1,t 2,t n,t is an n-dimensional vector of idiosyncratic components uncorrelated with f at all leads and t lags. This last assumption, which means that all the joint correlation between observables is explainedbythecommonfactors,isstrongandunrealistic,however,Dozetal.(2006)haveshown thattheeffectsofthismisspecificationontheestimationofthecommonfactorsisnegligiblefor alargesamplesize(T)andthecross-sectionaldimension(n). We consider only one state variable (or factor) and two lags in equation 5 and an AR(1) processfortheidiosyncraticcomponentsdescribedinequation6.9 In the case of Canada, we depart from the usual DFM representation. As explained earlier, Statistics Canada produces a monthly GDP series. To fully exploit this source of information, we propose a new modelling strategy to incorporate the quarterly GDP series: We impose restrictions on the factor loadings such that the monthly GDP becomes a state variable and the quarterly GDP only loads the monthly GDP through the aggregation proposed by Mariano and Murasawa (2003). The other state variable is instead extracted from all the other series in our data set and interacts with the monthly GDP trough the VAR, helping to forecast the future realizations of the monthly GDP and, therefore, of the quarterly GDP. In practice, our model 9We use Bai and Ng (2002) information criteria to select the number of factors in equation 4 and Akaike informationcriteriatoselectthelagorderofequation5. Seetheappendixfordetails. 11

hasthefollowingstatespacerepresentation:10   ym t      ym   t−1     ym   t−2         ym     t−3  yq λ 2λ 3λ 2λ λ 0 0 .. 0   (cid:15)q t q q q q q   t      ym      ym   =   1 0 0 0 0 0 0 .. 0    t−4 +   0  , (7)  t              f t     x 0 0 0 0 0 Λ 0 .. 0   e t m t   f  t−1       f  t−2      f   t−3    f t−4 whereyq isthequarterlyGDP;ym isthemonthlyGDP;x aretheremainingmonthlyvariables; t t t λq is the coefficient that links the quarterly GDP to the monthly GDP through the aggregation proposed by Mariano and Murasawa (2003); Λm is the vector of dimension (n-2)x1 of factor loadings connecting the factor to all the monthly variables but the monthly GDP; and 0 are vectors of zero of dimension (n-2)x1 where n is the number of variables in our data set. The transitionequationisthen 10Forsimplicity,weassumeherethattheidiosyncraticcomponentsarei.i.d. whitenoise. However,ourmodel isestimatedassumingthattheidiosyncraticcomponentsfollowanAR(1),thoughthismakethestatespacerepresentationcumbersome. ThestatespacerepresentationwithAR(1)idiosyncraticerrorsisavailableinAppendix. 12

       ym a a 0 0 0 a a 0 0 0 ym uy t 11 12 13 14 t−1 t                ym   1 0 0 0 0 0 0 0 0 0  ym   0   t−1    t−2            ym   0 1 0 0 0 0 0 0 0 0  ym   0   t−2    t−3            ym   0 0 1 0 0 0 0 0 0 0  ym   0   t−3    t−4                   ym   0 0 0 1 0 0 0 0 0 0  ym   0   t−4  =   t−5 + , (8)         f   a a 0 0 0 a a 0 0 0  f   uf   t   21 22 23 24  t−1   t                f 0 0 0 0 0 1 0 0 0 0 f 0  t−1    t−2                   f   0 0 0 0 0 0 1 0 0 0  f   0  t−2 t−3                f   0 0 0 0 0 0 0 1 0 0  f   0   t−3    t−4           f 0 0 0 0 0 0 0 0 1 0 f 0 t−4 t−5 where f is the common factor to all the variables in the data set but the quarterly and monthly t GDP, and a are the VAR(2) coefficients obtained by regressing the monthly GDP and the i,j factorontheirowntwolags. 4 Institutional benchmarks To assess the forecasting accuracy of our results, we compare them against two different institutionalforecasters: theOECDandtheBankofCanada. TheBankofCanada’sMonetaryPolicyReportisaquarterlyreportoftheGoverningCouncil presenting the Bank’s projections for inflation and growth in the Canadian economy and its assessmentofrisks. ItispublishedinJanuary,April,July,andOctober. The OECD Economic Outlook is the OECD’s twice-yearly analysis of major economic trends and prospects for the next two years. Prepared by the OECD Economics Department, the Outlook puts forward a consistent set of projections for output, employment, government spending, prices, and current balances based on a review of each member country and of the induced effect of each of them on international developments. It is published in March and 13

September. In addition, the other important benchmark we consider is Bloomberg, which conducts a surveyandcollectsforecastsfromanalystsandeconomiststoproducepredictionsforGDPand other market-relevant variables before their release dates. Bloomberg publishes predictions as soon as they have at least three respondents to their questionnaire, which is generally around twoweeksbeforethereleaseoftherelevantdataseries. Thereafterthepredictioniscontinually reviseduntil24hoursbeforetherelease. Thefinalnumberisusuallyclosetotheactualrelease value. 5 Model evaluation To evaluate the performance of the model, we report a “pseudo real time” historical reconstruction from 2006:Q1 to 2015:Q3. We estimate the model recursively (the estimation period starts in January 1992) and take into account information from each new data release (in real time), but we do not consider revisions (pseudo). This last point can in principle distort the results in favor of the model, given that the Bloomberg survey relies on real-time information. However, given the robustness of factor models to data revision errors (see Giannone et al., 2006;andBan´buraetal.,2013),weexpectthisnottobethecase. The results of the historical evaluation are reported in figure 1 below. The figure compares the QoQ GDP nowcast with the OECD and Bank of Canada forecasts and with a benchmark nowcasting model that does not include U.S. variables. Figure 1 shows that the nowcasting model mimics the actual realization of GDP very well and that it outperforms the benchmark model. ThisresultimpliesthattheU.S.variablesweincludeinourmodelimprovetheforecast performanceofthemodel. Figure 2 compares the root-mean-squared forecast error (RMSFE) of the model-on average for all of the calendar quarters in the historical reconstruction period- with the benchmark model, the short-term forecasts of the Bank of Canada, and the OECD, Bloomberg’s survey of independent forecasters (published the day before the preliminary GDP release), and an auto- 14

Figure1: GDPnowcast 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.5 60-naJ-3 60-nuJ-3 60-voN-3 70-rpA-3 70-peS-3 80-beF-3 80-luJ-3 80-ceD-3 90-yaM-3 90-tcO-3 01-raM-3 01-guA-3 11-naJ-3 11-nuJ-3 11-voN-3 21-rpA-3 21-peS-3 31-beF-3 31-luJ-3 31-ceD-3 41-yaM-3 41-tcO-3 51-raM-3 51-guA-3 out turn nowcast model nowcast model benchmark OECD Bank of Canada Notes. The graph reports the comparison between a GDP nowcast using U.S. variables(nowcastmodel),aGDPnowcastwithoutU.S.variables(nowcastmodel benchmark), theGDPactualvalue, andOECDandBankofCanadaprofessional forecasts. regressiveforecastthatchangesonlywhenGDPisreleased. Themodel’squarterlyGDPgrowthpredictionisfirstmade90daysbeforethestartofagiven quarter. ItisthenupdatedwitheachsuccessivedatareleaseuntilthereleaseofpreliminaryGDP, whichtakesplace150daysafterthestartofthecalendarquarter. Thusforeachcalendarquarter, thereisaperiodof240days(the“predictionperiod”)overwhichthepredictioniscontinuously updated. Thisperiodismeasuredbythexaxis. TheyaxismeasurestheRMSFEforeachseries of predictions. From figure 2, we can clearly learn two lessons. First, the U.S. data are crucial to obtaining an accurate nowcast of quarterly Canadian GDP growth; indeed, the RMSFE line produced by the model without U.S. data is always above the RMSFE line produced by the model with U.S. data. Second, our mechanical model produces nowcasts that are on average 15

Figure2: RMSFE 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 98- 48- 18- 57- 17- 06- 55- 35- 05- 44- 83- 92- 42- 12- 51- 11- 0 5 7 01 61 22 13 63 93 54 94 06 56 76 07 67 28 19 69 99 501 901 021 521 721 031 631 241 nowcast model nowcast model benchmark AR Bloomberg Bank of Canada OECD Notes. The y axis reports the root-mean-squared forecast error (RMSFE) over the period from 2006:Q1 to 2015:Q3. The forecast accuracy is evaluated from the first month of the previous quarter to the time when GDP is released. The x axisreportsthedistanceindaysfromthebeginningofthecurrentquarter. 16

Table2: AverageMSFEReductionbyVariable m1 m2 m3 U.S.ISMManufacturingPMI 4.66 -1.29 -0.48 U.S.ChangeinNonfarmEmployment -2.95 -0.02 -0.48 U.S.IndustrialProductionIndex -6.81 -0.66 -0.17 U.S.CapacityUtilization -6.81 -0.66 -0.17 CanadaIveyPMI -0.34 -0.24 0.54 CanadaBuildingPermits -0.59 0.18 0.21 CanadaEmployment 0.10 -0.95 -1.42 CanadaDwellingStarts -0.14 -0.22 -0.20 CanadaImportsofGoods -2.77 -0.24 0.04 CanadaExportsofGoods -2.62 -0.24 0.00 CanadaManufacturingShipments -0.11 0.23 -0.72 CanadaNewMotorVehicleSales -0.70 -0.15 -0.46 CanadaNewManufacturingOrders -0.11 0.23 -0.72 CanadaWholesaleTrade -0.55 0.19 -0.32 CanadaRetailSalesValue -0.18 0.08 -0.38 CanadaMonthlyGDP -0.91 -0.57 -7.37 CanadaRealGDP -0.34 Notes. These results are referred to as the first (m1), second (m2),andthird(m3)monthsofthenowcastperiod. moreaccuratethanthoseproducedbyinstitutionalforecastersandmarketparticipants. In table 2, we report the RMSFE reduction by release in each of the three months of the reference quarter. From table 2, we notice that U.S. data releases have the biggest effect in improving the accuracy of the model’s prediction in the first month. In the second month, Canadian employment, dwellings starts, exports and imports seem to be relevant together with U.S.PMI.Inthethirdmonth,monthlyGDPplaysamajorroleinimprovingtheaccuracyofthe model’sprediction. 17

5.1 Do U.S. variables matter? From the previous section, we have shown the role played by U.S. variables in improving themodel’snowcastingperformanceofthemodel. Inthissection,weformallytestwhetherthe RMSFE of the model that incorporates U.S. variables is statistically different than the RMSFE ofthemodelthatonlyincludesCanadiandata. In table 3, we report the Diebold and Mariano (2002) test (“DM test”) of equal predictive accuracy to check whether the difference in forecasting performance between models is significant. For each month, we report the sample average of the difference between the squared errors of the AR and the benchmark model forecast, nowcast, and backcast, both with respect to our model and coincident with the first Canadian release (Canada PMI). We report the value oftheDMtestanditsstandarddeviationestimatedusingheteroskedasticityandautocorrelation robuststandarderrors(seeappendixA.2fordetails). Thetestconfirmsthatourmodelperforms moreaccuratelythantheAR(onlyinthenowcastandbackcast)andslightlymoreaccuratethan thebenchmarkmodel(onlyinthenowcast). Fromfigure2,wecanseethatthemodel’sRMSFEdeclinesmoreorlesscontinuouslyover thepredictionperiod,whichmeansthatnewinformationhasamonotonicandnegativeeffecton uncertainty. Toformallytestthedeclineinuncertainty,asmoredataarriveweapplythetestfor forecast rationality proposed by Patton and Timmermann (2012). Table 4 reports the p-values ofthreemonotonicitytestsfor,respectively,theforecasterrors,themean-squaredforecast,and covariance between the forecast and the target variable (see the appendix for a description of thetests). Monotonicitycannotberejectedbyanyofthethreetestssconfirmingtheevidenceof figure 2 and proving the importance of incorporating new information into the forecast update asitarrives. 18

Table3: Diebold-Marianotestofequalforecastingaccuracy Forecast Nowcast Backcast AR BCB AR BCB AR BCB 1m 0.03 0.02 0.26 0.22 0.33 0.04 (0.03) (0.03) (0.14) (0.13) (0.14) (0.02) 2m 0.07 0.06 0.39 0.26 0.37 0.00 (0.07) (0.07) (0.22) (0.15) (0.15) (0.01) 3m 0.07 0.08 0.23 0.10 (0.08) (0.10) (0.14) (0.06) Notes. ThetablereportstheestimatedconstantandtheHACestimatorofitsstandarderrorinthefirst,second,andthirdmonth oftheforecast,nowcast,andbackcast,respectively. TheARand thebenchmarkarecomparedagainstthenowcastmodel. Table4: Monotonicitytests ∆e ≥ 0 ∆f ≥ 0 ∆c ≥ 0 nowcastmodel 0.4977 0.5049 0.5060 Notes. The table reports the p-values of three monotonicity tests for, respectively, the forecast errors, the mean-squared forecast, and covariance between the forecastandthetargetvariable. 19

5.2 The news The importance of calculating the news is twofold. First, given that the news is defined as the difference between the actual value of the data release and the value predicted by the model, it is possible to check whether the model is well specified in all its dimensions. The average of the news for each release should be close to zero, and the standard deviation should be such that |mean| < 2 standard deviations. Table 5 confirms this statement. In addition, the table also compares the model’s performance in predicting each of the series with that of the Bloombergsurvey. Weshowthat,formostseries,themodel’spredictionsarecomparabletothe Bloombergsurvey’spredictions. Finally,weincludeintable5themeanandstandarddeviation of the revisions for each of the series in the data set. As the means of the revisions are close to zero and the standard deviations are such that |mean| < 2 standard deviations, this result suggeststhatthemodel’srelativeperformancewouldhavebeensimilarinrealtime.11 Table5: Averagenewsandstandarddeviation Model Bloomberg Revisions Units/ Mean StD Mean StD Mean StD Transformation U.S.ISMManufacturingPMI D.I./Levels -0.21 2.24 0.01 1.82 -0.18 1.86 U.S.ChangeinNonfarmEmployment Thous/Levels -28.87 139.25 7.56 106.34 8.00 63.35 U.S.IndustrialProduction Index/MoM -0.09 0.62 0.12 0.53 0.06 0.42 U.S.CapacityUtilization Percentage/Diff 0.05 0.48 0.02 0.57 0.05 0.36 CanadaIveyPMI D.I./Levels -0.43 7.29 -0.93 6.82 -0.21 3.02 CanadaBuildingPermits ThousUnits/MoM 0.09 11.28 -0.37 10.84 0.00 0.10 CanadaEmployment Thous/Diff -4.34 33.39 CanadaDwellingStarts ThousUnits/MoM -0.11 9.52 0.1 12.02 -0.19 4.18 CanadaImportsofGoods Mil.C$/MoM 0.09 2.49 CanadaExportsofGoods Mil.C$/MoM -0.05 3.17 CanadaManufacturingShipments Thous.C$/MoM -0.12 1.75 0.18 1.47 0.04 0.91 CanadaMotorVehicleSales Thous/YoY 0.19 6.55 CanadaManufacturingOrders Thous.C$/MoM -0.10 4.83 CanadaWholesaleTrade Thous.C$/MoM -0.10 1.20 -0.05 1.17 -0.01 0.86 CanadaRetailSalesValue Thous.C$/MoM -0.05 0.98 -0.02 0.73 -0.04 0.53 CanadaMonthlyGDP Mil.C$/MoM -0.08 0.25 0.02 0.23 -0.03 0.18 CanadaRealGDP Mil.C$/QoQ -0.03 0.18 -0.02 0.29 -0.03 1.10 Notes.D.I.=diffusionindex;Diff.=differences;Thous.Units=ThousandUnits;Mil.C$=MillionCanadianDollars; MoM=MonthonMonth;QoQ=Quarteronquarter;YoY=yearonyear. The second important feature of the news within a nowcasting framework is that it allows for the interpretation of all the data releases in terms of the signals they give about current 11NotethattheBloombergsurveyisconductedinrealtimeandtherespondentswhoseforecastsitreflectsare attemptingtopredictthefirstreleaseofeachseries,whereasthereconstructionofthemodel’spredictionsisbased onthelastavailablevintageofdata,ignoringrevisions. 20

economic conditions (Ban´bura and Modugno, 2010). It is possible, with the use of equation 3, to decompose the forecast revision into contributions from the news in individual series. The impact that a given release has on the GDP nowcast is the product of two variables: the news (ortheunexpectedcomponentofthereleasevalue)andtherelevanceoftheseriesinrelationto GDP, which is expressed as its weight (that is, impact = news standard deviation x weight).12 Figure 3 shows the average impact of each variable in the first, second, and third months of the quarter. As expected, U.S. variables have a strong impact in the first month, U.S. PMI is also relevantinthesecondmonth. MonthlyGDPhasarelevantcontributiononlyinthethirdmonth. SeeappendixA.4forthedecompositionoftheaverageimpact. Figure3: Variables’relevance Canada Real GDP Canada Monthly GDP Canada Retail Sales Value Canada Wholesale Trade Canada Manufacturing Orders Canada Motor Vehicle Sales Canada Manufacturing Shipments Canada Exports of Goods Canada Imports of Goods Canada Dwelling Starts Canada Employment Canada Building Permits Canada Ivey PMI US Capacity Utilization US Industrial Production Index US Change in Nonfarm Employment US ISM Manufacturing PMI 0 5 10 15 20 m3 m2 m1 Notes. Variables’averageimpactinthefirst(m1),second(m2),andthird(m3) monthsofthenowcast. 12Weconsiderthestandarddeviationinsteadofthemeanbecausethelattershouldbeclosetozeroandalsoin ordertodiscardthesign. 21

6 Conclusion WeproposedaneconometricframeworkforinferringthestateoftheCanadianeconomy,in terms of real GDP, that can be easily updated any time new information is available. Canadian data are characterized by two peculiarities: the presence of a monthly measure of real GDP other than the quarterly one, and the lack of availability of timely industrial production data. We solve the first challenge by proposing a new modelling strategy that coherently takes into account the relation of the quarterly real GDP and its monthly counterpart. We deal with the lack of timely industrial production data by including in our information set U.S. industrial productiondata. Results show that our model can deliver nowcasts that are at least as accurate as those produced by institutional forecasters such as the Bank of Canada, the OECD, and the Bloomberg survey. This result suggests that a model-based nowcast that can quickly incorporate a potentiallylargesetofnewinformationcancomplementtojudgmentalnowcastinassessingthestate oftheCanadianeconomy. Moreover,weshowthatU.S.dataarecrucialtoobtaininganaccurate realGDPnowcastforCanada;thelackoftimelyproductiondatafortheCanadianeconomycan becompensatedforbyusingU.S.data,confirmingthattheinterconnectednessbetweenthetwo economiescanbeexploitedforproducingamoreprecisenowcastoftheCanadianrealGDP. 22

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Appendix A1. Selecting the number of factors and lags We select the optimal number of factors using an information criteria approach. The idea is to choose the number of factors that maximizes the general fit of the model, using a penalty function to account for the loss in parsimony. Bai and Ng (2002) derive information criteria to determine the number of factors in approximate factor models when the factors are estimated by principal components. They also show that their information criterion (IC) can be applied to any consistent estimator of the factors provided that the penalty function is derived from the correctconvergencerate. Table 6 reports the IC and the sum of the variance of the idiosyncratic components for the differentspecifications,whichallowforadifferentnumberoffactors. TheICselectsthemodel Table6: Modelselection(numberoffactors) Sample1 Sample2 Sample3 IC V IC V IC V 1 0.13 0.84 0.10 0.81 0.04 0.76 2 0.21 0.66 0.19 0.64 0.13 0.61 3 0.34 0.56 0.40 0.58 0.44 0.60 4 0.55 0.50 0.79 0.62 0.59 0.51 T 59 155 272 N 15 14 14 Notes. IC stands for information criteria, and V is the sum ofthevarianceoftheidiosyncraticcomponent. with one factor. Given that our data set is unbalanced at the top and some series are more recent than others, we report the test on three different samples. The first (sample 1) considers a balanced panel in the estimation period 2001:Q1 to 2005:Q4 (15 series and 59 observations), the second (sample 2) a restricted balanced panel in which we exclude one of the most recent series (14 series, 155 observations), and the third (sample 3) a balanced panel that incorporates the whole sample (estimation and forecasting period). The choice of one factor is confirmed acrossthedifferentsamples. 27

To select the number of lags in equation 5 of the model, we report in table 7 the values of theAkaikeIC,whichselectstwolags. Table7: Modelselection(numberoflags) Numberoflags Akaikeinformationcriteria 1 -1.35 2 -1.65 3 -1.60 4 -1.58 Notes. The lag is chosen in correspondence with the minimumAICvalue. A2. Diebold-Mariano test Denote the loss associated with forecast error e as L(e ) and the time-t loss differential t t between forecasts 1 and 2 as d = L(e )−L(e ). The Diebold-Mariano (DM) test requires 12t 1t 2t onlythatthelossdifferentialiscovariancestationary: E(d ) = µ,∀t 12t cov(d ,d ) = γ(τ),∀t 12t 12t−τ 0 < var(d ) = σ < ∞ 12t 2 The key hypothesis of equal predictive accuracy (that is, equal expected loss) corresponds toE(d ) = 0,inwhichcaseunderthemaintainedassumptionDM, 12t ¯ d 12 d DM = → N(0,1), 12 σˆ d¯ 12 where d ¯ = 1 (cid:80)T d is the sample mean loss differential and σ is a consistent estimator 12 T t=1 12t d¯ 12 ¯ ofthestandarddeviationofd . 12 DM is thus an asymptotic z −test of the hypothesis that the mean of a constructed but observedseries(thelossdifferential)iszero. However,forecasterrors,andhencelossdifferential, 28

may be serially correlated for various reasons. In this paper, we calculate the DM statistics by regression of the loss differential on an intercept, using heteroskedasticity and autocorrelation robust standard errors. In a fully articulated econometric model in which we have pseudo outof-sample forecasts, we follow West (1996) and define the test on the sample mean quadratic lossasfollows: (cid:80)T (e2 −e2 ) ¯ t=t∗+1 1,t|t−1 2,t|t−1 d = , 12 T −t∗ where e is a time-t pseudo out-of-sample one-step ahead forecast error. We do not cont|t−1 sider a rolling scheme, so results should be taken with caution, as the test ignores estimation uncertainty. A3. Monotonicity test WerelyonthefirstthreetestsofPattonandTimmermann(2012),basedonthemultivariate inequality tests in regression models of Wolak (1987). We report the p-values for the nowcast model. Test1: Monotonicityoftheforecasterrors Let us define y˜ = yk and e = y˜ −E[y˜|Ω ] as the forecast error obtained on the basis t t,1 t|Ωv t t v of the information set corresponding to the data vintage Ω , and e as that obtained on the v t|Ωv+1 basisofalarger,morerecentvintage,v +1andv = 1,...,V. Themeansquarederror(MSE)differentialis∆e = E[e2 ]−E[e2 ]. v t|Ωv t|Ωv+1 The test is H0 : ∆e ≥ 0 vs H1 :∆e (cid:11) 0, where the (V −1)×1 vector of MSE differentials isgivenby∆e ≡ (∆e,...,∆e )(cid:48). 1 V−1 Test2: Monotonicityofthemeansquaredforecast Definethemeansquaredforecast(MSF)foragivenvintageasE[y˜2 ] = E[E[y˜2|Ω ]]and t|Ωv t v considerthedifference,∆f = E[y˜2 ]−E[y˜2 ],anditsassociatedvector,∆f. v t|Ωv t|Ωv+1 29

ThetestisH0: ∆f ≤ 0vsH1: ∆f (cid:10) 0. Test3: Monotonicityofcovariancebetweentheforecastandthetargetvariable Here we consider the covariance between the forecast and the target variable for different vintages,v,andthedifference,∆c = E[y˜ y˜]−E[y˜ y˜]. Theassociatedvectorisdefined v t|Ωv t t|Ωv+1 t as∆c andthetestisH0: ∆c ≤ 0vsH1: ∆c (cid:10) 0. Wolak (1987) derived a test statistic whose distribution under the null is a weighted sum of chi-squaredvariables. A4. Impact of the releases on the nowcast Table8: Impactofthereleasesonthenowcast A B C m1 m2 m3 m1 m2 m3 m1 m2 m3 U.S.ISMManufacturingPMI 4.657 4.695 2.105 2.164 2.503 2.096 10.076 11.751 4.412 U.S.ChangeinNonfarmEmployment 0.047 0.051 0.026 148.212 130.015 136.422 7.035 6.573 3.606 U.S.IndustrialProductionIndex 13.330 14.116 9.933 0.810 0.489 0.540 10.803 6.906 5.359 U.S.CapacityUtilization 14.827 15.658 10.948 0.610 0.381 0.413 9.046 5.965 4.518 CanadaIveyPMI 0.315 0.314 0.202 7.797 5.293 7.849 2.458 1.663 1.585 CanadaBuildingPermits 0.099 0.110 0.106 10.246 12.246 10.365 1.017 1.347 1.096 CanadaEmployment 0.175 0.182 0.123 30.964 35.993 33.488 5.430 6.538 4.126 CanadaDwellingStarts 0.241 0.256 0.194 10.419 8.562 9.700 2.509 2.190 1.877 CanadaImportsofGoods 2.031 2.299 2.322 2.544 2.349 2.628 5.165 5.400 6.104 CanadaExportsofGoods 1.732 1.934 1.961 3.936 3.000 2.458 6.818 5.802 4.821 CanadaManufacturingShipments 2.440 2.876 2.827 1.955 1.640 1.614 4.771 4.716 4.565 CanadaNewMotorVehicleSales 0.187 0.199 0.206 7.347 6.238 6.168 1.374 1.241 1.268 CanadaNewManufacturingOrders 0.897 1.058 1.040 4.976 5.383 4.149 4.466 5.697 4.315 CanadaWholesaleTrade 1.229 1.371 1.464 1.184 1.262 1.169 1.456 1.730 1.711 CanadaRetailSalesValue 1.053 1.158 1.223 0.929 1.052 0.986 0.978 1.218 1.206 CanadaMonthlyGDP 23.697 36.688 78.031 0.253 0.261 0.226 6.000 9.557 17.609 CanadaRealGDP 19.542 0.176 3.446 Notes.Aistheaverageweight,Bisthenewsstandarddeviation,andCistheaverageimpactequaltoA·B. A5. Complete state space TheexactstatespacerepresentationofourmodelfornowcastingCanadianrealGDPgrowth isdescribedinthefollowingtwosystemsofequationswhere,asinequation9,yq isthequarterly t GDP;ymisthemonthlyGDP;thevectorx containstheremainingn−2(wherenisthenumber t t of variables in our dataset) monthly variables; λq is the coefficient that links the quarterly GDP 30

to the monthly GDP through the aggregation proposed by Mariano and Murasawa (2003); Λm isthevectorofdimensionn−2offactorloadingslinkingthefactortoallthemonthlyvariables but the monthly GDP; 0 are vectors of zero of dimension (n−2)x1; I is an identity matrix of dimension n − 2; f is the common factor to all the variables in the data set but the quarterly t and monthly real GDP; (cid:15)q is the idiosyncratic component that enters with its lags, aggregated a t la Mariano and Murasawa (2003), in the observation equation for the quarterly GDP; and e is t the(n−2)x1vectoroftheidiosyncraticcomponentsoftherestofthemonthlyvariables. Inthe transition equation 10 we have that a are the VAR(2) coefficients obtained by regressing the i,j monthly GDP and the factor on their own two lags; ρ is the autoregressive coefficient of the q idiosyncraticcomponentsofthequarterlyGDP;andPisadiagonalmatrixofdimensionn−2 containing the autoregressive coefficients for the idiosyncratic components of the remaining n−2variables. 31

  ym t      ym  t−1      ym   t−2     ym   t−3       ym  t−4      f   t     f   t−1       y t q λ q 2λ q 3λ q 2λ q λ q 0 0 .. 0 1 2 3 2 1 0 0(cid:48)   f t−2              ym  =  1 0 0 0 0 0 0 .. 0 0 0 0 0 0 0 0(cid:48)  f  (9)  t    t−3       x 0 0 0 0 0 Λ 0 .. 0 0 0 0 0 0 0 I  f  t m  t−4      (cid:15)q   t      (cid:15)q   t−1     (cid:15)q   t−2      (cid:15)q   t−3      (cid:15)q   t−4     0      e t 32

       ym a a 0 0 0 a a 0 0 0 0 0 0 0 0 0 0(cid:48) ym uy t 11 12 13 14 t−1 t                ym   1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0(cid:48)  ym   0  t−1 t−2                ym   0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0(cid:48)  ym   0   t−2    t−3            ym   0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0(cid:48)  ym   0   t−3    t−4                   ym   0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0(cid:48)  ym   0  t−4 t−5                f   a a 0 0 0 a a 0 0 0 0 0 0 0 0 0 0(cid:48)  f   uf   t   21 22 23 24  t−1   t          f   0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0(cid:48)  f   0   t−1    t−2                   f t−2   0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0(cid:48)  f t−3   0                 f  =  0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0(cid:48)  f + 0  (10)  t−3    t−4            f   0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0(cid:48)  f   0   t−4    t−5             (cid:15)q t     0 0 0 0 0 0 0 0 0 ρ q 0 0 0 0 0 0 0(cid:48)     (cid:15)q t−1     v t q                  (cid:15)q   0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0(cid:48)  (cid:15)q   0   t−1    t−2            (cid:15)q   0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0(cid:48)  (cid:15)q   0   t−2    t−3             (cid:15)q     0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0(cid:48)     (cid:15)q     0   t−3 t−4                (cid:15)q   0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0(cid:48)  (cid:15)q   0   t−4    t−5            0   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0(cid:48)  0   0                e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 P e v t t−1 t 33