Iterated Multi-Step Forecasting with Model Coefficients Changing Across Iterations. Michal Franta 1

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1 Ieraed Muli-Sep Forecasing wih Model Coefficiens Changing Across Ieraions Michal Frana 1 Absrac: Ieraed muli-sep forecass are usually consruced assuming he same model in each forecasing ieraion. In his paper, he model coefficiens are allowed o change across forecasing ieraions according o he in-sample predicion performance a a paricular forecasing horizon. The echnique can hus be viewed as a combinaion of ieraed and direc forecasing. The superior poin and densiy forecasing performance of his approach is demonsraed on a sandard medium-scale vecor auoregression employing variables used in he Smes and Wouers (2007) model of he US economy. The esimaion of he model and forecasing are carried ou in a Bayesian way on daa covering he period 1959Q1 2016Q1. JEL Codes: C11; C32; C53; Keywords: Muli-sep forecass, VAR, Bayesian esimaion, ieraed forecasing, direc forecasing, densiy forecasing 1 Michal Frana, Czech Naional Bank, michal.frana@cnb.cz I would like o hank Jan Bruha, Andrea Carriero, Simona Malovana, Howell Tong, and seminar paricipans a he Czech Naional Bank for useful commens. The views expressed here are hose of he auhor and no necessarily hose of he Czech Naional Bank.

2 1. Inroducion Muli-sep forecasing is one of he mos imporan asks in applied macroeconomics. Several approaches have emerged, and heir relaive accuracy ofen depends on he forecasing echnique and he circumsances of he forecasing exercise. In his paper he focus is primarily on ieraed muli-sep forecasing, i.e., on a forecasing echnique ha combines one-sep-ahead forecass ino forecass for several periods ahead. Ieraed forecass are someimes referred o as plug-in forecass, reflecing he sequenial naure of he forecasing process. Models used for forecasing are ofen esimaed by means of he likelihood funcion. Maximum likelihood esimaion effecively means ha he parameer esimaes minimize he one-sep-ahead predicion errors wihin he daa sample used for he esimaion. If he model describes he daa-generaing process correcly, hen he maximum likelihood esimaes are asympoically efficien and a single-plug-in model ha uses he same parameer esimaes for all forecasing ieraions is preferred. However, in pracice models are no correc. In-sample predicion errors a longer horizons can hen conain sysemaic informaion no included in he one-sep-ahead errors, and i is desirable o incorporae such informaion ino he esimaion procedure. The lieraure discussing such procedures sars probably wih Cox (1961), who provides explici formulas for predicors ha exploi informaion from in-sample predicion errors a longer horizons for a saionary AR process. Tiao and Xu (1993) exend such consideraions o ARIMA processes. Xia and Tong (2011) denoe he family of approaches o model fi oher han one-sep-ahead predicion errors as feaure maching. 2 Recenly, Schorfheide (2005) and Kapeanios e al. (2015) consider a vecor of predicion errors for differen horizons in esimaion and forecasing in he VAR and DSGE model frameworks, respecively. The approach o muli-sep forecasing inroduced in his paper follows his line of research and explois informaion from in-sample predicion errors a longer horizons. The model coefficiens are hus allowed o change in he direcion of minimizing he m-sep-ahead predicion errors. So, for he firs forecasing ieraion (he one-sep-ahead forecas) he esimaion mehod is basically maximum likelihood, excep ha some prior informaion on he model parameers is imposed. Oher forecasing ieraions hen ake ino accoun boh he esimaion resuls from he previous forecasing ieraions and he in-sample predicion errors for he respecive horizon. The approach is reminiscen of he direc forecasing mehod, in which he esimaion of a horizon-specific model is relaed o he predicion errors a he corresponding horizon. The presened procedure hus in a way represens a combinaion of ieraed and direc forecasing. The weighs of he forecasing echniques are deermined according o how close he resuling models are o he rue daa-generaing process. The combinaion of direc and ieraed forecasing could be an answer o he rade-off beween esimaion bias and esimaion variance ha is an inheren feaure of comparisons of he wo basic forecasing echniques (Findley, 1983). The bias resuling from possible 2 There can be oher moivaions for using m-sep-ahead predicion errors when fiing a model differen from he forecasing performance of he model. For example, Tonner and Bruha (2014) use he eigh-sep-ahead predicion error reflecing he lengh of moneary policy horizon. 2

3 misspecificaion of he benchmark model muliplied by he ieraing one-sep-ahead forecass can be correced by aking ino accoun he sysemaic relaions beween wo disan daa poins beyond wha can be inferred from he relaions beween adjacen daa poins. On he oher hand, he inefficiency of direc forecasing can be diminished by he ieraive naure of he procedure and by he fac ha he ieraed approach produces more efficien parameer esimaes. 3 The presened approach is close o Kapeanios e al. (2015). In conras o ha paper, he approach in his paper inroduces a fully fledged Bayesian perspecive and allows for differen model coefficiens for differen forecasing horizons when ieraing one-sep-ahead forecass. On he oher hand, he possibiliy of changing coefficiens in each ieraion precludes join esimaion of all he parameers appearing in he forecasing procedure because of he size of he parameer vecor. Therefore, he rade-off beween he forecasing performances a wo differen horizons and heir poenial opimizaion across forecasing ieraions is no a subjec of he esimaion procedure. The forecasing performance of he proposed mehodology is examined on a medium-scale vecor auoregression ha includes he same variables as he DSGE model of he US economy in Smes and Wouers (2007). More precisely, he daa sample sars in 1959Q1 and he forecasing performance is examined on he daa observed during he period 1998Q4 2016Q1. Boh poin and densiy forecass are discussed. For poin forecass i urns ou ha adjused ieraed forecass ouperform boh sandard ieraed and direc forecass. The resul is even more clear-cu for densiy forecass. Adjused ieraed forecass hus represen a echnique for dealing wih he rade-off beween poenially biased ieraed forecasing and inefficien direc forecasing. The res of he paper is organized as follows. Secion 2 presens he model and he sandard ieraed forecasing procedure. Secion 3 describes how he model coefficiens are adjused o minimize he m-sep-ahead predicion errors and how he adjused ieraed forecass are consruced. Secion 4 presens he daase and describes he specificaion of priors and he se-up of he forecasing performance exercise. Secion 5 discusses he resuls and offers some robusness checks. Finally, Secion 6 concludes and Appendix A provides a brief descripion of he daa. 2. Model and ieraed forecasing To demonsrae he general principle for he adjusmen of model coefficiens in an ieraed forecasing process, a sandard VAR model is considered: y C B1 y ~ N0,, 1... B y p p 3 A deailed empirical comparison of direc and ieraed forecass can be found in Marcellino e al. (2006). 3

4 where y is an n 1 vecor of endogenous variables, is an n 1 vecor of exogenous shocks, C, B 1,, Bp are an n 1 vecor and nn marices of consans and AR parameers, respecively, and is an nn T vecor of daa y y,..., y. Sysem can be rewrien as follows: 1 T marix of error covariances. Model is he model for he y X ~ N0,, where X I x, x 1, y,, y ], and vec[ C, B1,, B ]. n [ 1 p p The one-sep-ahead forecas (or predicive densiy) is hen: y X T1 T1 T 1 ~ 0,. T 1 N (3) Noe ha yt 1 is an n 1 vecor of random variables wih a disribuion given by he disribuion of he esimaed model parameers and. Ieraing (3) forward, forecass for oher forecasing horizons, For example, he wo-sep-ahead forecas is as follows: y T2 T2 T 2 T X ~ 0, 2 N h 2,..., H, can be generaed. (4) where he marix X I 1, yˆ, y,, y ] conains he observed daa and he resuls T 2 n [ T 1 T T 1 p from he firs forecasing ieraion. The ieraed forecass are usually based on he same esimaed model parameers ˆ and ˆ for each forecasing ieraion. Model is esimaed employing he Normal inverse Wishar prior: ~ N b, ~ iw k,. (5) The prior maerializes he prior belief ha he variables follow a process close o a random walk. Combining prior (5) wih he likelihood funcion yields he poseriors of he model parameers. The likelihood funcion (condiional on he iniial p observaions) can be expressed as a produc of he condiional probabiliy densiies: 4

5 p T T 1 y, py y,,. (6) p1 Equaion (6) can be inerpreed such ha he likelihood is he value of he probabiliy densiy in he case where he model produces a one-sep-ahead in-sample predicion equal o he observed values. Equivalenly, i represens he value of he probabiliy densiy ha he onesep-ahead predicion errors are zero. So, maximizing he likelihood means minimizing he insample one-sep-ahead predicion errors. If he prior disribuions are fla, he poseriors are proporional o he likelihood and he above-menioned inerpreaion of likelihood also applies o he poseriors. The Normal inverse Wishar prior belongs o he family of naural conjugae priors and hus he poseriors follow he same disribuions. Moreover, he formulas for he poserior disribuions of he model parameers can be expressed in closed form: T, y ~ N ˆ, ˆ T y ~ iw d, ˆ (7) where ˆ 1 xx 1 1 and ˆ ˆ x' y. Nex, x x p 1,..., xt, y y p,..., yt 1, and is consruced such ha he columns are creaed from he prior coefficiens for he parameers in each equaion. Finally, he scale parameer of he poserior for he error ˆ ˆ 1 SSR ˆ, where he erm SSR denoes he covariance marix equals sum of he squared residuals from he regression wih he poserior of he AR coefficiens. The degrees of freedom parameer d k T p. The full disribuion of he one-sep-ahead forecas yt 1 T T1 T1 T1 T1 is maricvariae-: y y ~ MT X ˆ, X ˆ X 1, kt p, ˆ. (8) The disribuions of he ieraed forecass for oher forecasing horizons have an analyical form and are simulaed. h 2,..., H do no 3. Adjused ieraed forecasing In his secion, we show how he forecasing ieraion process is adjused o ake ino accoun he in-sample predicion errors for higher forecasing horizons. In he firs forecasing ieraion, i.e., for he forecass for one period ahead ( h 1), we sick o he original formula (3) and he original parameer esimaes, which are now indexed by he forecasing ieraion, 5

6 i.e., ˆ and ˆ. Such esimaes ake ino accoun he one-sep-ahead predicion performance wihin he daa sample, as he likelihood funcion reflecing he probabiliy of zero forecasing errors is combined wih he prior disribuions in (5). Analogously, we can discuss he wo-sep-ahead predicion errors for he observed daa given he esimaion resuls from he firs forecasing ieraion. More precisely, insead of he 1 probabiliy densiy of he one-sep-ahead forecas py y,, probabiliy of zero wo-sep-ahead forecas errors in period : ˆ ˆ our focus moves o he 2 p y y,,,,, (9) which can be reformulaed using he fied values from he firs forecasing ieraion ˆ yˆ X as follows: ˆ ˆ y y,,,, py y, yˆ,, p. (10) 1 Pu differenly, he likelihood of he observed daa given he model parameers for he model of wo-sep-ahead ieraed forecass: y C ~ N B 0, 1 yˆ 1 B 2 y 2 B p y p (11) can be expressed as follows: p T T 2 y,, yˆ,, ˆ,,, ˆ T 1 y p1 p y y y 1 T p1 p1 2 ˆ,, ˆ p y y,,. (12) Formula (12) suggess ha he likelihood of model (11) expresses he probabiliy densiy of zero wo-sep-ahead forecas errors condiional on he firs forecasing ieraion. The likelihood can be combined wih he prior on he model parameers, yielding he poserior of he model parameers for he second forecasing ieraion: and When esimaing model (11) he imporan poin is ha fied values. yˆ 1 need o be reaed as a random variable. This fac affecs boh he specificaion of he priors for he second forecasing ieraion and he Bayesian inference iself. 6

7 For he second forecasing ieraion we reain our prior belief ha he process y follows a random walk in he form of he Normal inverse Wishar prior. In addiion, he uncerainy relaed o he fied values needs o be accouned for. Given he esimaion resuls from he firs forecasing ieraion, he fied values used in he second ieraion are disribued normally wih he following momens: where p 2,..., T 1. E yˆ var 1 ˆ ˆ yˆ X var X, 1 X (13) The fied value yˆ 1 includes esimaion uncerainy relaed o o he shock realized a ime -1. The fied value ˆ 1 ˆ bu no uncerainy relaed y and he observed value y 1 differ in he realized shock 1 for which we have formulaed a prior in he firs forecasing ieraion. The shock becomes a par of he disurbance erm a ime in he model for he second forecasing ieraion (11). The sum of wo i.i.d. normally disribued variables has he same mean and double he variance. Therefore, consisency of he priors on he error covariance implies doubled prior error covariance for he second forecasing ieraion. Nex, given ha he prior from he firs forecasing ieraion holds exacly, he disribuion of he fied value yˆ 1 is cenered on y 2 wih a variance proporional o he prior variance of he AR parameers. Model (11) is hen close o he model y CB y B y B y B y. The random walk prior belief implies p p ha ha B should be cenered on uniy, because if y 1 RW RW RW follows a random walk, hen i holds y y y. (14) The prior variance for he AR parameer B1 variance of he coefficien from he firs forecasing ieraion random variable yˆ 1 should be lowered in comparison o he prior B1 already imposes a degree of uncerainy ha because he presence of B1 is cenered on uniy. So, he ighness of he prior variance of he AR parameer a he firs lag is half of he assumed prior variance from he firs forecasing ieraion. This choice is discussed in deail in he secion describing prior specificaions. Finally, he Kronecker srucure of he prior variance on he AR erms in he Normal inverse Wishar disribuion implies ha muliplying he scale of he prior on he error covariance marix proporionally affecs he prior on he variance on he AR parameers. Such muliplicaion is compensaed for by muliplying he overall ighness by 1/ 2. 7

8 The presence of he random variable in he se of RHS variables in model (11) also affecs he Bayesian inference. Analyical formulas are no available and he marginal poseriors of he model parameers for he second forecasing ieraion are simulaed using MC sampling wih he following seps: 1) Iniialize he values of and. 2) Given he sample from he poserior disribuions of forecasing ieraion, ake a random draw 3) Given he observed daa ˆ 1 T y and he draw ˆ 1 esimaed in he firs y according o (13) for p 2,..., T 1. y, ake a random draw of, following he sandard formulas for he Normal inverse Wishar conjugae priors. 4 4) Repea seps 2 and 3 many imes and ake summary saisics of he draws of he model parameer subses. Ieraed forecass for horizon h 2 firs forecasing ieraion and hen by aking draws ieraion. are simulaed by using he draws of and and from he from he second forecasing For oher forecasing ieraions, he procedures for he adjusmen of model coefficiens and he simulaion of forecass are analogous o he case h Daa, priors, and se-up of forecasing exercise The daa se includes real GDP (RGDP), he GDP deflaor (PGDP), consumpion (Cons), invesmen (GDPInv), hours worked (Emp. Hours), wages (Real Comp/Hour), and he federal funds rae (FedFunds). The variables are of quarerly frequency covering he period 1959Q1 2016Q1. 5 A lis of variables can be found in Appendix A. All variables are in annualized log levels excep for he federal funds rae, which is in levels divided by 100. Figure 1 presens he daa se. I can be seen ha boh rending variables and ime series ha are presumably saionary are included. Such diversiy can indicae wheher or no he effec of he proposed mehodology on forecasing performance is dependen on he basic properies of he ime series. 4 The implemenaion is such ha he means of he poserior disribuions for and are aken. Knowledge of he analyical form of he poserior disribuions is hus exploied. As a robusness check, en random draws of, are aken insead of a value equal o he poserior mean. This change resuls in slighly more imprecise esimaes of and. 5 Source: Federal Reserve Bank of S. Louis Daabase (FRED). 8

9 Figure 1: Endogenous variables enering he model esimaion. The parameers of he inverse Wishar prior assumed for he error covariance marix are se so ha he degrees of freedom parameer d n 2, which is he minimum value ha guaranees he exisence of a mean of he disribuion. The scale marix,, is a diagonal marix wih he esimaed error variances of he AR regressions of he respecive LHS variable on is firs own lag. The specificaion of he prior disribuions of he AR parameers is such ha in all equaions he mean of he coefficien on he firs own lag of he LHS variable is one and ha for he res of he coefficiens is zero. The prior mean on he inerceps is zero as well. The variance of he prior disribuion for he AR parameers condiional on is such ha is elemens are defined in he following way: he coefficien in he i-h equaion for he s-h lag of he j-h variable is he following: 1 2 ii 2 s jj d n, (15) where parameer represens he overall ighness of he prior variance. The prior variance 4 on he coefficien a he inercep is 10. As discussed above, he priors change wih he forecasing ieraion h. Firs, he scale marix min h, p o accoun for he inclusion of fied from he inverse Wishar prior is muliplied by values in he models for higher forecasing ieraions. To filer ou he effec of such rescaling 9

10 on he prior variance of he AR parameers, he overall ighness is muliplied by min h, p. Finally, o accoun for he uncerainy imposed by he uncerain fied values in he regression, he overall ighness of he prior variance of he AR parameers a he fied values is divided by 2. The value of he ighness parameer for he firs forecasing ieraion is se equal o 0.2, which is a sandard value in he lieraure. The prior variance on he inercep does no change across forecasing ieraions. The inuiion behind ighening he prior on he AR parameers a fied values is capured in Figure 2. In he second forecasing ieraion wo disribuions are combined in he form of heir produc he disribuion of fied values from he firs forecasing ieraion ( y he prior on he respecive parameer a he fied value ( B 1 ˆ 1 ) and ). The produc canno be expressed analyically and Figure 2, panel c, shows is simulaion. If one compares he simulaed produc and prior imposed in he firs forecasing ieraion ( B 1 ) hey are very similar. And his is he purpose of prior ighening o ensure ha he prior uncerainy relaed o independen variables is similar in all forecasing ieraions (because we ry o model he same dependen variable in all forecasing ieraions). Figure 2. Prior ighening in he second forecasing ieraion. Noe: The simulaed fied value of real GDP from he firs forecasing ieraion (panel a), he prior on he coefficien a he firs lag of real GDP in he equaion for real GDP in he second forecasing ieraion (panel b), and he simulaed produc of he wo (panel c). Panel d shows he prior on he coefficien a he firs lag of real GDP in he equaion for real GDP in he firs forecasing ieraion. The esimaion is done on he full sample. 10

11 Following Giannone e al. (2015), he number of lags in is se o five. The sampler conains 5,000 ieraions. Convergence is esed using sandard measures: he auocorrelaion of parameer draws produced by he sampler, he inefficiency facor, and he measure of he number of draws needed o ge a saionary disribuion from he sampler (Rafery and Lewis, 1992). All measures sugges convergence of he sampler. The resuls are available upon reques. The pseudo-ou-of-sample forecasing exercise is based on 70 observaions beween 1998Q4 and 2016Q1. So, in he firs round he models are esimaed on he period 1959Q1 1998Q3 and forecass for up o 12 quarers ahead are simulaed. The ieraed forecass, he adjused ieraed forecass, and he direc forecass are hen compared in erms of poin and densiy forecasing accuracy. The second round hen uses daa covering he period 1959Q1 1998Q4, ec. The poin forecasing accuracy is compued using he sandard mean squared forecas error (MSFE). The Diebold-Mariano es of equal forecasing accuracy is carried ou, correcing for auocorrelaion of he residuals. The densiy forecasing accuracy is assessed using he Kullback-Leibler Informaion Crierion. Minimizaion of he crierion can be rewrien as maximizaion of he expeced logarihmic score, which is esimaed by he average logarihmic score: 1, (16) N ln f h, y i, h A where yi, h is he ex-pos realizaion of he variable and f h, densiy of ha variable compued a ime a forecasing horizon h. is he simulaed poserior 5. Resuls Table 1 demonsraes how he in-sample fi of models ha ake ino accoun predicion errors for longer forecasing horizons improves. The in-sample fi is measured as he mean squared error of he fied values consruced for a paricular horizon. The fis of models wih coefficiens adjused for higher-period predicion errors and he sandard model are compared. The posiive values in he able sugges ha he in-sample fi of he model wih adjused coefficiens is superior for all horizons (exhibiing lower mean square errors). This is no surprising, as simply a higher number of parameers is used o explain he observed daa. However, he increase in fi demonsraes ha he original model is no correcly specified. If he model for he firs forecasing period described he daa generaing process correcly, he improvemen in daa fi would no be observed. 11

12 Table 1: The mean difference of he in-sample fi of models used for ieraed and adjused ieraed forecasing. RGDP PGDP Cons GDPInv Emp. Hours Real Comp FedFunds /Hour Horizon: Noes: The in-sample fi is esimaed using he squared differences beween he fied and observed values. The models are esimaed on he full sample. Nex, he poin forecasing performance is examined. Figure 3 repors he MSFEs of he forecass produced by sandard ieraed forecasing (black solid line) and by adjused ieraed forecasing (red dashed line). The model which allows for changes in coefficiens exhibis lower MSFEs for almos all variables and horizons. The MSFEs are expressed in unis of he respecive variable, so hey are no direcly comparable across variables. However, Figure 3 suggess ha for some variables, he adjused forecasing procedure can lower he MSFE o half of he MSFE of ieraed forecass. The differences in MSFEs beween he wo approaches are repored in Table 2. The able also conains he resuls of he Diebold-Mariano es of equal forecasing accuracy of he wo forecasing echniques. 12

13 Figure 3. Mean square forecas errors a a paricular forecasing horizon for he adjused ieraed forecass and ieraed forecass. Table 2. The difference beween he MSFEs of ieraed and adjused ieraed forecass. RGDP PGDP Cons GDPInv Emp. Hours Real Comp FedFunds Horizon: /Hour * * *** *** * * * * *** ** ** ** *** * *** ** ** *** *** * *** *** *** ** *** *** *** * *** *** ** ** *** *** ** * *** *** ** Noe: *, **, and *** denoe significance a he 10%, 5%, and 1% levels of confidence for he Diebold- Mariano es of equal forecasing accuracy. Table 2 shows ha all excep one of he saisically significan differences in forecasing performance are observed only for he case where adjused ieraed forecasing is more accurae han sandard ieraed forecasing. Moreover, he magniude of he difference is subsanial and in some cases is close o he average difference in he variable beween wo adjacen periods. Finally, he improvemen in forecasing accuracy is observed regardless of wheher he variables exhibi rends or no. 13

14 Turning o he densiy forecasing performance, Table 3 suggess ha adjusing he coefficiens improves he accuracy of he densiy forecass in all cases (i.e., he average logarihmic score for adjused ieraed densiy forecass is higher han ha for sandard ieraed densiy forecass). While he median ieraed forecass are in some cases comparable o he median adjused forecass, he comparison of whole densiies suggess a clear preference for adjusing model coefficiens in ieraed forecasing. Table 3. The difference beween he average logarihmic scores of adjused ieraed and ieraed forecass. RGDP PGDP Cons GDPInv Emp. Hours Real Comp FedFunds Horizon: /Hour In producing adjused ieraed muli-sep forecass, a model wih differen coefficiens is used in each ieraion. As an example, Figure 4 shows he evoluion of he esimaes of he seleced AR parameers in he equaion for real GDP. I repors he evoluion of he inercep ( C 1 ), he coefficien on he firs lag of real GDP ( B 11 ), and he coefficiens on he firs lags of he GDP deflaor and consumpion ( 12 B and 13 B ). From he second forecasing ieraion, he coefficien on he own lag of real GDP moves close o uniy from is original value (denoed by he red dashed line). Similarly, afer hree forecasing ieraions, he coefficiens on he firs lag of he oher repored variables are close o zero. The value of he inercep converges o a posiive figure. No surprisingly, i urns ou ha he bes predicion a longer horizons is obained by aking he previous period fied value and adding he mean of real growh, which is esimaed by he inercep. Noe ha real GDP eners he model in log-level form. The purpose of he esimaion procedure is o choose he mos accurae way of moving from he informaion included in he observed variables when forecasing shor horizons o longer horizons, where he esimaed long-run value dominaes. Pu differenly, maximum likelihood represens a high-pass filer, whereas he procedure for adjusing he coefficiens for longer forecasing horizons represens a low-pass filer. 14

15 A slighly differen picure can be seen when one looks a he evoluion of he coefficiens in he equaion for he federal funds (FF) rae see Figure 5. The ineres rae eners he analysis in levels, bu does no exhibi clear rending behavior like he res of he variables. The inercep approaches zero and he evoluion of he parameers a heir own lag is such ha heir iling enables he model o accelerae owards he sample mean of he FF rae. Figure 4: Evoluion of he coefficiens in he equaion for real GDP. Noe: Panels indicae evoluion of he inercep ( C 1 ), he firs lag of real GDP ( B 11 ), he firs lag of he GDP deflaor ( B 12 ), and consumpion ( B 13 ). The red dashed line indicaes he median esimae for he firs forecasing ieraion. 15

16 Figure 5: Evoluion of he coefficiens in he equaion for he federal funds rae. Noe: Panels indicae evoluion of he inercep ( C 7 ), he firs lag of he federal funds rae ( B 77 ), he firs lag of real wages ( B 76 ), and hours worked ( B 75 ). The red dashed line indicaes he median esimae for he firs forecasing ieraion. The esimaion is done on he full sample. Finally, Figure 6 shows he evoluion of seleced coefficiens of he error covariance marix over he forecasing ieraions. The presened diagonal elemens increase, reflecing he increasing prior on he error variances. As discussed above, he prior is rescaled o reflec he fac ha higher forecasing horizons conain higher uncerainy semming from shocks. Figure 6: Evoluion of seleced elemens of he error covariance marix. 16

17 Noe: Panels indicae evoluion of he median of he error variance in he real GDP equaion ( 11 ) and he federal funds equaion ( 22 ) and he covariances beween he wo. The esimaion is done on he full sample. 5.1 Robusness issues The specificaion of priors follows sandard values from he lieraure. The only excepion is he parameer, represening he overall ighness. For he firs forecasing ieraion i equals 0.2, which is a sandard value. However, for oher forecasing ieraions no sandard values are available. The only clue follows from he fac ha he fied values used in he oher forecasing ieraions are uncerain and he overall ighness should reflec his fac by forcing he priors owards heir prior means. A robusness check regarding he overall ighness is, however, necessary. The robusness exercise consiss of wo exremes. The firs exercise assumes ha he overall ighness ignores he uncerainy in he fied values in he sense ha i does no decrease for coefficiens a fied values and he prior on he error variance is no re-scaled by 2. On he oher hand, he second exercise assumes a more profound drop in he overall ighness of he priors for parameers a fied values. The overall ighness is no ½ of ha from he firs forecasing ieraion, bu 1/3. The specific values of he overall ighness for parameers a fied values for differen forecasing ieraions are repored in Table 4. Table 4. Alernaive values of he overall ighness of he prior on AR parameers a fied values. horizon: Iniial calibraion Loose lambda Tigh lambda Figure 7 shows MSFEs for he sandard ieraed forecass and adjused ieraed forecass wih differen overall ighness for he prior on parameers a fied values. Ignoring he uncerainy relaed o fied values ofen leads o worse forecasing performance in comparison o boh sandard ieraed forecass and adjused ieraed forecass wih he iniial calibraion of. On he oher hand, more inensive ighening of prior variances resuls in very similar resuls o he iniial calibraion of. 17

18 Figure 7. Mean square forecas errors of ieraed forecass and adjused forecass wih differen values of he overall ighness parameer. Finally, resricing he daa se for he evaluaion of forecasing accuracy o he period 1998Q4 2016Q1 o examine he role of he Grea Recession suggess ha he Grea Recession does no affec he conclusions abou he superior forecasing performance of adjused ieraed forecass. The deailed resuls are available upon reques. 5.1 Adjused ieraed, ieraed, and direc forecasing This subsecion focuses on comparing ieraed forecass, ieraed forecass adjused according o he m-sep-ahead in-sample predicion error performance, and direc forecass. For direc forecass, he same prior as for sandard ieraed forecass is assumed. Figure 8 compares he MSFEs of he hree forecasing echniques. Regarding he performance of ieraed and direc forecass, i urns ou ha no clear-cu conclusion can be drawn. The GDP deflaor is forecased more accuraely by ieraed forecass for all horizons, bu for oher variables direc forecass ofen ouperform ieraed forecass. Ineresingly, for shor horizons of up o wo quarers, he majoriy of he variables are beer forecased using ieraed forecass. Long horizons are almos exclusively beer forecased using he direc forecasing echnique (he only excepion being he GDP deflaor). This resul suggess he presence of bias, which is muliplied by ieraing forecass in he ieraed forecasing echnique. 18

19 Figure 8: Mean square forecas errors a a paricular forecasing horizon for he adjused ieraed forecass, ieraed forecass, and direc forecass. Focusing on he comparison of adjused ieraed forecass and direc forecass, i can be concluded ha in he vas majoriy of saisically significan cases, adjused ieraed forecass ouperform direc forecass see Table 5. Table 5. The difference beween he MSFEs of direc and adjused ieraed forecass. RGDP PGDP Cons GDPInv Emp. Hours Real Comp FedFunds Horizon: /Hour ** ** * ** ** ** ** * * * ** * ** ** ** ** ** *** * ** *** ** Noe: *, **, and *** denoe significance a he 10%, 5%, and 1% levels of confidence. 19

20 So, adjused ieraed forecass are superior in almos all cases o ieraed forecass and in he majoriy of cases also o direc forecass. A similar conclusion can be drawn for densiy forecass see Table 6. Adjused ieraed densiy forecass are more accurae han direc densiy forecass in all cases. The accuracy of he densiy forecass suggess efficiency of he model parameer esimaes. Table 6. The difference beween he average logarihmic scores of direc and adjused ieraed forecass. RGDP PGDP Cons GDPInv Emp. Hours Real Comp FedFunds Horizon: /Hour Figure 9 repors he evoluion of seleced esimaed AR parameers in he equaion for he FF rae for direc forecasing. When we compare i o Figure 5, i urns ou ha he model for adjused ieraed forecasing produces more efficien parameer esimaes (as measured by he disance beween he 16 h and 84 h perceniles of he poserior disribuion of seleced parameers). Adjused ieraed forecass hus seem o enjoy he advanage of efficiency of parameer esimaes in comparison o direc forecass. Furhermore, adjused ieraed forecass also share he advanage of direc forecass in erms of forecasing robusness. 20

21 Figure 9. Evoluion of he coefficiens in direc forecasing in he equaion for he federal funds rae Noe: Panels indicae evoluion of he inercep ( C 7 ), he firs lag of he federal funds rae ( B 77 ), he firs lag of real wages ( B 76 ), and hours worked ( B 75 ). The esimaion is done on he full sample. 6. Conclusions The paper demonsraes how o adjus radiional ieraed muli-sep forecass o ge more accurae poin and densiy forecass. The adjusmen draws on in-sample predicion errors for higher forecasing horizons. So, he approach exends forecasing based on one-sep-ahead insample predicion errors. The poin and densiy forecasing accuracy is demonsraed on a sandard VAR model mimicking he Smes and Wouers (2007) DSGE model. The model employed is a medium-scale VAR. For small-scale VARs he problem of misspecificaion is more profound and he improvemen in accuracy would probably be greaer. Similarly, DSGE models impose cross-coefficien resricions, resuling in possible misspecificaion. So, he gain in forecasing performance discussed in his paper would presumably be higher for DSGE models han i is for heir VAR counerpars. The suggesed approach can be viewed as a combinaion of ieraed and direc forecasing. Ieraed forecasing is represened by using one-sep-ahead forecass o ge muli-sep forecass. Direc forecasing is presen in aking ino accoun he in-sample predicion error a a paricular forecasing horizon. The combinaion of he wo echniques is viable, as i could represen a response o he famous rade-off beween bias and efficiency involved in he heoreical comparison of he wo forecasing echniques. The resuls in he paper sugges ha his is so. Firs, adjused ieraed forecass are more precise in erms of mean squared forecasing errors. The bias is herefore lower. In addiion, he densiy forecasing performance exercise suggess ha adjused ieraed forecasing produces more accurae 21

22 densiy forecass, i.e., forecass ha are closer o he rue densiies of macroeconomic variables. Inefficien model parameer esimaes would lead o less accurae densiy forecass and hus he efficiency of ieraed forecasing seems o carry over o adjused ieraed forecasing. 22

23 References Cox, D. R. (1961): Predicion by Exponenially Weighed Moving Averages and Relaed Mehods, Journal of he Royal Saisical Sociey: Series B, 23, Findley, D. (1983): On he Use of Muliple Models of Muli-Period Forecasing, Proceedings of he Business and Saisics Secion, American Saisical Associaion, Giannone, D., Lenza, M., and G. E. Primiceri (2015): Prior Selecion for Vecor Auoregression, Review of Economics and Saisics, 97, Kapeanios G., Price, S., and K. Theodoridis (2015): A New Approach o Muli-Sep Forecasing Using Dynamic Sochasic General Equilibrium Models, Economics Leers, 136(C), Marcellino, M., Sock, J. H., and M. W. Wason (2006): A Comparison of Direc and Ieraed Mulisep AR Mehods for Forecasing Macroeconomic Time Series, Journal of Economerics, 135, Rafery, A. E., and S. Lewis (1992): How Many Ieraions in he Gibbs Sampler? In J. Bernardo, J. Berger, A. P. Dawid, and A. F. M. Smih (eds.), Bayesian Saisics, , Oxford Universiy Press Schorfheide, F. (2005): VAR Forecasing Under Misspecificaion, Journal of Economerics, 128, Smes, F., and R. Wouers (2007): Shocks and Fricions in US Business Cycles: A Bayesian DSGE Approach, American Economic Review, 97, Tiao G. C., and D. Xu (1993): Robusness of Maximum Likelihood Esimaes for Muli-sep Predicions: The Exponenial Smoohing Case, Biomerika, 80(3), Tonner, J., and J. Bruha (2014): The Czech Housing Marke Through he Lens of a DSGE Model Conaining Collaeral-Consrained Households, Czech Naional Bank Working Paper 9/2014. Xia, Y., and H. Tong (2011): Feaure Maching in Time Series Modeling, Saisical Science, 26,

24 Appendix A: Daa Table A1. Lis of variables. Variable Descripion Unis Seasonal adjusmen RGDP Real Gross Domesic Produc Index 2000:Q1=100 SAAR PGDP Gross Domesic Produc: Implici Price Deflaor Index 2000:Q1=100 SA Cons Real Personal Consumpion Expendiures Index 2000:Q1=100 SAAR GDPInv Real Gross Privae Domesic Invesmen Index 2000:Q1=100 SAAR Emp. Hours Nonfarm Business Secor: Hours of All Persons Index 1982:Q1=100 SA Real Comp/Hour Nonfarm Business Secor: Real Compensaion Per Hour Index 1982:Q1=100 SA FedFunds Effecive Federal Funds Rae %, quarerly average NSA Noe: SAAR seasonally adjused annual rae, SA seasonally adjused, NSA no seasonally adjused.