Estimating TVP-VAR models with time invariant long-run multipliers
EEstimating TVP-VAR models with time invariantlong-run multipliers
Denis Belomestny a,b , Ekaterina Krymova a,b, ∗ , Andrey Polbin b,c a University of Duisburg-Essen, Essen, Germany b Russian Academy of National Economy and Public Administration, Moscow, Russia c Gaidar Institute for Economic Policy, Moscow, Russia
Abstract
The main goal of this paper is to develop a methodology for estimating timevarying parameter vector auto-regression (TVP-VAR) models with a time-invariant long-run relationship between endogenous variables and changes inexogenous variables. We propose a Gibbs sampling scheme for estimation ofmodel parameters as well as time-invariant long-run multiplier parameters.Further we demonstrate the applicability of the proposed method by ana-lyzing examples of the Norwegian and Russian economies based on the dataon real GDP, real exchange rate and real oil prices. Our results show thatincorporating the time invariance constraint on the long-run multipliers inTVP-VAR model helps to significantly improve the forecasting performance.
Keywords: time-varying parameter VAR models, VARX models, long-runmultipliers, oil prices, GDP, exchange rate flexibility
JEL:
C11, C51, C52, C53, E32, E37, E52, F41, F47
Declarations of interest: none
1. Introduction
During the past two decades time-varying parameter estimation becamevery popular in macroeconomic modeling. The existing literature providesstrong evidence for a time varying behavior of volatility (Primiceri, 2005,Justiniano and Primiceri, 2008, McConnell and Perez-Quiros, 2000), long-run economic growth (Kim and Nelson, 1999, Cogley, 2005, Antolin-Diaz ∗ Corresponding author. E-mail address: ekaterina.krymova@epfl.chPresent address: SDSC EPFL, Lausanne, Switzerland a r X i v : . [ ec on . E M ] A ug t al., 2017), trend inflation (Cogley and Sbordone, 2008, Stock and Watson,2007, Clark and Doh, 2014), inflation persistence (Cogley et al., 2010, Kanget al., 2009), oil price persistence (Kruse and Wegener, 2019), dependenceof main macroeconomic variables on oil prices (Baumeister and Peersman,2013, Chen, 2009, Cross and Nguyen, 2017, Riggi and Venditti, 2015).After seminal papers (Primiceri, 2005, Del Negro and Primiceri, 2015,Cogley and Sargent, 2005) Bayesian time-varying parameter vector autore-gression (TVP-VAR) model with stochastic volatility became one of the mainmodeling tools to capture temporary changes in relations between the vari-ables. Time-varying parameters are believed to follow simple stochastic pro-cesses, parameters of which are estimated with the help of Monte Carlotechniques (see Gelfand and Smith, 1990, 1991, Carter and Kohn, 1994). Asdemonstrated in (Koop and Korobilis, 2013, D’Agostino et al., 2013, Clarkand Ravazzolo, 2015) Bayesian TVP-VAR models could be used for fore-casting. Nevertheless, Bayesian TVP-VAR models have not yet become anubiquitous forecasting tool due to a large number of parameters to estimate.In this paper we consider models with changing in time parameters mo-tivated by a change in economic policy regimes. According to Lucas critique(Lucas et al., 1976) rational economic agents take the structural changesin economy into account when making decisions. Therefore changes in eco-nomic policy should lead to the changes in parameters of such non-structuralmodels as, for example, large macroeconometric models consisting of simul-taneous equations or vector autoregression models. In a series of papers thehigh volatility of US macroeconomic indicators is related to poor monetarypolicy performance at the time before Paul Volcker became chairman of theFed (Clarida et al., 2000, Judd et al., 1998, Lubik and Schorfheide, 2004,Mavroeidis, 2010). However, empirical evidence for this hypothesis on thebasis of time varying parameter models is controversial. Primiceri (2005) pro-posed TVP-VAR model and developed a Bayesian method to estimate modelparameters. An example of TVP-VAR modelling of US economy failed todemonstrate the changes in the monetary policy transmission. Along withthat (Cogley and Sargent, 2001, 2005, Canova and P´erez Forero, 2015, Gam-betti et al., 2008) provided empirical evidences of a notable change in themonetary policy transmission mechanism using TVP-VAR and in (Sims andZha, 2006) with the help of Markov switching VAR model. At the same timethere is a strong empirical evidence in favour of a nominal exchange rateregime influence on the business cycle performance of developing countries.A floating exchange rate has a stabilizing effect on the output under the2nfluence of terms-of-trade shocks. The latter was shown by Broda (2004)with the help of VAR methods and by (Edwards and Yeyati, 2005) usingpanel regression techniques. In addition, exchange rate regimes in develop-ing countries demonstrate changeable behavior (Levy-Yeyati and Sturzeneg-ger, 2005). Thus TVP-VAR models are promising for modeling of economiesunder exchange rate regime shifts.We aim to analyze econometric models with time-varying short-term andinvariant long-term relationships in order to describe economic system whosecross-correlation relationships change due to changes in the monetary policyand exchange rate regimes. The long-term assumptions arise from a classicalhypothesis of the long-run money neutrality. Empirical support in favourof this hypothesis is exhaustively documented in the literature, we cite hereonly (Fisher and Seater, 1993, King and Watson, 1992, Weber, 1994), seealso references therein. Furthermore, the long-run neutrality of monetarypolicy shocks is a typical assumption in estimation of SVAR models (Altiget al., 2011, Canova and P´erez Forero, 2015, Peersman, 2005). We also pro-pose a methodology for estimation of TVP-VAR models with time-invariantlong-run relations of endogenous variables to changes in exogenous variables.VAR model with exogenous variables (VARX) is one of the main methods fordescribing the dynamics of small open economies (Cushman and Zha, 1997,Fern´andez et al., 2017, Uribe and Yue, 2006). Natural candidates for exoge-nous variables in VARX models are oil prices, terms of trade, world interestrates, external demand and many others. Hence, the proposed methodologymay find application in numerous practical examples.The paper is organized as follows. Section 2 describes a new methodol-ogy for modeling of economy based on TVP-VAR model (Primiceri, 2005)incorporating non-zero long-run restriction (time invariance of long-run mul-tipliers). We estimate long-run multipliers within a Monte Carlo procedure.Section 3 describes a particular case of modeling with real GDP, real ex-change rate as endogenous and real oil prices as exogenous variable for an oilexporting country. Section 4 contains the results of estimation of the modelon the Norwegian and Russian datasets. We demonstrate the model’s fore-casting performance in comparison with classical VARX and a modificationof TVP-VAR with exogenous variables. Our results show that the time in-variance constraint for long-run multiplier brings significantly improvementof TVP-VAR model performance in terms of forecasting accuracy.3 . Constrained TVP-VAR We recall first a framework of time varying parameter vector auto-regressionmodel (TVP-VAR) of Primiceri (2005), Del Negro and Primiceri (2015),where the endogenous time series vector y t ∈ R n is modeled by the followingmeasurement equation y t = c t + B ,t y t − + . . . + B k,t y t − k + u t , t = 1 , . . . , T, (1)where B i,t , i = 1 , . . . , k, are n × n matrices of time varying coefficients, arandom vector u t ∈ R n contains heteroskedastic unobserved shocks with acovariance matrix Ω t . The covariance matrix Ω t is defined via a decomposi-tion A t Ω t A (cid:62) t = Σ t Σ (cid:62) t , where A t is a lower triangle matrix and Σ t = diag( σ ,t , . . . , σ n,t ) is a diagonalmatrix. Then it follows that u t = A − t Σ t e t , (2)where e t ∈ R n is a vector with independent standard Gaussian components.In Primiceri (2005), Del Negro and Primiceri (2015) a Bayesian approachwas used for statistical inference in this model.Here we present an extension of the model (1). In particular, we introducean exogenous variable x t and a long run constraint on the VAR coefficients.Our generalized TVP-VAR model for the exogenous variables reads as y t = c t + B ,t y t − + . . . + B k,t y t − k + k (cid:88) i =0 D i,t x t − i + u t , t = 1 , . . . , T, (3)where D i,t ∈ R n are n -dimensional time varying vectors of coefficients and c t is a n -dimensional time-varying intercept term. Note that the exogenoustime series enter the right hand side of (3) with zero lag. We restrict ourselvesfor simplicity to the case of one exogenous variable. Our goal is to developa Bayesian estimation procedure for the extended model with exogenousvariables (3) under the following long-run time invariant constraints on thevectors of coefficients B i,t and D i,t θ = (cid:34) I n − k (cid:88) j =1 B j,t (cid:35) − k (cid:88) i =0 D i,t , (4)4here θ ∈ R n is a constant multiplier parameter. Thus we impose conditionthat the shocks in the exogenous variable lead to the same long-run responsein the endogenous vector independently of the time when the shock occurs.As was discussed in introduction, such modelling approach can be appropri-ate for economic systems whose cross-correlation relationships change due tochanges in the monetary policy and exchange rate regimes if the hypothesisof the long-run neutrality of money holds. In the modern New Keynesianmodels the particular form of the monetary policy rule matters for the shapeof transition path from one long-run equilibrium to another, however, theinfluence of the monetary policy rule on the long-run equilibrium is usuallyabsent. In Primiceri (2005), Del Negro and Primiceri (2015) the coefficients ofthe model (1) were modeled in the following way. Let all the vectors B i,t , i = 1 , . . . , k be stacked into a vector B t of length k · n , let α t be a vectorof non-zero and non-one elements of the matrix A t (stacked by rows) and σ t be a vector of the diagonal elements of the matrix Σ t . The dynamics of thetime varying parameters is specified as random walks: B t = B t − + ν t , α t = α t − + ζ t , log( σ t ) = log( σ t − ) + η t , (5)where all innovations are assumed to be jointly normally distributed and thelogarithm is applied to the vector σ t element-wise. In particular we assumethat V = Var (cid:15) t ν t ζ t η t = I n Q G
00 0 0 W , where I n is a n -dimensional identity matrix, Q, G and W are positive def-inite matrices. The prior distributions for the hyperparameters, Q , W andthe blocks of G , are assumed to be independent inverse-Wishart. The priorsfor the initial states of the time varying coefficients, simultaneous relationsand log standard errors, B , α and log( σ ), are assumed to be indepen-dent normally distributed, where the parameters of the prior distributionsare estimated by means of the ordinary least squares (OLS) from the first t observations using the regression model (3) (for the details see Section 4.1of (Primiceri, 2005)). These assumptions imply normal priors on the entire5equences of the B ’s, α ’s and log σ ’s (conditional on Q , W and G ). We useMarkov Chain Monte Carlo (MCMC) technique to generate a sample fromthe joint posterior of B, A, Σ , V , where B is a matrix in R kn × ( T − t ) , whichcontains the path of the coefficients B t , A contains a t , and Σ contains σ t for t = t + 1 , . . . , T . In particular, Gibbs sampling (Carter and Kohn, 1994)is used in order to exploit the blocking structure of the unknowns. Gibbssampling is carried out in four steps, returning draws of the time varying coef-ficients ( B ), simultaneous relations ( A ), volatilities (Σ) and hyperparameters( V ), conditional on the observed data and the rest of the parameters. Con-ditional on A and Σ, the inference for the state space model defined by (1)and (5) is carried out with the help of the Kalman filter (Hamilton, 1995).The conditional posterior of B is a product of Gaussian densities, therefore B can be sampled using a standard simulation smoother (Carter and Kohn,1994). For the same reason, the posterior distribution of A, conditionally on B and Σ , is also a product of normal distributions. Hence A can be drawnin the same way. Remind that the process A t u t is the product of Σ t and e t , which is a nonlinear system of measurement equations (see equation (2)).This system can be transformed into a non-Gaussian state space model bysquaring and taking logarithms for every t :2 log( A t u t ) = 2 log( σ t ) + log( e t ) , where log( σ t ) is a random walk (5). Despite being linear, this system hasinnovations log( e t ) distributed as log χ (1). We approximate the system withthe help of a mixture of Gaussians following (Primiceri, 2005, Kim et al.,1998, Carter and Kohn, 1994). We adopt this scheme for estimation inthe model (3) under the constraint (4). With a slight abuse of notationswe model parameters B t , A t , Σ t of (3) in the same way as described abovefor (1). Without the constraint (4) the extension of the model (1) to theexogenous observations is straightforward if one assumes D i,t = D i,t − + ¯ ν i,t , (6)where the innovations (¯ ν i,t ) are jointly normally distributed and independentof η t , ν t , ζ t . Imposing multiplier constraints (1) introduces a relation between D i,t , i = 0 , . . . , k , which allows us to express one of the coefficients as D ,t = (cid:34) I n − k (cid:88) j =1 B j,t (cid:35) θ − k (cid:88) i =1 D i,t .
6e estimate parameters of the prior distributions for B , D ,i , i = 0 , . . . , k ,log( σ ) from the first t observations using OLS ( see (3)). Denote˜ V = (cid:18) V
00 ˜ Q (cid:19) , where ˜ Q is a covariance of ¯ ν i,t , i = 1 , . . . , k with independent inverse-Wishartprior. We assume a prior distribution for multiplier θ to be Gaussian N ( µ , U ),where the parameter µ ∈ R n is estimated from a relation (4) for θ from es-timates of B , D ,i , i = 0 , . . . , k .The covariance matrix U ∈ R n × n is a diagonal matrix with large diag-onal elements (uninformative prior). We propose a Gibbs sampling schemeto generate sample paths from the joint posterior of B, A,
Σ as well as toestimate θ . The details are given in the next section for an example of mod-eling gross domestic product (GDP) and real effective exchange rate (ER)with exogenous oil price. We demonstrate the performance of the proposedmethod for the case of Russian and Norwegian economies.
3. Modeling of gross domestic product and real effective exchange
The main goal of the following setup is to model the gross domesticproduct (GDP) and the real effective exchange rate (ER) while treating oilprice as an exogenous variable under a long-run constraint. Let y t ∈ R bean endogenous vector, whose first component y ,t denotes the difference ofthe logarithms of real effective exchange rate S ,t , the second component y ,t stands for the difference of logarithms of GDP S ,t : y i,t = log( S i,t /S i,t − ) . Denote the difference of the logarithms of the exogenous oil price S x,t at time t by x t = log S x,t S x,t − . We use (3) to model y t : y t = c t + β t y t − + D ,t x t + D ,t x t − + u t , t = 1 , . . . , T, (7)were c t ∈ R , β t = (cid:20) B ( t ) B ( t ) B ( t ) B ( t ) (cid:21) , D i,t = (cid:20) D i,t D i,t (cid:21) , i ∈ { , } , u t is indepen-dent from x t . Under the constraint (4) we have θ = [ I n − β t ] − ( D ,t + D ,t ) , (8)where θ ∈ R is an unobserved multiplier parameter. As our variables enterthe model logarithmically, the parameter θ has an interpretation of the long-run elasticity. From (8) we derive D ,t = [ I n − β t ] θ − D ,t θ the corresponding measurement equation reads asfollows y t − θx t = c t + β t [ y t − − θx t ] − D ,t ( x t − x t − ) + u t , t = 1 , . . . , T. (9) First we use OLS to estimate parameters (means and variances) of theprior distributions for B , D i, , i = 0 , B t (vectorized β t ), D ,t , D ,t without elasticity constraints. We assume a priordistribution for θ to be N ( µ , U ), where the vector µ ∈ R n is estimated from(8) given estimates of the mean of prior distributions of B , D i, , i = 0 , U have large values. Denote by s a vectorwith indicator variables in Gaussian mixture approximation which takes partin estimating u t (see Section 2.1). Denote the trajectories of all parametersfor a fixed value of θ ( j − at j th MCMC simulation step by Z ( j ) θ = [ B ( j ) , D ( j )1 , Σ ( j ) , A ( j ) , ˜ V ( j ) ] . The steps of the proposed Gibbs sampling scheme are as follows.1. Draw Z ( j ) θ conditionally on θ ( j − and Y based on the model (9). De-note by p and ˜ p the likelihood and the approximated likelihood (usingGaussian mixtures), respectively, and υ = [ B, D , A, ˜ V ] . We proceedwith the following sampling steps (see (Del Negro and Primiceri, 2015)for details) by drawing: • Σ from ˜ p (Σ | Y, υ, s ), • υ from p ( υ | Y, Σ), • s from ˜ p ( s | Y, Σ , υ ) .
2. Draw θ ( j ) conditionally on Z ( j ) θ from N ( µ j , U j ) . The parameters of theposterior distribution µ j , v j are estimated from the observations˜ Y t ( j ) = C ( j ) t θ + u ( j ) t , t = t + 1 , . . . , T, where ˜ Y ( j ) t = y t − c t − β ( j ) t y t − − D ( j )1 ,t ( x t − − x t ) , C ( j ) t = x t [ I − β ( j ) t ] and u ( j ) t ∼ N (0 , H ( j ) t ) . Therefore the posterior distribution of θ ( j ) is definedby the covariance U − j = U − + T (cid:88) t = t +1 C ( j ) (cid:62) t [ H ( j ) t ] − C ( j ) t , µ j = U j (cid:32) U − θ + T (cid:88) t = t +1 C ( j ) (cid:62) t [ H ( j ) t ] − ˜ Y t ( j ) (cid:33) . Impulse response characterization demonstrates the behavior of the out-put after a small shock in the input variable. We shall be interested in 10%increase of oil prices obtained by a single shock x t = log(1 .
1) in the model(7) with (8). The shock evolves according to (7) as δy t +1 = log(1 . D ,t ,δy t +2 = log(1 . β t D ,t + D ,t ] ,. . .δy t + k = log(1 . β kt D ,t + β k − t D ,t ] . Hence, a change in the logarithm (element-wise) of the vector S t reads aslog S t + k − log S t = k (cid:88) j =1 δy t + j . Thus, when k → ∞ we getlog S t + k − log S t → [ I − β t ] − ( D ,t + D ,t ) log(1 .
1) = log(1 . θ. (10)Therefore θ defines a fully adjusted value of a response after a shock anddescribes the underlying permanent state of economy.
4. Numerical results
This section contains empirical analysis of the proposed constrained TVP-VAR method based on economic data for Norwegian and Russian economies.We have selected these countries for the analysis because they are among thetop oil-exporters. Furthermore both Russia and Norway underwent signifi-cant changes in exchange rate policy in historical retrospective. We compareperformance of the constrained TVP-VAR for modeling the Norwegian andthe Russian economies with the following benchmark methods: 1) method9rom (Del Negro and Primiceri, 2015, Primiceri, 2005) extended for estima-tion of the model with exogenous variables (7) with no elasticity restrictions , 2) VAR with constant parameters. For comparison we computed the abso-lute value of the deviation of out-of-sample forecasts from the correspondingobservations of GDP and real exchange rate for the number of steps aheadlying in the set { , , , , } . For Norway we had 157 quarterly observations of the real exchange rate S ,t , GDP S ,t , and oil price S x,t staring from the 1st quarter 1980 till 1stquarter 2019. Therefore the number of logarithm differences y t and x t ofobservations S x,t and S i,t , i = 1 , t = 40 observa-tions of y ti , i = 1 , x t . We use an uninformative prior for the elasticity θ ∼ N ( µ , U ) (see Section 2.1) with U = diag([0 . , . θ =[0 . , . (cid:62) , where the first component corresponds to real exchange rateand the second to GDP. In-sample forecasts by the constrained TVP-VARfor the time interval [1992Q , ] are shown in Figures 1, 2. Figures 3,4 contain IRF for years 1991, 2008, 2018 whereas Fig.5 contains 3D IRF forGDP and real exchange rate.The errors of 1-5 step ahead out-of-sample forecasts for the proposedmethod and benchmark methods for the time interval [1992Q , ] arecollected in Table 1. Long-run the impulse responses to a positive shock inoil prices are positive for both the real exchange rate and real GDP. Improve-ment in the terms of trade leads to the exchange rate strengthening, whichensures internal and external equilibrium. This means that for the samevolume of exports a country can buy a larger volume of imported goods.Therefore the prices of domestic non-tradable goods relative to prices ofimported goods should increase to ensure the increase in the share of im-ported goods in aggregated consumption (Edwards, 1988). Furthermore theoil prices rise leads to an increase in GDP through the capital accumulationchannel, namely, the higher oil prices result in new investment opportunities(Esfahani et al., 2014) and increase of domestic returns (Idrisov et al., 2015). The code for the proposed method and the first benchmark method is based on mod-ifications of a CRAN package (Krueger, 2015)
990 1995 2000 2005 2010 2015 2020
Norway, GDP t v a l ue datain−sampleout−of−sample Figure 1: Five steps ahead in-sample (red) and out-of sample (green) forecasts for Norwe-gian GDP by the model (7) with elasticity constraint (8)
The model indicates significant change in the short run transmission mecha-nism of oil prices shocks to the real exchange rate. IRFs for years 1991 and2018 are statistically different. Before the Norges Bank turned to inflationtargeting in 2001 the real exchange rate response had been strengtheninggradually towards its long-run equilibrium after a shock in oil prices. Underthe inflation targeting regime we see some overshooting of the real exchangerate. There is no sizable time variation in model parameters over the lastdecade, and the shape of the impulse response function for the real exchangerate stabilizes. Impulse response function for the real GDP changes veryslightly during the entire period under review. The results of pseudo out-of-sample forecasting experiment in Table 1 show that the proposed TVP-VARwith time-invariant long-run multipliers outperforms the benchmark TVP-VAR without constraints. Thus reduction of degrees of freedom in TVP-VARmodel helps to improve the forecasting accuracy. The proposed constrainedTVP-VAR delivers smaller forecast errors than constant-parameter VAR for1-3 steps-ahead forecasts and accuracy similar to VAR for 4-5 step-aheadforecasts.Next we consider an example of Russian economy, were a transition to-wards the inflation targeting regime has started in 2014.11
990 1995 2000 2005 2010 2015 2020
Norway, ER t v a l ue datain−sampleout−of−sample Figure 2: Five steps ahead in-sample (red) and out-of sample (green) forecasts for Norwe-gian real effective exchange rate by the model (7) with elasticity constraint (8)
10 15 20 . . . Percentage change in ER t v a l ue . . . . Percentage change in ER t v a l ue Figure 3: Norway. Impulse response functions for the proposed model: real exchangerate as a response variable . . . . Percentage change in GDP t v a l ue . . . . Percentage change in GDP t v a l ue Figure 4: Norway. Impulse response functions for the proposed model: GDP as a responsevariable igure 5: Norway. 3D impulse response functions for the proposed model: the impulsein oil prices and GDP (left) and real effective exchange rate (right) as response variablescorrespondingly Table 1: Norway. Mean and standard deviation of absolute error of out-of-sample fore-casts for the proposed method, TVP-VAR (7) with exogenous variables and VAR.
We use 93 quarterly observations of real effective exchange rate of S ,t ,GDP S ,t for Russia and oil price S x,t from the 1st quarter 1995 till the4th quarter 2018. A number of logarithm differences of observations of y t therefore was 92. For the estimation of prior parameters of constrained TVP-VAR we used the first t = 40 observations of y ti , i = 1 , x t . We selectedan uninformative Gaussian prior for the elasticity θ ∼ N ( µ , U ) (see Section2.1) with U = diag([0 . , . θ = [0 . , . (cid:62) , where the first component correspondsto real exchange rate and the second to GDP. Posterior median of VARpart of constrained TVP-VAR coefficients are shown in Figure 6, posteriormedians of D ,t and D ,t are in Figure 7. The five-step ahead in-sampleforecasts for the GDP and the real effective exchange rate (ER) along without-of-sample forecast for the time interval [2007Q , ] are shown in15igures 8, 9. Median of long-run growth rate with 60% confidence intervalsfor GDP, which is the second component of ( I − B t ) − c t , in percents is shownin Figure 10. − . . c t 2006 2012 . . . B t 2006 2012 . . B t2006 2012 . . c t 2006 2012 − . − . B t 2006 2012 . . B t Figure 6: VAR coefficients in (7) with (8): first column corresponds to c t , the next twocolumns show the behavior of the entries of B t . . . D t 2006 2010 2014 − . . D t2006 2010 2014 . . D t 2006 2010 2014 . . D t Figure 7: The dynamics of the entries of exogenous coefficients D ,t and D ,t . We compare impulse response functions for the years 2008 and 2018 forGDP and real exchange rate (ER) to the shock in exogenous logarithm of16
005 2010 2015
Russia, GDP t v a l ue datain−sampleout−of−sample Figure 8: Five steps ahead in-sample (red) and out-of sample (green) forecasts for GDPby the model (7) with elasticity constraint (8)
Russia, ER t v a l ue datain−sampleout−of−sample Figure 9: Five steps ahead in-sample (red) and out-of sample (green) forecasts for the realeffective exchange rate by the model (7) with elasticity constraint (8).
006 2010 2014 − Percentage change in GDP trend t v a l ue Figure 10: Percents of long-run growth rate of Russian GDP with 60% confidence intervals differences of oil prices. Results in Figs. 11 demonstrate the convergenceto the same limiting value defined by (10); 3D-plots of impulse responsefunctions are shown in Figs. 12. During the years before the crisis of 2008–2009 the Central Bank of Russia followed the policy of a managed nominalruble exchange rate. From IRF for this period one can observe a gradualstrengthening of the real exchange rate towards its long-run equilibrium af-ter an increase in oil prices. During the next years the Central Bank ofRussia switched to a floating exchange rate. After that the real exchangerate began to react to oil price shocks more sharply with the overshootingeffect. It should be noted that during periods of gradual reaction of the ex-change rate to the oil price shocks, real GDP reacted quite strongly to theshock. During the periods of sharp reaction of the real exchange rate the realGDP demonstrates gradual increase. Therefore our results are in line witha classical view: flexible exchange rates are shock absorbers for small openeconomies and the floating exchange rate regime of monetary policy reducesvolatility of the GDP growth.The mean absolute errors and standard deviations for of 1-5 steps out-of-sample forecasts of GDP and real exchange rate for the proposed methodand benchmark methods for the time interval [2007Q , ] are shownin Table 2. One may conclude that in terms of forecasts a classical VARgives better result than TVP-VAR (Del Negro and Primiceri, 2015, Prim-iceri, 2005) extended for exogenous variables case. Imposing the elasticityconstraint helps to improve the situation: the proposed method outperformsboth benchmark methods in forecasting of GDP 1-3 steps ahead. The pro-18
10 15 20 . . . . . Percentage change in GDP t v a l ue . . . . . Percentage change in ER t v a l ue Figure 11: Impulse response functions for the proposed model: the impulse in oil andGDP and real exchange rate as response variableFigure 12: 3D impulse response functions for the proposed model: the impulse in oil andGDP (left) and real exchange rate (right) as response variables correspondingly posed method gives smaller forecasting error that non-constrained TVP-VARfor the real exchange rate. Nevertheless, the uncertainty coming from thecoefficients model brings though delivers slightly less accuracy than VAR forthe real exchange rate and 4-5 steps ahead forecasts of GDP. The resultsdemonstrate that imposing the long run elasticity constraint allows to im-prove the quality of modeling. Figure 10 demonstrates significant decreasein the long-run growth rates for the Russian economy.19onstrained VAR TVP-VAR (7)steps mean std mean std mean stdGDP1 28.01 41.46 46.37 50.81 51.02 56.582 61.58 79.01 79.97 97.29 98.21 110.713 91.24 116.77 105.50 124.53 120.59 148.804 119.06 147.81 120.42 141.74 135.97 167.985 155.03 166.91 133.42 158.50 150.30 180.91ER1 4.99 7.37 4.67 4.69 5.48 5.372 6.16 5.95 6.00 5.69 7.50 6.973 6.96 5.95 6.65 4.61 7.19 5.754 6.77 6.77 6.52 5.49 6.39 6.525 7.48 7.92 6.91 6.05 7.36 6.39
Table 2: Russia. Mean and standard deviation of absolute error of out-of-sample forecastsfor the proposed method, TVP-VAR (7) with exogenous variables and VAR. . Conclusions In the paper we propose a TVP-VAR model with a time-invariant con-straint on the long-run multipliers of endogenous variables with respect tochanges in exogenous variable. We provide a Bayesian estimation methodfor TVP-VAR parameters and multipliers. The proposed methodology canbe used for a wide range of practical applications as an alternative to VARX,for example, in open economies modeling. Our approach is tailored to eco-nomic systems whose cross-correlation relationships change due to changesin the monetary policy and exchange rate regimes under the hypothesis oflong-run money neutrality. In the modern New Keynesian models the partic-ular monetary policy rule matters for the shape of transition path from onelong-run equilibrium to another. However, usually there is no influence of themonetary policy rule on the long-run equilibrium. We apply the proposedmethodology to model relationship between the real GDP, the real exchangerate and real oil prices for the Norwegian and the Russian economies. Resultsshow that incorporating the time invariance constraint for the long-run mul-tipliers significantly improves forecasting performance of TVP-VAR model.Impulse responses are interpretable. The oil price increase leads to statisti-cally significant real exchange rate appreciation and GDP increase in longrun. During periods of gradual reaction of the real exchange rate to the oilprice shocks, real GDP reacted strongly to the shock. During periods of thesharp reaction of the real exchange rate, the real GDP demonstrates gradualincrease. Therefore our results are in line with classical view that flexibleexchange rates are shock absorbers for small open economies and the float-ing exchange rate regime of monetary policy reduces volatility of the GDPgrowth.