Forecasting with Bayesian vector autoregressive models: Comparison of direct and iterated multistep methods

Sugita, Katsuhiro Article Forecasting with Bayesian vector autoregressive models: Comparison of direct and iterated multistep methods Asian Journal of Economics and Banking (AJEB) Provided in Cooperation with: Ho Chi Minh University of Banking (HUB), Ho Chi Minh City Suggested Citation: Sugita, Katsuhiro (2022) : Forecasting with Bayesian vector autoregressive models: Comparison of direct and iterated multistep methods, Asian Journal of Economics and Banking (AJEB), ISSN 2633-7991, Emerald, Leeds, Vol. 6, Iss. 2, pp. 142-154, https://doi.org/10.1108/AJEB-04-2022-0044 This Version is available at: https://hdl.handle.net/10419/334070 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Forecasting with Bayesian vector autoregressive models: comparison of direct and iterated multistep methods Katsuhiro Sugita Faculty of Global and Regional Studies, University of the Ryukyus, Okinawa, Japan Abstract Purpose –The paper compares multi-period forecasting performances by direct and iterated method using Bayesian vector autoregressive (VAR) models. Design/methodology/approach –The paper adopts Bayesian VAR models with three different priors – independent Normal-Wishart prior, the Minnesota prior and the stochastic search variable selection (SSVS). Monte Carlo simulations are conducted to compare forecasting performances. An empirical study using US macroeconomic data are shown as an illustration. Findings –In theory direct forecasts are more efficient asymptotically and more robust to model misspecification than iterated forecasts, and iterated forecasts tend to bias but more efficient if the one-period ahead model is correctly specified. From the results of the Monte Carlo simulations, iterated forecasts tend to outperform direct forecasts, particularly with longer lag model and with longer forecast horizons. Implementing SSVS prior generally improves forecasting performance over unrestricted VAR model for either nonstationary or stationary data. Originality/value –The paper finds that iterated forecasts using model with the SSVS prior generally best outperform, suggesting that the SSVS restrictions on insignificant parameters alleviates over-parameterized problem of VAR in one-step ahead forecast and thus offers an appreciable improvement in forecast performance of iterated forecasts. Keywords Forecasting, Bayesian econometrics, VAR model Paper type Research paper 1. Introduction Vector autoregressive (VAR) models have been widely used to forecast macroeconomic variables and to analyze macroeconomics and policy. For one-period ahead forecasting, one has to just estimate the model. However, it is often the case that more than one-period forecasting is of interest. In making a multi-period forecast, there are two methods –direct forecast method and iterated forecast method, and there have been several theoretical research about which method is better for multi-period forecasting such as Bhansali (1996, 1997),Clements and Hendry (1996),Kang (2003),Chevillon and Hendry (2005),Ing (2003) and among others. These literature tend to conclude that direct forecasts are more robust to model specification and more efficient asymptotically, and thus the direct forecast method is preferable compared with the iterated forecast method, while the iterated forecast method can AJEB 6,2 142 © Katsuhiro Sugita. Published in Asian Journal of Economics and Banking. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/ legalcode. This work was supported by JSPS KAKENHI grant number 20K01591. The current issue and full text archive of this journal is available on Emerald Insight at: https://www.emerald.com/insight/2615-9821.htm Received 21 April 2022 Revised 28 April 2022 Accepted 28 April 2022 Asian Journal of Economics and Banking Vol. 6 No. 2, 2022 pp. 142-154 Emerald Publishing Limited e-ISSN: 2633-7991 p-ISSN: 2615-9821 DOI 10.1108/AJEB-04-2022-0044 Downloaded from http://www.emerald.com/ajeb/article-pdf/6/2/142/358140/ajeb-04-2022-0044.pdf by ZBW German National Library of Economics user on 16 December 2025 be more efficient only if the one-period ahead model is correctly specified. However, some empirical research studies show that iterated forecasts outperform direct forecasts. Ang et al. (2006) find that the iterated forecasts of the US GDP growth perform better than the direct forecasts. Marcellino et al. (2006) show that iterated forecasts outperform direct forecasts, especially with longer lag and longer forecast horizon; this paper uses 170 US monthly macroeconomic time series for either univariate or multivariate models. Pesaran et al. (2011) state that whether direct or iterated method is better in multi-period forecasting depends upon the sample size, forecast horizon, the underlying data generating process (DGP) and the methods used to select lag length for the model, and thus it is ultimately an empirical matter. For multivariate VAR models, there exists an over-parameterization problem, which leads to imprecise inference and thus deteriorates the forecast performance. Some Bayesian approaches to VAR models have been increasingly popular since Bayesian method can shrink VAR models by restricting its prior distributions. In this paper we investigate whether restricted parsimonious VAR models can mitigate misspecification problem and thus improve the forecasting performance of iterated method. Here, an independent NormalWishart prior is used for the unrestricted VAR and the Minnesota prior (Minn) and the stochastic search variable selection (SSVS) prior are used for the restricted prior to compare multiperiod forecasting performance between the direct and iterated forecast method. We conduct numerical simulations using both stationary and nonstationary data generating processes (DGPs) to evaluate forecasting performances with 2-, 4-, 8and 12-step ahead horizons, and compute the mean squared forecast error (MSFE) to compare direct forecasts with iterated forecasts using Bayesian VAR models with unrestricted and restricted priors. Iterated forecasts are found to outperform direct forecasts for both unrestricted and restricted VAR models, particularly with long-lag model and with long forecasting horizon. Implementing SSVS in VAR is found to generally improve forecasting performance appreciably. With relatively long lag length and thus a large number of parameters in the model, it seems that SSVS can effectively restrict insignificant parameters in the model and thus improve forecasting performance. The plan of this paper is as follows. In Section 2 multi-period forecasting using VAR model is described, and method to evaluate forecasting performances. Section 3 reviews Bayesian VAR models with three different priors –the independent Normal-Wishart prior, the Minnesota prior and the SSVS prior. Section 4 illustrates numerical experiments with artificially generated data, and then examines the results of the numerical simulations. Section 5 illustrates an application to a simple three variables VAR of US macroeconomics. Section 6 concludes. This paper is based on preliminary working papers, Sugita (2018),Sugita (2019a) and Sugita (2019b).All results reported in this paper are generated using Ox version 7.2 for Linux (see Doornik, 2013). 2. Iterated and direct multi-period forecasts for VAR models This section describes iterated and direct forecasting methods for VAR models. Let y t be an n31 vector of observations at time t, then a VAR model with plag is written as y0 t¼ μ 0þX p i¼1 y0 t−iΘiþ ε 0 τ (1) for t51, ...,T, where μ is a n31 vector of an intercept term; Θ i are n3nmatrices of coefficients for i51, ...,p;« t are n31 independent Nn0;ΣðÞerrors; and the covariance matrix Σis an n3npositive definite matrix. The one-step ahead forecast b y0 tþ1jtof the VAR model is obtained by estimating the parameters in eq. (1) as b y0 tþ1jt¼b μ 0 ðIÞþPp i¼1y0 tþ1−ijtb ΘðIÞ;i,whereb μ ðIÞand b ΘðIÞ;iare the Forecasting with Bayesian VAR models 143 Downloaded from http://www.emerald.com/ajeb/article-pdf/6/2/142/358140/ajeb-04-2022-0044.pdf by ZBW German National Library of Economics user on 16 December 2025 estimators for μ and Θ i in eq. (1). To make forecasting further than one-period ahead into the future, there are two methods for making multi-period forecasts –iterated forecasts and direct forecasts methods. Iterated forecasts for the h-period forecasts are obtained recursively as b y0 tþh¼b μ 0 ðIÞþX p i¼1b y0 tþh−ib ΘðIÞ;i(2) where b yijt¼yjfor j≤t. Direct forecasts for the multi-period forecasting are obtained by estimating the model y0 t¼ μ 0þX p i¼1 y0 t−h−iΘiþ ε t;(3) Then using the estimated coefficients directly to make the forecast of b ytþh¼b μ ðDÞþX p i¼1 yt−ib ΘðDÞ;i(4) where b μ ðDÞand b ΘðDÞ;iare the estimators for μ and Θ i in eq. (3). Thus, the relative forecast accuracy depends on how accurate b ΘðIÞ;iand b ΘðDÞ;iare estimated. If b ΘðIÞ;iis badly estimated with large errors, then its powered values diverge increasingly from Θ i . Since the iterated method depends on one-period ahead coefficients b ΘðIÞ;i, the direct method is preferable when the one-period ahead model is not specified correctly. Chevillon and Hendry (2005) evaluate the asymptotic and finite-sample properties of direct forecasting method, and show that, compared with iterated method, the direct method is more efficient asymptotically, more precise in finite samples and more robust against model misspecification. The theoretical advantages of the direct forecasting method over the iterated method are shown by Bhansali (1996,1997), Clements and Hendry (1996),Kang (2003) and Ing (2003) among others. However, Marcellino et al. (2006) evaluates a large-scale empirical comparison of iterated and direct forecasts using US macroeconomic time series data, and finds that iterated forecasts tend to have smaller MSFEs than direct forecasts, contrary to the theoretical preference of direct forecasts. To evaluate the forecasting performances among several different models, the MSFE is the most widely used. Let y0 τ þhis a vector of observations at time τ þhfor τ 5 τ 0 ,...,Th 1, and h52-, 4-, 8and 12-step ahead forecasts. Then, b Φ¼b μ 0;b Θ01;...;b Θ0p  0is estimated for both the direct and iterated method, using information up to τ 1 to forecast values b y τ þh starting from τ 5 τ 0 up to τ 5Th1, and calculate the MSFE defined as: MSFE ¼1 Th τ 0þ1X T−h τ ¼ τ 0 y τ þhb y τ þhjb Φ;Y τ 1 hi 2 :(5) where Y τ −1¼X τ −1;X τ −2;...;X1 ðÞ. 3. Bayesian VARs This section presents Bayesian VAR models with three different priors –independent Normal-Wishart prior, the Minnesota prior and the SSVS prior. The VAR model in eq. (1) can be written in matrix form as follows: Y¼XΦþ ε (6) where the T3nmatrix Yis defined as Y¼ðy1;...;yTÞ0; the T3(1 þnp) matrix Xis defined as X¼ðx1;...;xTÞ0; the (1 þnp)31 vector is defined as xt¼1;y0 t−1;...;y0 t−p  0, the (1 þnp)3nmatrix Φis defined as Φ¼ð μ 0;Θ0 1;...;Θ0 pÞ0; and the «is a T3nmatrix with AJEB 6,2 144 Downloaded from http://www.emerald.com/ajeb/article-pdf/6/2/142/358140/ajeb-04-2022-0044.pdf by ZBW German National Library of Economics user on 16 December 2025 ε ¼ð ε 1;...; ε TÞ0. Based on the VAR model in eq. (1) or eq. (6), we describe briefly the three priors in the following subsections. 3.1 Independent Normal-Wishart prior The VAR model in eq. (6) with the independent Normal-Wishart prior vec ΦðÞ∼MN vec Φ0 ðÞ;V0 ðÞ (7) Σ∼IW Σ0; ν 0 ðÞ (8) where MN refers to a multivariate normal with mean vec(Φ 0 ) and covariance-variance matrix V 0 :IW refers to an inverted Wishart distribution with parameters Σ 0 and degrees of freedom, ν 0 . Unlike the natural conjugate priors, prior for Φin eq. (7) and Σin eq. (8) are independently specified. With the joint prior and the likelihood, the conditional posterior densities of vec(Φ) and Σare derived as follows: vecðΦÞjΣ;Y∼MNðvecðΦ*Þ;V*Þ(9) ΣjΦ;Y∼IW Σ*; ν *  (10) where V*¼V−1 0þΣ⊗X0XðÞ  −1and vec B+ ðÞ¼V*V−1 0vec Φ0 ðÞþΣ⊗Iκ ðÞ −1vec X0YðÞ hi , Σ*¼Y−XΦðÞ 0Y−XΦðÞþΣ0, and ν * 5Tþ ν 0 . Given these conditional posterior specifications above, the Gibbs sampler generates sample draws. Note that, with zero prior mean Φ 0 50 and large prior variance V 0 in eq. (7), the posterior mean for Φis almost identical to the Maximum likelihood estimator. In this paper the hyperparameters are set at vec(Φ 0 )50 and V 0 5100 in eq. (7),Σ 0 50.1I, and ν 55ineq. (8). 3.2 Minnesota prior Litterman (1986) proposes what we call the Minnesota prior which is shrinkage prior for a Bayesian VAR model with random walk components. For a VAR model with p-the lag in eq. (1), the Minnesota prior for the coefficients assumes that the importance of the lagged variables is shrinking with the lag length, so that the prior is tighter around zero with lag length such that Θi∼Nð Θi;VðΘiÞÞwhere the expected values of Θ i is defined as  Θ1¼Inand  Θ2¼¼  Θp¼0n, and the variance of Θ i is given as: VðΘiÞ¼λ2 i2 1θb σ 1 2b σ 2 2 θb σ 1 2b σ n 2 θb σ 2 2b σ 1 21 θb σ 2 2b σ n 2 . . .. . . 1. . . θb σ n 2b σ 1 2θb σ n 2b σ 2 2 1 2 6 6 6 6 4 3 7 7 7 7 5 ;(11) where 0 < θ< 1, and Σ¼diag b σ 2 1;...;b σ 2 n  . In this paper, the hyperparameters in eq. (11) are set at λ50.05 and θ50.1. 3.3 SSVS prior Without any restriction on the regression coefficients and the covariance matrix in eq. (1), VAR models usually has over-parameterization problem. They contain a very large number of parameters, leading to imprecise inference and deterioration of the forecast performance. To overcome this problem, George et al. (2008) apply the Bayesian SSVS method in a VAR. The SSVS method, proposed by George et al. (2008) and George and McCulloch (1997), restricts the parameters of the model by using a hierarchical prior on the parameters. Forecasting with Bayesian VAR models 145 Downloaded from http://www.emerald.com/ajeb/article-pdf/6/2/142/358140/ajeb-04-2022-0044.pdf by ZBW German National Library of Economics user on 16 December 2025 SSVS defines the prior for the VAR coefficient Φfor each element in Φ.Let f j be each element in Φ, then the prior for f j is a hierarchical prior with mixture of two normal distributions with different variance conditional on an unknown dummy variable γ j that takes zero or one: f jjγj∼1γjN0; τ 2 0;jþγjN0; τ 2 1;j(12) where τ 2 0;jis small and τ 2 0;j< τ 2 1;j. This implies that if γ j 50, that is, the element f j is restricted to be close to 0 as f jjγj∼Nð0; τ 2 0;jÞ, the prior for f j jγ j is virtually zero with small variance, on the other hand, if γ j 51, that is, the element f j is unrestricted as f jjγj∼Nð0; τ 2 1;jÞ, the prior is relatively noninformative with large variance. The priors on γ j are assumed to be independent Bernoulli p i ∈(0, 1) random variables as follows: Pγj¼1¼pj Pγj¼0¼1pj(13) where p j is the prior parameter and p j 50.5 for a natural default choice. George and McCulloch (1997) and George et al. (2008) use a default semiautomatic approach that sets τ kj ¼ckb σ f jfor k50, 1, where b σ f jis the OLS estimates of the standard error of f j in an unrestricted VAR and pre-selected constants c 0 and c 1 must be c 0 <c 1 e.g. c 0 50.1 and c 1 510 as used by George et al. (2008),Jochmann et al. (2010) and Jochmann et al. (2013).In this paper, we follow these values for the hyperparameters. 4. Monte Carlo simulations This section presents Monte Carlo simulations to illustrate forecasting performances for both iterated and direct forecast methods using VAR models. Two DGPs are considered: one follows non-stationaryprocess andthe other follows stationary process. For each DGPs, 100 samples of size T5150 were simulated, and then for each sample, three types of priors are compared: (1) inverted Normal-Wishart (INW) prior, (2) Minnesota (Minn) prior and (3) SSVS prior. The following two DGPs for VARs are considered for this experiment. Both DGPs contain intercept term. DGP 1 is a four-variable VAR with four lags, containing unit roots with parameters DGP1 :ΦðDGP1Þ¼ 0:20:20:20:2 0:80 0 0 00:40 0 000:40 0000 0:20 0 0 0000 000:30 0000:4 0000 00:30 0 0000 0000:3 0000 00:30 0 000:30 0000:3 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ;and ΨðDGP1Þ¼ 10:50:50:5 01 0 0 00 1 0 00 0 1 2 6 6 4 3 7 7 5 where Ψis upper-triangular of the Choleski decomposition of Σ 1 5ΨΨ 1 . AJEB 6,2 146 Downloaded from http://www.emerald.com/ajeb/article-pdf/6/2/142/358140/ajeb-04-2022-0044.pdf by ZBW German National Library of Economics user on 16 December 2025 Next, DGP 2 is also a four-variable VAR with four lags, but stationary data with parameters DGP 2 :ΦðDGP2Þ¼ 0:50:50:50:5 0:60 0 0 0:30:60 0 00:30:60 000:30:6 0000 0:20 0 0 00:20 0 000:20 0:30 00:2 00:30 0 000:30 0000:3 0:30 0 0 00:30 0 000:30 0000:3 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ;and ΨðDGP2Þ¼ΨðDGP1Þ: Each DGP is repeated 100 times to obtain 100 samples. As for determination of the lag length p, three different methods are used as (1) p54 (fixed), (2) p58 (fixed), and (3) pchosen by the Akaike information criterion (AIC) with 0 ≤p≤12. The first method has the fixed lag length as p54 is the true lag length. We do not use the Bayesian information criterion (BIC) for the lag length determination since the BIC is generally choosing short lag length, and the use of SSVS means that short lag model is not required to consider. For the selection of lag by the AIC, the AIC is computed at each date τ , where τ 0 ≤ τ ≤Th, based on the one-step ahead regression for the iterated forecasts, and on the h-step ahead regression eq. (4) for the direct forecasts. For each τ in eq. (5), MCMC is run with 20,000 draws after 5,000 burn-in from τ 5 τ 0 up to τ 5Th1 to compute the MSFEs in eq. (5) for each estimator by a recursive forecasting exercise of both an iterated and a direct multi-period forecasting method. The Monte Carlo simulations for the multi-step forecasting are examined. Table 1 summarizes the MSFEs of both iterated and direct forecasts methods with forecast horizon 2-, 4-, 8and 12-steps ahead. The MSFE in the table are the sum of the MSFE for each variable. For all series, pseudo-out-of-sample forecastsb y τ þhare computed for τ 580 to τ 5150 h1, then we calculate the MSFE defined as eq. (5). Each figure in Table 1 is the average over 100 sample MSFEs. Inspection of Table 1 suggests the following: (1) Among the three estimators by the INW, the Minn and the SSVS, the SSVS produces the lowest MSFE in most cases, though in a very few cases of direct forecasts the Minn shows barely better performances than the SSVS. (2) The forecast performances by SSVS prior tends to be insensitive to the choice of the lag length, while the INW estimator considerably deteriorates the performances as the lag length is longer. That is, even if the lag length is more than 4 (that is the true lag length), the SSVS treats the coefficients on longer lags to be zero, while the forecast performances of other two models are largely depend upon the selection of the lag length. The Minnesota prior effectively provides shrinkage in parameters of the longer lags. Forecasting with Bayesian VAR models 147 Downloaded from http://www.emerald.com/ajeb/article-pdf/6/2/142/358140/ajeb-04-2022-0044.pdf by ZBW German National Library of Economics user on 16 December 2025 (3) For the INW and the SSVS, the iterated method of forecasts is better than the direct method, though for the Minn the results by iterated method are better for DGP2 than those by the direct method, but in some cases worse for DGP1. (4) For these DGPs, the SSVS model with iterated forecast performs best for any forecast horizon. Table 2 illustrates the distributions of the relative MSFE, that is the ratios of the MSFE of the direct forecast to the MSFE of the iterated forecast for different forecast horizons, MSFE directðÞ MSFE iterated ðÞ . The table shows the mean, standard deviations, 95% highest posterior density intervals (HPDI) of the relative MSFE, and pr.(<1), which is probability that the ratio is less than 1 (the direct forecasts performs better than the iterated forecasts). The following results are found: (1) For the INW and the SSVS, the mean values of the relative MSFEs are always greater than 1 (means that the iterated forecasts outperform the direct forecasts), while for the Minn the ratios are either greater or less than 1. (2) For the INW and the SSVS, the mean values of the relative MSFEs are getting large as the forecast horizons are longer, meaning that the relative performance of the iterated forecasts improves with the forecast horizon. (3) The MSFE ratios by the INW are quite sensitive to the choice of the lag length. As the lag length is longer, the relative MSFEs by the INW are getting larger. However, the relative MSFE by the SSVS is not affected by the choice of the lag length due to the insensitivities of the SSVS to the lag length. (4) For all three estimators, the standard deviations of the relative MSFE are larger as the forecasts horizon is longer. DGP 1 DGP 2 Forecast horizon Forecast horizon Model Method 2 4 8 12 2 4 8 12 Lag 54 INW Direct 7.352 11.62 25.33 40.50 8.536 13.48 19.40 24.76 Iterated 7.047 10.29 19.46 29.39 8.273 12.34 15.80 19.39 Minn Direct 7.103 10.50 21.77 34.75 8.858 13.58 18.40 23.15 Iterated 7.114 10.52 21.31 35.10 8.349 12.27 16.16 20.37 SSVS Direct 6.517 10.14 21.92 34.28 8.091 12.47 17.74 22.72 Iterated 6.381 9.405 17.43 26.03 7.729 11.72 15.38 18.87 Lag 58 INW Direct 9.579 15.49 34.15 55.85 11.11 17.67 25.50 32.35 Iterated 9.035 13.18 24.59 36.95 10.54 15.61 19.02 22.29 Minn Direct 7.768 11.49 24.26 39.23 9.825 14.83 20.17 24.58 Iterated 8.013 12.43 27.12 40.94 9.452 13.77 18.43 23.84 SSVS Direct 6.788 10.80 23.87 37.53 8.407 13.05 18.50 23.81 Iterated 6.621 9.838 18.23 27.15 7.962 12.04 15.75 19.26 Lag by AIC INW Direct 9.766 16.74 44.74 79.95 11.86 18.98 27.88 40.70 Iterated 9.183 13.31 25.55 42.84 10.43 15.31 19.46 24.07 Minn Direct 7.340 11.25 25.19 41.81 9.280 14.00 18.48 23.99 Iterated 7.550 11.50 24.15 45.29 8.841 13.01 17.25 21.99 SSVS Direct 6.681 10.78 24.16 37.88 8.437 13.15 18.65 24.22 Iterated 6.595 9.750 18.23 27.36 7.937 11.97 15.71 19.28 Table 1. Monte Carlo simulation: average MSFEs AJEB 6,2 148 Downloaded from http://www.emerald.com/ajeb/article-pdf/6/2/142/358140/ajeb-04-2022-0044.pdf by ZBW German National Library of Economics user on 16 December 2025 DGP 1 DGP 2 Forecast horizon Forecast horizon Model 2 4 8 12 2 4 8 12 Lag 54 INW Mean 1.043 1.125 1.277 1.335 1.031 1.090 1.221 1.269 St dev 0.027 0.079 0.206 0.320 0.029 0.063 0.117 0.172 HPDI (L) 0.991 0.938 0.912 0.821 0.974 0.982 1.031 0.915 HPDI (H) 1.100 1.272 1.790 2.162 1.090 1.242 1.486 1.676 Pr.(<1) 0.06 0.05 0.07 0.12 0.07 0.09 0.02 0.05 Minn Mean 0.999 0.997 1.016 0.999 1.062 1.107 1.137 1.132 St dev 0.031 0.071 0.193 0.294 0.035 0.063 0.097 0.154 HPDI (L) 0.944 0.850 0.660 0.507 0.993 0.987 0.980 0.895 HPDI (H) 1.052 1.123 1.462 1.607 1.141 1.249 1.363 1.410 Pr.(<1) 0.50 0.49 0.49 0.51 0.03 0.04 0.04 0.22 SSVS mean 1.021 1.074 1.241 1.293 1.047 1.061 1.136 1.173 St dev 0.026 0.073 0.208 0.289 0.041 0.063 0.122 0.180 HPDI (L) 0.964 0.951 0.862 0.803 0.971 0.938 0.961 0.892 HPDI (H) 1.073 1.248 1.722 2.066 1.135 1.203 1.469 1.583 Pr.(<1) 0.23 0.12 0.08 0.13 0.11 0.17 0.06 0.12 Lag 58 INW Mean 1.060 1.166 1.357 1.446 1.054 1.132 1.345 1.467 St dev 0.040 0.096 0.253 0.421 0.034 0.072 0.180 0.312 HPDI (L) 0.982 0.988 0.942 0.803 0.986 1.021 1.022 1.097 HPDI (H) 1.146 1.396 1.976 2.871 1.119 1.304 1.772 2.059 Pr.(<1) 0.07 0.04 0.08 0.10 0.06 0.01 0.01 0.01 Minn Mean 0.970 1.166 0.905 0.834 1.040 1.079 1.092 1.030 St dev 0.034 0.096 0.224 0.328 0.035 0.061 0.108 0.176 HPDI (L) 0.899 0.750 0.537 0.307 0.966 0.975 0.915 0.705 HPDI (H) 1.044 1.126 1.324 1.570 1.105 1.206 1.323 1.381 Pr.(<1) 0.81 0.84 0.69 0.75 0.15 0.10 0.20 0.41 SSVS Mean 1.024 1.091 1.289 1.349 1.055 1.081 1.159 1.206 St dev 0.027 0.073 0.211 0.299 0.043 0.062 0.122 0.207 HPDI (L) 0.974 0.965 0.942 0.821 0.981 0.986 0.961 0.876 HPDI (H) 1.080 1.242 1.722 2.054 1.137 1.235 1.461 1.774 Pr.(<1) 0.14 0.09 0.05 0.09 0.05 0.05 0.05 0.12 Lag by AIC INW Mean 1.068 1.275 1.729 1.875 1.140 1.253 1.456 1.687 St dev 0.163 0.313 0.596 0.861 0.155 0.290 0.519 0.695 HPDI (L) 0.760 0.791 0.905 0.845 0.871 0.794 0.690 0.787 HPDI (H) 1.466 1.974 3.136 4.556 1.459 1.957 2.626 3.614 Pr.(<1) 0.35 0.20 0.04 0.09 0.22 0.20 0.22 0.12 Minn Mean 0.975 0.990 1.073 1.066 1.052 1.080 1.076 1.091 St dev 0.055 0.130 0.305 0.454 0.061 0.104 0.128 0.190 HPDI (L) 0.871 0.753 0.607 0.373 0.928 0.926 0.839 0.750 HPDI (H) 1.075 1.235 1.718 2.218 1.211 1.373 1.363 1.500 Pr.(<1) 0.72 0.54 0.47 0.50 0.16 0.17 0.29 0.34 SSVS Mean 1.014 1.106 1.313 1.367 1.062 1.097 1.171 1.224 St dev 0.041 0.110 0.237 0.353 0.051 0.077 0.139 0.237 HPDI (L) 0.946 0.927 0.979 0.795 0.965 0.973 0.982 0.914 HPDI (H) 1.087 1.348 1.877 2.251 1.187 1.273 1.460 1.883 Pr.(<1) 0.37 0.15 0.07 0.12 0.10 0.12 0.04 0.12 Table 2. MSFE ratio Forecasting with Bayesian VAR models 149 Downloaded from http://www.emerald.com/ajeb/article-pdf/6/2/142/358140/ajeb-04-2022-0044.pdf by ZBW German National Library of Economics user on 16 December 2025