Macroeconomic and credit forecasts during the Greek crisis using Bayesian VARs

Abstract

We investigate the ability of small- and medium-scale Bayesian VARs (BVARs) to produce accurate macroeconomic (output and inflation) and credit (loans and lending rate) out-of-sample forecasts during the latest Greek crisis. We implement recently proposed Bayesian shrinkage techniques based on Bayesian hierarchical modeling, and we evaluate the information content of forty-two (42) monthly macroeconomic and financial variables in terms of point and density forecasting. Alternative competing models employed in the study include Bayesian autoregressions (BARs) and time-varying parameter VARs with stochastic volatility, among others. The empirical results reveal that, overall, medium-scale BVARs enriched with economy-wide variables can considerably and consistently improve short-term inflation forecasts. The information content of financial variables, on the other hand, proves to be beneficial for the lending rate density forecasts across forecasting horizons. Both of the above-mentioned results are robust to alternative specification choices, while for the rest of the variables smaller-scale BVARs, or even univariate BARs, produce superior forecasts. Finally, we find that the popular, data-driven, shrinkage methods produce, on average, inferior forecasts compared to the theoretically grounded method considered here.

Appendices

Appendix 1

See Tables 7, 8, and 9

Table 7 Variables used in the baseline BVAR

Table 8 Additional economy-wide variables used in the BVARs

Table 9 Additional financial variables used in the BVARs

Appendix 2: Heuristic methods of selecting hyperparameters

Banbura et al. (2010) (BGR) method. The BGR method selects $\lambda$ on the basis of the in-sample fit of large-scale VARs, which has to be as close as possible to the in-sample fit of a parsimonious small-scale VAR. In particular, we define as ‘Small’ a VAR model with the five variables of Table 7 and 3 lags. The implicit assumption here is that the small-scale VAR does not suffer from overfitting. After defining an in-sample period, the “Fit” for each model m is defined as follows:³⁰

$$\begin{aligned} Fit_{m} (\lambda )=\frac{1}{n}\sum \limits _{i=1}^{n} \frac{{ MSFE}_{m} ( {i,\lambda } )}{{ MSFE}_{Small} ( {i,0} )} \end{aligned}$$

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where ${ MSFE}_{m} ( {i,\lambda } )$ is the one-step ahead mean squared forecast error (MSFE) for model m. The MSFE is computed for each of the n variables of interest, i, and for a given level of $\lambda$. Then, it is normalized by the ${ MSFE}_{Small} ( {i,0} )$, which is the MSFE produced by the small-scale model using the prior (i.e., for $\lambda =0$). Finally, the normalized MSFE measure is averaged across the n target variables. We employ grid search methods in order to choose the value of $\lambda$, which minimizes the following criterion:

$$\begin{aligned} \lambda _{m}^{*} =\arg \mathop {\min }\limits _{\lambda } \left| {Fit_{m} ( \lambda )-Fit_{Small} ( \infty )} \right| \end{aligned}$$

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with $\lambda$ incrementing by 0.0001.

Carriero et al. (2009) (CKM) method. This method is based on a real-time process that can be described as follows. First, we choose a range of values for $\lambda$ over which we estimate the respective VAR models. Next, we produce one-step ahead forecasts at $\tau$ point in time, with $\tau$ belonging to the out-of-sample period, and compute the sum of squared forecasting errors (SFE) over the n variables of interest, i.e., ${ SFE}_{m} (\tau ,\lambda )=\sum _{i=1}^{n} {FE_{m}^{i} (\tau ,\lambda )}$. Finally, in the next period ($\tau +1)$, we chose the $\lambda$ that minimizes the SFE in the previous period ($\tau$), i.e.:

$$\begin{aligned} \lambda _{m}^{*} (\tau +1)=\arg \mathop {\min }\limits _\lambda \left\{ {{ SFE}_m (\tau ,\lambda )} \right\} \end{aligned}$$

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Compared to the BGR, the CKM method is dynamic, since $\lambda$ is optimized for each sample estimated in the forecasting procedure.³¹ It is also based on the out-of-sample accuracy instead of the in-sample fitting of the BGR method. For comparability reasons we use the same grid of values for $\lambda (t)$, as in the BGR method.

Moreover, following common practice and for both methods, we set $\sigma _{0,j} =\left( {d-n-1} \right) ss_j$, for $j=1,\ldots ,n$, where $ss_j$ is the residual variance of an AR(1) model estimated with OLS (e.g., see Kadiyala and Karlsson 1997; Carriero et al. 2015a, b). Finally, for the BVAR models using data in levels, the “sum of coefficients” and “dummy initial observation” hyperparameters, $\mu$ and $\delta$, respectively, are set equal to the default value of 1 as in the study of Carriero et al. (2015b).

Appendix 3: Marginal likelihood

According to Giannone et al. (2015), the marginal likelihood (ML) is defined as:

$$\begin{aligned} p( Y )=\left( {\frac{1}{\pi }} \right) ^{\frac{n( {T-p} )}{n}}\frac{\varGamma _{n} \left( {\frac{T-p+d}{2}} \right) }{\varGamma _{n} \left( {\frac{d}{2}} \right) }\cdot \left| {\varOmega _{0}} \right| ^{-\frac{n}{2}}\cdot \left| {\varSigma _{0}} \right| ^{\frac{d}{2}}\cdot \left| {{X}^{\prime }X+\varOmega _{0}^{-1} } \right| ^{-\frac{n}{2}}\cdot \nonumber \\ \left| {\varSigma _{0} +{\hat{E}}^{\prime }}{\hat{E}}+\left( {\hat{B}}- B_{0} \right) \varOmega _{0}^{-1} \left( {\hat{B}}-B_0 \right) \right| ^{-\frac{T-p+d}{2}} \end{aligned}$$

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The ML is also used to select the optimal lag length as in Carriero et al. (2015a, (2015b):

$$\begin{aligned} \hbox {p}^{*}=\mathop {\arg \max }\limits _{\mathrm{p}} \ln p( Y ) \end{aligned}$$

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where we optimize over the grid $\hbox {p} = 1,{\ldots }, 12$ or 13 depending on whether we use models with variables in differences or levels.