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Empirical Estimation

In this section we describe the estimation of equation (8.12) and how our specific model assumptions help in addressing econometric issues that previous work has neglected. Interacted panel VAR models can be estimated with OLS (Towbin and Weber 2013). But if the model is complex, due to the presence of an unobserved common factor, for instance, estimation via OLS may not be feasible. For that reason we use Bayesian methods, and in particular the Gibbs sampler, to estimate our model. Gibbs sampling permits us to break down the estimation of this complex model into several stages, which reduces the difficulty of this task drastically. All of the main estimation stages are discussed below.

We allow all of the coefficients that are associated with lagged dependent variables in equation (8.12), Bw, k, c, to be country-specific. All previous work that estimated interacted panel VAR models on annual data, such as Broda (2004), Raddatz (2007) or Towbin and Weber (2013), estimates Bw, k, c by pooling, therefore assuming identical dynamics for all coefficients in the panel, i.e. Bw , k , c = Bw , k. A violation of that assumption will typically lead to an upward, frequently referred to as dynamic heterogeneity, bias in the VAR coefficients (Pesaran and Smith 1995), resulting in a substantial increase in the persistence of the impulse responses. Indeed, the Monte Carlo simulations presented in Canova (2007) show that this bias can be large even for a relatively small degree of dynamic heterogeneity. This would make it difficult to test our hypothesis of interest. Pesaran and Smith (1995) propose the mean group estimator as a solution to this problem (See Sa et al. 2013) for an application of the mean group estimator in interacted panel VAR models in quarterly data.). This approach is implemented by estimating the VAR model country by country and then averaging the country-specific VAR estimates to obtain the panel estimate. But Rebucci (2003) points out that with annual data, where the time series dimension is small, mean group panel VAR estimates may be subject to serious small sample bias. This is lack of precision is probably the most important reason why all previous work that estimated interacted panel VAR models on annual data chooses to pool the coefficients across countries.

In contrast to these previous studies, we do not impose the assumption of pooling on the data. Instead, we only impose the prior of a common mean, but still allow all of the lagged dependent variable coefficients to be country-specific, as in Jarocinski (2010). In particular, we assume that the following prior for Bw , k , c:

where Bw , k is the pooled mean across countries with the variance Лс determining the tightness of this prior. We follow Jarocinski (2010) and parameterize Лс = XLc. X is treated as a hyperparameter and is estimated from the data. In other words, this approach allows us to directly estimate and control for the degree of dynamic heterogeneity bias in the data, which in turn allows to obtain a bias free pooled estimate of Bw , k. The greater X the larger the degree to which the country-specific coefficients are allowed to differ from the common mean. If X ^ , this approach will lead to country-by-country estimates, while X = 0 implies pooling across all countries. Lc, as explained in Sect. 8.7 is calibrated pre-estimation. The parameterization of Лс in this manner has the econometrically convenient property that it is necessary only to estimate one hyperparameter to determine the degree of heterogeneity in the coefficients. But there is, of course, one drawback: the coefficients in Bw , k , c may have different magnitudes. In specifying a single parameter that determines the degree of heterogeneity, there is therefore the risk that some coefficients are allowed to differ from the common mean by a small fraction of their own size, while others can differ by orders of magnitude. An analogous procedure for the Litterman (1986) prior, Lc is a matrix of scaling factors used

to address this problem. In particular, Lc (k,n) = c-, where c is the coun-

act

try, n the equation and k the number of the variable regardless of lag. Th

is the estimated variance of the residuals of a univariate autoregression of the endogenous variable in equation n, of the same order as the VAR, and is obtained pre-estimation. а2л is the corresponding variance for variable k and obtained in an identical manner. To the extent that unexpected movements in variables will reflect the difference in the size of VAR coefficients, scaling by this ratio of variances allows us to address this issue.

The pooled estimate, Bw, k, is estimated by a weighted average of Bw,к, c with the weights as the inverse of Лс, meaning that coefficients of countries closer to the pooled mean get a greater weight and vice versa. In contrast to previous work, our approach therefore allows inference of the degree of dynamic heterogeneity, X, directly from the data and since we allow for dynamic heterogeneity explicitly, the pooled coefficients estimates Bw , k will not be subject to dynamic heterogeneity bias.

A separate, but equally important, econometric issue is the potential presence of cross-sectional dependence. An important maintained assumption in applied panel data studies is the independence of individual units in the cross-section. As first noted by Stephan (1934), this is unlikely to hold in economic applications. This issue, commonly referred to as cross-sectional dependence, has been the subject of a rapidly growing academic literature in recent years. To our knowledge, all previous panel VAR studies make the assumption of cross-sectional independence implicitly. This is, however, difficult to know for certain, since none of the previous studies discuss this issue. This problem is likely to be particular severe when estimating panel VARs on macroeconomic data, since shocks are likely to spill over across countries. Indeed, in a similar, but not interacted, panel VAR model, Gilhooly et al. (2012) show that falsely assuming cross-sectional independence can lead to drastically different results. To ensure that our estimates are not subject to potential bias from this source, and only reflect shocks of domestic origin, we follow the suggestion of Bai (2009) who first proposed the idea of addressing crosssectional dependence in short panels with unobserved common factors, one for each equation.

This is, of course, not the only way of addressing cross-sectional dependence when the number of cross-sectional units is greater than the number of time-series observations in each country. For this case, Pesaran (2006) proposed the common correlated effects estimator. But the corresponding version of the mean group estimator is likely to again suffer from small sample bias in our application. Similarly, estimators other than the one presented in this chapter, which have been specifically designed to address cross-sectional dependence in short dynamic panels, such as the GMM in Sarafidis and Wansbeek (2010) or the maximum likelihood estimator in Bai (2009), do not allow for dynamic heterogeneity. In addi?tion to cross-sectional dependence, these factors will likely reflect other important exogenous control variables which are common to all of the countries, such as global oil price and financial shocks. We assume that these two factors, contained in the matrix F, are independent with distribution N(0, IM) at each point in time and that the VAR residuals Uc are uncorrelated across countries, as the unobserved factors will absorb this cross-country correlation. Finally, it is assumed that E[ UCF] = 0, the VAR residuals and the factors are orthogonal. As with any factor model, there are issues of indeterminacy that need to be addressed ahead of estimation. First, there is a question of scale. One can multiply the matrix of factor loadings, Dc, by a constant d for all i, which gives Dc = dD. We can also

F

divide the factor by d, which yields F = . The scale of the model FDC is

d

thus observationally equivalent to the scale of the model FDC. In order to address this problem the scale of each factor is set to unity. Even then a choice remains as to the sign of F. To identify the sign of the factors we restrict all of the factor loadings in one particular country to be positive. Finally, to identify multiple factors, additional assumptions may need to be made on the matrix of coefficients Dc.

Thus far, the approach presented here is identical to the one in Gilhooly et al. (2012), who were the first to propose an estimator for Bayesian panel VARs that allow for both dynamic heterogeneity and cross-sectional dependence. But they did not consider interacted panel VARs. The additional complication that arises in this case is that it is necessary to estimate the impact matrix of the VAR to allow the impact coefficients to vary with the economic structure as well. The presence of zeros in Gw , c, to avoid perfect multicollinearity, creates an asymmetry among the equations, which neither the framework presented in their paper nor the one in Jarocinski (2010) can handle. To address this problem, we exploit the fact that our model is estimated with Gibbs sampling. The advantage of this Bayesian technique is that estimation can be broken down into multiple steps: for example, Bw, k, c is estimated conditional on knowing Bw , k and X; X is estimated conditional on knowing Bw, k , c and Bw, k and Bw , k is estimated conditional on knowing Bw , k, c. It is therefore possible to estimate each parameter of the model in a separate step. We therefore add one additional step to the Gibbs sampler presented in

Gilhooly et al. (2012) as follows: we assume that Gw,c = Gw, meaning that we pool the data to estimate these coefficients, since the impact matrix is not subject to dynamic heterogeneity bias. For the second equation only,

the vector of coefficients [p]{h ? &2 ? Цз р?з PtlcA ? Ц5] is then drawn from a standard normal distribution, conditional on knowing all of the remaining parameters of the model. The coefficients in Hw are drawn in an analogous way, with the difference that these are drawn for both equations of course.

We estimate this proposed model with Bayesian methods by repeating the Gibbs sampling chain, described in great detail in Sect. 8.7. 400,000 times, with 300,000 iterations as burn-in and retaining every 100th draw, leaving us with 1000 draws for inference. The model is estimated with two lags. Ex-ante lag length selection criteria, such as the Akaike, Hanan-Quinn, and Schwartz-Bayesian criteria, suggest a lag length of one. However, one of the main assumptions of the VAR model is that residuals behave like white noise. Estimated with one lag, the residuals were autocorrelated of order 1, which is obviously inconsistent with white noise behaviour and suggested that in at least one of the equations 8.2 lags would be necessary. Since the bias from omitting a lag is typically worse than that from including an extra lag, we estimate the model with 2 lags.

 
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