Discussion of empirical results

In the first step, we estimated the system in Equation (6.8) given two alternative sets of stochastic assumptions resulting in the specifications M_o and M_x. Table 6.3 shows point estimates, standard errors, i-statistics, and p-values for zero restriction tests of parameters describing the relationship in the analysed system of regressions. Estimation is performed based on annual data for NUTS-2 regions covering the period from 2005 to 2018. We report estimation results in the case of each region, that is, for i - l...,16. We also compare the impact of the explanatory variables on the related variable in the system in Equation (6.8), as shown in Figure 6.3.

In both models M_o and Af, the relationship between a specified set of explanatory variables and the regional GDP remains qualitatively the same irrespective of the existence of restrictions imposed on the matrix of contemporaneous covariances S (M_o) or to the unconstrained stochastic structure (Mf. Both settings, leading to different versions of the FGLS estimator, provide empirically the same picture about the strength of relationships stated in Equation (6.8) for each region. Another critical issue is related to the substantial heterogeneity of parameters in the system in Equation (6.8) describing the GDP in regions. For a particular explanatory variable, estimates of the accompanying regression parameters vary across NUTS-2 regions as i varies across the whole system. These results are specific to both M_o and M_x models. Hence, we conclude that the analysed system of regressions in Equation (6.8) is empirically relevant compared to the simple panel regression in Equation (6.1 ) that assumes no variability of parameters with respect to i.

Despite the common features discussed here, differences in the stochastic structure of models M_o and yield much different pictures regarding the level of statistical uncertainty in the problem of estimation of /Is. Within the unconstrained SURE model (MJ standard errors of estimation of parameters in Equation (6.8) are always smaller. A detailed inspection of Table 6.3 yields the conclusion that the standard errors obtained in the case of model are at least two times smaller than their counterparts obtained from the simple stochastic structure in M_o. The FGLS procedure utilising contemporaneous correlations for error terms in Equation (6.8) reduces the statistical uncertainty of parameters in model M_x, compared to the simple OLS method applied for each equation separately in model M_o.

In the next step, we performed spatial and temporal disaggregation of GDP and GDP annual growth rates based on estimated models M_o and Since all regional characteristics are linear functions of parameters contained in Equation (6.8), it is possible to compare results with respect to differences in stochastic structure among M_o and M_x.

Similarities in point estimates of /3's in M_o and M_t are the primary reason that the dynamic pattern of disaggregated regional GDP (Figures 6.4 and 6.6) and growth rates (Figures 6.5 and 6.7) represent the same dynamic pattern when comparing results for a particular region among models M_o and M_x. The system of regressions in Equation (6.8) also generates aggregation of analysed indicators of substantial diversity among regions, with relatively smaller variability of GDP growth rates than in the case of simple regression in Equation (6.1). The main difference between M_o and

is presented in Figures 6.5 and 6.7, where the bands of 95% confidence intervals are depicted for quarterly regional annual GDP growth rates. Again, just like in the case of original parameters, the unconstrained stochastic structure of M_t reduces substantially statistical uncertainty, making the spread of confidence intervals on M_x at least two times smaller than in the case of model M_o.

When building our most complicated model we paid much attention to avoid restrictions on the stochastic structure as well as keeping the system as diverse as possible across regions. Surprisingly, the resulting rich parameterisation, where most free parameters describe the stochastic properties of the error terms, reduces statistical uncertainty. In the case of an appropriately chosen set of explanatory variables, the system of the form in Equation (6.8) may serve as a very flexible tool in the disaggregation problem, generating highly diverse and - from the statistical point of view - precise estimates of the variables of interest.

Finally, we discuss some additional elements of regional analysis that are possible to obtain by applying Equation (6.8) in the unconstrained version (Af/). Figure 6.8 presents maps with estimates of contemporaneous correlations of the error terms in Equation (6.8), that is, the elements of the matrix R given by the formula in Equation (4.6). For comparison, in the case of model M_o analogous maps would look the same for all NUTS-2 regions, indicating a correlation equal to unity for a particular i and zero correlations with others. Formally, elements of matrix R, that is, p_w inform how the error term from the z-th equation in Equation (6.8) is correlated with the error term from the/-th equation. Thus, the maps presented in Figure 6.8 show the empirical estimates of the strength of the impact of shocks existing beyond the deterministic part of regression for the z-th region to the variability of the GDP measure in the case of the j-th region. Consequently for a chosen i and all7=1,..., 16, such jV z the estimated values of p_fj can be interpreted as measures of the influence of economic shocks that occurred in the z-th region to the economic conditions observed in the other regions.

According to the results presented in Figure 6.3, the most influential regions are the Mazowieckie, Wielkopolskie, and Zachodniopomorskie voivodships. In these cases, estimates of correlations depict the strong spatial structure of propagation of economic shocks with positive spatial dependence among regions. Generally, each region has some local impact as correlations are mainly positive locally. However, there is one interesting exception, namely the Slaskie voivodship. In this case, shocks influencing economic conditions in the region have a very limited impact on the rest of voivodships. Moreover, in this case, some of the estimated correlations are negative indicating the positive influence of adverse shocks from Silesia to the rest of the country.