LIKELIHOOD-BASED ANALYSIS IN MIXTURE GLOBAL VARs

Abstract

In this paper we propose a mixture global vector autoregressive model with exogenous variables to study country/region interdependencies that exist between national and international variables in the world economy. This model generalizes the linear one introduced by Pesaran et al. (Journal of Business & Economic Statistics 22(2):129–162, 2004) taking into account the possibility of contagion effects. Then, we give a complete solution of the above mixture global model, discuss its stability properties, and provide a likelihood-based analysis of the country/region specific models. Finally, a simulation experiment is presented to demonstrate the utility of the proposed algorithm and the nature of the problems under consideration.

Appendix

Treating (weakly) exogenous variables. The instrumental variable (IV) estimation is a general technique to handle problems associated with (weak) exogeneity. More precisely, IV is used when an explanatory variable of interest is correlated with the error term, in which case the standard estimation methods give biased results. Even in the simple univariate case, the choice of IV is not easy to make and it depends on the field of application. A simple check is to compute the linear correlation of IV with the suspected variable, and its correlation with the dependent variable. If the linear correlation is weak, then expect results of poor reliability, if the sample is not very large. The IV is correlated with ${\textbf{x}}_{it}^{*}$, but uncorrelated with the model error term ${\varvec{\epsilon }}_{it}^{(m)}$ by assumption or by construction. IV is used to replace the problematic variable ${\textbf{x}}_{it}^{*}$. More formally, the IV ${\textbf{v}}_{it}$ for the variable of concern ${\textbf{x}}_{it}^{*}$ satisfies ${\text {cov}} ({\textbf{v}}_{it}, {\textbf{x}}_{it}^{*}) \ne {\varvec{0}}$, ${\text {cov}} ({\textbf{v}}_{it}, {\varvec{\epsilon }}_{it}^{(m)}) = {\varvec{0}}$, and also it is not correlated with any unobservable factors influencing the dependent variable ${\textbf{x}}_{it}$. In the multivariate setting, the selection of IV seems to be more involved. Selecting IV requires three conditions: (1) IV is associated with the exposure; (2) IV only affects the outcome through the exposure; and (3) the effect of IV on the outcome is uncofounded. If the preceding rules are satisfied, the IV technique completely eliminates the simultaneous equation bias only in large samples. In finite samples, the larger the excess of observations over IV, the more the bias is reduced. General methodologies for selecting IVs in the multivariate setting have been provided by Hu and Schennach [17], Lin et al. [23], and Windmeijer et al. [29].

Proof of Theorem 4.1

The maximum likelihood estimates are obtained forming the Lagrangean

$$\mathcal{J}({\varvec{\theta }}_{i }, {\varvec{\pi }}_{i }) = \mathcal{L} ({\varvec{\theta }}_{i }, {\varvec{\pi }}_{i }) + \lambda (1 - \pi _{ i 1} - \cdots - \pi _{i M})$$

and setting the derivatives with respect to the elements of ${\varvec{\theta }}_{i m}$ and $\pi _{i m}$ equal to zero (i.e., first order conditions FOC). From (8) we get

$$\begin{aligned} \frac{\partial \, \mathcal{L}}{\partial \, {\textbf{a}}_{i }^{(m)}}&= \sum _{t = 1}^T \, \frac{1}{f({\textbf{x}}_{i t} | {\textbf{x}}_{i, t-1}; {\varvec{\theta }}_{i }, {\varvec{\pi }}_{i }) } \, \frac{\partial \, f({\textbf{x}}_{i t} | {\textbf{X}}_{i, t-1}; {\varvec{\theta }}_{i }, {\varvec{\pi }}_{i })}{\partial \, {\textbf{a}}_{i }^{(m)}}\\&= \sum _{t = 1}^T \, [{\varvec{\Sigma }}_{i i}^{(m)} ]^{- 1} \, {\varvec{\epsilon }}_{i t}^{(m)} \, \xi _{i m t | t} = 0 \end{aligned}$$

where ${\varvec{\epsilon }}_{i t}^{(m)}$ comes from (1). Then we obtain

$$\sum _{t = 1}^T \, ({\textbf{x}}_{i t} - {\widehat{\textbf{a}}}_{i}^{(m)} \, - \, \sum _{j = 1}^{p_i} \, {\widehat{\Phi }}_{i j}^{(m)} \, {\textbf{x}}_{i, t - j} \, - \, \sum _{h = 0}^{q_i}\, {\widehat{\varvec{\Lambda }}}_{i h}^{(m)} \, {\textbf{x}}_{i, t - h}^{*} ) \, \xi _{i m t | t} \, = \, 0. \qquad (A1)$$

Furthermore, from (8) we get

$$\frac{\partial \, \mathcal{L}}{\partial \, [{\varvec{\Sigma }}_{i i}^{(m)}]^{- 1}} \, = \, \sum _{t = 1}^T \, \frac{1}{2} \, \left( {\varvec{\Sigma }}_{i i}^{(m)} \, - \, {\varvec{\epsilon }}_{i t}^{(m)} \, [{\varvec{\epsilon }}_{i t}^{(m)}]^{'} \right) \, \xi _{i m t | t} \, = \, 0$$

from which (13) follows. We also have

$$\frac{\partial \, \mathcal{L}}{\partial \, {\varvec{\Phi }}_{i j}^{(m)}} \, = \, \sum _{t = 1}^T \, [{\varvec{\Sigma }}_{i i}^{(m)}]^{- 1} \, {\varvec{\epsilon }}_{i t}^{(m)} \, {\textbf{x}}_{i, t - j}^{'} \, \xi _{i m t | t} \, = \, 0 \qquad \qquad \qquad \qquad \qquad \qquad (A2)$$

for $j = 1, \dots , p_i$, and ${\varvec{\epsilon }}_{i t}^{(m)}$ comes from (1). Finally, we get

$$\frac{\partial \, \mathcal{L}}{\partial \, {\varvec{\Lambda }}_{i h}^{(m)}} \, = \, \sum _{t = 1}^T \, [{\varvec{\Sigma }}_{i i}^{(m)}]^{- 1} \, {\varvec{\epsilon }}_{i t}^{(m)} \, {\textbf{x}}_{i, t - h}^{* '} \, \xi _{i m t | t} \, = \, 0 \qquad \qquad \qquad \qquad \qquad (A3)$$

for $h= 0, \dots , q_i$. From (A1)–(A3) we obtain the linear system

$$\begin{aligned} {\textbf{a}}_{i }^{(m)} \, S_{i m} \, + \, {\varvec{\Phi }}_{i}^{(m)} \, {\textbf{B}}_{i m}^{'} \, + \, {\varvec{\Lambda }}_{i}^{(m)} \, {\textbf{B}}_{i m}^{* '} \,&= \, {\textbf{f}}_{i m} \\ {\textbf{a}}_{i }^{(m)} \, {\textbf{B}}_{i m} \, + \, {\varvec{\Phi }}_{i}^{(m)} \, {\textbf{A}}_{i m} \, + \, {\varvec{\Lambda }}_{i}^{(m)} \, {\textbf{D}}_{i m}^{ '} \,&= \, {\textbf{H}}_{i m} \\ {\textbf{a}}_{i }^{(m)} \, {\textbf{B}}_{i m}^{*} \, + \, {\varvec{\Phi }}_{i}^{(m)} \, {\textbf{D}}_{i m} \, + \, {\varvec{\Lambda }}_{i}^{(m)} \, {\textbf{A}}_{i m}^{ *} \,&= \, {\textbf{K}}_{i m} \end{aligned}$$

which can be written, more compactly, as

$$\begin{pmatrix} {\textbf{a}}_{i}^{(m)}&\,&{\varvec{\Omega }}_{i}^{(m)} \end{pmatrix} \, \begin{pmatrix} {S}_{i m} &{} {\textbf{U}}_{i m} \\ {\textbf{U}}_{i m}^{'} &{} {\textbf{W}}_{i m} \end{pmatrix} \, = \, \begin{pmatrix} {\textbf{f}}_{i m}&{\textbf{J}}_{i m} \end{pmatrix}.$$

This gives (12). Finally, we have

$$\frac{\partial \, \mathcal{J}}{\partial \, \pi _{i m}} \, = \, \frac{\partial \, \mathcal{L}}{\partial \, \pi _{i m}} \, - \, \lambda \, = \, 0$$

where

$$\frac{\partial \, \mathcal{L}}{\partial \, \pi _{i m}} = \pi _{i m}^{- 1} \, \sum _{t = 1}^T \, \xi _{i m t | t} \, = \, \pi _{i m}^{- 1} \, S_{i m}.$$

Thus we obtain

$$\pi _{i m} \, \lambda \, = \, \sum _{t = 1}^T \, \xi _{i m t | t}.$$

Summing up over $m = 1, \dots , M$ produces

$$\lambda = \lambda \, \sum _{m = 1}^M \, \pi _{i m} \, = \, \sum _{t = 1}^T \, \sum _{m = 1}^M \, \xi _{i m t | t} \, = \, \sum _{t = 1}^T \, 1 \, = \, T$$

hence

$${\widehat{\pi }}_{i m} = \lambda ^{- 1} \, S_{i m} \, = \, T^{-1} \, S_{i m}$$

as in the last sentence of Theorem 4.1.