# Structural Panel VARs

## Abstract

**:**

## 1. Introduction

## 2. Representation of Model Dynamics and Estimation

**Structural Shock Representation.**Associated with the $M\times 1$ vector of demeaned panel data, ${z}_{it}$, let ${\xi}_{it}={({\overline{\u03f5}}_{t}^{\prime},{\tilde{\u03f5}}_{it}^{\prime})}^{\prime}$ where ${\overline{\u03f5}}_{t}$ and ${\tilde{\u03f5}}_{it}$ are $M\times 1$ vectors of common and idiosyncratic white noise shocks respectively. Let ${\Lambda}_{i}$ be an $M\times M$ diagonal matrix such that the diagonal elements are the loading coefficients ${\lambda}_{i,m}$, $m=1,...,M$. Then

**Relationships Between Reduced Forms and Structural Forms.**The primary equations that relate the reduced form to the structural form in the absence of any identifying restrictions are as follows:

**Typical Structural Identifying Restrictions on Dynamics.**Let $\Delta {z}_{it}={A}_{i}\left(L\right){\u03f5}_{it}$, ${B}_{i}\left(L\right)\Delta {z}_{it}={\u03f5}_{it}$ and $\Delta {\overline{z}}_{t}=\overline{A}\left(L\right){\overline{\u03f5}}_{t}$, $\overline{B}\left(L\right)\Delta {\overline{z}}_{t}={\overline{\u03f5}}_{t}$ represent the structural forms in terms of the composite and common shocks respectively. Then the typical forms of structural identifying restrictions on the dynamics can be represented as:

**(a.)**- on $A\left(0\right)$ decompositions for:$${\Omega}_{\mu ,i}={A}_{i}\left(0\right){A}_{i}{\left(0\right)}^{\prime}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\Omega}_{\overline{\mu}}=\overline{A}\left(0\right)\overline{A}{\left(0\right)}^{\prime}$$
**(b.)**- Short run timing/information restrictions on $B\left(0\right)$ decompositions for:$${\Omega}_{\mu ,i}={B}_{i}{\left(0\right)}^{-1}{B}_{i}{\left(0\right)}^{-{1}^{\prime}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\Omega}_{\overline{\mu}}=\overline{B}{\left(0\right)}^{-1}\overline{B}{\left(0\right)}^{-{1}^{\prime}}$$
**(c.)**- Long run response restrictions on $A\left(1\right)$ decompositions for:$${\Omega}_{\mu ,i}\left(1\right)={A}_{i}\left(1\right){A}_{i}{\left(1\right)}^{\prime}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\Omega}_{\overline{\mu}}\left(1\right)=\overline{A}\left(1\right)\overline{A}{\left(1\right)}^{\prime}$$

**Summary of Estimation Algorithm for Panel SVARs.**The following is a summary of the estimation algorithm for an unbalanced panel $\Delta {z}_{it}$ with dimensions $i=1,...,N$, $t=1,...,{T}_{i}$, $m=1,...,M$:

**(1.)**- Compute the time effects, $\Delta {\overline{z}}_{t}={N}_{t}^{-1}{\sum}_{i=1}^{{N}_{t}}\Delta {z}_{it}$, and use these along with $\Delta {z}_{it}$ to estimate the reduced form VARs, $\overline{R}\left(L\right)\Delta {\overline{z}}_{t}={\overline{\mu}}_{t}$ and ${R}_{i}\left(L\right)\Delta {z}_{it}={\mu}_{it}$ for each member, i, using an information criteria to fit an appropriate member-specific lag truncation, ${P}_{i}$.
**(2.)**- Use appropriate identifying restrictions as per Equations (5) through (7) combined with the reduced form estimates and the mapping Equations (2) through (4) to obtain the structural shock estimates for ${\u03f5}_{it}$ and ${\overline{\u03f5}}_{t}$.
**(3.)**- Compute the diagonal elements of the loading matrix, ${\Lambda}_{i}$, as the correlations between ${\u03f5}_{it}$ and ${\overline{\u03f5}}_{t}$ for each member, i (use these with Equation (1) to compute ${\tilde{\u03f5}}_{it}$ if the raw idiosyncratic shock series are desired).
**(4.)**- Compute the member-specific impulse responses to unit shocks, ${\overline{A}}_{i}\left(L\right)$, ${\tilde{A}}_{i}\left(L\right)$ as per Equation (9).
**(5.)**- Use the sample distribution of estimated ${A}_{i}\left(L\right)$, ${\overline{A}}_{i}\left(L\right)$ and ${\tilde{A}}_{i}\left(L\right)$ responses to describe properties of the sample, such as the median or the confidence interval quantiles, or to create fitted values for member-specific impulse response estimates as described below.

## 3. Small Sample Performance

**Summary of Simulated Panel Structure.**The following equations summarize the heterogeneous structure of the artificially simulated panel:

**2**, ${\Lambda}_{i}$ is also equivalent to the correlation between the composite and common structural shocks for each member, i, and, as such, must be bounded between 0.0 and 1.0. Accordingly, the choice of ${L}_{max}$ values is also bounded. Toward this end, in our simulations that follow, we set ${L}_{max,1}=0.25$ and ${L}_{max,2}=0.25$, so that ${\Lambda}_{i}(1,1)$ is drawn from $U(0.15,0.65)$ and ${\Lambda}_{i}(2,2)$ is drawn from $U(0.05,0.55)$. Similar to our treatment of the parameter matrix ${C}_{i}$, we recenter the ${\Lambda}_{i}$ values after sampling, so that the values we assign to the Blanchard and Quah responses become the true mean responses of the panel DGP. Correspondingly, the common and idiosyncratic responses have an added layer of heterogeneity beyond the ${C}_{i}$ values, due to the heterogeneity in the ${\Lambda}_{i}$ loadings.

**Figure 8.**Panel fitted individual member response estimates to idiosyncratic shocks for N=30, T=100.

**Figure 11.**Panel fitted individual member response estimates to idiosyncratic shocks for N=50, T=12.

**Figure 12.**Mean squared error (MSE) for panel fitted individual member response estimates to composite shocks for N=30, T=100.

**Figure 13.**MSE for panel fitted individual member response estimates to common shocks for N=30, T=100.

**Figure 14.**MSE for panel fittend individual member response estimates to idiosyncratic shocks for N=30, T=100.

## 4. Concluding Remarks

## Acknowledgements

## Conflicts of Interest

## References

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^{1.}A copy of the computer code designed to implement the techniques developed in this paper is available upon request from the author.^{2.}In principle, the dimensionality for the vector of common shocks may be smaller than that of the idiosyncratic shocks, which can be implemented by setting ${\lambda}_{m,i}=0\phantom{\rule{3.33333pt}{0ex}}\forall i$ for some composite shocks.)^{3.}A referee raised the interesting point that since for an unbalanced panel, ${N}_{t}$ is time varying, depending on one’s interpretation, $\overline{A}\left(L\right)$ might also be taken to be time varying. To avoid complications associated with this, we prefer to consider that over the full time span of the panel, for a given sample of $i=1,..,N$ total members, the DGP for $\overline{A}\left(L\right)$ has constant (non-time varying) moving average coefficients. In this context, the fact that ${N}_{t}$ is time varying for an unbalanced panel simply reflects the idea that we have a different number, ${N}_{t}$, of sample realizations of the total number, N, of members for different time periods, such that ${N}_{t}\le N\forall t$.^{4.}An interesting implication is that if the restrictions were assumed to differ, it would imply that a conventional time series SVAR analysis without the benefit of the panel structure would likely incorrectly identify the various composite shocks.^{5.}We obtained the estimates using the replication code "bqexample.rpf" available for download from www.estima.com [18].

© 2013 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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Pedroni, P.
Structural Panel VARs. *Econometrics* **2013**, *1*, 180-206.
https://doi.org/10.3390/econometrics1020180

**AMA Style**

Pedroni P.
Structural Panel VARs. *Econometrics*. 2013; 1(2):180-206.
https://doi.org/10.3390/econometrics1020180

**Chicago/Turabian Style**

Pedroni, Peter.
2013. "Structural Panel VARs" *Econometrics* 1, no. 2: 180-206.
https://doi.org/10.3390/econometrics1020180