Estimation, Inference, and Empirical Analysis for Time-Varying VAR Models

Abstract

Vector autoregressive (VAR) models are widely used in practical studies, for example, forecasting, modeling policy transmission mechanism, and measuring connection of economic agents. To better capture the dynamics, this article introduces a new class of time-varying VAR models in which the coefficients and covariance matrix of the error innovations are allowed to change smoothly over time. Accordingly, we establish a set of asymptotic properties including the impulse response analyses subject to structural VAR identification conditions, an information criterion to select the optimal lag, and a Wald-type test to determine the constant coefficients. Simulation studies are conducted to evaluate the theoretical findings. Finally, we demonstrate the empirical relevance and usefulness of the proposed methods through an application on U.S. government spending multipliers.

Keywords:

• Instrumental variable approach
• Parameter stability
• Time-varying impulse response

Supplementary Materials

Supplementary materials contain further mathematical symbols, as well as the proofs of the main results and the necessary lemmas.

Acknowledgments

The authors of this article would like to thank the Co-Editor, the Associate Editor and three referees for their constructive comments. Thanks also go to George Athanasopoulos, Rainer Dahlhaus, David Frazier, Oliver Linton, Gael Martin, Peter CB Phillips, and Wei Biao Wu for their helpful comments on earlier versions of this article.

Notes

1 Estimating the covariance matrix using the local linear kernel method (such as the second estimator of (3)) is a nontrivial problem, and even has its own literature. We refer interested readers to Zhang and Wu (2012) for more details.

2 One may regard that ${\mathit{x}}_t^{*}$ is a very special case of the following data generating process:

${\mathit{x}}_t^{*}={\mathit{a}}^{*}({\tau }_t)+{\sum }_{j=1}^p{\mathit{A}}_j^{*}({\tau }_t){\mathit{x}}_{t-j}^{*}+{\mathit{\omega }}^{*}({\tau }_t){\mathit{ϵ}}_t^{*},$

where ${\mathit{a}}^{*}(\tau )=0,{\mathit{A}}_j^{*}(\tau )=0$ and ${\mathit{\omega }}^{*}(\tau )={\mathit{I}}_d$. Using the proof identical to that of Theorem 3, we can show that ${\tilde{Q}}_{\mathit{C},\widehat{\mathit{H}}}^b$ and ${\widehat{Q}}_{\mathit{C},\widehat{\mathit{H}}}$ yield the same asymptotic distribution.

3 We report the average coverage rates by averaging across parameters and time just for ease of presentation. In our own experiment, the empirical coverage rate of the given individual element for different time period is quite close to each other. In addition, the empirical coverage rates for each element of VAR coefficients and impulse responses are also close to each other. However, the empirical coverage rates for the diagonal elements of the errors’ covariance matrix are much lower than those of the off-diagonal elements of the errors’ covariance matrix when the sample size is relatively small.

4 Following (Ramey and Zubairy 2018), the trend GDP is estimated as a sixth-degree polynomial for the logarithm of GDP.

5 Certainly, one may examine each element of these matrices. However, it will lead to a quite lengthy presentation. In order not to deviate from our main goal, we no longer conduct more testing along this line.

Additional information

Funding

Gao and Peng acknowledge financial support from the Australian Research Council Discovery Grants Program under Grant Numbers: DP200102769 and DP210100476, respectively.