Model Selection and Estimation in Structural VARs with External Instruments

Description

A vector autoregression (VAR) is a simple and common way to model the dynamics ofstationary multivariate time series. One can construct VAR models purely based onstatistical properties of economic variables without imposing any economic theory, soeconomists refer to this type of models as the reduced-form VARs. To learn the impactsof a policy change or economic shock on a system of economic variables over time, onehas to impose restrictions on the VAR models so that they have economic interpretation.The VARs with economic restrictions are referred to as the structural VARs, and theexogenous policy changes or economic innovations are known as the structural shocks.Usually, structural shocks are not directly observable but are linear transformations ofthe reduced-from VAR's error terms, which are referred as the reduced-form shocks.Hence, economic restrictions are imposed on VAR models so that the structural shockscan be recovered from the reduced-form shocks.This project focuses on the specification issues of the identification scheme usingexternal instruments developed by Stock and Watson (2012) and Mertens and Ravn(2013). This approach uses variables outside the VAR system as the instruments for theunderlying structural shocks. These external instruments have to be correlated with thestructural shock of interest but uncorrelated with all other structural shocks. Theimpacts of the structural shock, known as the impulse response functions, can be thenestimated by generalized method of moments (GMM). However, the external instrumentapproach relies on a crucial normalization assumption, the violation of which can lead toa failure of identification and inconsistent estimators. Also, since the lags of the externalinstruments are potentially valid instruments, there could be many candidateinstruments. If we use many irrelevant instruments in the second stage GMMestimation, then the estimator will have larger finite-sample bias. Moreover, the lagorder of the VAR model is also an important parameter. Underestimating the lag orderwill lead to inconsistent estimates for the impulse response functions, whileoverestimating the lag order will reduce the estimation efficiency. We propose a unifiedapproach that can test the normalization assumption, select the relevant instrumentsand determine the lag order simultaneously using a shrinkage estimator. This projectwill establish the theory for the proposed estimator, investigate the finite-sampleproperties by simulations, and conduct empirical applications to illustrate the usefulnessof the proposed estimator.