Vector Autoregression

Vector Autoregressions

March 2001 (Revised July 2, 2001)

James H. Stock and Mark W. Watson

James H. Stock is the Roy E. Larsen Professor of Political Economy, John F. Kennedy School of Government, Harvard University, Cambridge, Massachusetts. Mark W. Watson is Professor of Economics and Public Affairs, Department of Economics and Woodrow Wilson School of Public and International Affairs,Princeton, New Jersey. Both authors are Research Associates, National Bureau of Economic Research, Cambridge, Massachusetts.

Macroeconometricians do four things: describe and summarize macroeconomic data, make macroeconomic forecasts, quantify what we do or do not know about the true structure of the macroeconomy, and advise (and sometimes become) macroeconomic policymakers. In the 1970s,these four tasks – data description, forecasting, structural inference, and policy analysis – were performed using a variety of techniques. These ranged from large models with hundreds of equations, to single equation models that focused on interactions of a few variables, to simple univariate time series models involving only a single variable. But after the macroeconomic chaos of the 1970s, noneof these approaches appeared especially trustworthy. Two decades ago, Christopher Sims (1980) provided a new macroeconometric framework that held great promise: vector autoregressions (VARs). A univariate autoregression is a single-equation, single-variable linear model in which the current value of a variable is explained by its own lagged values. A VAR is a n-equation, nvariable linear model inwhich each variable is in turn explained by its own lagged values, plus current and past values of the remaining n-1 variables. This simple framework provides a systematic way to capture rich dynamics in multiple time series, and the statistical toolkit that came with VARs was easy to use and interpret. As Sims (1980) and others argued in a series of influential early papers, VARs held out thepromise of providing a coherent and credible approach to data description, forecasting, structural inference, and policy analysis. In this article, we assess how well VARs have addressed these four macroeconometric tasks. Our answer is “it depends.” In data description and forecasting, VARs have proven to be powerful and reliable tools that are now, rightly, in

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everyday use. Structuralinference and policy analysis are, however, inherently more difficult because they require differentiating between correlation and causation; this is the “identification problem” in the jargon of econometrics. This problem cannot be solved by a purely statistical tool, even a powerful one like a VAR. Rather, economic theory or institutional knowledge is required to solve the identification (causationversus correlation) problem.

A Peek Inside the VAR Toolkit1
What, precisely, is the effect of a 100 basis point hike in the Fed Funds rate on the rate of inflation one year hence? How big an interest rate cut is needed to offset an expected half percentage point rise in the unemployment rate? How well does the Phillips curve predict inflation? What fraction of the variation in inflation in thepast forty years is due to monetary policy as opposed to external shocks? Many macroeconomists like to think they know the answer to these and similar questions, perhaps with a modest range of uncertainty. In the next two sections, we take a quantitative look at these and related questions using several three-variable VARs estimated using quarterly U.S. data on the rate of price inflation (πt),the unemployment rate (ut,), and the interest rate (Rt, specifically, the federal funds rate) from from 1960:I – 2000:IV.2 First we construct and examine these models as a way to display the VAR toolkit; criticisms are reserved for the next section.

Three Varieties of VARs VARs come in three varieties: reduced form, recursive, and structural.

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A reduced form VAR expresses each variable...