Vector Autoregressions

Questions & answers

Q1. What are the “four tasks” of applied macroeconometrics the paper uses to organize its assessment of VARs?

Stock and Watson organize their review around VARs’ performance at four tasks: data description (summarizing comovements among macroeconomic variables), forecasting, structural inference (estimating the causal effect of a shock, holding other structural relationships fixed), and policy analysis (evaluating the consequences of a change in a policy rule, not just a one-time shock). They use a deliberately simple illustrative three-variable VAR — inflation, the unemployment rate, and the federal funds rate, quarterly U.S. data 1960:I-2000:IV — to walk through each task concretely rather than surveying results abstractly.

Q2. What are the three “varieties” of VAR the paper distinguishes, and how do they differ?

A reduced-form VAR expresses each variable as a function of lagged values of all variables, with contemporaneously correlated (but serially uncorrelated) error terms, estimated equation-by-equation by OLS; a recursive VAR imposes a specific causal ordering (equivalent to a Cholesky factorization of the reduced-form covariance matrix) so that errors become uncorrelated across equations, with results that can depend on which of the n! possible orderings is chosen; and a structural VAR uses economic theory or institutional knowledge — rather than an arbitrary ordering — to impose identifying restrictions on the contemporaneous relationships among variables, either fully (all causal links specified) or partially (a single equation identified via instruments).

Q3. What does the illustrative VAR reveal about data description — Granger causality, variance decomposition, and impulse responses?

In the illustrative system, unemployment Granger-causes inflation (p=0.02) and the federal funds rate Granger-causes unemployment (p=0.01), while the funds rate does not directly Granger-cause inflation (p=0.27); at a 12-quarter horizon, a recursive variance decomposition attributes about 75% of the funds rate’s own forecast-error variance jointly to inflation and unemployment shocks, while inflation’s forecast-error variance is overwhelmingly (82%) explained by its own shocks. An inflation shock is shown to decay slowly, taking roughly 24 quarters to dissipate, and to be associated with persistent increases in both unemployment and the interest rate — illustrating the kind of multivariate comovement information a univariate model cannot capture.

Q4. How sensitive are the paper’s structural impulse responses to the assumed policy rule, and why does this matter?

Using a backward-looking Taylor rule to identify a structural federal-funds-rate shock, the illustrative system implies inflation falls about 0.3 percentage points over roughly three years and unemployment rises about 0.2 points, concentrated mostly in the third year; switching to a forward-looking Taylor rule instead implies a sharper roughly 0.5-point inflation decline within about a year and a roughly 0.5-point unemployment increase peaking around one year before eventually falling about 0.5 points below baseline by year six. The authors describe the responses as “very sensitive” to this seemingly modest change in the identifying assumption, concluding that structural VAR impulse responses in this application “hinge on detailed institutional knowledge of how the Fed sets interest rates” rather than being a robust, assumption-free description of the data.

Q5. What does the pseudo out-of-sample forecasting exercise show about VARs’ forecasting performance?

Comparing root mean squared forecast errors over a 1985:I-2000:IV pseudo out-of-sample period, the small illustrative VAR generally matches or improves on both a univariate autoregression and a naive random-walk forecast across 2-, 4-, and 8-quarter horizons for inflation, unemployment, and the federal funds rate — for example, the VAR’s 8-quarter-ahead RMSE for the funds rate (1.70) is noticeably lower than the random walk’s (2.18). The authors summarize this as “the VAR either does no worse than or improves upon the univariate autoregression, and both improve upon the random walk forecast,” positioning small VARs as a standard, credible benchmark against which more elaborate forecasting methods are judged.

Q6. What are the three criticisms the paper raises against structural VAR identification?

First, omitted-variable bias: factors left out of the model get absorbed into the estimated “shock,” illustrated by the price puzzle, which the paper attributes to the Fed having used forward-looking information (such as commodity prices) that simple VARs omitted, biasing the estimated response of prices to a policy shock — a problem with the standard fix of including commodity prices in the system. Second, parameter instability: several studies (Bernanke and Blinder 1992; Bernanke and Mihov 1998; Clarida, Galí, and Gertler 2000; Boivin 2000) document that monetary policy rules have not been stable over long samples, undermining constant-coefficient VARs estimated over many decades. Third, implausible timing conventions: zero within-period restrictions (e.g., assuming output cannot respond contemporaneously to a policy shock) may be defensible at a daily frequency but become “less plausible” at a monthly or quarterly frequency, and the authors warn that researchers sometimes construct a convenient theoretical story — a “Wold causal chain” — to justify a particular recursive ordering after the fact, cautioning that “rarely does it add value to repackage a recursive VAR and sell it as structural.”

Q7. What constructive alternatives to timing-restriction-based identification does the paper highlight?

The paper points to three constructive approaches for more credible structural VAR identification: exploiting detailed institutional knowledge (Blanchard and Perotti’s use of statutory tax-code features to identify fiscal shocks; Bernanke and Mihov’s reserves-market model for monetary shocks), imposing long-run restrictions grounded in theory (King, Plosser, Stock, and Watson’s use of long-run monetary neutrality), and agnostic or robust identification using inequality or sign restrictions (Faust 1998; Uhlig 1999) that avoid pinning down the exact contemporaneous timing of effects.

Q8. Why does the paper consider policy analysis the most difficult of the four VAR tasks?

Policy analysis — evaluating the consequences of a change in a policy rule, rather than the effect of a single unanticipated shock — runs into the Lucas critique: if the model’s structural equations involve forward-looking expectations, then all of the VAR’s estimated coefficients will in general depend on the policy rule in place, so a VAR estimated under one rule cannot be mechanically used to forecast outcomes under a different rule. The paper notes that “the practical importance of this critique for VAR-based policy analysis is a matter of debate,” and observes that analyzing a one-time policy innovation while holding the rule fixed is comparatively straightforward (a function of the estimated impulse responses), whereas analyzing an actual change in the rule requires a fully and correctly specified structural VAR with every equation properly identified.