Mark W. Watson | Macro Paper Warehouse

ABCs (and Ds) of Understanding VARs

This 2007 American Economic Review paper by Fernández-Villaverde, Rubio-Ramírez, Sargent, and Watson asks when the structural economic shocks in a DSGE model’s state-space representation can be recovered from the one-step-ahead forecast errors (“innovations”) of a VAR estimated on the model’s observables — the “invertibility” problem. Writing the model as a state equation x_{t+1} = Ax_t + Bw_{t+1} and observable equation y_{t+1} = Cx_t + Dw_{t+1}, with w_t an i.i.d. Gaussian vector of structural economic shocks, they show that in the square case (number of observables k equals number of shocks m, and D is nonsingular) the VAR innovations equal the structural shocks if and only if the eigenvalues of A − BD^{-1}C are strictly less than one in modulus — a “poor man’s invertibility condition” that can be checked directly from the model’s own matrices without deriving a full VARMA representation. When this eigenvalue condition fails, the VAR instead recovers the model’s “innovations representation,” a distinct state-space system built from the Kalman-filtered state estimate x̂_t = E(x_t|y^t) rather than the true state x_t; because the state cannot then be fully inferred from current and past observables (Σ = var(x_t|y^t) > 0), the variance of the VAR’s innovations strictly exceeds that of the true structural shocks (D̂D̂’ > DD’), and the VAR’s estimated impulse responses can differ sharply — even in sign — from the model’s true responses. The paper illustrates this failure analytically in a permanent-income consumption model (Sargent 1987, chap. XII) calibrated with gross interest rate R = 1.2 and income shock scale σ_w = 1: when only the consumption-income surplus y_t − c_t is observed, A − BD^{-1}C = R > 1, so the eigenvalue condition fails and Σ = σ²_w(1 − R^{-2}) > 0. The resulting VAR — an AR(1) for the surplus — has impulse responses that are “markedly different” from the true model’s: consumption responds with the opposite sign to a VAR shock than it does to the true structural shock, and the surplus response has a positive present value in the VAR representation versus a present value of exactly zero in the true model (which imposes budget balance). The authors note that observing additional variables (such as consumption, income, or the value of accumulated assets) can restore invertibility, and conclude that despite this problem VARs remain informative about the shapes of impulse responses that theories should be disciplined to match, even when they cannot recover every structural shock exactly. The analysis is purely theoretical and methodological — it presents no empirical VAR estimation or Monte Carlo evidence — and is restricted to the square case (k = m) with Gaussian shocks and the time-invariant (steady-state) limits of the Kalman filter.

Disentangling the Channels of the 2007-09 Recession

This 2012 Brookings Papers on Economic Activity paper by James Stock and Mark Watson asks whether the unusual severity of the 2007-09 recession and the weakness of the recovery that followed required some new economic mechanism – a “financial crisis factor” – or whether the episode can be understood as a larger, historically typical draw of familiar shocks propagating through the economy in its usual way. They address this with a high-dimensional dynamic factor model (DFM) fit to 200 quarterly U.S. macroeconomic series (1959Q1-2011Q2, 132 of them used to estimate the factors), extracting six common factors – a choice consistent with Bai-Ng (2002) information-criteria tests (which themselves gave mixed guidance, ranging from 3-4 to 12 factors depending on the criterion), visual inspection of a scree plot, and the number of distinct structural shocks examined – with factor loadings and dynamics estimated by principal components over the pre-crisis 1959Q1-2007Q3 subsample (the “old” factors) and then extended into the crisis period using a four-lag factor VAR fit over the full 1959Q1-2011Q2 sample. To identify structural shocks (oil, monetary policy, productivity, uncertainty, liquidity/financial risk, and fiscal), the paper develops what it treats as its most-cited contribution: a general “external-instruments” (proxy SVAR) identification strategy in which a shock is recovered as the population regression of an outside instrument – correlated with that shock and, under an exogeneity condition, uncorrelated with the other structural shocks – onto the reduced-form factor innovations, applied one instrument at a time across 18 candidate instruments spanning the six shock categories. The paper’s three headline findings are: (1) a DFM estimated only on pre-2007Q4 data, when fed the actual post-2007Q4 realizations of the “old” factors, tracks the 2007-09 downturn well, and formal tests find little evidence of a break in the factor loadings (rejected at the 5 percent level for only 15 percent of series against full-sample loadings, 12 percent against 1984Q1-2007Q3 loadings) or of a missing factor (a subsampling test gives p=0.59 at an 8-quarter post-2007Q3 horizon and p=0.90 at 15 quarters) – so the crisis reflects unusually large innovations to the same six factors that drove earlier postwar recessions, not a new factor or new dynamics; (2) those innovations were concentrated in financial and uncertainty-related series – the TED spread, VIX, and housing starts saw roughly 8-standard-deviation innovations in 2008Q4, while oil prices moved 1.7 standard deviations in 2007Q1 and 3.4 in 2008Q2 – and a composite uncertainty-liquidity shock (the first principal component of five estimated uncertainty and financial-risk shocks) is attributed roughly two-thirds of the 2007Q4-2009Q2 decline: 6.2 of a 9.2-percentage-point GDP shortfall and 4.5 of a 7.3-point employment shortfall, with oil and monetary-policy shocks contributing moderately and productivity and fiscal shocks contributing little; and (3) the slow recovery is mostly a story of secular trend decline rather than an unusually weak cyclical response – of the roughly 3-point shortfall in post-trough GDP growth relative to pre-1984 recoveries, about four-fifths (2.4 points) reflects slower trend growth rather than a weak cyclical rebound, and for employment the larger part of a roughly 6-point shortfall (3.3 points of trend versus 2.7 points cyclical) is trend-driven, traced to a decades-long decline in trend employment growth (trend GDP growth itself fell by roughly 1.2 percentage points from 1965 to 2005) attributed mainly to the plateauing of female labor-force participation and an aging-driven decline in male participation. Two scope conditions travel with these results and qualify how they should be used: the external-instrument shock estimates are frequently weakly identified (first-stage F-statistics below 5 in 10 of the 18 instrument cases, below 10 in all but 3) and are often substantially correlated with one another both within and across nominally distinct shock categories – most strikingly a -0.93 correlation between two ostensibly separate fiscal shocks whose underlying instruments correlate only -0.06 – which the authors say precludes treating the individual named shocks as a clean, mutually orthogonal decomposition; and because the DFM is linear and does not impose a zero lower bound, its monetary-policy shock estimates counterfactually permit negative interest rates and do not capture unconventional monetary policy, so the finding that monetary policy was “neutral or contractionary” during the crisis and recovery must be read subject to that caveat rather than as a claim about the effect of the actual, unconventional monetary-policy response.

Vector Autoregressions

This 2001 Journal of Economic Perspectives paper by James Stock and Mark Watson reviews how vector autoregressions (VARs) have performed at the four core tasks of applied macroeconometrics — data description, forecasting, structural inference, and policy analysis — roughly twenty years after Christopher Sims’s original 1980 proposal, illustrated throughout with a simple three-variable system (inflation, unemployment, and the federal funds rate) estimated on quarterly U.S. data, 1960-2000. The authors find VARs perform strongly at data description (Granger-causality tests, impulse responses, and variance decompositions reveal, for example, that inflation and unemployment shocks jointly account for about 75% of the federal funds rate’s forecast-error variance at a three-year horizon) and provide a solid forecasting benchmark (a small VAR modestly outperforms both a univariate autoregression and a random walk at most horizons in a pseudo out-of-sample exercise), but they are considerably more skeptical about structural inference and policy analysis. Structural VAR identification is criticized on three grounds — omitted-variable bias (illustrated by the “price puzzle,” which arises when variables the Fed actually used to forecast inflation, like commodity prices, are left out of the model), parameter instability in monetary policy rules over long samples, and implausible zero-restriction timing conventions that are sometimes dressed up as “structural” theory without real economic content — and the paper shows that structural impulse responses can be “very sensitive” to seemingly minor changes in the assumed policy rule (switching from a backward-looking to a forward-looking Taylor rule roughly doubles the estimated inflation and unemployment responses to a funds-rate shock). The authors conclude that VARs’ “structural implications are only as sound as their identification schemes,” and that combining good economic theory and institutional detail with flexible statistical methods like VARs remains the central ongoing challenge for the field.