Stock-Watson (1993) A Simple Estimator of Cointegrating Vectors in Higher Order Integrated Systems

Summary

This paper introduces the Dynamic OLS (DOLS) estimator of cointegrating vectors — one of the most widely used tools in applied cointegration analysis because of its simplicity. In the leading I(1), single-cointegrating-vector case, one simply regresses one variable on the contemporaneous levels of the others plus leads and lags of their first differences and a constant, by OLS (or GLS), and the resulting estimator is asymptotically efficient — asymptotically equivalent to the Johansen / Ahn–Reinsel maximum-likelihood estimator — with a mixed-normal limiting distribution that yields Wald statistics with standard chi-squared null distributions. The framework handles general I(d) variables, differing orders of integration, and deterministic components. The estimators are applied to long-run U.S. M1 money demand over 1900–1989, which is found to be stable. (Econometrica 61(4): 783–820.)

Key Claims

• The DOLS estimator. For an I(1) system with a single cointegrating vector, add leads and lags of the first differences of the regressors to the static cointegrating regression: $y_t = \mu + \beta' x_t + \sum_{j=-p}^{p}\delta_j'\,\Delta x_{t-j} + u_t$, estimated by OLS (DOLS) or GLS (DGLS). The leads-and-lags terms soak up the correlation between the equilibrium error and the regressor innovations, removing the second-order (endogeneity and serial-correlation) bias that plagues the static OLS "levels" regression.
• Asymptotic efficiency and standard inference. Motivated as the Gaussian MLE for a particular triangular representation, DOLS is asymptotically efficient in the Saikkonen (1991) / Phillips (1991a) sense: its estimator of $\beta$ has an asymptotic distribution that is a random mixture of normals, so $t$- and Wald statistics (using a long-run-variance-corrected covariance) have the usual normal / $\chi^2$ asymptotics — one can do standard inference on cointegrating coefficients.
• Generality. Unlike much prior work restricted to I(1) variables with no drift, the estimators are developed for cointegrating regressions among general I(d) variables with general deterministic components and variables of differing higher orders of integration.
• Relation to other efficient estimators. DOLS is asymptotically equivalent to Johansen's VECM MLE and to the Phillips–Hansen fully-modified (FM-OLS) and Park estimators; it reaches the same efficiency bound but is computed by a single least-squares regression, needing no kernel/spectral long-run-variance estimation for the point estimate.
• Finite-sample evidence. In Monte Carlo experiments, all the asymptotically efficient estimators do well under simple short-run dynamics, but for designs mimicking U.S. money/income/interest-rate dynamics they differ substantially; DOLS performs well relative to the alternatives.
• Application — long-run U.S. M1 money demand, 1900–1989. Money demand is empirically a stable cointegrating relation among real balances, real income, and an interest rate over the century: 95% confidence intervals of (0.88, 1.06) for the income elasticity and about (−0.13, −0.08) for the interest-rate semi-elasticity. Estimates from the postwar subsample alone are unstable with large sampling uncertainty (the postwar data are dominated by a single trend).

Concepts Introduced or Extended

• Cointegration
• Common Stochastic Trends

Entities Mentioned

• James H. Stock
• Mark W. Watson

Quotes

"In the I(1) case with a single cointegrating vector, one simply regresses one of the variables onto contemporaneous levels of the remaining variables, leads and lags of their first differences, and a constant, using either ordinary or generalized least squares."

"Ml demand is found to be stable over 1900–1989; the 95% confidence intervals for the income elasticity and interest rate semielasticity are (.88, 1.06) and (−.13, −.08), respectively."

My Take

DOLS is the estimator practitioners reach for because it turns "estimate a cointegrating vector and test hypotheses about it" into a single augmented least-squares regression with textbook $t$- and Wald inference — no VECM system, no kernel long-run-variance estimation for the coefficient. Conceptually it is the time-domain twin of the Phillips–Hansen fully-modified approach: both purge the second-order bias of the static cointegrating regression, DOLS by leads-and-lags augmentation rather than a frequency-zero correction. The paper also modeled how to argue a long-run relationship credibly — pairing the estimator with a serious money-demand application and Monte Carlo evidence tailored to that application's dynamics. Its main practical caveats are the choice of lead/lag truncation and the need for a HAC covariance for inference (the point estimate is bias-corrected, the standard errors still require a long-run-variance estimate).