Dynamic Seemingly Unrelated Regression

Definition

Dynamic Seemingly Unrelated Regression (DSUR) is a fully parametric, asymptotically efficient estimator for a system of N cointegrating regressions whose equilibrium errors are cross-sectionally correlated. It extends equation-by-equation dynamic ordinary least squares (DOLS) to the multi-equation setting by applying seemingly unrelated regression (SUR)-style generalized least squares (GLS) across a system augmented with leads and lags of first differences of regressors from all equations simultaneously.

Key Ideas

Triangular representation: $y_{it} = x_{it}'\beta_i + u^y_{it}$ ; $\Delta x_{it} = e_{it}$ — each $x_{it}$ is I(1), $y_{it}$ cointegrates with $x_{it}$ .
Full-system leads/lags: each equation is augmented with $\Delta x_{jt\pm p}$ for ALL $j = 1, \ldots, N$ (not just own-equation regressors). This cross-equation augmentation is what achieves the efficiency bound — DOLS omits this and loses efficiency when equilibrium errors are correlated.
SUR GLS step: after augmenting, residuals from step-1 ordinary least squares (OLS) give $\hat\Omega$ ; then GLS across the stacked system with weight $\hat\Omega^{-1} \otimes I_T$ achieves the semiparametric efficiency bound.
Feasible DSUR: uses estimated $\hat\Omega$ from step-1 residuals; consistent and asymptotically as efficient as oracle DSUR.
Restricted DSUR (RDSUR): imposes $\beta_1 = \cdots = \beta_N = \beta$ via Wald $\chi^2$ test; when homogeneity holds and error variances differ substantially across equations, RDSUR achieves dramatically larger gains than unrestricted DSUR (up to 73% mean squared error (MSE) reduction vs panel DOLS at T=50).
System DOLS (SDOLS): stacks equations and estimates jointly but ignores cross-equation Δx leads/lags; does not achieve the efficiency bound.
Lag selection: Bayesian information criterion (BIC) applied equation-by-equation to DOLS residuals.
Scalability limit: $\Omega$ has $N(N+1)/2$ free parameters; DSUR is practical for $N \leq 8$ – $10$ with $T \gg N$ ; infeasible for large panels.

How It Works

Let the system be stacked as $y_i = X_i \beta_i + Z_{-i} \delta_i + \varepsilon_i$ , where $Z_{-i}$ collects leads/lags of $\Delta x_j$ for all $j = 1, \ldots, N$ . The two-step estimator:

Estimate each equation by OLS on the augmented regression → get residuals $\hat{e}_i$ .
Estimate $\hat\Omega_{ij} = (1/T)\sum_t \hat{e}_{it} \hat{e}_{jt}$ .
Apply GLS across equations using $(\hat\Omega^{-1} \otimes I_T)$ as the weighting matrix.

Proposition 4 (Mark-Ogaki-Sul 2003): $\hat\beta_{\text{DSUR}}$ is asymptotically normal and achieves the semiparametric efficiency bound — no consistent estimator can have smaller asymptotic variance.

The relationship to classical SUR (Zellner 1962) is exact: DSUR IS an SUR estimator applied to augmented regressions. The "D" prefix signals dynamic augmentation with leads/lags of $\Delta x$ to handle I(1) regressors and endogeneity; the SUR part is the GLS step exploiting $\Omega$ . Without the cross-equation $\Delta x$ leads/lags, the estimator reverts to SDOLS, which is less efficient when errors are correlated across equations.

Monte Carlo efficiency summary (relative MSE vs DOLS=1)

Case	Correlation $\rho$	T=50	T=100	T=300
I	low	1.13	1.00	0.97
II	moderate	1.11	0.93	0.88
III	high $\approx 0.95$	0.54	0.45	0.44

RDSUR under homogeneous $\beta$ (Case VI, high $\rho$ + variance heterogeneity): Rel.MSE $\approx 0.27$ at T=50.

Why It Matters

When cointegrating equations share correlated equilibrium errors — as in related bilateral exchange rates or saving-investment ratios of similar economies — ignoring the correlation leaves efficiency on the table. DSUR recaptures it parametrically, without nonparametric long-run covariance estimation (as fully modified SUR (FM-SUR) requires). Park-Ogaki (1991) seemingly unrelated canonical cointegrating regression (SU-CCR) is the asymptotically equivalent nonparametric precursor, but it had poor small-sample performance: the spectral estimator of $\Omega$ adds noise. DSUR avoids this by augmenting leads/lags directly into the regression — the same idea that makes DOLS outperform fully modified OLS (FM-OLS) in single-equation cointegration.

Open Questions

How does DSUR perform when the true lag length is unknown and BIC selects different lags across equations?
What is the size and power of DSUR-based Wald tests under heteroskedastic or fat-tailed errors?
Extensions to time-varying $\beta_i$ ?
Panel DSUR with large N and T both growing?