A One-Factor Multivariate Time Series Model of Metropolitan Wage Rates

Summary

Engle and Watson formulate a single-factor state-space model in which an unobserved metropolitan wage component evolves as a second-order autoregressive (AR(2)) process and is extracted from five Los Angeles sectoral wage series via the Kalman filter. Maximum likelihood is obtained by a scoring algorithm that requires only first derivatives of the log-likelihood and K additional Kalman filter passes to compute the information matrix. Model B — AR(2) metro factor with AR(1) sector-specific errors — passes serial-correlation Lagrange multiplier (LM) diagnostics and outperforms ordinary least squares (OLS) regression in out-of-sample forecasting for 1976 and 1977.

Key Claims

General state-space form: $x_t = \phi x_{t-1} + \gamma z_t + v_t$ (hidden factor); $y_t = \alpha x_t + \beta z_t + e_t$ (observed sectoral wages); innovations $(v_t, e_t) \sim \mathrm{NID}(Q, R)$ (normally and independently distributed).
Economic wage model (Eq. 4.1): $w_{it} = \alpha_i m_t + \beta_i n_{it} + e_{it}$ ; $m_t$ = unobserved LA metro wage component; $\alpha_i$ are factor loadings normalized to $\alpha_\text{CC} = 1.0$ ; $n_{it}$ = observed industry-national trend.
Model A (AR(1) metro, i.i.d. sector errors) fails serial-correlation diagnostics; Model B (AR(2) metro: $\hat\phi_1 = 1.606$ , $\hat\phi_2 = -0.619$ ; AR(1) sector errors) accepted; $\mathrm{LM} = 9.1 < \chi^2(6)$ critical value 12.6.
Near unit root in metro component: $\hat\phi_1 + \hat\phi_2 \approx 0.987$ — persistence consistent with nearly integrated wage level.
Factor loadings (Table 3, Model B): Construction/Contract (CC) = 1.000 (normalization), Retail Trade (RT) = 0.663, Durable Manufacturing (DM) = 0.549, Nondurable Manufacturing (NDM) = 0.380, Wholesale Trade (WT) = 0.302; traded-goods sectors load less on the common metro factor.
Scoring algorithm (Eq. 3.3): $\theta^{k+1} = \theta^k + \lambda_k J_k^{-1}(\partial L/\partial\theta)|_{\theta^k}$ ; the information matrix $J$ (Eq. 3.12) is computed via $K$ extra Kalman filter passes using only first derivatives — second derivatives are never needed.
LM / score diagnostics for serial correlation are computable under the null without estimating the alternative — a practical advantage.
Out-of-sample: factor model beats OLS regression in sum of squared errors (SSE) for both hold-out years (1976 and 1977).

Concepts Introduced or Extended

State-Space Representation — dynamic factor model as a state-space special case; scoring algorithm for Kalman-filter ML; LM diagnostics computable under null
Random Coefficient Model — related time-varying-parameter (TVP) precursor; same Kalman-filter likelihood infrastructure

Entities Mentioned

Quotes

"The information matrix is computed from K additional Kalman filter recursions, each using only first derivatives of the log likelihood."

"The factor loadings reveal that the construction sector, being a non-traded local industry, loads most heavily on the metropolitan component, while wholesale and manufacturing sectors are more integrated with national labor markets."

My Take

A landmark application of state-space ML to economics before dynamic stochastic general equilibrium (DSGE) models made such methods standard. The scoring algorithm — computing the Fisher information from K Kalman passes rather than Hessian inversion — is elegant and is still a standard technique in state-space estimation. The near-unit-root finding (sum of AR coefficients $\approx 0.987$ ) anticipates the large persistence literature. The economic interpretation of the factor loadings (non-traded local industries load more on the common metro factor) is intuitive and confirms the model structure. One limitation: the single-factor constraint may not capture all cross-sector correlation; the diagonal R matrix imposes zero residual covariance across sectors conditional on the factor.