Analyzing Regression with Lagged Variables

When a regression equation includes lagged values of the dependent variable (e.g., $Y_{t-1}$) on the right-hand side, it models the current value ($Y_t$) based on its past values. This setup is common in time series analysis.

The key issue is the potential correlation between the lagged dependent variable ($Y_{t-1}$) and the regression's error term ($\epsilon_t$).

• If $Y_{t-1}$ and $\epsilon_t$ are uncorrelated (often assuming errors are white noise), Ordinary Least Squares (OLS) estimates are typically unbiased and consistent.
• However, if $Y_{t-1}$ and $\epsilon_t$ are correlated (commonly due to autocorrelation in the error terms, such as $\epsilon_t = \rho \epsilon_{t-1} + u_t$), the OLS estimates suffer from endogeneity.

Impact on Coefficient Estimates

Endogeneity causes the OLS estimates to be biased (not equal to the true value in finite samples) and inconsistent (do not converge to the true value as the sample size grows).

The question asks what including lagged variables *could* lead to. Let's examine the options:

• Biased but consistent coefficients' estimates: This implies estimates have inaccuracies in smaller samples (bias) but converge towards the true values as the sample size increases (consistency). This reflects estimators that are asymptotically correct but may have finite-sample bias. Including $Y_{t-1}$ can introduce such finite-sample bias, while consistency depends on other model properties (like the absence of autocorrelation). This option represents a possible outcome.
• Biased and inconsistent coefficients' estimates: This is the typical result when $Y_{t-1}$ is included *and* the error terms are autocorrelated.
• Unbiased and inconsistent coefficients' estimates: This combination is less standard; inconsistency usually implies bias, at least asymptotically.
• Unbiased and consistent coefficients' estimates: This occurs when $Y_{t-1}$ is included but is uncorrelated with the error term ($\epsilon_t$).

Considering the possibilities, estimates being 'Biased but consistent' is a plausible outcome, highlighting the distinction between finite-sample properties (bias) and asymptotic properties (consistency).