Time Series Regression – Difference Between Serially Correlated Errors & Autocorrelation of Residuals

Question

If I have a time series regression model with serially correlated errors:

$Y_t=\beta_0+\beta_1X_t+\epsilon_t$

$\epsilon_t=\rho\epsilon_{t-1}+v_t$

why is the estimate of the correlation ($\rho$) in the second equation NOT the autocorrelation of the residuals from the first equation?

The autocorrelation of the residuals (from the first equation using OLS regression) is easy to get, BUT the rho is estimated with more difficulty [e.g. nonlinear least squares]. But I'm not following why.

Answer 1

It's the difference between sequential and joint estimation of parameters, fundamentally. If you estimate $\{\beta_0, \beta_1, \rho\}$ jointly you'll get a different set of estimates for all three parameters (almost always) than if you estimate $\{\beta_0, \beta_1 | \rho=0\}$ then estimate $\rho$ based on the residuals, which is what sequential estimation does. By not taking the estimate of $\rho$ into account when estimating $\beta$, you lose efficiency, which propagates into the residuals, which in turn affects the quality of the estimate of $\rho$ based on those residuals.

Note that if $\rho$ actually is 0, or very close to it, and your sample size is small(ish), the sequential approach might actually be better, because it gets rid of the effect of randomness in the estimate of $\rho$ on the estimates of $\beta$. But if you knew you were in that situation, you'd probably be better off still just setting $\hat{\rho} = 0$ and going with OLS to estimate $\beta$. The trouble is, you don't know, unless you have some external information, so sticking with maximum likelihood or some other joint estimation methodology is still your best bet.