Time Series Regression: serially correlated errors vs autocorrelation of residuals?

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.

• When you refer to the "autocorrelation of the residuals", are the residuals from estimating $Y_t = \beta_0 + \beta_1 X_t + e_t$ or from the joint estimation of equations 1 & 2? – jbowman Feb 27 '12 at 0:28
• @jbowman: just the first equation, using simple OLS regression. – zbicyclist Feb 27 '12 at 1:04
• Your second equation seems to allow for non-constant variance in the errors (except in the case where ${\rm var}(v_{t}) = 1-\rho$), which implies non-constant correlation (although ${\rm cov}(\epsilon_{t}, \epsilon_{t-1}) = \rho$ is constant). So, if the true data generating model is given by these two equations, the autocorrelation of the residuals from an OLS model is not the estimator you're looking for. – Macro Mar 2 '12 at 3:39

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.
• If the sample size is big enough, sequential and non-sequential estimates should be close enough, since non-sequential procedure produces consistent estimates. The sequential procedure has its advantages, since even if the exact structure of error variance is not known, you will get consistent estimates of the parameters $\beta$. If the model for the error turns out not to be AR(1), using maximum likelihood might in turn produce worse estimates than sequential procedure. – mpiktas Mar 2 '12 at 4:16