Regression to obtain autocorrelation measure (AR(1))

Question

This is not homework. I am a frequent user on math.stackexchange, but I am learning a bit about time series models and came across this example. Any ideas would be greatly appreciated.

A linear regression model was fit to some time-series data by ordinary least squares. The residuals from the fit were then used to create two new variables, namely $E$ with values $\hat{e}_2,...,\hat{e}_n$ and $E_1$ with values $\hat{e}_1,...,\hat{e}_{n-1}$. A linear “regression through the origin” was then run with E as the dependent variable and $E_1$ as the predictor. The slope estimate was 0.412 with a standard error of 0.133. Assume that the $e_t$ follow a standard AR(1) model.

1. Estimate the first order autocorrelation $\rho$ of the AR model.

2. Can the output be used to obtain a valid standard error for the estimate in 1?

Is the answer to #1 above just the slope of the regression E~E_1?

Answer 1

Yes to all three questions. But it is more efficient to estimate the AR error model at the same time as the linear regression model using GLS or MLE.

Answer 2

The $\rm AR(1)$ model is given as $X_n= \rho X_{n-1} + e_n.$ The parameter $\rho$ is normally estimated by conditional least squares. If the model is correct the eis have mean $0$ and variance $\sigma_e^2.$

The parameter $\rho$ is $$\rho=\frac{{\rm Cov}(X_n, X_{n-1})}{{\rm Var}(X_n)}.$$

When the estimate of $e_n$ is paired with the estimate of $e_{n-1}$ the slope is theoretically $0$ but will not be exactly zero because the estimates of residuals are based on the estimate of $\rho$.

The answer to 2 is yes.