# autoregressive time series estimation with OLS vs Yule-Walker

$$AR(p)\text{: }X_t = \beta_0 + \sum_{i=1}^p \beta_i X_{t-i} + \epsilon_t$$

It’s autoregression, so some kind of regression. I can write it in a data frame like I would any other regression. We assume (often) Gaussian errors. Why do we estimate the parameters using the Yule-Walker equations? Why not then use OLS?

Example of an AR(2) data frame for $$1,2,3,4,5,6,7$$:

X_t | X_{t-1} | X_{t-2}
========================
3 |    2    |    1
4 |    3    |    2
5 |    4    |    3
6 |    5    |    4
7 |    6    |    5


Consider, for example, the AR(1) case. The population moment condition is $$\beta_1 = \frac{Cov(X_t, X_{t-1})}{Var(X_{t-1}, X_{t-1})}.$$ Both Yule-Walker and OLS substitute the sample moments into the above. For Yule-Walker, in computing the sample variance of "$$X_{t-1}$$", the sample mean from the entire sample is subtracted from $$X_{t-1}$$. For OLS, the sample mean of "$$X_{t-1}$$" (i.e. first observation omitted) is subtracted from $$X_{t-1}$$. Then there is also the difference between $$n$$ and $$n-1$$. That is the only difference.
• I think the equation of $\beta_1$ is not a moment condition but a solution of thereof. – Richard Hardy Aug 14 '20 at 5:51