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

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

The difference between OLS and Yule-Walker estimates are trivial---only difference is the sample means that enter into the sample moment calculations. With OLS, the initial lagged part of sample is omitted in the sample mean calculation. This difference vanish as sample size become large.

They are both method of moment estimators implementing the same population moment conditions. (Distributional assumptions like Gaussianity are not relevant.)

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.

For AR(p), the same comments apply verbatim, to the sample variance-covariance matrix and the vector of autocovariances.

  • $\begingroup$ So why is Yule-Walker the go-to method? $\endgroup$ – Dave Aug 14 '20 at 1:38
  • $\begingroup$ "...go-to method" according to whom...? Generate any series with sample size 1000, fit any AR model by OLS and YW, and see how the estimates differ, or not. $\endgroup$ – Michael Aug 14 '20 at 1:41
  • 1
    $\begingroup$ I think the equation of $\beta_1$ is not a moment condition but a solution of thereof. $\endgroup$ – Richard Hardy Aug 14 '20 at 5:51

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