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Will the impulse response function and autocorrelation function of an AR(2) coincide? I say that they will not, because $IRF_{t}(t+k)$ (that is, the effect of a unit shock at time $t$, on the process at $t+k$) will be 1 when $k=0$. However, $corr(y_{t}, y_{t})$ is the variance of the $y_t$ process, and will not necessarily be 1. Thank you.

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  • $\begingroup$ Hi: The autocorrelation at lag zero is, by definition, the scaled variance of $y_t$ which is 1.0. Recall that, in general, the auo-correlations at any lag are just scaled covariances; i.e: covariance divided by the variance and the covariance at lag zero is the variance. $\endgroup$
    – mark leeds
    May 22, 2021 at 13:31
  • $\begingroup$ assume that there is no constant, so that the autocovariance at lag 2 is $E[y_{t+2}y_t]$ and verify your claim $\endgroup$ May 22, 2021 at 13:42

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