# Bias of autocorrelation function in finite sample for ARMA processes

I remember reading that, when estimating the autocorrelation function of a univariate ARMA time series using finite samples, the estimate is biased and specifically the lag-1 ACF is negatively biased in the case of iid observations. However, I can't recall the source and I am therefore not sure this statement is true. Does anyone know whether this is true or not? Thanks.

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What version of the sample autocorrelation function are you using? – cardinal Sep 12 '11 at 22:35
In the case of iid observations, the true correlation at lag 1 is zero, and by a symmetry argument it's hard to see how an estimator could be biased negatively (w/o having it subtract a positive number from the estimate :-) Certainly the sample lag-1 correlation coefficient is unbiased. – jbowman Dec 12 '11 at 23:45
I would like to see a more rigorous demonstration of that, @jbowman. After all, although OLS regression is unbiased, it introduces a negative expected correlation in its residuals, so it's not far-fetched to think that a regression-based ACF estimator might do something similar. The problem with the present question is that no specific estimator of the lag-1 ACF is proposed, making it unanswerable without further assumptions. – whuber Jan 11 '12 at 22:22
@whuber - I was thinking of the ACF of an iid sequence, but on reading your comment I wonder if you are thinking of the ACF of the residuals of a model. I also wonder if that wasn't the actual question? Also, the usual lag-1 sample autocorrelation is biased, but towards 0, not negatively, because $E[1/s^2] > 1/\sigma^2$ (Jensen's Inequality). For an iid process it's still unbiased as the expectation of the numerator is zero. – jbowman Jan 12 '12 at 0:17