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Theoretical Autocorrelation Function (ACF):

For a weakly stationary time series {$r_t$}, the definition of ACF is (from Ruey Tsay's "Analysis of Financial Time Series")

$ \rho_l=\frac{Cov(r_t,r_{t-l})}{\sqrt{Var(r_t)Var(r_{t-l})}}=\frac{Cov(r_t,r_{t-l})}{Var(r_t)} $

It calculates the correlation of two random variables: $r_t$ and $r_{t-l}$

sample ACF calculates the correlation of a time series and a lag $l$ of it, it is two different random variables from $r_t$ and $r_{t-l}$

So what is the point of comparing these two different quantities?

E.g., we have calculated the theoretical ACF value between $r_1$ and $r_5$ of a time series, it is actually a random process.

We want to check if the theoretical calculation is good, so we instantiate the random process numerous times. For each instantiation, we pick out the value of $r_1$ and $r_5$. Finally, we obtain samples of random variable $r_1$ and $r_5$. Then we use the samples to calculate the sample ACF between $r_1$ and $r_5$. This is the correct way I believe to calculate the sample ACF, and the value to compare with the theoretical ACF.

In a word, in my opinion, the correlation between a time series and a lag 5 of it, is NOT the correct way of calculating the sample ACF between $r_1$ and $r_5$. And it is meaningless to compare this value with the theoretical ACF between $r_1$ and $r_5$.

Where am I wrong?

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The whole idea is to efficiently characterize the original data with as few coefficients as possible. For alternative models we can compute the implied acf or the theoretical acf given the data.

It is logical to numerically examine the alternative/trial acf's with the actual one and then select the one theoretical acf that best mimics the actual acf. This can be done visually or via an iterative search process like the one I programmed in AUTOBOX which allows for latent deterministic structure as in https://autobox.com/pdfs/ARIMA%20FLOW%20CHART.pdf which also considers time varying error variance and time varying parameters.

A one-step approach simply tries a partial list of pre-specified models without deterministic structure and uses a modified error variance as the basis of model selection.

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