# Solved – Why do we compare sample ACF and theoretical ACF in time series analysis

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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As to your objection to the definition for the ACF, I'm afraid I don't follow your argument. This is merely the definition of a particular function used to measure auto-correlation of stationary time-series. (Incidentally, the difference between $$r_1$$ and $$r_5$$ would be a lag of four, not five.)