Why do you subtract the mean when calculating autocorrelation? Autocorrelation is defined as this:
$$\widehat{\rho}_{k} = \frac{\sum_{t=k+1}^{T}{\left(r_{t}-\overline{r}\right)\left(r_{t-k} - \overline{r}\right)}}{\left(\sum_{t=1}^{T}{r_{t} - \overline{r}}\right)^{2}}$$
Why is it necessary to subtract from the mean?
Is it not sufficient to divide the value at $k+1$ by the value at $k$ instead?
$$\rho = \text{mean}\left(\frac{r_{k + 1}}{r_{k}}\right)$$
 A: Let's start from the basics. Variance tells us about the variability around the mean
$$
\operatorname{Var}(X) = E[(X - E[X])^2]
$$
You can generalize this concept to two variables, the covariance
$$
\operatorname{Cov}(X, Y) = E[(X - E[X]) (Y - E[Y])]
$$
where variance is a special case of it
$$
\operatorname{Cov}(X, X) = E[(X - E[X])^2]
$$
Correlation is just a normalized covariance so that it is bounded between -1 and 1,
$$
\operatorname{Corr}(X, Y) = \frac{\operatorname{Cov}(X, Y)}{\sigma_X \sigma_Y}
$$
Autocorrelation is just a special case of correlation.
Yes, you can calculate the expected value of the ratio of two variables and in some case, it might be a meaningful statistic, but it doesn't anymore measure the "spread" or "co-spread" of the variables.
You may be interested in reading the How would you explain covariance to someone who understands only the mean? thread.
A: Similar to simple covariance and correlation, the mean is subtracted while estimating the autocorrelation (autocovariance or cross-correlation and cross-covariance).
The following is the covariance of $X$ and $Y$ for example, where the means of the random variables are subtracted from the random variable itself:
$$\operatorname{cov}(X,Y)=\mathbb E[(X-\mathbb E[X])(Y-\mathbb E[Y])]$$
Your suggestion corresponds to something like $\mathbb E[X/Y]$, which is fundamentally different than the covariance or the correlation.
In time-series literature, we usually see the generative equation similar to
$$x_t=\rho x_{t-1}+\epsilon_t$$
However, not all time series are generated this way, so ignoring the noise term and having a rough estimate like $\rho\approx x_t/x_{t-1}$ may not work well, even in this type of series, let alone the general family of random processes.
