I just wrote two papers on a related topic. Let us not use the log method right now as it was originally intended as an approximation from the time we used punch cards. You can, but we will come back to why you may not want to.
If we begin with a simple AR(1) model $$x_{t+1}=\beta{x}_t+\varepsilon_{t+1},$$ then we know that we are buying assets with the subjective intent of making money. Whether we do or not is a different question. Because of this, it is irrational for $\beta\le{1}$, asymptotically. If there were a local event where that existed, it would not be an issue unless more than half of all trades were by people who purposefully wanted to take a loss. By theorem, there does not exist a non-Bayesian solution for this problem.
You can verify this at
White, J.S. (1958) The Limiting Distribution of the Serial Correlation Coefficient in
the Explosive Case. The Annals of Mathematical Statistics, 29, 1188-1197.
I show that a Bayesian solution does exist. In fact, depending on assumptions, there may be different solutions under differing cases. In general, however, this will hold. There is a different issue that the S&P 500 is constantly changing membership, and so $\beta$ is also constantly changing with each rebalance. It might be a small change at each step, but it is changing composition, and so this may be a poor model. Technically, this is not stationary since by definition $\beta$ updates quarterly.
The Bayesian solution is to solve using a likelihood of $$f(\mathbf{X}|\beta;\alpha;\sigma)=\frac{1}{\pi}\frac{\sigma}{\sigma^2+(x_{t+1}-\beta{x}_t-\alpha)^2},$$ where $\mathbf{X}$ is the matrix of data.
If $\pi(\alpha;\beta;\sigma)$ is your prior and $\pi'(\alpha;\beta;\sigma|\mathbf{x})$ is your posterior, then you solve for the posterior as $$\pi'(\alpha;\beta;\sigma|\mathbf{x})=\frac{\prod_{t=0}^{T-1}f(\mathbf{X}|\beta;\alpha;\sigma)\pi(\alpha;\beta;\sigma)}{\int\int\int\prod_{t=0}^{T-1}f(\mathbf{X}|\beta;\alpha;\sigma)\pi(\alpha;\beta;\sigma)\mathrm{d}\sigma\mathrm{d}\alpha\mathrm{d}\beta}$$
For predictive work, there is a predictive density available. Do note that the prior mass on $\beta<1$ is zero and that the prior mass on $\sigma\le{0}$ is zero. This will improve the quality of your estimate. There are time frames where the MLE is less than 1. As this is impossible, asymptotically, the spurious result of an unusual sample is avoided by the regularization of the prior.
If we denote the predictive distribution for future values of $\tilde{x}_\tau$ as $\pi''(\tilde{x}_\tau|\mathbf{X})$, then we can solve for a prediction using $$\pi''(\tilde{x}_\tau|\mathbf{X})=\int\int\int{f(\tilde{x}_\tau|\beta;\alpha;\sigma)}\pi'(\alpha;\beta;\sigma|\mathbf{X})\mathrm{d}\sigma\mathrm{d}\alpha\mathrm{d}\beta$$
If you take the logarithm, the mean of the logs will overstate return by 2% per year on the disaggregated returns for all annual trades in the CRSP universe from 1925-2013 and understate risk by 4%.
If you perform spectral analysis on the S&P the period is about 40-41 years. This implies that one swing of the proverbial pendulum takes 40-41 years of data and that any other amount will generate biased return and scale estimators. The full spectrum of returns in the density covers this period.
Logarithms overstate return and understate risk because the distribution of log returns, ignoring bankruptcy and mergers, follows a hyperbolic secant distribution and the mean of the logs works out to be the median of the data. The center of location, $\mu$, however, is at the mode. The limitation on liability truncates the distribution and shifts the median 2% away from the mode.
I do not know what it does to S&P data; I only know what it does to disaggregated data. I did perform a test to determine whether standard models or this model was better and the Bayes factors exclude the standard solution.
There isn't a proper reversion to the mode concept here as capital is a source and not a sink. Nonetheless, Slutzky's paper
Slutzky, Eugen (1937) The Summation of Random Causes as the Source of Cyclic Processes. Econometrica,5(2), 105-146
would imply a swing around the mode. This would look like mean-reversion in log space.
Interestingly, there does not exist an admissible non-Bayesian estimator for financial returns, in the general case. There are some special cases where it would exist.
There are two serious caveats for this method using the S&P 500. The first was mentioned above, changes in composition imply that you do not know what you are really measuring. To imply simple substitution would result in no change in the slope would imply that all stocks provide the same return. The second is that even if there are no issues with composition, we do not know whether returns or the scale parameter is scale invariant.
Two minor notes, whereas the relationship of the standard deviation with respect to time for the normal distribution is $\sqrt{t}\sigma$, it is ${t}\sigma$ with this distribution; and, there does exist a case where such an AR(1) process can behave close to a normal and that is when endpoint prices are far from equilibrium. In that case, you get a bizarre distribution that has no defined mean, but is the convex combination of two densities, one with a mean and one without. As prices move far from equilibrium, the percentage composition of the one with a mean gets close to unity.
$\sigma$ is a scale parameter and not a standard deviation. No mean exists for this density. As a result, the sample variance is undefined and appears as a random number. It should look like it is heteroskedastic with volatility clusters when there are runs. It is askedastic. While returns cannot be thought of in terms of variances, prices can and the $\sigma$ in return is a measure of the heteroskedasticity in prices.
Finally, there is no squared error for this type of problem. The appropriate cost function is the linear absolute loss function.