If you believe that market timing is impossible, then using a window equal to the entire dataset is the best possible approach. This is the most unbiased estimator possible to sample the true underlying mean of the return distribution.jimbomahoney wrote: ↑Sun Feb 23, 2020 1:39 pmThat's a very fair point, and I do agree that choosing a window of a fixed period is equivalent to market timing.Uncorrelated wrote: ↑Sun Feb 23, 2020 11:25 amIt appears that we disagree on the definition of market timing, and also whether it is possible. If you are varying your asset allocation as a result in economic conditions, that is market timing.

It's not that I don't believe you or BtM, just that the arguments are either nonexistent or poor. This makes it nearly impossible to evaluate the strategy.

However, even a "window" of the entire dataset equates to the same thing. I'm not happy using a 30 or 100 year "window" to form an expectation of future returns. I've no idea what future reaturns are going to be, so any "window" is, at best, a guide.

I'm just going to have to accept the risk either way. A "fixed" expectation will not respond to changes. A moving window will be "biased" to whatever the dataset has tended towards.

The reason I like a moving window of returns, and the reason I'm fairly confident about it is two-fold:

1) My returns are similar to BTM, who as far as I can tell, is using a fixed return value for ~70+ years of data. I'm using a ~3-4 year moving window and getting similar returns, albeit with more volatility.

2) The fact that there is a pattern, similar to the rebalance frequency, that tends to be noisy for short periods (and therefore unreliable) but clearly trails off at longer periods. Yes, the short periods tend towards chance and the particularities of that dataset / time period. But there is, I believe, a trend. Short = random, but higher likelihood of being "optimal". Long = less noisy, but higher likelihood of lower returns. See some of the plots in my long post.

This is my justifcation anyway. I totally appreciate that itcouldbe chance that 2 - 8 years just happens to work best for this time period.

I don't find your reasons to be convincing. If a limited window is better than an estimate based on the full dataset, then the only explanations I can come up with are chance and momentum.

I expected that was the case based on your estimated window size. This is a pretty big error, the returns should be calculated in excess of the risk free rate. The are three reasons for this: a log utility agent only cares about the return and volatility(!) in excess of the risk free rate. It is consistent with the theory of an equity risk premium, and it would make the strategy independent of the currency inflation. Since the dollar has seen widely varying inflation in the last 50 years, this greatly impacts the results.I'm taking the arithmetic mean daily return for the period and converting it to an annual return. i.e. this should be the arithmetic annual return. I guess this is the nominal return.Uncorrelated wrote: ↑Sun Feb 23, 2020 11:25 amThe "optimal" window for calculating the arithmetic return with your data is 2.5-8 years. Are you calculating this based on the nominal returns, real returns, or the excess returns? What is the theoretical basis for this window size? What are the objections against using a window spanning the entire data period? I fear that using a short window may (accidentally) result in exploitation of momentum effects, you don't seem to believe in momentum.

There is some discussion over what the correct risk-free rate is, but the most commonly used is either the 1 month or the 3-month t-bill.

That doesn't fully answer my question. I don't see any real evidence that estimating the SD based on a 20-50 day window is better than estimating the SD based on the entire dataset. Maybe it is. But I don't see any evidence.Again, there is a trend. The results from SD window length become more stable as they get longer, but there is, as BTM says, not much difference once it's sufficiently long (e.g. 20 - 50 days). See my analysis in the long post.Uncorrelated wrote: ↑Sun Feb 23, 2020 11:25 amRegarding the SD and correlation window length, the longer the correlation window is, the better the results. Doesn't that imply that the best estimate for correlation is an estimate over the full sample? (i.e. correlation is not time-varying). The results when varying the SD look pretty random and statistically insignificant. If you can't find a theoretical explanation why an 23-day SD window is better than a 10-day or 40-day SD window, I think there is a large chance that this is a result of overfitting.

Based on these images I am almost certain that virtually all of the returns are explained by the accidental momentum exposure present in the algorithm. Furthermore, the data shows substantial serial correlation which means that the results are extremely sensitive to overfitting. With a return estimation window of 5 years, there are only 3 independent data samples present in the images you posted.I'd also considered that and can show some results. I'd deliberately designed in various checks as the code executes so that I can see what it's doing.

That's one data sample each for the train, test and validation set (that was a machine learning joke, but it's not far from the truth).