Moving Average Window Size Determination

Question

Is there a "correct" way of determining a moving average window/smoothing parameter (or at least a starting guess for a financial time-series?

I understand of course that in some sense, this could be considered a "hyperparameter" for let's say - a trading strategy - and that one could use some kind of cross-validation to optimize it, but frankly this has little interpretability and starts to veer into what I'd consider overfitting territory. Moreover, if one has a strong initial guess, they can tune it with the same methods in a Bayesian sense, by concentrating the prior at the given initial guess.

Is there a way to such a guess using physical or economic fundamentals/reasoning? Something I considered was computing the Fourier Transform of the time-series in question (or of its autocorrelation function?), and then take the mode of the magnitude, e.g. $$w_{\text{opt}} = \mathrm{argmax}_{\xi}|\hat{f}(\xi)|$$

but frankly I only understand Fourier Analysis from a mathematical perspective, and not its application to discretely sampled signals (e.g. time-series), so I am not sure if this is a sensible idea.

Answer 1

I don't think that there is one right way to approach this problem. However, I will give an example which I found quite interesting. The JP-Morgan risk-metrics approach was (or still is I don't know) quite popular in the industry. They use an EWMA $$ \sigma_{t}^2=(1-\lambda)r_{t-1}^2+\lambda\sigma_{t-1}^2 $$ to predict daily or monthly volatility. For daily returns they use $\lambda=0.94$ as "optimal" decay factor for every time series. As a reason for that they wrote in their 1996 technical document that it is too elaborately to calculate an optimal decay factor for every time series for different time periods.

What they did instead is that they modelled the log returns $r_t$ via $$ r_t= \sigma_t \epsilon_t \quad , \epsilon_t \overset{iid}{\sim} {\cal N}(0,1) $$ where $\sigma_t^2$ is modelled via the EWMA above. Now for $N$ different time series, they defined the root mean squared error (RMSE) as: $$ RMSE=\sqrt{\frac{1}{T}\sum_{t=1}^T(r_t^2-\sigma_t^2)^2} $$ Now let $\hat{\lambda}_i$ denote the optimal decay factor for time series $i$ (that one which minimises the RMSE) and $\tau_i$ the corresponding value of the RMSE. They calculated the sum of the minimum RMSE's $$ \sum_{i=1}^N\tau_i $$ used this quantity to calculate an relative RMSE $$ \theta_i= \frac{\tau_i}{\sum_{i=1}^N\tau_i} $$ then used this quantity to derive the weights $$ \omega_i = \frac{\theta_i}{\sum_{i=1}^N\theta_i} $$ and finally got $$ \lambda = \sum_{i=1}^N\omega_i\hat{\lambda}_i \approx 0.94 $$ as optimal decay factor over $N$ time series.

As you can see, this is one possible way to determine an "optimal" decay factor for an EWMA. I am sure that there is a lot of literature out there dealing with this type of problems and that there are for sure more sophisticated approaches. However, in my opinion the main problem is to find a good balance between simplicity and accuracy. Depending on your intended use, one may be weighted more than the other.