Both example residual sets pass tests for normality under S-W test. Generally speaking, I agree it is a good idea to test for residual normality, serial correlation (durbin-watson), etc... in fits. One could use a tighter moving average than the example I gave, however, there are advantages with regard to bias-variance tradeoff and other metrics, for using smoothing spline fits. MAs are just not a good method for time series fits; there are plenty of better modern methods available.
Arbitrary MAs like the ones people use from TA packages, you mean? What happens when you go back and fit using an MA(q) model fit using the innovations algorithm or Hannan-Rissanen? Does the residual profile look comparable to that of the spline? My initial guess as to why you're saying what you're saying is because a spline isn't going to go and look at Cov( x_t, x_(t-lag) ), it's just going to fit in the presence of what's there. If at every step along the way when you are deducing how much to incorporate into your average, you go back and look at the covariance matrix for various lags, and then dynamically adjust how much information gets kept (q parameter varies in the MA(q) model), you might get better results -- comparable to the spline. [Not that I have tried.] My initial argument was just that running around assuming you are going to get a clean normal distribution on residuals from a spline is asking for a beating at some point in time in the future. The other guy still didn't say how he wants to explain forecasting methods for data points in the future. I'm guessing linear interpolation and assuming that the covariance structure between points doesn't change all that much from point to point; however, it's important to know what his predictor is. Whether it's the standard ARMA prediction operator, or just a guess. The prediction operator's forecast gives you a result that has a ridiculously wide confidence interval also. Also, I am becoming a fan of your posts. We should discuss R and math and become pals.
Bingo! Fit past data does not guarantee future performance, in fact the closer the fit, the less robust the performance will likely be. Linear interpolation and ASSUMING the covar doesn't change is a big assumption IMHO although there are techniques to at least attempt to profit from this assumption with relatively low risk...
Prediction is what gets many people lost in the process of developing good trading models. While you make good points, the "step 1" into the future part is where the buck stops so to speak. Forecasting volatility per GARCH has its potential uses, however, one is still using a fitting process (MLE for GARCH I think) to produce a forecast and its associated confidence. IMO, the GARCH process is the best one out there to do this, but, its is entirely useless on its own. Here's why: the use of a normal distribution. I've found Cauchy distributions to be much more effective when properly incorporated. All that mathematical mumbo-jumbo aside, the real heart of any good mean reversion system is the point at which a market is determined to be far enough away from value. The value (i.e. the mean or whatever you chose to use) is not really important, the important part is how far away from value the current price is and how much time the price is spending there. Trading mean reversion isn't about pinpointing value and trading forecasts, its about statistically defining what an overvalued or undervalued price trades like. Mike
I am confused. We need to say that the price is far enough from value, but you also said that it is not important to know where the value is. Do you mean that it is not important to know =exactly= where the value is, just to know where in general the value is and then concentrate on how far the price has moved from there and how the price is behaving at that point? This sounds allot like the Dalton MP teachings. I hope my question makes any sense.
Yes. Focus on how price behaves when it is away from value - i.e. when it is extended in any sense of the word. Creating systems that rely on market specific and/or precise valuations are inherently flawed. Why? Because value is always a relative term. When trading the open, the prior day's close has significance mainly because yesterday's close is the most important and up-to-date piece of information (pending any new events). There are a lot of good mean-reversion systems that trade the open as a result, the open represents an immediate pricing inefficiency (gaps to be specific). Does the same hold true for last week's close? I'm not sure. Would it make a difference if we average the past three daily closes and derived value that way for the open that way? I don't know but my research says no... How about linear regression? Not sure... What I can say though is that anytime lag is introduced in a value forecast, one inherently assigns a *lower* weight to the most recent, and most accurate, assessment of value.
So do you enter when price gets a certain distance away from the spline? And does your system require averaging down when price does not immediately go in your favor?
No and no. I trigger a potential buy condition when the price moves far enough way from the mean. That buy condition may or may not be satisfied depending on the price action after the price has crossed below or above that distance-from-mean. Never average down.
am i correct then in understanding that you do not use a spline as your mean? what do you use as mean? it was my understanding that RTM systems required averaging down because otherwise if you only take one position, you run the risk of price going against you and then the mean going below your entry price (assuming you're long) and then, when price reverts, you still have a loss.
To be mathematically precise, you only know the mean for the distribution of residuals from your start time to your end time, say t. What exactly is your "mean" in the future, at time t+1? If you argue that for t+1 , we can use the mean we calculated at t, that's one thing; however, the idea behind regressions and splines is that you are constructing a best guess for the mean at various levels, or increments of time that you know about -- not what the future mean is. A better mean is a best guess based on the lagged correlation matrix, not just assuming the same mean at time t. Let's say S(t) is stock price at time t, and and S(t+1) is the price at t+1. Based on your posts, you're telling me to only worry about the fitted E[S(t)] and that, if at time t, the residual is far enough away from E[S(t)], that you should assume that E[S(t+1)] will be reasonably close to E[S(t)] and that the reversion can be captured. Basically, you're just gambling that E[S(t)] ~= E[S(t+1)]. There's nothing that really supports this in theory in any deterministic sense. A stochastic shock can dislocate E[S(t+1)] much much further away from E[S(t)]. You end taking the entire risk. With pairs, the idea is just that some of the risk is statistically hedged, but in terms of just trading residuals on a spline, you have not hedged out your risk statistically at all. This is a valid trading strategy, but the trader is definitely exposed to risk. One 3+ sigma event and you're done for.