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Fitting Models to Short Time Series

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Fitting Models to Short Time Series

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Fol­low­ing my post on fit­ting mod­els to long time series, I thought I’d tackle the oppo­site prob­lem, which is more com­mon in busi­ness environments.

I often get asked how few data points can be used to fit a time series model. As with almost all sam­ple size ques­tions, there is no easy answer. It depends on the num­ber of model para­me­ters to be esti­mated and the amount of ran­dom­ness in the data. The sam­ple size required increases with the num­ber of para­me­ters to be esti­mated, and the amount of noise in the data.

Using least squares esti­ma­tion, or some other non-​​regularized esti­ma­tion method, it is pos­si­ble to esti­mate a model only if you have more obser­va­tions than para­me­ters.  (If you use the LASSO, or some other reg­u­lar­iza­tion tech­nique, it is pos­si­ble to esti­mate a model with fewer obser­va­tions than para­me­ters.) How­ever, there is no guar­an­tee that a fit­ted model will be any good for fore­cast­ing, espe­cially when the data are noisy.

Some text­books pro­vide rules-​​of-​​thumb giv­ing min­i­mum sam­ple sizes for var­i­ous time series mod­els. These are mis­lead­ing and unsub­stan­ti­ated in the­ory or prac­tice. Fur­ther, they ignore the under­ly­ing vari­abil­ity of the data and often over­look the num­ber of para­me­ters to be esti­mated as well. There is, for exam­ple, no jus­ti­fi­ca­tion what­ever for the magic num­ber of 30 often given as a min­i­mum for ARIMA modelling.

The only rea­son­able approach is to first check that there are enough obser­va­tions to esti­mate the model, and then to test if the model per­forms well out-​​of-​​sample. With short series, there is not enough data to allow some obser­va­tions to be with­eld for test­ing pur­poses. How­ever, the AIC can be used as a proxy for the one-​​step fore­cast out-​​of-​​sample MSE (see here). The AIC allows both the num­ber of para­me­ters and the amount of noise to be taken into account.

What tends to hap­pen with short series is that the AIC sug­gests very sim­ple mod­els because any­thing with more than one or two para­me­ters will pro­duce poor fore­casts due to the esti­ma­tion error.  I applied the auto.arima() func­tion from the fore­cast pack­age in R to all the series from the M-​​competition with fewer than 20 obser­va­tions. There were a total of 144 series, of which 32 had mod­els with zero para­me­ters (ran­dom walks), 95 had mod­els with one para­me­ter, 15 had mod­els with two para­me­ters and 2 series had mod­els with three para­me­ters. For what it’s worth, here is the code.

n <- unlist(lapply(M1,function(x){length(x$x)}))
n <- n[n<20]
series <- names(n)
nparam <- numeric(length(n))
for(i in 1:length(n))
  fit <- auto.arima(M1[[series[i]]]$x)
  nparam[i] <- length(fit$coef)

Sea­sonal mod­els bring their own dif­fi­cul­ties because the sea­son­al­ity usu­ally takes up m-1 degrees of free­dom where m is the sea­sonal period (e.g., m=12 for monthly data). Fourier terms are one way to reduce the prob­lem — use­ful when­ever the ratio of m to sam­ple size is large. Fur­ther com­ments on sea­son­al­ity and sam­ple size are in my short Fore­sight paper with Andrey Kostenko: “Min­i­mum sam­ple size require­ments for sea­sonal fore­cast­ing mod­els”, although I wrote that for a sta­tis­ti­cally unso­phis­ti­cated audi­ence, so there is no men­tion of the LASSO or AIC as pos­si­ble solutions.

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