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The following graph shows the ACF (sample autocorrelation function) and PACF (partial autocorrelation function) of the residuals in a linear regression. There is a sinusoidal decay in the ACF and two spikes at lag 1 and 4 in the PACF. Considering the spike at lag 4, can we still assume AR(1) for these residuals? I should add that, the best model based on AIC is ARIMA(2,0,1) and based on BIC is AR(1) using the auto.arima function in package forecast. Here are the codes:

auto.arima(resid(lm.3),ic=c("aic"))
Series: resid(lm.3) 
ARIMA(2,0,1) with zero mean     

Coefficients:
     ar1     ar2     ma1
  0.0579  0.5125  0.8500
s.e.  0.2228  0.1981  0.1753

sigma^2 estimated as 0.0008776:  log likelihood=207.36
AIC=-406.71   AICc=-406.29   BIC=-396.33


auto.arima(resid(lm.3),ic=c("bic"))
Series: resid(lm.3) 
ARIMA(1,0,0) with zero mean     

Coefficients:
     ar1
  0.7771
s.e.  0.0616

sigma^2 estimated as 0.0009202:  log likelihood=205.11
AIC=-406.23   AICc=-406.1   BIC=-401.04

enter image description here

The time series plot of the original residuals is enter image description here

And the ACF and PACF plot for the first difference is enter image description here

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The short answer is to fit an AR(1) model & check it. If what you're left with after that is pretty much white noise, you might well be safe to assume they're AR(1) - if that's a reasonable model a priori, & depending on what it is you're wanting to do with them.

The ACF & PACF suggest, however, that there's perhaps more structure there than a simple AR(1) model. You shouldn't necessarily be bothered about the fourth lag in the PACF being just over the 5% significance level (assuming that's what the blue line is - you didn't say) - there's no correction for multiple testing, so in 20-odd lags you'd expect that. But the wavy ACF could indicate you need either to difference or to put in at least an extra AR term. Given how slowly the ACF is decaying, most likely the former.

AIC is helpful, but if you're using it in an automatic fashion, you'll often find a number of models with not much difference in AIC (a difference of less than 2 is often taken as equivalent to "just as good").

In response to the comments:

(1) Is the series stationary or not? It's hard to tell for a short, highly autocorrelated series like this. Unit root tests (KPSS & augmented Dickey-Fuller) might help (but in my experience they rarely tell you anything that isn't obvious from the correlograms & the time series plot itself). A random walk & an AR(1) model with a high AR parameter can both look plausible & pass any diagnostic tests you might perform. Only over the long term are you likely to be able to tell. NB You may have good a priori reasons to pick one or the other.

(2) If it's stationary, AR(1) or more complex model? The ACF hints at other possibilities that are worth testing, but doesn't rule out an AR(1) - remember that real ACFs from short series can look quite different from the theoretical ones. Most people would go for the simplest, at least for the time being, provided that it fits well enough (& see above about AICs). NB A priori considerations can be important here too.

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  • $\begingroup$ I added the results for the AIC and BIC as the criteria separately above. It seems to me that the two model are not significantly different since both AIC and AICc are almost the same. For BIC, there is a difference of almost 5. Considering the log likelihood, I again came to this conclusion that the two models are not that much different. $\endgroup$ – Stat Dec 10 '12 at 15:55
  • $\begingroup$ Which two models? $\endgroup$ – Scortchi Dec 10 '12 at 16:02
  • $\begingroup$ AR(1) and ARIMA(2,0,1) $\endgroup$ – Stat Dec 10 '12 at 16:02
  • $\begingroup$ All right - but are either any good? If you fit an AR(1) model to the original residuals & plot the ACF & PACF of the new residuals, do you get white noise like for the differenced series ? $\endgroup$ – Scortchi Dec 10 '12 at 16:07
  • $\begingroup$ I got almost like white noise, there will be a spike (slightly greater than the 95% confidence limit at lag 3 for both ACF and PACF) when fitting AR(1). $\endgroup$ – Stat Dec 10 '12 at 16:15
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If there are no pulses, level shifts, time trends then take first differences and evaluate the acf and pacf of that series. If your paclage doesn't allow that the YOU difference the series AND submit the "new series"

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  • $\begingroup$ Thank you so much for the answer. I have added the time series plot of the original residuals together with the ACF and PACF of the 1st difference. I appreciate any comments. $\endgroup$ – Stat Dec 10 '12 at 15:50
  • $\begingroup$ Looks like differencing turned them to white noise. Compare with the ACF & PACF from an AR(1) model if you like. $\endgroup$ – Scortchi Dec 10 '12 at 15:56
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    $\begingroup$ I agree , it looks like either a first differnce or an AR(1) model with coefficient near 1.0. The plot of the orifiginal residuals suggests (in general) a tendency for a positive residual to follow a positive and a negative to follow a negative. Based upon these results I suggest constructing a TRansfer Function using the original data as important lags in one or more of the predictors may have been omitted. In this case the residuals can suggest autocorrelation whereas the more correct fix is to extract more structure from the X's. $\endgroup$ – IrishStat Dec 11 '12 at 9:47
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Also check if a transformation helps you:

require(forecast)
lambda = BoxCox.lambda(serie)
fit = auto.arima(serie, stepwise = FALSE, lambda = lambda)
summary(fit)
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