Solved – Is ARMA model possible for series with non significant ACF/PACF

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I am playing around with SP500 data from the MASS package. From observation of the ACF and PACF there seem to be no significant autocorrelation. Now I want to model the volatility of the returns but I also want a mean equation. I'm not sure how I will get my mean equation as the ACF and PACF suggest that I can't use MA or AR terms in my mean equation.

Also, applying the auto.arima function from the forecast package yields an ARIMA(3,0,5) model. I am confused with such results as the ACF/PACF do not even show significance.

I am using the Box-Jenkins approach in my project and it wouldn't make sense to say that there are 3 AR and 5 MA terms when the ACF and PACF don't support what I say. I need to be able to explain why I am choosing this rank.

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Here are the ACF and PACF plots:

library(MASS) library(forecast) tsdisplay(SP500)

enter image description here

So contrary to what the question states, there are significant correlations at many lags. Naturally auto.arima() tries to model the correlation structure as best it can. Because the default maximum number of AR and MA parameters are both set to 5, it cannot handle all the significant correlations seen here, but it does quite a good job as is seen when the residuals are plotted.

fit <- auto.arima(SP500) tsdisplay(residuals(fit))

enter image description here

You can get a better model by making auto.arima() work harder:

fit <- auto.arima(SP500, approximation=FALSE, stepwise=FALSE, max.order=10)

This returns an ARIMA(5,0,2) with better residual plots than those shown above and which passes a Ljung-Box test:

> Box.test(residuals(fit), lag=20, fitdf=8, type="Ljung")      Box-Ljung test  data:  residuals(fit) X-squared = 17.365, df = 12, p-value = 0.1364

The model chosen has the smallest AICc value for all ARIMA($p,0,q$) models with $ple 5$ and $qle 5$.

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