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.

**Contents**hide

#### Best Answer

Here are the ACF and PACF plots:

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

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)) `

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 $qle 5$.

### Similar Posts:

- Solved – Is ARMA model possible for series with non significant ACF/PACF
- Solved – Is ARMA model possible for series with non significant ACF/PACF
- Solved – ACF and PACF of residuals to determine ARIMA model
- Solved – If you have an ARIMA model i.e. ARIMA(1,0,1) or ARIMA(2,0,3), how do you read the ACF and PACF
- Solved – Understanding ACF – PCF Plots of ARIMA model