# Re: [R] modeling time series with ARIMA

Date: Tue, 04 Dec 2007 09:34:43 +1100

eugen pircalabelu wrote:
> Good afternoon!
>
> I'm trying to model a time series on the following data, which represent a monthly consumption of juices:
>

```>> x<-scan()
```

> 1: 2859 3613 3930 5193 4523 3226 4280 3436 3235 3379 3517 6022
> 13: 4465 4604 5441 6575 6092 6607 6390 6150 6488 5912 6228 10196
> 25: 7612 7270 8617 9535 8449 8520 9148 8077 7824 7991 7660 12130
> 37: 9135 9512 9631 12642 11369 12140 13953 12421 11081
> 46:
>
```>> arima(x,order=c(2,1,2), seasonal=list(order=c(0,1,0), period=12))->l
>> acf(l\$resid)
>> sd(l\$resid)
>> Box.test(l\$resid)
```

>
> Now, my problem:
> 1. All the analysis that i have seen regarding ARIMA modeling, had the residuals acf, within the confidence interval, while my residual acf at first lag is very close to one (and going out of the confidence interval), even if the Box.test can not reject the null hypothesis of a significant acf for all my residuals.
> I imagine that i am doing something wrong with my model. Is the acf at lag 1 a sign that my residuals are not white noise, or what is wrong here?

Perhaps you misread the plot? I've tried your code and the there is no significant correlation at lag 1, but at lag zero, which is 1 by definition.

Are you referring to the 95% prediction limits (as in Box 1994 "Time series analysis: forecasting and control" pp 139–145)? If so, a bad model would mean that the variance of the shocks (errors) is higher, therefore the prediction limits would be wider. (In other words, you've explained less of the variance of the time series.)

I've found that for a given time series you can fit several different ARIMA models with very similar results. Out of a group of "sensible" models (judged by residuals and cross-validated forecast MSE), I'd choose the simplest model(s).

```--
Department of Mathematics and Statistics
The University of Melbourne
Parkville 3010, Victoria, Australia
email: g.abraham_at_ms.unimelb.edu.au
web: http://www.ms.unimelb.edu.au/~gabraham

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