[R] modeling time series with ARIMA

[Gad Abraham]

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: 
> Read 45 items
> 
>> 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.

> 
> 2. What would be the impact of an inappropriate model on the confidence interval for a future prediction? (disregarding the fact  that an inappropriate model would give a bad forecast on future value, could it have also an impact on enlarging the interval?)

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.)

> 
> 3. As a rule of thumb, do you chose your model by selecting the lowest AIC, or by the lowest standard deviation of the residuals ?

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).


-- 
Gad Abraham
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