# ARMA parameters

I am currently doing a short term forecast using ARIMA model.I have been following Box and Jenkins method and to choose the best ARIMA parameters to do do my forecast I tested various(p,q) combinations and took the one with the lowest aic(akaike information criteria). I assigned a d value as the number of differenciation required. My problem is that my forecast is converging into a straight line after 2 hours. Did I choose a wrong method? Is it because I have to try higer AR(p) and MA(q) parameters. I put a threshold for AR(p) at 4 and for MA(q) at 2. I saw in some articles, they were going upt to 24 to do there forecast. Is there a limit for choosing parameters? Can you give me some references please?

You cannot say that you choose a wrong model just because the forecasts are converging to a straight line. But it may be due to the fact that you didn't pick up the right order. Note that higher orders are much more difficult to estimate.

The search limits for the AR and MA parts most of the time comes from your own knowledge. But in the forecast package in R (that uses different criteria like "aicc","aic", "bic" to find the best model), they set by default max.p=5, max.q=5, max.P=2, max.Q=2, max.order=5, max.d=2, max.D=1, where the capital letters (P, D, Q) refer to seasonal orders. However, as mentioned in the book R Cookbook by Paul Teetor, P. 384, if you think that your models needs more coefficients, then you need to expand the search limit. On the other hand, if you look at the book Introduction to Time Series and Forecasting (2nd Ed.) by Peter J. Brockwell, Richard A. Davis, P. 161, the maximum range for both p and q is from 0 to 27. Note that they use AICC rather than AIC. Therefore , to my knowledge, there is no universal agreement on these limits.

One last thing, in practice, you normally end up with some competitive models and not just one model based on one criteria. Then at the end of the day, you will again double check those models and pick up one of them to use.

My problem is that my forecast is converging into a straight line after 2 hours.

This is not an abnormal behavior for ARMA models, they're stationary. You're probably talking about the mean forecast, i.e. $$E[y_{t}]$$. For instance, in AR(1) model: $$y_{t+1}=c+\phi y_t+\varepsilon_t$$, the mean forecast is $$E[y_t]=\frac{c}{1-\phi}$$, i.e. a constant.

This MATLAB help may be helpful in understanding the convergence of ARIMA models.

Take a look at a typical AR(p) forecast, it's basically a smooth line in long horizon forecast, it has some interesting structure only in the short range forecast horizon:
(source: mathworks.com)

• What can I do to improve the model. Do I have to choose higher values of AR(p) and MA(q) parameters. Actually,I am doing the forecast using ARIMA with I(d=1) – Kunta May 15 '14 at 14:05
• What are you trying to accomplish? All ARIMA models' mean forecasts converge to something in a long run. Do you want to see more "structure" in the forecast? Increasing the lags will only help in this regard in short term. – Aksakal May 15 '14 at 14:26
• Think of this: at 10 mins interval, you have 36 intervals to forecast. Your ARIMA must have comparable number of lags, if you want to produce interesting forecast, i.e. the one which is not a straight line. I can't tell you how many lags, but it's got to be more than handful, maybe 18 or more. Then you run into different issues. Did you consider ARIMAX or VAR? I bet that the temperature or pressure could be important exogenous or endogenous variables. – Aksakal May 15 '14 at 15:47
• You might also look into exponential smoothing methods. See the forecast package in R. – jseabold May 15 '14 at 16:54
• I'm sure there are topics in SE on Kalman filters. – Aksakal May 16 '14 at 13:38