Forecasting Models – ARIMAX Forecasting in SPSS vs. R: A Comparative Analysis

arimadynamic-regressionforecastingrspss

I'm using time series data containing both trend and seasonality. I also have 2 endogenous predictor variables that I would like to include in my model.

In R I've used the forecast package to develop a dynamic regression model with use of auto.arima() and the xreg argument from the forecast package. I understand this procedure takes a regression and then attempts to fit the residuals with an ARMA Model.

I've also developed what seems to be an appropriate model using the forecasting Module in SPSS by specifying a Seasonal ARIMA model and including my covariates. However, one of the coefficients on one of my endogeneous predictors has a negative sign which makes no sense intuitively.

I've read Dr. Hyndman's article The ARIMAX model muddle and found it to be extremely insightful and useful. However, I have not been able to find any documentation on what type of statistical procedure SPSS uses to fit an ARIMA model with covariates, so I'm not sure how I should interpret the coefficients or how concerned I should be with a flipped sign. Any help clarifying the modelling procedure used by SPSS would be tremendously appreciated.

Best Answer

See the Algorithms doc available from the Help menu for computational details. If you have Statistics version 23, you might also be interested in the TCM procedure (Analyze > Forecasting > Create Temporal Causal Models)

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Solved – ARIMAX with a specified nonlinear model using the arima function in R

Do the parameters $\beta_3$ and $\beta_4$ have any particular interpretation or meaning in the context of your model? You can consider reparameterizing the model merging $\beta_3 t_i /\beta_4$ as $\theta\,t_i$. In this way the model becomes linear in the parameters and you can pass the regressors $t_i$ and $d_t$ through argument xreg in function arima.

You should include $t_i$ only once in the reparameterized model. In your model, the regressor $t_i$ is proportional to the regressor $2\pi t_i$. With this type of collinearity among the regressors the optimization algorithm won't be able to invert the system matrix.

Edit (updated answer for the new model defined in the question)

The function stats::arima does not allow you to specify this non-linear parameterization. You should modify the function arima, which is not straightforward since would require among other things recompiling the package. You can also try to create your own function, the functions that you would need are essentially: makeARIMA, KalmanLike and nlm, but this is also quite a hard work. You may ask for further guidance at Stack Overflow.

Did you find this model in any paper? If you know some application of this model published in a paper and the data are available, it would be a nice exercise to do and someone could be willing to replicate it.

Arimax Prediction – Using Forecast Package for Arimax Prediction

Prediction intervals are based on residual variance as estimated by the maximum likelihood optimization.

I like forecast function from the forecast package:

fc <- forecast(forecast.arima,h = 5, xreg=tf[114:length(tf)])

which gives me the following predictions:

  Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
Jun 2005       17.61393 17.47222 17.75565 17.39720 17.83067
Jul 2005       17.51725 17.35753 17.67697 17.27299 17.76152
Aug 2005       17.51725 17.35753 17.67697 17.27299 17.76152
Sep 2005       17.51725 17.35753 17.67697 17.27299 17.76152
Oct 2005       17.51725 17.35753 17.67697 17.27299 17.76152

the way prediction interval works is by following formula:

$y_t \pm z\sigma$ where $z$ is a multiplier which takes values such as 1.96 for 95% prediction interval and 1.28 for 80% prediction interval $\sigma$ is the standard deviation of the residual also forecast distribution which is also the square root of sigma^2 in the maximum likelihood estimate. As you show sigma^2 ( 0.01223) is identical in both your models, prediction intervals will also be same and will match.

If you want to check it,

Upper Limits:

> fc$mean[1]+sqrt(forecast.arima$sigma2)*1.96
[1] 17.83068

Lower Limits:

> fc$mean[1]-sqrt(forecast.arima$sigma2)*1.96
[1] 17.39719

which matches the prediction interval provided by the forecast function. To answer your question, yes leveraging forecast package will work in this case and the prediction intervals will be correct.

Best Answer

Related Solutions

Solved – ARIMAX with a specified nonlinear model using the arima function in R

Arimax Prediction – Using Forecast Package for Arimax Prediction

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