Goal: This case introduces Box-Jenkins modeling in the context of forecasting the federal budget deficit for the period 1997M1-1997M4 as we did in Chapter Six, Case #1. Specifically, this case examines

• How to Identify an ARIMA Forecasting Model
• Estimating and Generating Forecasts of an ARIMA model
• Seasonal Differencing using FORECASTX^TM
• Comparing ARIMA forecasts of BUDGET with Time-Series Decomposition forecasts.

Recall that in Chapter Six, Case #1 you produced forecasts of the monthly federal budget surplus/deficit for the period 1997M1-1997M4 using linear-trend time-series decomposition. Recall that your forecasts for 1997M1-1997M4 and actual values were:

From this Table we calculated RMSE to be 28.48, i.e., the average error expected each month is approximately 28 billion dollars using linear trend decomposition. In this case we see if we can improve upon these forecasts applying Box-Jenkins models.

The spreadsheet for this problem is C7_Case1.xls. It contains the following data:

The series BUDGET is the non-seasonally adjusted monthly federal budget surplus (deficit). The series is the "official" federal budget numbers from the Congressional Budget Office.

To examine the behavior of monthly deficit data (BUDGET) over the historical period (1990M1-1996M12), we generated a time-series plot of the data using Excel.

Question #1: Viewing the time-series plot of the data, what behavior do you need to model?

ANSWER:

The following ACF and PACF for BUDGET were generated using FORECASTX^TM.

Question #2: What is a candidate ARMA-type model for generating monthly forecasts?

ANSWER:

Using the ACF and PACF estimates to identify a "candidate" ARMA model, remember our ARMA identification rules of thumb: (1) If the autocorrelation function abruptly stops at some point -- say, after q spikes -- then the appropriate model is an MA(q) model; (2) If the partial autocorrelation function abruptly stops at some point -- say, after p spikes -- then the appropriate model is an AR(p) type; (3) If neither function falls off abruptly, but both decline toward zero in some fashion, the appropriate model is an ARMA(p, q). By reference to the ACF and PACF correlograms, we see the data appears to have MA components at lags 1, 3, 11, and 12, whereas the AR components arise at lags 1,2, 5, 9, 11, and 12. This suggests up to an ARMA(12,12) model to generate forecasts into the holdout period and forecast horizon.

We can estimate an ARMA model in FORECASTX^TM by simply editing the parameters that appear once you have selected Box Jenkins as the forecast method. Using FORECASTX^TM we find that the highest order lags allowed are nine for the MA and AR terms. Accordingly, we estimated an ARIMA(9,0,9)*(0,0,0) model, which is an ARMA(9,9) model in the notation of the text, as a first pass at the data and generated the following reports. Note this requires we set all differencing and seasonal parameters to zero in the edit parameters box. Accordingly, our ARIMA(9,0,9)*(0,0,0) model has no data transformation aimed at eliminating trend and seasonality from the data.

Estimation results for the 1996 holdout period and the 1997 forecast period are shown below.

The following are the estimation results of the ARIMA(9,0,9)*(0,0,0) model.

Finally, to see how well the ARIMA(9,0,9)*(0,0,0) model performs, we generated a correlogram of the estimated residuals as reported below.

Question #3: By examining the summary statistics above and the correlogram of the estimated residuals, how well does the ARIMA(9,0,9)*(0,0,0) model fit the data?

ANSWER:

In addition, the ACF function has significant autocorrelation at lags 3, 6, and 12. Similarly, the PACF has a spike at lag 3, and is marginally significant at lag 12. This suggests the residuals are not white noise and the ARIMA(9,0,9)*(0,0,0) model is suspect. The problem appears to be seasonality in BUDGET data, which can easily be handled in FORECASTX^TM using its seasonal differencing capabilities.

Using FORECASTX^TM we re-estimated the model using the optimal parameter selection option, which automatically determines the degree of data level and seasonal differencing. Here simply leave all parameters blank in the edit parameters box and FORECASTX^TM will select the optimal parameters.

The Expert Selection option in FORECASTX^TM selected an ARIMA(1,0,1)*(0,1,0) model as the best. This model employs first-degree seasonal differencing with first-order non-seasonal MA and AR terms. Estimation results for the 1996 holdout period and the 1997 forecast period, as well as summary reports are shown below.

The following estimation results of the ARIMA(1,0,1)*(0,1,0) model were obtained from FORECASTX^TM.

To see how well the ARIMA(1,0,1)*(0,1,0) model fit the data, we generated a correlogram of the estimated residuals:

Question #4: By examining the summary statistics above and the correlogram of the estimated residuals, how well does the ARIMA(1,0,1)*(0,1,0) model fit the data?

ANSWER:

Questions #5: Report your ARIMA forecasts for the forecast period 1997M1-1997M4, and compare the accuracy with the results from Chapter Six, Case #1.

ANSWER:

Question #1: Try re-estimating this case with updated BUDGET data into 2001. Contrast and compare your results with those of this case. Explain!