Statgraphics for Windows
Analysis summary report for S&P 500 series
Geometric random walk model with growth

Analysis Summary

Data variable: SP500

Number of observations = 270
Start index = 1/71
Sampling interval = 1.0 month(s)
Length of seasonality = 12

Forecast Summary
Math adjustment: Natural log
Nonseasonal differencing of order: 1

Forecast model selected: ARIMA(0,1,0) with constant
Number of forecasts generated: 12
Number of periods withheld for validation: 24

Estimation Validation
Statistic Period Period
MSE 83.9654 543.469
MAE 5.78478 13.2415
MAPE 3.4784 3.13166
ME 0.197613 0.963639
MPE -0.117144 0.102417

ARIMA Model Summary
Parameter Estimate Stnd. Error t P-value
Mean 0.00560047 0.00298078 1.87886 0.061349
Constant 0.00560047
Backforecasting: yes
Estimated white noise variance = 0.00217684 with 268 degrees of freedom
Estimated white noise standard deviation = 0.0466567
Number of iterations: 1

In this model (geometric random walk with growth), the constant term (0.00560047) represents the average percentage return from one period to the next (i.e. 0.56% per month).

The estimated white noise standard deviation (0.046657) is the standard deviation of the monthly percentage changes, which measures the volatility of returns (4.66% per month).

Also, note that because of the effects of inflation and compound growth, the Mean Squared Error (MSE) and Mean Absolute Error (MAE) statistics are not directly comparable between the estimation and validation periods, but the Mean Absolute Percentage Error (MAPE) is comparable.