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.