**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.**