Anon

186 The Basics of financial economeTrics

table 9.7 Autoregressive Moving Average Model, Weekly Sample Returns of S&P
500 Index from January 1998 through December 2012

Variable Coefficient t-Statistic p-Value

c 0.00 1.25 0.21

ρ 1 −1.10 −7.44 0.00

ρ 2 −0.83 −5.44 0.00

ρ 3 −0.03 −0.68 0.50

δ 1 0.97 6.79 0.00

δ 2 0.72 5.72 0.00

from the χ^2 distribution, we are unable to reject the null hypothesis that
there is no autocorrelation. Hence, the ARMA(3,2) model appears to be
adequate in modeling the weekly S&P 500 index return series.
While the model seems adequate, we don’t yet know if it does a good
job of forecasting the data. In order to judge the forecasting performance of
the ARMA(3,2) model, we need a set of competing models. For the purpose
of illustration, we used an AR(1) and an MA(1) model and compared the
forecasting performance of ARMA(3,2) against these two models. A good
approach of model evaluation is to divide the sample into an in-sample
estimation period and a holdout sample. The in-sample period is used to
estimate the model parameters and the holdout sample will be used to con-
struct out-of-sample forecasts. For our illustration, we use an estimation
period for the three models from January 1998 through December 2011
(730 observations), holding back the last 52 observations to construct out-
of-sample forecasts.
It is possible to forecast 52 weeks forward, but long-term time series
forecasts tend to be less reliable than short-term forecasts. One way to get
around this problem is to compute long-term forecasts by iterating forward
by one-step forecasts. This involves estimating the models from January
1998 through December 2011 and forecasting the S&P 500 returns for the
first week of January 2012. Then we add the first week’s realized return
to the estimation period and then forecast the second week’s return. This
iterative process continues until we use up the entire holdout sample. This
exercise will yield 52 forecasts with 52 realizations (i.e., observed values).
The results of these forecasts and the S&P weekly returns are presented
in Figure 9.3. The FORECAST_AR1 shown in the figure denotes the fore-
casts generated by the AR(1) model. The FORECAST_MA1 indicates the
forecasts generated by the MA(1) model, while the FORECAST_AR3MA2
denotes the forecasts generated by the ARMA(3,2) model. In 2012, the S&P