Anon

188 The Basics of financial economeTrics

500 returns fluctuated considerably more than the prediction made by the
models. However, all three models seem to have predicted the direction of
the change fairly well.
A formal way of testing the accuracy of the forecasts is to compare the
forecast errors (the difference between the realized values and the forecasted
values of the holdout sample) and select a model that generates the lowest
aggregate forecast error. One cannot just add the forecast errors of the hold-
out sample because such aggregation leads to positive differences offsetting
negative differences, leaving a small forecast error. One way to overcome
this problem is to either square the errors or to take the absolute value
of the errors and then aggregate them. For illustrative purposes, we squared
the errors and then divided the aggregated value by the number of forecast
errors. The resulting metric is called the mean squared error (MSE). The
MSE of the three models are then compared and the model with the smallest
MSE would be the most accurate model. In the case of the weekly S&P 500
returns, the MSE of the AR(1) model is 0.000231. The MSE of the MA(1) is
0.000237 while that of the ARMA(3,2) model is 0.000247. The MSE of the
AR(1) is slightly better than the other two models, but the differences are
very small. Overall, the MSE measures suggest that the three models should
provide adequate forecasts.

Vector Autoregressive Models

So far in the chapter we examined how to model and forecast one single time
series variable. A natural extension would be to model and forecast multiple
time series variables jointly. In 1980, Christopher Sims introduced vector
autoregression (VAR) modeling or vector autoregressive models to analyze
multiple time series.^8 A first-order two-variable VAR model takes the form

ybby bz

zccy cz

tttyt

ttt

=+ ++

=+ ++

−−

−−

12131

12131

ε

εεzt

(9.7)

where yt and zt are two variables of interest. The parameter estimates are
shown by the letters b and c, and ε shows white noise errors that are assumed
to be uncorrelated with each other.
Notice that yt and zt influence each other. For example, b 3 shows the
effect of a unit change in zt− 1 on yt while c 2 shows the influence of yt− 1 on
zt. For example, yt could be daily returns on some U.S. stock index while zt

(^8) Christopher Sims, “Macroeconomics and Reality,” Econometrica 48 (1980): 1−48.