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C h a p t e r Additional Topics in Time

16 Series Regression

T his chapter takes up some further topics in time series regression, starting with
forecasting. Chapter 14 considered forecasting a single variable. In practice,
however, you might want to forecast two or more variables, such as the growth rate
of GDP and the rate of inflation. Section 16.1 introduces a model for forecasting
multiple variables, vector autoregressions (VARs), in which lagged values of two or
more variables are used to forecast future values of those variables. Chapter 14 also
focused on making forecasts one period (e.g., one quarter) into the future, but
m­ aking forecasts two, three, or more periods into the future is important as well.
Methods for making multiperiod forecasts are discussed in Section 16.2.

Sections 16.3 and 16.4 return to the topic of Section 14.6, stochastic trends. Section
16.3 introduces additional models of stochastic trends and an alternative test for a unit
autoregressive root. Section 16.4 introduces the concept of cointegration, which arises
when two variables share a common stochastic trend—that is, when each variable
contains a stochastic trend, but a weighted difference of the two variables does not.

In some time series data, especially financial data, the variance changes over
time: Sometimes the series exhibits high volatility, while at other times the volatility
is low, so the data exhibit clusters of volatility. Section 16.5 discusses volatility cluster-
ing and introduces models in which the variance of the forecast error changes over
time, that is, models in which the forecast error is conditionally heteroskedastic. Mod-
els of conditional heteroskedasticity have several applications. One application is
computing forecast intervals, where the width of the interval changes over time to
reflect periods of high or low uncertainty. Another application is forecasting the
uncertainty of returns on an asset, such as a stock, which in turn can be useful in
assessing the risk of owning that asset.

16.1 Vector Autoregressions

Chapter 14 focused on forecasting the growth rate of GDP, but in reality eco-
nomic forecasters are in the business of forecasting other key macroeconomic
variables as well, such as the rate of inflation, the unemployment rate, and interest
rates. One approach is to develop a separate forecasting model for each variable,

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