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14.3 Autoregressions 577

was approximately constant, with the exception of a single devaluation in 1968, in
which the official value of the pound, relative to the dollar, was decreased to
$2.40. Since 1972 the exchange rate has fluctuated over a very wide range.

The index of industrial production for Japan (Figure 14.2c) measures
Japan’s output of industrial commodities. The logarithm of the series is plotted in
Figure 14.2c, and changes in this series can be interpreted as (fractional) growth
rates. During the 1960s and early 1970s, Japanese industrial production grew
quickly, but this growth slowed in the late 1970s and 1980s, and industrial pro-
duction has grown little since the early 1990s.

The Wilshire 5000 stock price index is an index of the share prices of all firms
traded on exchanges in the United States. Figure 14.2d plots the daily percentage
changes in this index for trading days from January 2, 1990, to December 31, 2013 (a
total of 4003 observations). Unlike the other series in Figure 14.2, there is very little
serial correlation in these daily percentage changes; if there were, then you could
predict them using past daily changes and make money by buying when you expect
the market to rise and selling when you expect it to fall. Although the changes are
essentially unpredictable, inspection of Figure 14.2d reveals patterns in their volatil-
ity. For example, the standard deviation of daily percentage changes was relatively
large in 1998–2003 and 2007–2008, and it was relatively small in 1994 and 2004. This
“volatility clustering” is found in many financial time series, and econometric models
for modeling this special type of heteroskedasticity are taken up in Section 16.5.

14.3 Autoregressions

How fast will GDP grow over the next year? Will growth be strong, so it will be a
good year for the U.S. economy, or weak—perhaps even negative—signaling that
the economy will be in a recession? Firms use growth forecasts when they forecast
sales of their products, and local governments use growth forecasts when they
develop their budgets for the upcoming year. Economists at central banks, like the
U.S. Federal Reserve Bank, use growth forecasts when they set monetary policy.
Wall Street investors rely on growth forecasts when deciding how much to pay for
stocks and bonds. In this section, we consider forecasts made using an autoregression,
a regression model that relates a time series variable to its past values.

The First-Order Autoregressive Model

If you want to predict the future of a time series, a good place to start is in the
immediate past. For example, if you want to forecast the rate of GDP growth
in the next quarter, you might see how fast GDP grew in the last quarter.