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14.2 Introduction to Time Series Data and Serial Correlation 571

Figure 14.1 The Logarithm and the Growth Rate of Real GDP in the United States, 1960–2012

Logarithm
9.75
9.50
9.25
9.00
8.75
8.50
8.25
8.00

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015
(a) US GDP ($1996, Billions)

Percent at an annual rate
20
15
10
5
0
–5
–10
1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015

(b) Growth Rate in US GDP

GDP increased from $3 trillion per year in 1960 to over $15 trillion per year in 2012, when measured in inflation-
adjusted $1996. This five-fold increase corresponds to an increase of 1.6 log points. The growth rate of GDP was not
constant, and it varied considerably from quarter to quarter.

The rate of growth was not constant, however, and the figure shows
declines in GDP during the recessions of 1960–1961, 1970, 1974–1975, 1980, 1981–1982,
1990–1991, 2001, and 2007–2009, episodes denoted by shading in Figure 14.1.

Lags, First Differences, Logarithms, and Growth Rates

The observation on the time series variable Y made at date t is denoted Yt, and the
total number of observations is denoted T. The interval between observations—
that is, the period of time between observation t and observation t + 1—is some