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574 Chapter 14 Introduction to Time Series Regression and Forecasting

places), as obtained by directly computing the percentage growth. These calcula-
tions can be summarized as

Annualized rate of GDP Growth = GDPGRt ≅ 4003ln(GDPt) - ln(GDPt - 1)4
= 400∆ln(GDPt), (14.2)

where GDPt is the value of GDP at date t. The factor of 400 arises from converting
fractional change to percentages (multiplying by 100) and converting quarterly
percentage change to an equivalent annual rate (multiplying by 4).

The final column of Table 14.1 illustrates lags. The first lag GDPGR in
2012:Q2 is 3.64%, the value of GDPGR in 2012:Q1.

Figure 14.1b plots GDPGRt from 1960:Q1 through 2012:Q4. It shows sub-
stantial variability in the growth rate of GDP. For example, GDP grew at an
annual rate of over 15% in 1978:Q2 and fell at annual rate of over 8% in 2008:Q4.
Over the entire period, the growth rate averaged 3.1% (which is responsible for
the increase of GDP from $3.1 trillion in 1960 to $15.5 trillion in 2012), and the
sample standard deviation was 3.4%.

Autocorrelation

In time series data, the value of Y in one period typically is correlated with its
value in the next period. The correlation of a series with its own lagged values is
called autocorrelation or serial correlation. The first autocorrelation (or

Key Concept Autocorrelation (Serial Correlation) and Autocovariance

14.2 The j th autocovariance of a series Yt is the covariance between Yt and its j th lag,
Yt - j, and the j th autocorrelation coefficient is the correlation between Yt and Yt - j.
That is,

jth autocovariance = cov(Yt, Yt - j) (14.3)

jthautocorrelation = rj = corr(Yt,Yt - j) = cov(Yt,Yt - j) . (14.4)
2var(Yt)var(Yt - j)

The j th autocorrelation coefficient is sometimes called the j th serial correlation

coefficient.