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572 Chapter 14 Introduction to Time Series Regression and Forecasting

unit of time such as weeks, months, quarters (3-month units), or years. For
example, the GDP data studied in this chapter are quarterly, so the unit of time
(a “period”) is a quarter of a year.

Special terminology and notation are used to indicate future and past values
of Y. The value of Y in the previous period is called its first lagged value or, more
simply, its first lag, and is denoted Yt - 1. Its jth lagged value (or simply its jth lag) is
its value j periods ago, which is Yt - j. Similarly, Yt + 1 denotes the value of Y one
period into the future.

The change in the value of Y between period t - 1 and period t is Yt - Yt - 1;
this change is called the first difference in the variable Yt. In time series data, “∆” is
used to represent the first difference, so ∆Yt = Yt - Yt - 1.

Economic time series are often analyzed after computing their logarithms or
the changes in their logarithms. One reason for this is that many economic series
exhibit growth that is approximately exponential; that is, over the long run, the
series tends to grow by a certain percentage per year on average. This implies that
the logarithm of the series grows approximately linearly, and is why Figure 14.1a
plots the logarithm of U.S. GDP. Another reason is that the standard deviation
of many economic time series is approximately proportional to its level; that is,
the standard deviation is well expressed as a percentage of the level of the series.
This implies that the standard deviation of the logarithm of the series is approxi-
mately constant. In either case, it is useful to transform the series so that changes
in the transformed series are proportional (or percentage) changes in the original
series, and this is achieved by taking the logarithm of the series.1

Lags, first differences, and growth rates are summarized in Key Concept 14.1.
Lags, changes, and percentage changes are illustrated using the U.S. GDP data
in Table 14.1. The first column shows the date, or period, where the first quarter
of 2012 is denoted 2012:Q1, the second quarter of 2012 is denoted 2012:Q2, and so
forth. The second column shows the value of the GDP in that quarter, the third
column shows the logarithm of GDP, and the fourth column shows the growth rate
of GDP (in percentage points at an annual rate). For example, from the first quar-
ter to the second quarter of 2012, GDP increased from $15,382 to $15,428 billion,

1The change of the logarithm of a variable is approximately equal to the proportional change of that
variable—that is, ln(X + a) - ln(X) ≅ a>X, where the approximation works best when a/X is small
[see Equation (8.16) and the surrounding discussion]. Now, replace X with Yt - 1 and a with∆Yt and
note that Yt = Yt - 1 + ∆Yt. This means that the proportional change in the series Yt between peri-
ods t - 1 and t is approximately ln(Yt) - ln(Yt - 1) = ln(Yt - 1 + ∆Yt) - ln(Yt - 1) ≅ ∆Yt>Yt - 1. The
expression ln(Yt) - ln(Yt - 1) is the first difference of ln(Yt), that is ∆ln(Yt). Thus ∆ln(Yt) ≅ ∆Yt>Yt - 1.
The percentage change is 100 times the fractional change, so the percentage change in the series Yt is
approximately 100∆ln(Yt).