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14.2 Introduction to Time Series Data and Serial Correlation 573

Lags, First Differences, Logarithms, and Growth Rates Key Concept

• The first lag of a time series Yt is Yt - 1; its jth lag is Yt - j. 14.1
• The first difference of a series, ∆Yt, is its change between periods t - 1 and t,

that is ∆Yt = Yt - Yt - 1.
• The first difference of the logarithm of Yt is ∆ln1Yt2 = ln1Yt2 - ln1Yt - 12.
• The percentage change of a time series Yt between periods t - 1 and t is

approximately 100∆ln1Yt2, where the approximation is most accurate when
the percentage change is small.

which is a percentage increase of 100 * (15428 - 15382) >15382 = 0.30%. This is
the percentage increase from one quarter to the next. It is conventional to report

rates of growth in macroeconomic time series on an annual basis, which is the

percentage increase in GDP that would occur over a year if the series were to

continue to increase at the same rate. Because there are four quarters in a year,

the annualized rate of GDP growth in 2012:Q2 is 0.30 * 4 = 1.20, or 1.20%.
This percentage change can also be computed using the differences-of-

logarithms approximation in Key Concept 14.1. The difference in the logarithm

of GDP from 2012:Q1 to 2012:Q2 is ln(15428) - ln(15382) = 0.0030, yielding the
approximate quarterly percentage difference 100 * 0.0030 = 0.30%. On an
annualized basis, this is 0.30 * 4 = 1.20, or 1.20%, the same (to two decimal

TABLE 14.1 GDP in the United States in 2012 and the First Quarter of 2013

Quarter U.S. GDP (billions Logarithm of Growth Rate of GDP at an Annual First Lag,
2012:Q1 of $1996), GDPt GDP, ln(GDPt) Rate, GDPGRt 400 : ∆ln (GDPt) GDPGRt 1

15382 9.641 3.64 4.75

2012:Q2 15428 9.644 1.20 3.64

2012:Q3 15534 9.651 2.75 1.20

2012:Q4 15540 9.651 0.15 2.75

2013:Q1 15584 9.654 1.14 0.15

Note: The quarterly rate of GDP growth is the first difference of the logarithm. This is converted into percentage points at an
annual rate by multiplying by 400. The first lag is its value in the previous quarter. All entries are rounded to the nearest decimal.