Basic Econometrics, Gujarati and Porter
CHAPTER 21:
TIME SERIES ECONOMETRICS: SOME BASIC CONCEPTS
21.1 A stochastic process is said to be weakly stationary if its mean
and variance are constant over time and if the value of the
21.2 If a time series has to be differenced d times before it becomes
21.3 Loosely speaking, the term unit root means that a given time
21.5 The DF test is a statistical test that can be used to determine if a time
21.6 The EG and AEG tests are statistical procedures that can be used to
21.7 Two variables are said to be cointegrated if there is a stable long-run
21.8 Tests of unit roots are performed on individual time series.
21.9 If a nonstationary variable is regressed on another nonstationary
21.10 See the answer to the preceding question.
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21.11 Most economic time series exhibit trends. If such trends are
21.12 If a time series exhibits a deterministic trend, the residuals from
21.13 A random walk is an example of a nonstationary process. If a
variable follows a random walk, it means its value today is equal to
21.15 Cointegration implies a long term, or equilibrium, relationship
Empirical Exercises
21.16 (a) The correlograms for all these time series very much resemble
21.17 The regression results are as follows:
∆log PCE
t
=0.1899 +0.0002t−0.0261 log PCE
t−1
Basic Econometrics, Gujarati and Porter
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∆log DPI
t
=0.1235 +0.0001t−0.0161log DPI
t−1
* This tau value is not statistically significant, suggesting that the
log PDI time series contains a unit root, that is, it is nonstationary.
* This tau value is not statistically significant, suggesting that this
time series has a unit root.
21.18
If the error terms in the model are serially correlated, ADF is the
more appropriate test. The
τ
statistics for the appropriate coefficient
from the ADF regressions for the three series are:
21.19
(a) Probably yes, because individually the two time series
are nonstationary.
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21.20
The scattergrams of the first differences of log DPI, log Profits, and
21.21
In theory there should not be an intercept in the model. But if there
was a trend term in the original model, then an intercept could be
To see this, we first regressed log Dividends on log Profits and the
trend variable, which gave the following results:
Dependent Variable: log DIVIDEND
Variable Coefficient Std. Error t-Statistic
C 0.4358 0.1053 4.14
Now regressing the first differences of log Dividends on the first
differences of log Profits and the intercept, we get the following
results:
Basic Econometrics, Gujarati and Porter
Variable Coefficient Std. Error t-Statistic
C 0.0193 0.0021 9.24
21.22
From the first difference regression given in the preceding exercise,
we can obtain the residuals of this regression (
ˆ
t
u
) and subject them
21.23
(a) Since
τ
is less than the critical
τ
value, it seems that the
(b) Ordinarily, an absolute t value of as much as 2.35 or greater
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21.25
(a) & (b)
Here is the graph of the actual and fitted Y values:
From the given regression results you might think that this
is a “good” regression in that it has a high R
2
and significant
21.26
(a) Regression (1) shows that the elasticity of M1 with respect to
GDP is about 1.60, which seems statistically significant, as the t
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Basic Econometrics, Gujarati and Porter
(b) In the first difference form, there is still positive relationship
(c) & (d) From regression (3) it seems that the two variables are
(e) Equation (2) gives the short-run relationship between the logs
of money and GDP. The equation given here takes into account the
21.27
(a) & (b) The time graph of CPI very much resembles Fig. 21.12.
This graph clearly shows that generally there is an upward trend in
(c) Since Equation (1) omits two variables, we have to use the
F test.
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