CHAPTER 3. STATISTICS AND TIME SERIES SOLUTIONS University of Alabama EC 410

Gloria González-Rivera Forecasting For Economics and Business 2013


CHAPTER 3.
STATISTICS AND TIME SERIES

SOLUTIONS
by
Wei Lin and Yingying Sun
(University of California, Riverside)



Exercise 1
a. Let RP CE and RDP I denote “real personal consumption expenditure” and “real disposable
personal income” respectively. Their growth rates are calculated as follows,

G RP CEt = 100 × [log(RP CEt ) − log(RP CEt−1 )]
G RDP It = 100 × [log(RDP It ) − log(RDP It−1 )].

Figure 1 and Figure 2 plot G RP CEt and G RDP It respectively. From visual inspection of the
graphs, we can see that the growth rate of consumption has a lower volatility when compared
with the volatility of the growth rate of disposable income. G RP CE fluctuates mainly within
±2%, while G RDP I within ±4%. This phenomenon can be explained by the permanent income
hypothesis, which argues that people, preferring a smooth path for consumption, will base their con-
sumption on an average of their income over time rather than on their current income. Therefore,
a large fluctuation in the current disposable income will only translate into a smaller fluctuation in
consumption expenditure.

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1
Percent




0

-1

-2

-3
60 65 70 75 80 85 90 95 00 05 10

G_RPCE (%)


Figure 1: Time Series Plot of G RP CE




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,Gloria González-Rivera Forecasting For Economics and Business 2013



6

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2

Percent 0

-2

-4

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60 65 70 75 80 85 90 95 00 05 10

G_RDPI (%)


Figure 2: Time Series of G RDP I


b. Estimate the following regression model in EViews,

G RP CEt = β0 + β1 G RDP It + ut .

Table 1 reports the estimation results. In the model, both estimates of the intercept and the
coefficient of the growth rate of disposable income are statistically significant (their p-values are 0).
The adjusted R-squared is approximately 0.052, meaning that about 5% of total sample variation
of the dependent variable G RP CE is explained by the independent variable G RDP I. Observe
that a very statistical regressor does not imply necessarily a great fit. The estimate β̂1 = 0.17
means that, on average, 1% monthly increase in the growth rate of real disposable income results
in 0.17% increase in the growth rate of real personal consumption, giving some support for the
permanent income hypothesis.

Dependent Variable: G RPCE
Method: Least Squares
Sample (adjusted): 1959M02 2012M04
Included observations: 639 after adjustments
Newey-West HAC Standard Errors & Covariance (lag truncation=6)
Variable Coefficient Std. Error t-Statistic Prob.
C 0.225422 0.020710 10.88443 0.0000
G RDPI 0.174567 0.037659 4.635497 0.0000
R-squared 0.053117 Mean dependent var 0.271757
Adjusted R-squared 0.05163 S.D. dependent var 0.546044
S.E. of regression 0.531761 Akaike info criterion 1.57788
Sum squared resid 180.1243 Schwarz criterion 1.591839
Log likelihood -502.133 F-statistic 35.73339
Durbin-Watson stat 2.377045 Prob(F-statistic) 0.000000

Table 1: Regression Results for Exercise 1b



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, Gloria González-Rivera Forecasting For Economics and Business 2013


c. Add a lag of the growth in disposable income to the equation estimated in b, and estimate the
following regression model,

G RP CEt = β0 + β1 G RDP It + β2 G RDP It−1 + ut .

Table 2 reports the estimation results. The estimate of the coefficient of the newly added lagged
term (G RDP It−1 ) is statistically significant with p-value less than 0.18%. Therefore, there may be
a response of consumption growth to changes in income growth over time: 1% increase in growth
in disposable real income in the last period on average results in a 0.08% increase of growth in real
personal consumption in the current period. If we add the impact effect (0.187) and the one-month
lag effect (0.082), we have a total marginal effect on consumption growth of 0.27%, which is larger
than that in Table 1.
The student may want to experiment with additional lags in the regression model and check whether
there is statistical evidence for a one-to-one effect of income on consumption.
Dependent Variable: G RPCE
Method: Least Squares
Sample (adjusted): 1959M03 2012M04
Included observations: 638 after adjustments
Newey-West HAC Standard Errors & Covariance (lag truncation=6)
Variable Coefficient Std. Error t-Statistic Prob.
C 0.19887 0.02333 8.524343 0.0000
G RDPI 0.187269 0.036346 5.152463 0.0000
G RDPI(-1) 0.08286 0.026384 3.140473 0.0018
R-squared 0.064791 Mean dependent var 0.270540
Adjusted R-squared 0.061846 S.D. dependent var 0.545605
S.E. of regression 0.528464 Akaike info criterion 1.567006
Sum squared resid 177.339 Schwarz criterion 1.587970
Log likelihood -496.875 F-statistic 21.99642
Durbin-Watson stat 2.409702 Prob(F-statistic) 0.000000

Table 2: Regression Results for Exercise 1c

Exercise 2
Let CP I denote the monthly Consumer Price Index. The monthly inflation rate IN F LRAT E is,

IN F LRAT Et = 100 × [log(CP It ) − log(CP It−1 )].

Let N OM RAT E AN N denote the 3-month T-bill interest rate downloaded from the FRED. Note
that the interest rate is annualized, therefore, the corresponding monthly interest rate N OM RAT E
should be, " 1 #
N OM RAT E AN Nt 12
N OM RAT Et = 100 × 1+ −1 .
100
Then, the ex post monthly real interest rate REALRAT E is the difference between monthly
nominal interest rate N OM RAT Et and monthly inflation rate IN F LRAT Et ,

REALRAT Et = N OM RAT Et − IN F LRAT Et .

Add the real interest rate to the regression model in Exercise 1b,

G RP CEt = β0 + β1 G RDP It + β2 REALRAT Et + ut .

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