Welcome to Econ 420
Applied Regression Analysis
Study Guide
Week Fourteen
Answer Key to
Asst 11
(40 points)
• # 10, Page 157
• The Durbin-Watson statistic does not tell you
anything when applied to a cross-section
regression. Data in a cross-section regression
are often in alphabetical order, but that order has
nothing to do with the underlying characteristics
of the data. You could change the order of the
data in a cross-section, and you would get a
different value for the Durbin-Watson statistic.
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
#11, Page 157
The results of the regression are:
Dependent Variable is DVDEXP
Variable
Coefficient
Standard Error
t-Statistic p-Value
C
81.46
23.47
3.47
0.00
INCOME
0.061
0.0098
6.27
0.00
PRICE
-3.12
0.88
-3.54
0.00
RAINFALL
7.52
2.37
3.17
0.00
Observations: 24
R2 = 0.79
Adjusted R2 = 0.75
Residual Sum of Squares = 5038.14
F-statistic = 24.49
Durbin-Watson statistic: 2.32
The Durbin-Watson statistic comes out to 2.32. Since it is greater than 2, a test for
negative autocorrelation should be conducted. From the Durbin-Watson statistic table,
when k=3 and N=24, for a 5% error level, dU=1.10 and dl=1.66. Since we need to check
for negative (rather than positive) autocorrelation, these values must be subtracted from 4.
4-1.66=2.34
4-1.10=2.90
Since the Durbin-Watson statistic from our results, 2.32, is lower than 2.34, we do not reject
the null hypothesis of no autocorrelation (with a 5% error level). There is not enough
evidence of autocorrelation to justify taking any corrective action, but it is close though.
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
#12, Page 157
The results are:
Dependent Variable is DVDEXP
Variable
Coefficient
Standard Error t-Statistic
p-Value
C
71.18
24.99
2.85
0.01
INCOME
0.062
0.0088
7.05
0.00
PRICE
-2.78
0.98
-2.82
0.01
RAINFALL
8.29
2.30
3.60
0.00
AR(1)
-0.22
0.23
-0.95
0.36
Observations: 23
R2 = 0.78
Adjusted R2 = 0.73
Residual Sum of Squares = 4597.67
F-statistic = 15.82
Durbin-Watson statistic: 1.67
Comparing these results to those of the last question, they are not exactly the
same, but similar. Note that the AR(1) term is statistically insignificant.
This is consistent with the fact that we did not find autocorrelation in the
previous question.
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
b.
#13, Page 157
Results for Microsoft Revenue Model Including T
Dependent Variable is REVENUE
Variable
Coefficient
Standard Error
T-Statistic
P-Value
Constant
-49.99
88.32
-0.57
0.57
MARKETING 8.07
0.73
10.97
0.00
SECONDQ
10.84
80.77
0.13
0.89
THIRDQ
124.24
84.21
1.48
0.15
FOURTHQ
75.97
82.01
0.93
0.36
T
-25.81
8.06
-3.20
0.00
Observations: 55
R2 = 0.96
Adjusted R2 = 0.96
Residual Sum of Squares = 2,220,798
F-statistic = 247.41
Durbin-Watson statistic: 0.68
Yes, it is statistically significant at even a 1% error level. It only adds to our
understanding slightly. It tells us that Microsoft’s real revenues have been falling over
time when MARKETING and the quarter dummy variables are accounted for. It
would really add to our understanding if we could figure out why this is happening, so
the T variable doesn’t really tell us that much.
c. Using a 5% error level, dU=1.77, dL=1.34. The Durbin-Watson statistic from our
results, 0.68, is below dL=1.34. The null hypothesis of no autocorrelation is rejected
at a 5% error level; there is evidence of autocorrelation. This is the same result that
we got originally (Table 7.B) when we did not include T in the model.
• In this graph, there are 5
different observations on each
X (each representing a
different value of Y)
• Note that the errors on each
observation of X have a mean
of zero and the same variance
across all values of Xs
y
Chapter 8: Homoskedasticity
(up to Page 173)
10
9
8
7
6
5
4
3
2
1
0
0
2
4
x
6
Heteroskedasticity
•
•
Suppose X is income and Y is
consumption and we have cross
sectional data.
Note that at low levels of income
there is not much variation in
consumption but at higher levels if
income there is more variation in
consumption.
The reason is that families that
have little income spend a large
portion (maybe all) of their
income. But families that have a
large amount of income vary
greatly on how much of their
income the spend.
– That is, the error on each
observation of X comes from a
distribution with a mean of zero
but a different variance
– In this case the variance of error
increases as the level of income
increases.
•
Heteroskedacity is problem that is
more common in cross sectional
data sets.
Censored from below
10
8
6
y
•
4
2
0
0
2
4
x
6
Consequences
Heteroskedasticity
• Unbiased estimates (if the functional
form is correct and there are no missing
variables) but wrong standard errors
– OLS tend to underestimate the standard
errors
– You should know how this affects the
results of the t-test of significance.
A casual way to look for
heteroskedasticity
• Run the regression
• Look at the graph of residuals
• Is there a sign of heteroskedasticity?
– If it looks like the residuals are not evenly distributed
around a mean of zero, then yes.
• But we need a more formal test
• Note: We will not cover the Park Test
The White Test
•
•
•
•
•
Set the null and alternative hypotheses:
– Ho: homoskedasticity
– Ha: heteroskedasticity
Estimate the original regression
Use the squared residuals as a dependent
variable in a second equation that includes
Xs, X2s and product of each pair of Xs
• Find nR2
• Find Critcal chi-squared on page 324 (df =
number of of independent variables in the
second equation)
• If nR2 > critical chi-squaredreject Ho
EViews does the estimations
automatically
•
•
•
•
Estimate the original regression as usual
On the regression output click on “View”
Then on “Residual Test”
The choose White Heteroskedastcity
(cross terms included)
• On the top of your output, you will see
observations times R2
Asst 12
Due: Sunday, December 2
before 10 PM
(20 points)
Questions #13 and 14 on Page 181
Note
• We will have classes during the week of
December 3
• Our first class is on Tuesday, December 4
at 8 AM in Thomas 223.
– Don’t miss it.