page 1
EC 385
9.2
Problem Set 8 Answers
(a)
Considering each of the coefficients in turn, we have the following interpretations.
Intercept: At the beginning of the time period over which observations were taken,
on a day which is not Friday, Saturday or a holiday, and a day which has neither a
full moon nor a half moon, the average number of emergency room cases was 94.
T: The average number of emergency room cases has been increasing by 0.0338 per
day.
HOLIDAY: The average number of emergency room cases goes up by 13.9 on
holidays.
FRI and SAT: The average number of emergency room cases goes up by 6.9 and
10.6 on Fridays and Saturdays, respectively.
FULLMOON: The average number of emergency room cases goes up by 2.45 on
days when there is a full moon. However, a null hypothesis stating that a full moon
has no influence on the number of emergency room cases would not be rejected.
NEWMOON: The average number of emergency room cases goes up by 6.4 on days
when there is a new moon. However, a null hypothesis stating that a new moon has
no influence on the number of emergency room cases would not be rejected.
(b) See SAS output below.
(c) The null and alternative hypotheses are
H 0 : 6  7  0
H1 : 6 or 7 is nonzero.
The test statistic is
F
( SSER  SSEU ) 2 (27424  27109 ) / 2
=
SSEU (229  7)
27109 / 222

157.5
 1.29
122.11
where SSE R = 27424 is the sum of squared errors from the estimated equation with
FULLMOON and NEWMOON omitted and SSEU = 27109 is the sum of squared
errors from the estimated equation with these variables included. The calculated
value of the F statistic is 1.29. The critical F value at a 5% level of significance is
approximately 3.00  we fail to reject the null hypothesis that new and full moons
have no impact on the number of emergency room cases.
The REG Procedure
Model: MODEL1
Dependent Variable: calls
Analysis of Variance
Sum of
Mean
DF
Squares
Square
Source
Model
Error
Corrected Total
6
222
228
5693.37691
27109
32802
Root MSE
Dependent Mean
11.05042
100.56769
948.89615
122.11182
R-Square
Adj R-Sq
F Value
Pr > F
7.77
<.0001
0.1736
0.1512
page 2
Coeff Var
Variable
DF
Intercept
t
hol
fri
sat
full
new
10.98804
Parameter Estimates
Parameter
Standard
Estimate
Error
1
1
1
1
1
1
1
93.69583
0.03380
13.86293
6.90978
10.58940
2.45445
6.40595
t Value
Pr > |t|
60.09
3.06
2.15
3.27
5.00
0.62
1.50
<.0001
0.0025
0.0326
0.0012
<.0001
0.5382
0.1338
1.55916
0.01105
6.44517
2.11132
2.11843
3.98092
4.25689
******************************************************************************
The REG Procedure
Model: MODEL2
Dependent Variable: calls
Analysis of Variance
DF
Sum of
Squares
Mean
Square
4
224
228
5378.00978
27424
32802
1344.50245
122.42942
Root MSE
Dependent Mean
Coeff Var
11.06478
100.56769
11.00232
Source
Model
Error
Corrected Total
R-Square
Adj R-Sq
F Value
Pr > F
10.98
<.0001
0.1640
0.1490
Parameter Estimates
Variable
DF
Parameter
Estimate
Standard
Error
t Value
Pr > |t|
Intercept
1
94.02146
1.54585
60.82
<.0001
t
1
0.03383
0.01107
3.06
0.0025
hol
1
13.61679
6.45107
2.11
0.0359
fri
1
6.84914
2.11367
3.24
0.0014
sat
1
10.34207
2.11533
4.89
<.0001
*****************************************************************************
Chapter 11, Exercise 11.7
(a)
The least squares estimates of equation (11.7.5) are
y t = 2.243 + 0.164 xt + 1.145 nt
(2.669) (0.035) (0.414)
R2 = 0.45
These results suggest that an increase in income of $1000 will increase food
expenditure by $164; an additional person in the household will increase food
expenditure by $1,145. Both the estimated slope coefficients are significantly
different from zero.
page 3
(b) See Figures 11.2 and 11.3. Overall, the residuals tend to increase in absolute value as
x increases and as n increases. Thus, the plots suggest the existence of
heteroskedasticity that is dependent on both xt and nt.
10
5
RESID
0
-5
-10
-15
20
40
60
80
100
X
Figure 11.2 Residuals Plotted Against Income.
10
5
RESID
0
-5
-10
-15
0
2
4
6
8
N
Figure 11.3 Residuals Plotted Against Number of Persons
(c)
(i) To perform the first Goldfeld-Quandt test we order the observations according
to decreasing values of xt. Then, we find the least squares regression of
yt  1   2 xt   3 nt  et for both the first and second halves of the
observations, to obtain estimates  12 and  22 , respectively. We find that  12 =
31.129 and  22 = 5.8819. Although we are not hypothesizing constant error
variances within each subsample, to perform the Goldfeld-Quandt test we
proceed as if H0 and H1 are given by H0: 12   22 and H1:  22  12 . The test
statistic value is:
GQ 
ˆ 12 31.129

 5.2923
ˆ 22 5.8819
page 4
The 5% critical value for (16, 16) degrees of freedom is approximately Fc =
2.35. Thus, because GQ = 5.2923 > Fc = 2.35, we reject H0 and conclude that
heteroskedasticity exists, and is dependent on xt.
(ii) When we order the observations with respect to nt , there is not a unique
ordering because nt takes on repeated integer values. There are 8 observations
where nt = 3. One of these values must be included in the first 19 observations,
the other 7 in the last 19 observations.
GQ 
 12 28.233

 2.88
 22 9.799
This value is greater than 2.35, and so we reject a null hypothesis of
homoskedasticity and conclude that the error variances are dependent on nt.
These test outcomes are consistent with the evidence provided by the residual
plots in part (b).
(d) The alternative variance estimators (White) yield the following standard errors,
which we can compare to the Least squares standard errors.
Standard Errors
Coefficients
White
Least Squares
2
0.00082231  0.0287
0.0354
3
0.1898  0.4357
0.4140
The results from White's variance estimator suggest the usual least squares results
would underestimate the reliability of estimation for 2 and overestimate the
reliability of estimation for 3.
(e)
1) t2 = 2Xt
2) t2 = 2nt2
First we estimate this model:
Yt
x
n
e

 1   2 t  3 t  t
xt
xt
xt
xt
xt
See the output below, and confirm that we got the following parameter estimates. It is important
that you read the SAS output (and code) to understand what is what with these
parameter estimates.
b1 = 1.94785
b2 = 0.16210
b3 = 1.26610
page 5
Q: What is the predicted effect of an increase in household size of one person on food
expenditure?
A: If the number of persons increases by 1, food expenditures are predicted to increase by
$1,266.
Then we estimate this model:
Yt 1
x
n
e

  2 t  3 t  t
nt nt
nt
nt nt
See the output below, and confirm that we got the following parameter estimates. It is important
that you read the SAS output (and code) to understand what is what with these
parameter estimates.
b1 = 3.03932
b2 = 0.17086
b3 = 0.75978
Q: What is the predicted effect of an increase in household size of one person on food
expenditure?
A: If the number of persons increases by 1, food expenditures are predicted to increase by $760.
The REG Procedure
Model: MODEL1
Dependent Variable: y
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Corrected Total
2
35
37
526.07506
644.35366
1170.42872
263.03753
18.41010
Root MSE
Dependent Mean
Coeff Var
4.29070
15.95282
26.89619
R-Square
Adj R-Sq
F Value
Pr > F
14.29
<.0001
0.4495
0.4180
Parameter Estimates
Parameter
Standard
Variable
DF
Estimate
Error
t Value
Pr > |t|
Intercept
1
2.24323
2.66878
0.84
0.4063
x
1
0.16446
0.03540
4.65
<.0001
n
1
1.14506
0.41442
2.76
0.0091
******************************************************************************
The REG Procedure
Model: MODEL1
page 6
Dependent Variable: y
Consistent Covariance of Estimates
Variable
Intercept
x
n
Intercept
x
n
5.15290338
-0.053238036
-0.784846872
-0.053238036
0.0008223112
0.0045275225
-0.784846872
0.0045275225
0.1898108421
******************************************************************************
The REG Procedure
Model: big1
Dependent Variable: y
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Corrected Total
2
16
18
132.83854
498.06478
630.90332
66.41927
31.12905
Root MSE
Dependent Mean
Coeff Var
5.57934
18.46368
30.21792
Variable
DF
R-Square
Adj R-Sq
Parameter Estimates
Parameter
Standard
Estimate
Error
F Value
Pr > F
2.13
0.1509
0.2106
0.1119
t Value
Pr > |t|
Intercept
1
-4.04154
11.56096
-0.35
0.7312
x
1
0.26377
0.14898
1.77
0.0957
n
1
0.68938
0.70523
0.98
0.3429
******************************************************************************
The REG Procedure
Model: small1
Dependent Variable: y
Analysis of Variance
Source
DF
Sum of
Squares
Mean
Square
Model
Error
Corrected Total
2
16
18
205.84503
94.11088
299.95591
102.92252
5.88193
Root MSE
Dependent Mean
Coeff Var
2.42527
13.44195
18.04254
R-Square
Adj R-Sq
F Value
Pr > F
17.50
<.0001
0.6863
0.6470
Parameter Estimates
Variable
DF
Parameter
Estimate
Standard
Error
t Value
Pr > |t|
page 7
Intercept
x
n
1
1
1
-2.40541
0.23198
1.75852
3.02705
0.06394
0.36424
-0.79
3.63
4.83
0.4385
0.0023
0.0002
******************************************************************************
The REG Procedure
Model: big2
Dependent Variable: y
Analysis of Variance
Sum of
Squares
Source
DF
Model
Error
Corrected Total
2
16
18
158.90617
451.72298
610.62915
Root MSE
Dependent Mean
Coeff Var
5.31344
17.57326
30.23595
Variable
Intercept
x
n
DF
1
1
1
Mean
Square
F Value
Pr > F
2.81
0.0897
79.45309
28.23269
R-Square
Adj R-Sq
Parameter Estimates
Parameter
Standard
Estimate
Error
1.27007
7.50820
0.12806
0.06321
1.72145
1.19474
0.2602
0.1678
t Value
0.17
2.03
1.44
Pr > |t|
0.8678
0.0598
0.1689
******************************************************************************
The REG Procedure
Model: small2
Dependent Variable: y
Analysis of Variance
Sum of
Squares
Source
DF
Model
Error
Corrected Total
2
16
18
303.23945
156.77783
460.01728
Root MSE
Dependent Mean
Coeff Var
3.13027
14.33237
21.84059
Variable
Intercept
x
n
DF
1
1
1
F Value
Pr > F
15.47
0.0002
151.61973
9.79861
R-Square
Adj R-Sq
Parameter Estimates
Parameter
Standard
Estimate
Error
-1.32179
0.21091
1.70374
Mean
Square
3.55349
0.03792
1.00876
0.6592
0.6166
t Value
Pr > |t|
-0.37
5.56
1.69
0.7148
<.0001
0.1106
******************************************************************************
The REG Procedure
Model: MODEL1
Dependent Variable: ystar
page 8
Source
NOTE: No intercept in model. R-Square is redefined.
Analysis of Variance
Sum of
Mean
DF
Squares
Square
F Value
Model
Error
Uncorrected Total
3
35
38
173.18330
10.03953
183.22283
Root MSE
Dependent Mean
Coeff Var
0.53558
2.11243
25.35369
Variable
DF
x1star
x2star
nstar
57.72777
0.28684
R-Square
Adj R-Sq
Parameter Estimates
Parameter
Standard
Estimate
Error
1
1
1
1.94785
0.16210
1.26610
201.25
Pr > F
<.0001
0.9452
0.9405
t Value
Pr > |t|
0.86
5.04
3.44
0.3938
<.0001
0.0015
2.25597
0.03215
0.36858
******************************************************************************
The REG Procedure
Model: MODEL1
Dependent Variable: ynew
Analysis of Variance
Sum of
Mean
Source
DF
Squares
Square
F Value
Pr > F
Model
Error
Corrected Total
2
35
37
562.06414
50.33957
612.40370
Root MSE
Dependent Mean
Coeff Var
1.19928
5.73668
20.90547
Variable
Intercept
x1new
xnew
DF
1
1
1
281.03207
1.43827
R-Square
Adj R-Sq
Parameter Estimates
Parameter
Standard
Estimate
Error
0.75978
0.36127
3.03932
1.59049
0.17086
0.02081
195.40
<.0001
0.9178
0.9131
t Value
2.10
1.91
8.21
Pr > |t|
0.0427
0.0642
<.0001