|
Q4 Take home exam |
|
|
|
|
|
|
a. |
a) Compute the OLS regression of the number of crimes on population
and |
|
|
|
|
|
|
population density for all 51 observations. Test (using both the |
|
|
|
|
|
|
Goldfeld-Quandt test and the test used by Micro-Fit) whether the
null |
|
|
|
|
|
|
hypothesis that the residuals of the estimated equation are
homoscedastic |
|
|
|
|
|
|
can be accepted. Why might the
two tests give different results? |
|
|
|
|
|
|
Dependent
variable is CRIM93 |
|
|
|
|
|
|
51
observations used for estimation from
1 to 51 |
|
|
|
|
|
|
******************************************************************************* |
|
|
|
|
|
|
Regressor
Coefficient Standard
Error T-Ratio[Prob] |
|
|
|
|
|
|
CONSTANT
-37411.8
15369.2
-2.4342[.019] |
|
|
|
|
|
|
POP93 62.3154 2.0370 30.5915[.000] |
|
|
|
|
|
|
******************************************************************************* |
|
|
|
|
|
|
R-Squared
.95025 R-Bar-Squared .94923 |
|
|
|
|
|
|
S.E. of
Regression 81486.7 F-stat. F( 1, 49)
935.8409[.000] |
|
|
|
|
|
|
Mean of
Dependent Variable 277567.9 S.D. of Dependent Variable 361646.9 |
|
|
|
|
|
|
Residual
Sum of Squares 3.25E+11 Equation Log-likelihood -648.0637 |
|
|
|
|
|
|
Akaike
Info. Criterion -650.0637 Schwarz Bayesian Criterion -651.9955 |
|
|
|
|
|
|
* A:Serial Correlation*CHSQ( 1)=
3.2907[.070]*F( 1, 48)=
3.3108[.075]* |
|
|
|
|
|
|
* B:Functional Form *CHSQ( 1)= 5.6735[.017]*F( 1, 48)= 6.0081[.018]* |
|
|
|
|
|
|
* C:Normality *CHSQ( 2)=
139.6596[.000]* Not
applicable * |
|
|
|
|
|
|
* D:Heteroscedasticity*CHSQ( 1)=
2.7690[.096]*F( 1, 49)=
2.8131[.100]* |
|
|
|
|
|
|
Microfit test: |
|
|
|
|
|
|
H0:errors have an increasing variance |
|
|
|
|
|
|
H1:errors have the same variance |
|
|
|
|
|
|
For X2 p value is 0.096 |
|
|
|
|
|
|
For F test it is 0.1 |
|
|
|
|
|
|
At 5% level accept H0, there is no heteroscedacity |
|
|
|
|
|
|
|
|
|
|
|
|
|
Goldfeld-Quant |
|
|
|
|
|
|
H0:errors have an increasing variance |
|
|
|
|
|
|
H1:errors have the same variance |
|
|
|
|
|
|
I let c=11 and order the population |
|
|
|
|
|
|
for the first 20 |
|
|
|
|
|
|
Regressor
Coefficient Standard
Error T-Ratio[Prob] |
|
|
|
|
|
|
CONSTANT
970.1958 9217.7 .10525[.917] |
|
|
|
|
|
|
POP93 46.2374 6.8773 6.7232[.000] |
|
|
|
|
|
|
R-Squared |
|
0.71519 |
|
|
|
|
for the last 20 |
|
|
|
|
|
|
Regressor
Coefficient Standard
Error T-Ratio[Prob] |
|
|
|
|
|
|
CONSTANT
-103894.2
49308.7
-2.1070[.049] |
|
|
|
|
|
|
POP93 66.8360 4.2202 15.8370[.000] |
|
|
|
|
|
|
******************************************************************************* |
|
|
|
|
|
|
R-Squared |
|
0.93304 |
|
|
|
|
lamda=RSS2/RSS1=(1-r2(2))/(1-r2(1))= |
|
|
|
0.23510 |
|
|
df=(51-11-4)/2= |
|
18 |
Fcrit= |
2.2 |
|
|
Thus there is likely homoscedacity |
|
|
|
|
|
|
The second model is so much more restrictive. It depends on the c
value used etc |
|
|
|
|
|
|
The first model uses much more complicated techniques to spot
heterosced. Not |
|
|
|
|
|
|
just assuming that the error term variance depends on the square of |
|
|
|
|
|
ii |
b) Plot scatter graphs of the squared residuals from the
estimated |
|
|
|
|
|
|
equation against population and population squared, Do these plots |
|
|
|
|
|
|
provide additional help to enable you to decide whether
heteroscedasticity |
|
|
|
|
|
|
is present in your estimated equation? |
|
|
|
|
|
|
RES2 |
POP93 |
POP2 |
|
|
|
|
766719095.1 |
470 |
220900 |
|
|
|
|
595208004.6 |
576 |
331776 |
|
|
|
|
4820165130 |
579 |
335241 |
|
|
|
|
1118492260 |
598 |
357604 |
|
|
|
|
245872698.4 |
637 |
405769 |
|
|
|
|
779652136.5 |
698 |
487204 |
|
|
|
|
195252979.3 |
716 |
512656 |
|
|
|
|
639508012.1 |
841 |
707281 |
|
|
|
|
403465632.3 |
1000 |