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

1000000

124543776.3

1100

1210000

464.3909849

1124

1263376

1439589654

1166

1359556

561947.5822

1240

1537600

1346859740

1382

1909924

10918731.32

1613

2601769

1441629340

1616

2611456

889683797.1

1818

3305124

357718488.3

1860

3459600

8700467.328

2426

5885476

30898941.04

2535

6426225

109527336

2640

6969600

893060053.9

2821

7958041

542088929.5

3035

9211225

50427931.87

3233

10452289

302103622.8

3278

10745284

151343624.5

3564

12702096

649534469.6

3630

13176900

5674345602

3794

14394436

7185974738

3945

15563025

366251383.2

4181

17480761

4072383680

4290

18404100

2123390606

4524

20466576

975773446.1

4958

24581764

2472580407

5044

25441936

171531825.7

5094

25948836

487781730.7

5235

27405225

515791803.6

5259

27657081

4017875529

5706

32558436

1855981642

6018

36216324

9905582453

6473

41899729

1207709802

6902

47637604

8280285.049

6952

48330304

5627096725

7859

61763881

1313168202

9460

89491600

2426505575

11061

122345721

1175526129

11686

136562596

1015955886

12030

144720900

1078000806

13726

188403076

5595257545

18022

324792484

7417663339

18153

329531409

1161983188

31217

974501089

There are some values that are way out. Heteroscedacity is not present, just

There are some very weird constituents with high (or low) crime rate.

iii

One should exclude the outliers. Ie countys with unusually high or low pop or crime

Result is an OLS model that only applies to "normal" areas.

Alternatively, there are some more complicated estimation techniques that take

heteroscedacity into account. (GARCH). However, the resulting equation wont

be BLUE.

Finaly one should use population density instead of population to predict crime.

Population density might be a better explanatory variable for crime

iv

Correlating crime rate and population density (pop/area)

 Dependent variable is CRIM93

 51 observations used for estimation from    1 to   51

*******************************************************************************

 Regressor              Coefficient       Standard Error         T-Ratio[Prob]

 CONSTANT                 299992.9            53053.6             5.6545[.000]

 POPDEN                  -189.9849           143.7568            -1.3216[.192]

*******************************************************************************

 R-Squared                    .034417   R-Bar-Squared                  .014711

 S.E. of Regression          358976.9   F-stat.    F(  1,  49)    1.7465[.192]

 Mean of Dependent Variable  277567.9   S.D. of Dependent Variable    361646.9

 Residual Sum of Squares     6.31E+12   Equation Log-likelihood      -723.6874

 Akaike Info. Criterion     -725.6874   Schwarz Bayesian Criterion   -727.6192

* D:Heteroscedasticity*CHSQ(   1)=   .59156[.442]*F(   1,  49)=   .57503[.452]*

doing the density gets rid of heteroscedacity, however, popdensity is not

significant. But this model is restricted. Instead we can use the 2 variables,

pop and area, separately:

 Dependent variable is CRIM93

 51 observations used for estimation from    1 to   51

*******************************************************************************

 Regressor              Coefficient       Standard Error         T-Ratio[Prob]

 CONSTANT                 -48292.5            17359.5            -2.7819[.008]

 POP93                     62.0446             2.0326            30.5253[.000]

 AREA                      .068207            .051916             1.3138[.195]

*******************************************************************************

 R-Squared                     .95197   R-Bar-Squared                   .94997

* D:Heteroscedasticity*CHSQ(   1)=   2.2226[.136]*F(   1,  49)=   2.2327[.142]*

As seen, area is not signifficant, and heteroscedacity has increased, although it

is not critical. This is the best i can do, i am afraid