(2.b) Check the normality assumption of the residuals. Interpret the regression estimates and the DW statistics.
All three normality tests fail to reject the null hypothesis of normality, i.e., the normality assumption is satisfied.
Durbin-Watson statistic = 1.10721 < 1.6 (when DW > 2.4 or DW < 1.6, we conclude that the residuals are not independent). There is a positive autocorrelation between residual of the current year and the residual one-year ago. 注：在reg过程的model 语句里加DWPROB，可算出DW统计量对应的P值。
GDP has a positive significant effect on Violent crime rate, every 1 billion dollars increase in GDP is associated 1ith 0.292 violent crime cases increase per 100,000 people.
Consumption has a significant negative effect on violent crime. Every 1 billion dollars increase in consumptions is associated with 0.439 violent crime cases drop per 100,000 population.
Private investment has a significant negative effect on violent crime. Every one billion dollars increase in private investment would reduce the violent crime cases by 0.2358 per 100,000 population.
Fixed investment and government expenditures both have positive insignificant effect on crimes State-local expenditures has a positive significant effect on violent crime. Every 1 billion dollars increase in state-local expenditure is associated with 0.4249 reduction in violent crimes per 100,000 persons. Unemployment rate has a significant positive effect on crime. Every 1 percentage point increase in unemployment rate is associated with 9.377 crime increase per 100,000 persons.
Poverty rate has a significant negative effect on crime. Every 1 percentage point increase in total poverty rates is associated with 12.22 crime drop per 100,000 persons.
(2.c) Create the lag-crime rate.
(2.d) Run a regression model with violent crime rate as the response variable and Lag-crime, Consumption,
6
private investment, Fixed investment, Government expenditure, State and local expenditures,
Unemployment rate, GDP, and Total poverty rate as input variables. Save Residuals and Durbin-Watson (DW) statistics.
Analysis of Variance Sum of Mean
Source DF Squares Square F Value Pr > F
Model 9 1241703 137967 604.17 <.0001
Error 33 7535.81511 228.35803 Corrected Total 42 1249239
Root MSE 15.11152 R-Square 0.9940 Dependent Mean 491.40930 Adj R-Sq 0.9923 Coeff Var 3.07514 Parameter Estimates
Parameter Standard
Variable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 66.04552 34.73774 1.90 0.0660
GDP GDP 1 0.03880 0.02704 1.43 0.1608
consumption consumption 1 -0.13838 0.03784 -3.66 0.0009
private_in private invest 1 -0.09101 0.04365 -2.08 0.0449
Fixed_invest Fixed invest 1 0.09774 0.04210 2.32 0.0266
Government_exp Government_exp 1 0.05685 0.01883 3.02 0.0049
unemployment_rate unemployment rate 1 -13.31340 3.42796 -3.88 0.0005
State_local State_local 1 0.39126 0.10404 3.76 0.0007
povertyrate_total povertyrate_total 1 1.39968 2.33362 0.60 0.5527
7
lagR_Violent_crime 1 0.88047 0.08570 10.27
<.0001
Durbin-Watson D 2.225
(2.e) Check the normality assumption of the residuals. Interpret the regression estimates and the DW statistics.
All three normality tests fail to reject the null hypothesis of normality, i.e., the normality assumption is satisfied.
When lagged values of the dependent variable are used as explanatory variables, Durbin-Watson statistic is not appropriate to be used to test first-order autocorrelation of residuals. Use the following program command:
proc autoreg data=Sasuser.Q2; GOVERNMENT_EXP STATE_LOCAL UNEMPLOYMENT_RATE GDP
POVERTYRATE_TOTAL/lagdep=LAGRCRIMERATE; quit;
we can derive Durbin h statistics.
Durbin h -0.9400 Pr < h 0.1736 So the residuals are independent at the 5% significant level.
With the inclusion of last year’s crime in the model, GNP and total poverty rate become insignificant. Last year’s crime is positively associated with crimes of current year. The fixed investment and government expenditures have significant positive effects after the lag-crime adjustment. The effects of other variables remain the same.
(2.f) Test the hypothesis that both GDP and total poverty rate have no effects on violent crime when variables of
Lag-crime, Consumption, private investment , Fixed investment, Government expenditure, State and local expenditures, Unemployment rate are included in the model at 5% level of significance.
Average explaining power by the variables under test=SS (GDP, total poverty rate / Lag-crime, Consumption,
private investment, Fixed investment, Government expenditure, State and local expenditures, Unemployment rate) / 2= (926 + 82 )/2=1008/2=504
8
model R_VIOLENT_CRIME = LAGRCRIMERATE CONSUMPTION PRIVATE_INVEST FIXED_INVEST
run;
Average variation of errors= SSE /(33)= 228;
F-statistics = 504/228 = 2.2105 < 3.28492 (95 percentile of F2,33 is 3.28492). 也可用test语句，”test GDP,povertyrate_total;”来得到F值和P值
Conclusion: Fail to reject the null hypothesis. The two variables have no significant additional effect on response variable, violent crime when the other 7 variables are included in the model at 5% level.
3. Use the same data of question 2.
(3.a) Run a regression model with total Poverty rate as the response variable and GDP, Consumption, private
investment, Fixed investment, Government expenditure, State and local expenditures, Unemployment rate and Violent crime rate as input variables. Save Residuals and Durbin-Watson (DW) statistics.
The REG Procedure Model: MODEL1
Dependent Variable: povertyrate_total povertyrate_total
Number of Observations Read 58 Number of Observations Used 44 Number of Observations with Missing Values 14
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 8 274.39398 34.29925 19.84 <.0001
Error 35 60.49398 1.72840 Corrected Total 43 334.88795
Root MSE 1.31469 R-Square 0.8194 Dependent Mean 14.00682 Adj R-Sq 0.7781 Coeff Var 9.38604
9
Parameter Estimates
Parameter Standard
Variable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 15.79484 1.27538 12.38 <.0001
GDP GDP 1 0.00520 0.00215 2.42 0.0209
consumption consumption 1 -0.00146 0.00382 -0.38 0.7057
private_invest private invest 1 -0.00791 0.00374 -2.12 0.0413
Fixed_invest Fixed invest 1 0.00131 0.00363 0.36 0.7201
Government_exp Government_exp 1 0.00105 0.00158 0.66 0.5125
unemployment_rateunemployment rate 1 0.84402 0.18014 4.69 <.0001
State_local State_local 1 -0.02709 0.00824 -3.29 0.0023
R_Violent_crime 1 -0.02225 0.00615 -3.62
0.0009
Durbin-Watson D 0.735 Pr < DW <.0001
(3.b) Check the normality assumption of the residuals. Interpret the regression estimates and the DW statistics.
All three normality tests fail to reject the null hypothesis of normality, i.e., the normality assumption is satisfied.
Durbin-Watson statistic = 0.735168 < 1.6. There exists positive autocorrelation
10