计量经济学econometricsreport-crimerate

1、The Effects of Metropolis, Police & Prison Population Size on Violent Crime Rate- Using cross-sectional data in 1980By Ye TAOMarch. 2012.1. Introduction This report investigates the correlation between violent crime rate and its potential significant variables. The sample used is from 51 states of U

2、S in 1980. In previous literatures, Levitt (1996) found every incremental prisoner implies a reduction of crime rate by 15%, and police force is positively correlated to crime rate due to the higher crime rate increase demand for police force. Raphael (2001) and West (1986) state that unemployment p

3、ositively contributes to crime. The age-crime curve rises to its peak in the teen-ages and then falls, arising significance of age-crime relationship (Farrington, 1986; Greenberg, 1985). Also, urban cities are observed with much higher crime rate than rural areas (Boggs, 1965).2. Initial AnalysisRef

4、er to the literatures, I expect crime rate is a function of incarceration level (-), metropolis (+) unemployment (+), teenagers age-interval (+) and police level (+), etc. Therefore the regression model is:Criv =0 - 1Pris + 2Metro +3Unem +4Ag15-24 +5Polpc +6X + Initially, I transform the raw data of

5、 Ag 15_17 and Ag 18_24 to a new variable Ag 15_24, in order to represent the age-interval for teenagers. Afterwards, the following procedures are undertaken by OLS method:1)I set the dependent variable as violent crime rate, and use the other related data (such as polpc, ag, incpc, pris, etc) as ind

6、ependent variables; 2) in order not to omit significant variables, all variables (except data from property crime variables) are tested at 10% initially. Afterwards, the most insignificant variables are dropped and then repeat this procedure; 3) a regression equation is made from the statistical sig

7、nificant variables and variable with special interest (Unem and Ag15-24). After initial regression which tests at 10%, the obviously unreasonable and insignificant variables (AG0-14, incpc,GOVELEC, state, etc) are dropped. After dropping the redundant variables, I observe the F-statistic improve sub

8、stantially from 23 to 70, indicating the likelihood to reject H0 increase for F-test. ANOVAbModelSum of SquaresdfMean SquareFSig.1Regression446.455589.29170.000aResidual57.435451.276Total503.89050a. Predictors: (Constant), unem, polpc, Ag15_24, metro, prisb. Dependent Variable: crivUsing a F-test: H

9、0: 1=2=3=4=5=0 H1: At least one mentioned in H0 is non-zero. Based on the SPSS output: Fc =2.34 (V1 =6, V2 =40 at 5% s.l.). Fc =3.29 (V1 =6, V2 =40 at 1% s.l.). Clearly, the F-statistic of 70 is much bigger than the critical F-values for both. Therefore, I reject H0 at 1% significance level, and it

10、is 99% confident that at least one in H0 is non-zero. In addition, a two-tailed test is used for the t-test: For H0: = 0, for HA: 0Variable Sign & t-value As expected sign tC10% 5%1%df45 Conclusion Metro + 3.554 YES 1.684; 2.021; 2.704 reject H01%Ag15-24 1.268 NO as above cannot reject H0Polpc + 7.5

11、73 YES as above reject H01%Pris + 6.487 NO as above reject H01%Unem + 0.767 YES as above cannot reject H0Refer to the SPSS output and critical t-values, coefficient of Metro, Polpc and Unem have expected signs, however for Unem and Ag 15_24, H0 cannot be rejected due to its low t-value. The other ma

12、cro-data incpc is insignificant as well, which implies that the economic data has limited influence on crime rate. This observation is consistent with Levitts assumption. There is an inverse relationship between the pace of economic development and violence crime rate in USA during 1960s (Levitt, 20

13、05) Pris, Metro and Polpc are all statistically significant at 1%. Based on the SPSS output, I find a regression equation which is:Criv = -4.914 + 0.019Pris + 3.184Metro +0.021Polpc CoefficientsaModelUnstandardized CoefficientsStandardized CoefficientstSig.BStd. ErrorBeta1(Constant)-.2864.222-.068.9

14、46metro2.839.799.2123.554.001Ag15_24-26.62921.001-.067-1.268.211polpc.021.003.5157.573.000pris.020.003.4096.487.000unem7.85910.251.040.767.447a. Dependent Variable: crivAdjusted R2 is 0.873, which implies that 87.3% of the variation of violence crime rate around its mean is explained by the regressi

15、on equation, after adjusting for degree freedom. This indicates the model performs a good fitness. 3. Diagnostic TestsAccording to Gauss-Markov Theorem, OLS method is BLUE (Best Linear Unbiased Estimator), only if the first six of The Classical Assumptions are satisfied. 1) The regression model is l

16、inear, is correctly specified, and has an additive error term;2) The error term has a zero population mean;3) All explanatory variables are uncorrelated with the error term;4) Observations of the error term are uncorrelated with each other (no serial correlation);5) The error term has a constant var

17、iance (no heteroskedasticity);6) No explanatory variable is a perfect linear function of any other explanatory variable(s) (no perfect multicollinearity);7) The error term is normally distributed (optional).As a result, it will conduct diagnostic tests in this section, to check if these assumptions

18、are met.The SPSS scatter graphs show that CRIV (on Y-axis) basically has linear relationships with Metro, Polpc and Pris (on X-axis). Specification: As mentioned previously, it is not clear defined what the specific variables should be in a model, and there is contradiction between theories regardin

19、g Unem (Raphael, 2001; Levitt, 2005). Apply the 4 specification criteria to decide whether keep or drop Unem and Ag15-24: Theory: unemployment is a controversy to violent crime; teenagers are immature and may positively contribute violent crime (Farrington, 1986); t-test: both these variables have l

20、ow t-values and are statistically insignificant at 10%; Adj. R2: it decreases by 0.001 when the variables are dropped; Bias: No change in Polpc coefficient, however, Metro and Pris coefficients change by 12% and 5% respectively. Furthermore, it finds that Ag15-24 further improves the fitness when it

21、 is added alone and the changing effect become stronger. It suggests there may be bias in the original result, and Ag15-24 was an omitted variable. Unem is probably an irrelevant variable. Therefore, Ag15-24 is re-added to the model even it has low t-value. Ramseys RESET is used to test for omitted

22、variables or incorrect functional form.ANOVAbModelSum of SquaresdfMean SquareFSig.1Regression447.522674.58758.221.000aResidual56.369441.281Total503.89050a. Predictors: (Constant), yhat3, yhat4, AG15_24, metro, pris, polpc, yhat2b. Dependent Variable: crivANOVAbModelSum of SquaresdfMean SquareFSig.1R

23、egression445.7054111.42688.092.000aResidual58.185461.265Total503.89050a. Predictors: (Constant), AG15_24, polpc, metro, prisb. Dependent Variable: crivH0: the effects of yhat2, yhat3 and yhat4 are zero, the model is correctly specified HA: at least one of the effects of yhat2, yhat3 and yhat4 are no

24、n zero, the model is not correctly specified.F= (RSSM-RSS)/M / RSS/ (N-K-1) = (58.185 - 56.369) / 3/ 56.369/51-7-1 = 0.46F critical (5%, 3, 43) = 2.84 0.46, so cannot reject H0 and the model is correctly specified. AIC (Akaike Information Criterion) and SC (Schwarz Criterion) measure RSS and takes i

25、nto account the additional parameters. The model selection criteria will select models that minimize the AIC or SC. Refer to the equations, AIC = ln (RSS/N) +2(K+1)/N; and SC = ln(RSS/N) + ln(N)(K+1)/N . AICs, SCs and adjusted R2 for different models are:1. Criv=f (Pris, Metro, Ag15_24, Polpc, Unem)

26、: AIC=0.346, SC=0.463, adjR2=0.873;2. Criv=f (Pris, Metro, and Polpc): AIC= 0.336, SC= 0.487 and adjR2= 0.872;3. Criv=f (Pris, Metro, Polpc and Ag15_24): AIC= 0.328, SC=0.517 and adjR2=0.874.Base on the AIC, SC and adjusted R2 above, the third model with Ag15_24 is preferred, as it has the lowest AI

27、C and the highest adjusted R2. Therefore, the model is: Criv=0.628+2.902Metro+0.021Polpc+0.020Pris-28.782Ag15_24 (in Appendix).Assumption 2 & 3: the population mean of error tem can never be observed, however, Assumption 2 is assured as long as a constant term is included in the equation and all oth

28、er classical assumptions are met (Studenmund, 2011). In this case, it assumes that Assumption 2 is satisfied. Assumption 3 is expected to be satisfied as there are no simultaneous models in this case.Pearson correlation: there is the problem of imperfect multicollinearity. Obviously, police forces a

29、re positive correlated to cities and prisons population. This is also supported in SPSS output (s.c1%). Hence, it indicates the estimates can be sensitive to changes in specification, and the variance is high. However, I observe all the r values 0.80, which is below the benchmark in the UE textbook

30、(Studenmund, 2011). Furthermore, the VIFs are: 1.389 (Metro); 1.807 (Polpc); 1.532 (Pris), 1.095 (Ag15_24) which are all 0) or negative (p0) autocorrelation. According to the SPSS output, the DW statistic is 2.086, which is around 2. This indicates that there is probably no serial correlation in thi

31、s model. Using the sample size N=51 and the number of explanatory variables K=4 to obtain the critical interval, which is dL=1.38 and dU=1.72, so 4- dL=2.62 and 4- dU=2.28. Therefore, the DW statistic is in the interval of dU and 4- dU, so H0 cannot be rejected and there is no serial correlation in

32、the model.Detecting Heteroskedasticity: The problem of heteroskedasticity may exist because the sample variables are from different states, and the larger states may have greater variance than the small states.If heteroskedasticity exists OLS is no longer efficient, and assumption 5 is violated.Ther

33、efore, it uses Park, White and Auxiliary White tests to detect heteroskedasticity.Park test: initially, the needed data are computed or transformed for the test, and then run the regression with dependent variable (lnERSQ) and each other independent variable in log terms (as shown in the Appendix).H

34、0: =0 and homoskedastic residuals; HA: not = 0 and heteroskedastic residuals. Refer to SPSS output: t-statistics are 2.978 (lnMetro), 2.231 (lnPolpc), 1.083 (lnPris), and 0.435 (lnAg15_24). Using 2-sided test at 1% and 49df, tc=2.704. NR2 are 51*0.153=7.803 (lnMetro), 51*0.092=4.692 (lnPolpc), 51*0.

35、023=1.173 (lnPris) and 51*0.004=0.204 (lnAg15-24). Chi square critical value (1%, 1df) =6.63. Therefore, for lnPolpc, lnPris and lnAg15_24, the White statistics are less than the critical value and fail to reject H0 with homoskedastic residuals, but for lnMetro, both t statistic and White statistic

36、are greater than the critical values, so H0 is rejected and heteroskedasticity exists. White test: first, transforming raw data to target variables (MSQ, PoSQ, PrSQ, AgSQ, MetroPolpc, MetroPris, MetroAg, PolpcAg, PrisPolpc and PrisAg). Furthermore, it runs a regression with ERSQ (dependent variable)

37、 and other transformed variables (independent variables).H0: the variance of the disturbance term is constant; HA: the variance of the disturbance term is heteroskedastic of unknown form. Refer to SPSS output, F-statistic=1.268, which is smaller than FC(10,51-10-1,1%)=2.80, NR2=51*0.241=12.291, whic

38、h is smaller than the critical value of Chi square (1%, 10df)=23.2. Therefore, it fails to reject H0, and does not detect heteroskedastic residuals.ANOVAbModelSum of SquaresdfMean SquareFSig.1Regression60.792106.0791.268.280aResidual191.713404.793Total252.50450a. Predictors: (Constant), PrAg, AGSQ,

39、MAg, PoSQ, PrSQ, MSQ, MPr, PoAg, MPo, PrPob. Dependent Variable: ERSQAuxiliary White test: it computes a new variable PRE_1 used as independent variable and then runs regression with ERSQ (dependent variable).Model SummarybModelRR SquareAdjusted R SquareStd. Error of the Estimate1.187a.035.0152.2298

40、7a. Predictors: (Constant), Unstandardized Predicted Valueb. Dependent Variable: ERSQCompute NR2=51*0.187=9.537, which is greater than the critical value of Chi square 6.63(1%, 1df). Therefore, it rejects H0 and heteroskedasticity exists. As a result, Park test and Auxiliary White test detect the pr

41、oblem of heteroskedasticity, so it will use WLS (Weighted Least Squares) method to change the weights. For a proportionality factor Z, SPSS WLS weights take the form (1/Z*2). In this case, it estimates the weighted least squares via WLS Weight using Metro as the proportionality factor: computing a n

42、ew variable named INVMetroSQ (1/Metro2). ANOVAb,cModelSum of SquaresdfMean SquareFSig.1Regression1000.6554250.16483.277.000aResidual138.183463.004Total1138.83850a. Predictors: (Constant), Ag15_24, pris, metro, polpcb. Dependent Variable: crivc. Weighted Least Squares Regression - Weighted by INVMetr

43、oSQThe original model violates Assumption 5 and it indicates OLS is not efficient and generating inaccurate estimates of the standard error of the coefficient. Therefore, the model by WLS is preferred. Refer to the SPSS output, the regression equation by WLS is: Criv = 0.974 + 2.705Metro + 0.020Polp

44、c + 0.019Pris 28.076Ag15_24According to the result, teenagers proportion is negatively correlated to violent crime rate, which is inconsistent with the theory. The other 3 explanatory variables are all positively correlated to violent crime rate. Compare with the original OLS model in section 2, coe

45、fficient of Metro substantially change but the others do not. WLS model has adjusted R2 0.868, which performs a good fitness. Test of normality for the error term, as shown in the graph, the regression errors are subject to a relatively symmetric distribution but the peakedness problem is severe. Us

46、ing Jarque Bera test: H0: the errors are normally distributed HA: the errors are not normally distributed. JB= N/6 S2+ (K-3)2/4 =51/6 0.7322+2.1222/4= 14.12Refer to the table, the critical value for JB50, 5% = 4.26 14.12, So H0 is rejected and the errors are not normally distributed. For WLS errors,

47、 JB=16.99, the errors are not normally distributed as well (shown in Appendix). Therefore, Assumption 7 is violated and hypothesis tests such as t and F tests are no longer valid. However, it is expected that the first 6 Assumptions are satisfied by the WLS model, so parameter estimates are unbiased

48、, consistent and efficient. ConclusionIn conclusion, increase of prisoners is unable to deter violent crime, so it recommends the authority may increase the legal costs and punishments to criminals, in order to deter crimes. Also, it is important to improve security system in metropolis areas. Altho

49、ugh some estimated signs are different from the theoretical expectations, I would not reject my original expectations based on the literatures. As I use cross-sectional data in 1980 and the sample size is low, the findings cannot be generalized to other time periods or the overall trend. Messner and

50、 Sampson (1991) examine positive effects on criminal violence when male sex ratio is high. Furthermore, family disruption is positively related to the violent crime rates, and the ratio of single-parent positively contributes to teenagers violent crimes. Finally, for further analysis, it recommends

51、to add more potential relevant variables such as male sex ratio and single-parent ratio.Reference- Boggs, S., (1965). “Urban Crime Patterns”, American Sociological Review, Vol. 30, No. 6 pp. 899-908 /stable/2090968 .- Farrington, D., (1986). Age and Crime”, Crime and Justice, Vol.

52、 7, pp. 189-250 /stable/1147518.- Greenberg, D., (1985). “Age, Crime, and Social Explanation”, American Journal of Sociology, Vol. 91, No. 1 pp. 1-21 /stable/2779878 .- Levitt, S., (1996). “The Effect of Prison Population Size on Crime Rates: Evidence from Prison Overcrowding Litigation”, The Quarterly Journal of Economics, Vol. 111, No. 2 (May, 1996), 319-