Multiple Regression - Essay Example

We use a State-wide data set that includes a record of property crimes rates (CRIMES) as well as a record on per capita income (PINCOME), school dropout rates (DROPOUT), precipitation amounts (PRECIP), percentage of public aid recipients (PUBAID), population density (DENSITY), public aid for families with kids in terms of dollars received (KIDS), percentage of unemployed workers (UNEMPLOY), percentage of population living in urban areas. The methodology that we use is that of multiple regression analysis to obtain the magnitude and signs of the coefficients and t and F-tests obtain whether the respective coefficients are significant, individually, or jointly.

The regression equation we estimate is the following: (1) Results In this section we present the results of the analysis. Table 1 presents the results of the estimation of equation (1). Table 1: Results of simple OLS regression, all variables included Before interpreting the coefficients we look at the individual and joint significances of the estimated coefficients. From the upper right hand panel we find that F(8, 41) = 11.43 and prob>F =0.000. Recall that the null hypothesis of the f-test is that all coefficients are jointly zero.

From the computed statistic and the associated p-value we reject the null hypothesis. Thus, at least one of the parameters is non-zero. Now, from the lower panel in table 1 looking at the 3rd column (t-values) and the 4th column (p values) we can identify which coefficients are significantly different from zero. The null hypothesis of the t-test is that the coefficient in question is equal to zero while the alternative hypothesis is that it is non-zero. Recall that the 5% critical value for the two-sided t-test is 1.96. Looking at the elements from column 3 and 4, we find that only the variables DROPOUT, DENSITY and and URBAN are associated with coefficients that are statistically significantly different from zero.

We fail to find evidence in the data that suggests that the null hypothesis is false for the rest of the coefficients as well as the intercept. The upper right hand corner also presents the R-squared and the adjusted R-squared values which show that the fit is decent (anything greater than 50% on these scales is considered decent). In table 2 below, we take a look at whether our results may have been affected by the presence of multi-collnearity. The t-tests of significance showed that there were two significant coefficients and this was further confirmed by the f-test.

If multicollinearity is present to any worrisome degree, typically, although the f-test rejects the null, we fail to identify any statistically significant coefficients in the t-test. Thus, there is little evidence of multi-collinearity. Additionally, we also look at the VIFs in table 2 below. Observe that the mean VIF is only 2.46 and the VIF for PINCOME is a bit large (although significantly smaller than 10, the standard indicator of severe and worrisome multicollinearity). Table 2: Testing for presence of multi-colinearity, VIFs Now, we rerun the regression incorporating only the significant variables in the specification.

The model is specified as follows: The results of this regression are presented in table 3. Table 3: Results of OLS estimation including only the significant variables Observe first from the table above that the signs and the significances of the included variables remain the same. That is