Use the data in CRIMEI to answer this question. (i) For the...

Use the data in CRIMEI to answer this question. (i) For the OLS estimates reported in Table 17. 5 , find the heteroskedasticity-robust standard errors.In terms of statistical significance of the coefficients, are there any notable changes? (ii) Obtain the fully robust standard errors - that is, those that do not even require assumption $(17.35)-$ for the Poisson regression estimates in the second column. (This requires thatfully robust 95$\%$ confidence interval for $\beta_{\text {pcmv}}$ with that obtained using the standard error in Table $17.5 .$ (iii) Compute the average partial effects for each variable in the Poisson regression model. Use the formula for binary explanatory variables for black, hispan, and bom60. Compare the APEs for qemp86 and inc86 with the corresponding OLS coefficients. (iv) If your statistical package reports the robust standard errors for the APEs in part (ii), compare the robust $t$ statistic for the OLS estimate of $\beta_{p c m}$ with the robust $t$ statistic for the APE of pcm in the Poisson regression.

Use the data in CRIMEI to answer this question.
(i) For the OLS estimates reported in Table 17. 5 , find the heteroskedasticity-robust standard errors.In terms of statistical significance of the coefficients, are there any notable changes?
(ii) Obtain the fully robust standard errors - that is, those that do not even require assumption
$(17.35)-$ for the Poisson regression estimates in the second column. (This requires thatfully robust 95$\%$ confidence interval for $\beta_{\text {pcmv}}$ with that obtained using the standard error in
Table $17.5 .$
(iii) Compute the average partial effects for each variable in the Poisson regression model. Use the
formula for binary explanatory variables for black, hispan, and bom60. Compare the APEs for
qemp86 and inc86 with the corresponding OLS coefficients.
(iv) If your statistical package reports the robust standard errors for the APEs in part (ii), compare
the robust $t$ statistic for the OLS estimate of $\beta_{p c m}$ with the robust $t$ statistic for the APE of pcm in the Poisson regression.

Key Concepts

Confidence Intervals

Confidence intervals provide a range of values within which the true model parameters are expected to lie with a given level of confidence (typically 95%). They are a crucial tool in statistical inference, offering insights into the precision of the estimated parameters and the reliability of hypothesis tests.

Statistical Significance and t Statistics

Statistical significance evaluates whether the observed relationships in the data are unlikely to have occurred by chance. The t statistic, calculated as the ratio of the coefficient to its standard error, is used to test hypotheses about the coefficients. Robust versions of the t statistic use adjusted standard errors to provide valid inference when standard assumptions, such as constant variance, do not hold.

Average Partial Effects (APEs)

Average Partial Effects quantify the average change in the predicted outcome when an explanatory variable changes, holding other variables constant. In nonlinear models like Poisson regression, APEs provide a more intuitive measure of the impact of predictors compared to coefficient estimates, accounting for the non-linearity inherent in the model.

Ordinary Least Squares (OLS)

OLS is a regression method used to estimate the relationships between a dependent variable and one or more explanatory variables by minimizing the sum of the squared differences between observed and predicted values. It serves as a fundamental technique in econometrics, providing a basis for inference about the relationships within the data under standard assumptions.

Poisson Regression

Poisson regression is used for modeling count data and events occurring over a fixed period or space. Unlike OLS, Poisson regression accounts for the discrete and non-negative nature of count outcomes. It typically assumes that the mean and variance of the dependent variable are equal, a property that distinguishes it from continuous regression models.

Heteroskedasticity and Robust Standard Errors

Heteroskedasticity refers to the condition in which the variance of the errors in a regression model is not constant across observations. When this occurs, traditional standard errors can be misleading. Robust standard errors adjust for heteroskedasticity, providing more reliable statistical inference even when the error variance is unequal across the data.

Fully Robust Standard Errors

Fully robust standard errors extend the idea of robust inference by relaxing more stringent assumptions that underlie traditional error estimates. They do not require certain distributional assumptions such as those used in standard Poisson models, ensuring that the derived confidence intervals and hypothesis tests remain valid even in the presence of distributional misspecifications or other violations.

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