1. In a multiple regression with independent variables X1 and X2, we can write the precision of the estimated effect of X1 on the dependent variable as:VAR(bˆ2 )=s 2.This form of the expression for VAR(bˆ2 ) shows that, all else equal, the precision of the estimated effect of X1 is inversely proportional to its variation. Describe two other properties of VAR(bˆ2 ) illustrated by this expression. 10 pts.
2. Given the following (incomplete) regression information (a regression model of 1978 automobile price on their miles per gallon (mpg), headroom (in inches), a dummy variable indicating whether the car is foreign-made (domestic is the reference category), engine displacement (a measure of the car’s power), weight (in lbs.), and length (in inches), answer the following questions with appropriate statistical support: a. H0 : b2 = 0; H1 : b2 < 0 (where b2 is effect of mpg). Is there evidence in support of the research question? 10 pts.
b. is there a difference in the price of foreign and domestic cars (adjusted for other covariates)? 5 pts.
c. is there a difference in the effects of headroom and length? 5 pts.
Hint for b. and c.: use your knowledge of the variance of a sum/difference when two variables are correlated. We used extensions of this statistical property in Assignment 4.price | Coef. Std. Err. t P>|t| [95% Conf. Interval]
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a. Regress the Pineo-Porter occupational prestige on income, education, and percent of incumbents who are women. Create a table in which you report the estimated coefficients, their estimated standard errors; and t-ratios (and indicate whether statistically significant at p<.05 using a 2-tailed test) 10 pts.
b. Determine whether a model that specifies prestige as only a function of percent women is as good as the model that also includes education and income; report the statistical evidence that justifies your answer. 10 pts.
4. Select the Leinhardt data frame (from carData). Below I am pasting the contents of the Leinhardt data description so that you can understand the source and the measurement units.
Leinhardt and Wasserman's Data on Infant-Mortality
[1] Nation
[2] Per-capita income in US dollars
[3] Infant-mortality rate per 1000 live births, around 1970
[4] Region:
Americas Africa Europe
Asia = Asia and Oceania
[5] Oil-exporting country: yes no
According to your theory, you want to model infant mortality as a function of per capita income, region, and oil-exporting (i.e., use infant as your dependent variable and income, region, and oil as your independent variables); region and oil are factor variables. In order to do this, you need to apply dummy variable coding to region and to oil. Since region has 4 categories, you will need 3 dummy variables to represent each region in your multiple regression (use Americas as your reference category); oil has two categories, so one dummy variable will capture each of the two categories (use “no” as your reference category).
Note: the code in Deane5.Rmd for dummy variable coding.
Your theory expects that, net of the effect of income, the effect of oil differs within and between regions. To test this aspect of your theory, you will need to create interactions between 2 qualitative variables (again see Deane5.Rmd).
a. Is there evidence in support of your theory? Explain your answer. 10 pts.
Another aspect of your theory is that the level of income determines the extent of difference between oil-exporting countries vs. non-oil exporting countries. To test this aspect of your theory, you need to create an interaction term between income and oil and interpret the interaction in accordance with your theory.
Note: you can eliminate the the complexity of regional differences in oil-exporting vs. non-oil exporting countries for this aspect of your theory.
b. Is there evidence in support of this aspect of your theory? Justify your answer. 10 pts.
5. Again using the Prestige data frame (use Q3a. model specification):
a. Using visual and statistical diagnostics, is there evidence of heteroscedasticity in the effect of women?
Note: use the Deane7.Rmd
b. Regardless of your conclusion concerning Q5a., reconstruct the test statistic (t-ratio) for women using the “sandwich standard error” proposed by White (1980) and compare this to the t-ratio constructed using the conventional estimator.
6. In Exercise 1 and Assignment 2 you were asked to use the GeorgiaCo_Mortality data file (first in its
.csv format and then saved into an R data frame). For this question, you will use the Model 1 specification from Assignment 2 (wherein avemort ~ f(gini fmlhhd povrate nhistwhe nhistblk hispanic and metro). Suppose you are concerned about collinearity between your repetitive measures of socio- economic conditions (gini fmlhhd povrate) and between variables capturing race/ethnicity structure (nhistwhe nhistblk hispanic).
a. Assess evidence of collinearity using VIF
b. by specifying and interpreting auxiliary regressions.
Do these diagnostics evidence a degree of collinearity that should be addressed in the Model 1 specification?
Extra credit: use the princomp( ) or prcomp( ) function to combine the three socio-economic measures and then again for the three race/ethnicity measures; refit your multiple regression with these two components and metro and use the function to create a table showing the original Model 1 estimates and the new reduced model estimates.
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