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Section A

A researcher is interested in the relationship between two variables, X and Z and their effect on Y . They collect a random sample of individuals and estimate the following model:

Yi = β 1Xi + β2Zi + ϵi

where Yi  is the outcome of individual i, Xi  is individual i’s first characteristic, Zi  is individual i’s second characteristic and ϵi  is the error term.  The researcher suspects that Y may cause Z in such a way that: Zi = γYi + νi .

Suppose that: νi|Y ∼ N (0,σν(2))

νi  and ϵi  are statistically independent.

(a) Show that the error term ϵi = ( ) Zi − β1Xi −  .

(b) Express E(ϵi|Z) in terms of Zi,Xi,νi  and the model parameters. Explain all calculation steps.1        (c) With reference to relevant OLS assumption(s), explain if the OLS estimators ( 1 , 2 ) are unbiased.

Section B

Download the dataset weight.dta from Moodle.  The dataset contains the following variables with information on a random sample of 17,870 individuals:

❼ identifier: individual identifier number

❼ sex: 1=Male, 0=Female

❼ weight: weight without shoes (in pounds)

❼ height: height without shoes (in inches)

Use the dataset to answer parts (d) to (g). Include your Stata commands in an Appendix. Consider the following regression model:

weight = δ0 + δ1 heighti + νi

where weighti  is the weight of individual i in pounds, heighti  is individual’s i height in inches, and νi is the error term.

(d) Explain if you deem, it is appropriate to interpret the OLS estimator 1  as the causal effect of height on the weight of individuals, using relevant OLS assumption(s) and your intuition.

(e) Estimate the regression model with both homoscedastic and heteroscedastic-robust standard errors. Present the estimation results in a table. Interpret the estimated intercept, slope coefficient, and R2 , with reference to this particular regression model.  Can you make any conclusion regarding the errors of the model?

(f) Compute the OLS residuals from the regression in part (e) and plot them against height.  Explain if any OLS assumption(s) appear(s) violated.

(g) Estimate the regression model again for (i) women, and (ii) men.  Present the estimation results as separate columns in a table. Interpret and compare the slope coefficients with your answer in part (e).