Introductory Econometrics I – Spring 2022 Problem Set 3

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Introductory Econometrics I – Spring 2022

Problem Set 3

1. Consider the following regression:

y = β0 + β1 d + β2 z + β3 d . z + u,

where

● y is the personal income;

● d is a dummy (binary) variable for female (d = 1 when the person is female, and d = 0 if the person is male);

● z is a dummy variable for rural (z = 1 if the person lives in a rural area, and z = 0 if the person lives in an urban area).

We have a random sample {(yi , di , zi ) : 1 < i < n}. The OLS regression estimators are denoted by β 0 , β 1 , β 2 and β 3 (assuming the no perfect collinearity condition holds). (a) Write the rst-order conditions for the least squares regression problem. (b) De ne sample averages n n i=1 i=1 n n i=1 i=1 where nkl denotes the number of persons with both di = k and zi = l, for k, l e {0, 1}. How do you interpret the sample averages de ned above (c) Show that n di zi (yi _ b0 _ b1 di _ b2 zi _ b3 . di . zi ) = 0 i=1 where b0 = 00 , b1 = 10 _ yˉ00 , b2 = 01 _ yˉ00 , b3 = (11 _ yˉ10 ) _ (yˉ01 _ yˉ00 ). [Hint: use the fact that di(2) = di , zi(2) = zi and n11 = di zi .] (d) Now, ←ee』飞← the zero conditional mean condition: 俭[u|d, z] = 0. In part (c), you actually show b0 , b1 , b2 , and b3 satisfy one of the rst-order conditions for OLS. In fact, it can be shown that the other rst-order conditions are also satis ed. That means β j = bj for j = 0, 1, 2, 3. Use this fact to show β0 = 俭[y|d = 0, z = 0], β 1 = 俭[y|d = 1, z = 0] _ 俭[y|d = 0, z = 0], β2 = 俭[y|d = 0, z = 1] _ 俭[y|d = 0, z = 0], β3 = (俭[y|d = 1, z = 1] _ 俭[y|d = 1, z = 0]) _ (俭[y|d = 0, z = 1] _ 俭[y|d = 0, z = 0]). [Hint: Under the imposed conditions, we know β j is unbiased for βj for j = 0, 1, 2, 3. Take expectation of bj conditional on {(di , zi ) : 1 < i < n}.] (e) Use your answer to part (d) to explain the statistical meaning of the OLS estimator β 3 (what does it really estimate ). (f) Describe how to test the null hypothesis that the (population) average income of rural females does not di er from that of rural males at the 5% signi cance level. (g) Describe how to test the null hypothesis that the (population) average income of females does not di er from that of males in both rural and urban areas at the 5% signi cance level. 2. (Including Control Variables) Suppose we want to estimate the causal e ects of alcohol consumption (alcohol) on college grade point average (colGPA). In addition to collecting information on grade point averages and alcohol usage, we also obtain attendance information (say, percentage of lectures attended, called attend). A standardized test score (say, gaokaoScore) and high school GPA (hsGPA) are also available. (a) Should we include attend along with alcohol as explanatory variables in a multiple regression model (Think about how you would interpret βalcohol .) (b) Should gaokaoScore and hsGPA be included as explanatory variables Explain. 3. (Data exercise) Policy makers are interested in examining factors a ecting the smoking behavior. They collect a data set about individuals smoking behavior, including the following variables: ● id: individual index ● age: age of an individual ● agesq: age square ● cigs: number of cigarettes smoked per day ● restaurn: whether the individual lived in a city which requires no smoking in restaurants (0=no, 1=yes) ● educ: years of education Please answer the following questions using the dataset smoking.dta: (a) Create a new variable indicating age group named agegrp, which takes the following value: ,．0, if age < 30 ． agegrp = ． ． ．．3, if age > 70.

Calculate the average of cigs for each age group. Do you think age and cigs has a monotonic relationship [Hint: use the Stata command tabstat cigs, by(agegrp) stat(mean).] For the following questions,

estimate the following regression model using OLS:

cigs = β0 + β1 age + β2 age2 + β3 restaurn + u,

(b) According to to regression results, at what point does the marginal e ect of age on cigs change from

positive to negative (Round your answer to the nearest integer.)

(c) Explain the meaning of β3 .

(d) Policy makers are interested in examining whether the partial e ect of education on smoking is di erent for individuals living in cities with no-smoking mandate. Estimate the following regression model using OLS:

cigs = γ0 + γ1 educ + γ2 restaurn + γ3 restaurn . educ + e.

Write out the expression for when restaurn = 0 and restaurn = 1. How to understand the meaning of γ3

(e) Is γ3 signi cant at 5% level