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A1 Consider a three-period panel regression model
where and denote the values of an outcome and a regressor repectively for the ith individual in period t, with i= 1, ..., n and t=1, 2, 3; is an individual-specific unobserved effect, and is an unobserved error term. Assume that we have n individuals randomly sampled from the population.
Suppose we pool all the observations so that there are 3 observations per individual corresponding to the 3 time periods and thus a total of 3n observations altogether. Now, suppose we regress Yi on Xi (including an intercept) using all 3n observations. Under what assumption will this pooled OLS estimator give us consistent estimates of ?
Now consider the second -differenced estimator obtained by regressing
(Yi3 Yi2) (Yi2 Yi1) on the regressor (Xi3 Xi2) (Xi2 Xil). Under what conditions will this estimator be unbiased for the causal effect of X on Y ?
Under what condition would you prefer the estimator in part (a) over the estimator in part (b)?
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A2 Consider the regression model for an I.I.D. sample with N=1000 observations. Suppose I.I.D. (0,2) and are I.I.D. for and that the is independent of . Let /denote the OLS estimator of , and consider another estimator , constructed in the following way:
You can assume that Xi are continuously distributed and that never takes the value 0.
Is an unbiased estimator of ? Why?
Can / be a better estimator than the OLS estimator? Why?
Is a consistent estimator of ? Why?
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A3 Consider the regression model for whether a student gets a first class mark in the final examination:
for an I.I.D. sample with 331 observations, where is a dummy variable which equals 1 if student is male and is zero otherwise, and is a dummy variable for whether student i got a first class mark, and is zero otherwise. Table 1 at the end of this question gives the distribution of First by gender.
Suppose the unobserved component Ui is independent of and follows a normal distribution with mean 0 and variance 1. What is the probability of getting a first class degree for male and for female students in terms of ?
Given your answer to part (a), how would you estimate based on the information provided in Table 1? (Hint: remember that the cumulative distribution function for the normal distribution is strictly increasing).
Is it reasonable to assume that Ui is independent of M alei?
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A4 The following problem concerns the phenomenon of "job-lock" in the United States, where employees cannot leave their present job because of employer provided health-insurance. If they leave the present job and take up a new job, then the new health insurance will not cover them for pre-existing health conditions, leading to a job-lock. In order to study this phenomenon empiri- cally, a researcher estimates the following probit model on male workers in the construction industry who are between 35-45 years old and ethnically white:
where Yi is a dummy variable which equals 1 if individual i has changed jobs in 2013 and is zero otherwise, HIi is a dummy for whether individual i was covered by employer provided health insurance in 2012 and is zero otherwise, and ϕ (.)is the standard normal cumulative distribution function.
How would you test whether there is any job-lock, based on this equa- tion? Can you think of a reason why this test may not be a satisfactory indicator of job-lock?
Now let be a dummy variable taking the value 1 if individual has a chronic medical condition which will not be covered by the new insurance plan if he changed jobs, and is zero otherwise. Consider the equation
How would you test the presence of job-lock using estimates of parameters appearing in this equation? Why?
Would you prefer the method based on the equation in part (b) or the one in part (a)? Why?
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A5 A researcher considers the following expectations augmented Phillips curve
where is inflation, is expected inflation with the expectation formed in year t - 1, Ut is the unemployment rate, µ is the natural rate of unemployment, and et is a supply shock. The researcher has data on UK annual CPI inflation and the unemployment rates for the period 1989-2014. OLS regression of on Ut gives
with standard errors in parentheses.
Explain how the model estimated by OLS is related to the expectations augmented Phillips curve. What would be your estimate of µ?
Under what assumptions is the OLS estimator of the coefficient on Ut unbiased? Discuss the validity of these assumptions.
Test the hypothesis that Ut does not affect using a 5% significance level test. Give two reasons why the results of your test might be unreliable.
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A6 A researcher studies demand for cash in the UK. She runs a regression of the logarithm of the cash in circulation, log Mt, on the logarithm of the nominal household final consumption expenditure, log Ct, using quarterly data from 1985q1 to 2006q1. The OLS results (standard errors in parentheses) are:
How would you interpret the estimated coefficient on log?
A colleague points out that both log Mt and log Ct contain a clear time trend. What might be consequences of this fact for the validity of the above regression results?
Reestimating the regression with time trend gives
(1.01) (0.0015) (0.11)
so that the researcher becomes very puzzled about the sign of the esti- mated coefficient on log Ct. The colleague gets the residuals uAt from (1), and obtains the following OLS result:
Where What does this result tells us about the validity of (1)? Can you propose a better specification for the regression model describing the demand for cash?
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A7 In order to understand the relation between TV watching and obesity among children, a researcher estimates the following equation by OLS, with standard errors reported in parentheses:
(0.02) (0,001) (0.011) (0.005) (0,012)
where denotes log of hours spent watching TV by the ith child on the day before the survey was taken, is a dummy which equals 1 if the child is female and is 0 otherwise, is the square of the child's age, is the child's body mass index (weight for height) and Ui is an unobserved error term.
Assuming the Gauss-Markov assumptions hold, what is the interpretation of the coefficient -0.034 on female?
What do the above estimates imply about how watching TV varies with age, all else being equal?
Provide an example of an omitted variable which would imply that the estimated coefficient of bmi in the above equation is biased for the causal effect of bmi on watching TV.
B1 The following question pertains to the effect of background characteristics on the probability that an applicant is admitted to study economics at a selective UK university. TSA is an aptitude test with two components - Critical Thinking and Problem Solving - in each of which the maximum possible mark is100. We draw a random sample of 800 applicants and record their characteristics and whether they were admitted. The summary statistics are reported in Table 2, at the end of this question. Finally, a logit regression of admission on these background characteristics yields the output reported in Table 3 which appears on the next page. Based on these output, please answer the following questions:
A hypothesis test for the joint significance of the two dummy variables and their interaction yields a chi-square test-statistic with p-value equal to 0.0028. What can we infer from this?
What is the predicted probability of being admitted for a male, independent school applicant who has scored exactly the average mark in the two TSA components? What is the predicted probability of being admitted for a female, non-independent school applicant who has scored exactly the average mark in the two TSA components?
How would you test whether the differences in probabilities in part (b) are zero?
If your test suggests that the admission probability is lower for male, independent school applicants, would this imply discrimination against this demographic group?
What intercept and slope-coefficient estimates would we get if our dummy regressors were female, independent and their interaction, instead of male, independent and their interaction?
1 if admitted, 0 otherwise
TSA Critical score
TSA Problem-Solving score
1 if from indep school, 0 otherwise
1 if male, 0 otherwise
Maximized log-likelihood = -372.1331, N=800, LR chsq(5)=307.91, pvalue=0.00001.
B2 A researcher estimates the following ADL model using OLS:
(138.7) (0.11) (0.006) (0.007)
where is the monthly number of violent crimes in Cambridgeshire and is the number of people in East England who claim unemployment benefits during month t. The estimates of the standard errors are given in parentheses. The OLS estimates are based on 61 observations, starting from December 2010 up to December 2015. The average number of violent crimes and average number of unemployment benefit claims over this period were 738 and 89647, respectively.
What is the estimated value of the long-run change in the expected value of the violent crimes given a permanent increase in the number of unemployment claims by 1000? Is this estimate economically significant?
Under what assumptions are the OLS estimators of the coefficients of the ADL model consistent and asymptotically normal?
Let be the residuals from the above regression. The OLS regression of on a constant, yields
(224.4) (0.176) (0.008) (0.008) (0.217)
What does this tell us about the validity of the original OLS results?
The researcher next estimates the following OLS regression:
(45.3) (0.67) (0.074) (0.03)
What does this result imply about the validity of the assumptions that you discussed in (b)?
To forecast vct, the researcher estimates the following regression
Where The values of for December, November, and October of 2015 were, respectively 997, 1007, 1068. What is the forecast of for January 2016?
B3 We are interested in understanding how married couples' hours of work are related using data from n randomly sampled households. Consider the simul- taneous structural equations
Here denotes the husband's weekly hours of work in the ith sampled household, denotes the wife's weekly hours of work, denotes the number of children the couple has, while Ui and Vi denote the error terms. Please answer the following questions.
How would you interpret the coefficient ? You do not need to discuss any economic model of utility maximization here.
Solve for and in terms of , Ui and Vi by solving the two equations. The resulting equations are called the "reduced form" equations. Explain why an OLS of on and (and a constant) give us biased estimates of the causal effect of the wife's hours of work on the husband's hours of work.
Under what conditions can we use kids as an instrument for whours and estimate by two-stage least squares? Can you write down one situation where these assumptions may fail?
Finally, suppose that the covariance between U and V across households is zero, and suppose kids is a valid IV for whours in equation (1). Can you describe a way to consistently estimate , the cofficient of hhours in the second equation, i.e., equation (2)? You will need to consult the reduced form for hhours in part (b) of this question.
Is it reasonable to assume that the covariance between U and V is zero?