Advanced Microeconometrics Homework Assignment 2
1. (20 marks) (equal marks for each part)
Consider the model:
1 with probability Λ(β0 + β1x)
y = 0 with probability 1 − Λ(β0 + β1x)
where Λ() is the logistic cdf. The only regressor, x, is a dummy variable.
The data comprise 200 observations as follows: y=0 y=1
x=0 52 85 x=1 51 12
You must show your workings in each part. Do not use a computer. (i) Obtain the Maximum Likelihood Estimator of β0 and β1.
(ii) Obtain the estimated asymptotic standard errors.
(iii) Test the hypothesis that β1 = 0 using a Wald test
(iv) Test the hypothesis that β1 = 0 using a likelihood ratio test.
(v) Compute the marginal effect of β1 evaluated at x = 1. 2: Structural Models (20 marks)
Consider the following structural model: Y = AY + ε
E[ε]=0, E[εε′]=σ2I2, σ2 >0 1
where Y = (Y1,Y2)′ and ε = (ε1,ε2)′ are 2×1 vectors, I2 is the 2×2 identity matrix,
0 α A=00
and α and σ2 are parameters. Note also that a 2 × 2 matrix: B11 B12
B= B21 B22
is invertible if B11B22 − B21B12 ̸= 0 in which case:
B−1 = (B11B22 − B21B12)−1 B22 −B12 −B21 B11
(i) Show that I2 − A is an invertible matrix. (2 marks)
(ii) Show that Y = (I2 − A)−1ε. (3 marks)
(iii) ShowthatE[Y]=0andVAR[Y]=Πwhere
Π = σ2(I2 − A)−1(I2 − A′)−1
(4 marks)
(iv) Show that
σ2(1 + α2) ασ2 Π= ασ2 σ2
Are the structural parameters α and σ2 identified? (5 marks)
(v) Suppose now that we have an independent random sample (Y i)Ni=1.
Propose a consistent estimator of Π. (3 marks)
(vi) Use your consistent estimator of Π to propose consistent estimators of α and σ2. (3 marks)
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3. (20 marks) (1 mark for (i) and (ii), 3 marks for the others)
Consider the dynamic panel data model
yit =βyit−1 +uit i=1,…,N,t=1,…,T (1)
where uit = αi +εit, E[εit] = 0, E[ε2it] = σε2, E[εitεis] = 0 for t ̸= s, E[αiεit] = 0, E[αi2]=σα2 and|β|<1andT ≥3.
(i) Write down yi1 assuming that yi0 = 0.
(ii) Compute yi2 as a function of αi, εi1 and εi2.
(iii) Show that
yit =αi 1−β +
1−βt t s=1
βt−sεis i=1,...,N,t=1,...,T
(iv) Compute E[yit−1(αi+εit)]. Is the OLS estimator of β applied to equation
(1) consistent?
(v) Are the assumptions of the Random Effects model satisfied?
(vi) Apply the within-transformation to (1). Is the OLS estimator of β applied to the within transformed equation consistent?
(vii) Apply the first differences transformation to (1). Is the OLS estimator of β applied to the first differenced equation consistent?
(viii) Show that E[yit−2(εit − εit−1)] = 0. Propose a consistent estimator of β. Data Analysis
4. (40 marks)
This exercise uses data from Ziliak (1997) (available on blackboard). The dataset is MOM.dta., which is a balanced panel of continuously working,
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continuously married men aged 22-51 observed between 1978 and 1987. The variables include individual (id) and year (yr) identifiers, log annual hours worked (lnhrs), log hourly wage (lnwg), age (age), age squared (agesq), number of children in the household (chld) and an indicator for bad health (bdhlth).
An important issue in labour economics is the responsiveness of labour supply to wages. The standard textbook model of labour supply suggests that for people already working the effect of a wage increase on labour supply is ambiguous, with competing income effect (less work) and substitution effect (more work). We will use our data to estimate the labour supply curve for individual i in year t:
lnhrsit = βlnwgit + x′itγ + αi + εit
where xit includes age, age squared, number of children and health status.
(i) Can β be directly interpreted as a labour supply elasticity? Explain your answer. (2 marks)
(ii) For the following estimators: (1) population average, (2) between, (3) within, (4) first-differences, (5) random effects GLS give (i) the esti-
mated coefficient on log wage β, (ii) the default standard error and (iii) the standard error clustered on individual (use bootstrap with 200 replications if clustering option is not available). (10 marks)
(iii) Are the estimates of β similar? What could account for any differences you observe? (3 marks)
(iv) Is there a systematic difference between the default and clustered stan- dard errors for β? What could account for any difference? (5 marks)
(v) Perform a Hausman test of the difference between the fixed effects and random effects estimators. What do you conclude? Which model is favoured? (5 marks)
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(vi) Now suppose that lnwgit is endogenous due to correlation with both αi and εit. Propose a consistent estimator of β. State any assumptions that you make. (10 marks)
(vii) Implement your estimator from (vi) and interpret the results. (5 marks)
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