Advanced Microeconometrics Homework Assignment 2
1. (20 marks) (equal marks for each part)
Consider the model:
y = (1 with probability ⇤( 0 + 1x)
0 with probability 1 ⇤( 0 + 1x)
where ⇤() 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 0 and 1.
(ii) Obtain the estimated asymptotic standard errors.
(iii) Test the hypothesis that 1 = 0 using a Wald test
(iv) Test the hypothesis that 1 = 0 using a likelihood ratio test.
(v) Compute the marginal e↵ect of 1 evaluated at x = 1. 2: Structural Models (20 marks)
Consider the following structural model: Y = AY + ✏
E[✏]=0, E[✏✏0]= 2I2, 2 >0 1
where Y = (Y1,Y2)0 and ✏ = (✏1,✏2)0 are 2⇥1 vectors, I2 is the 2⇥2 identity matrix,
A=✓0 ↵◆ 00
and ↵ and 2 are parameters. Note also that a 2 ⇥ 2 matrix: B = ✓B11 B12◆
B21 B22
is invertible if B11B22 B21B12 6= 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) 1✏. (3 marks)
(iii) ShowthatE[Y]=0andVAR[Y]=⇧where
⇧ = 2(I2 A) 1(I2 A0) 1
(4 marks) (iv) Show that
⇧ = ✓ 2(1 + ↵2) ↵ 2◆ ↵ 2 2
Are the structural parameters ↵ and 2 identified? (5 marks)
(v) Suppose now that we have an independent random sample (Y i)Ni=1.
Propose a consistent estimator of ⇧. (3 marks)
(vi) Use your consistent estimator of ⇧ to propose consistent estimators of ↵ and 2. (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 = yit 1 +uit i=1,…,N,t=1,…,T (1)
where uit = ↵i +✏it, E[✏it] = 0, E[✏2it] = ✏2, E[✏it✏is] = 0 for t 6= s, E[↵i✏it] = 0, E[↵i2]= ↵2 and| |<1andT 3.
(i) Write down yi1 assuming that yi0 = 0.
(ii) Compute yi2 as a function of ↵i, ✏i1 and ✏i2.
(iii) Show that
✓1 t◆ Xt
yit =↵i 1 + t s✏is i=1,...,N,t=1,...,T s=1
(iv) Compute E[yit 1(↵i+✏it)]. Is the OLS estimator of applied to equation (1) consistent?
(v) Are the assumptions of the Random E↵ects model satisfied?
(vi) Apply the within-transformation to (1). Is the OLS estimator of applied to the within transformed equation consistent?
(vii) Apply the first di↵erences transformation to (1). Is the OLS estimator of applied to the first di↵erenced equation consistent?
(viii) Show that E[yit 2(✏it ✏it 1)] = 0. Propose a consistent estimator of . 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 e↵ect of a wage increase on labour supply is ambiguous, with competing income e↵ect (less work) and substitution e↵ect (more work). We will use our data to estimate the labour supply curve for individual i in year t:
lnhrsit = lnwgit + x0it + ↵i + ✏it
where xit includes age, age squared, number of children and health status.
(i) Can 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-di↵erences, (5) random e↵ects GLS give (i) the esti- mated coe cient on log wage b, (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 similar? What could account for any di↵erences you observe? (3 marks)
(iv) Is there a systematic di↵erence between the default and clustered stan- dard errors for ? What could account for any di↵erence? (5 marks)
(v) Perform a Hausman test of the di↵erence between the fixed e↵ects and random e↵ects 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 ↵i and ✏it. Propose a consistent estimator of . State any assumptions that you make. (10 marks)
(vii) Implement your estimator from (vi) and interpret the results. (5 marks)
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