Advanced Microeconometrics Homework Assignment 3
1. Hypothesis testing (25 marks) A researcher considers the following model with one exogenous regressor:
y = β0 + xβ1 + ε ε ∼ N(0,σ2)
in which x is independent of ε.
(i) Write down the conditional distribution of y given x. (3 marks)
(ii) Suppose that we have an independent random sample (yi, xi)Ni=1. Write down the likelihood and log likelihood functions. (4 marks)
(iii) Find the maximum likelihood estimators of β0 and β1. (8 marks)
(iv) You obtain the following STATA output for two specifications: one with regressors x and a constant, and another with a constant only. Use this information to test H0 : β1 = 0 against a two sided alternative at the 0.05 level. (6 marks)
(v) In addition, you obtain the following estimator of the variance covari- ance matrix of the parameters from the first model. Denoting the parameters by θ = (β0, β1, σ2)′ you obtain:
0.0116 −0.0004 0.0000
V AR[θ] = −0.0004 0.0091 0.0000 0.0000 0.0000 0.0058
Use this information and the ouput from part (iv) to test H0 : β1 = 0 against a two sided alternative at the 0.05 level. (4 marks)
1
2. Panel Data (20 marks) Consider the panel data model with random slopes:
yit = αi + βixit + εit
where βi ∼ N(β,σβ2), αi ∼ N(α,σα2) and εit ∼ N(0,σε2) are independent of
one another and of xit. Suppose that we have an iid sample (yit,xit)N,T . i=1,t=1
(i) Write down the likelihood function and the corresponding log likelihood function. (5 marks)
(ii) Comment on the feasibility of maximum likelihood estimation of the parameters (5 marks)
(iii) Describe in detail how you would obtain a simulated maximum likeli- hood estimator of the parameters. (10 marks)
2
3. Quantile regression (25 marks) Using data from the US Medi- cal Expenditure Panel Survey, consider a quantile regression model of total medical expenditure by the elderly. The dependent variable is log of the total expenditure (ltotexp). The explanatory variables are an indicator for pri- vate insurance status (suppins), one health-status variable (totchr) which measures the number of chronic conditions, and three socio-demographic variables (age, female, white). You obtain the following STATA output.
(i) Write down the corresponding quantile regression model, using the no- tation ln(y) for ltotexp, x for totchr, age, female, white and the constant, and d for suppins. Your answer should take the form
Qq(ln(y)|x,d) = …
where Q(·) is the conditional quantile function and q is a quantile. (4 marks)
3
(ii) Interpret the regression results, focussing on suppins. (5 marks)
(iii) Use the STATA output to obtain an estimate of the quantile treatment
effect
τ (x, q) = Qq (y|x, 1) − Qq (y|x, 0)
for a 65 year old white female with no chronic conditions and q = 0.9. (Note: The treatment effect τ(x,q) is for total medical expenditure, not log total medical expenditure.) (10 marks)
(iv) Explain in detail how you would construct a 0.95 confidence interval for the quantile treatment effect in part (iii). (6 marks)
Data Analysis
4. (30 marks) Consider the model with one exogenous covariate x:
y∗ =βx−ε
(i) Show that the above model is the probit model when ε follows the standard normal distribution (i.e. it yields the same distribution of y|x as the probit model). (5 marks)
(ii) Suppose now that we have an iid sample (yi, xi)Ni=1. Conduct a Monte- Carlo experiment to study:
(a) The properties of the probit estimator of β when ε follows the standard normal distribution. Explain your results. (10 marks)
4
1 if y∗≥0 0 otherwise
y=
where y∗ is a latent (i.e. unobserved) variable and ε and x are independently
distributed.
(b) The properties of the probit estimator of β when ε follows the standard uniform distribution. Explain your results. (10 marks)
Throughout your experiment use the parameter values N = 100, β = 1 and suppose that x follows a normal distribution with mean 0 and variance 1.
(iii) Repeat part (ii) with N = 10,000. Explain any change(s) in your results. (5 marks)
5