Problem Set # 2

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Problem Set # 2

Master in Business and Quantitative Methods

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2. If we consider only the nonzero observations, then the truncated re- gression model applies. The sample mean of the nonlimit observations is the least squares estimator in this context. Compute it and then comment on whether this sample mean should be an overestimate or underestimate of the true mean.

Problem 2 We now consider the tobit model that applies to the full data

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Problem 4 A top university requires all students to write an entry exam.

Students that obtain a score less than 100 are not admitted. Students with score above 100, the scores are registered. Then, the university selects stu- dents, from this group, for admittance. For each student, they observe the result of the exam:

reject, if the score is less than 100 or the score, if it is 100 or more.

Imagine that the dean is interested in the relationship between the students’

background characteristics and the score of the entry exam. He specifies the following model:

y^∗_i =β₀+x_i⁰β₁+ε_i, ε_i∼N(0, σ²) y_i =y^∗_i if y_i^∗≥100

y_i =rejected if y_i^∗ <100,

where y_i is the observed score of student i and x_i the vector of background characteristics (excluding an intercept).

1. The dean does a regression ofy_i onx_i and a constant, using the scores of 100 and more. Show that this approach does not lead to consistent estimators of β₁.

0.2 Exercises on panel data

Problem 5 The file panel.xls constains data on investment (y) and profit (x) forn= 3 firms over T = 10periods.

1. Pool the data and compute the least squares regression coefficients of the model y_it=α+βx_it+ε_it.

2. Estimate the fixed effects model and test the hypothesis that the con- stant term is the same for all three firms.

3. Carry out Hausman’s specification test for the random versus fixed effect model.

Problem 6 Suppose the fixed effects model y_i=X_iβ+iα_i+ε_i

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is formulated with an overall constant term and n−1 dummy variables instead of n. Investigate the effect that this supposition has on the set of dummy variable coefficients and on the least squares estimates of the slopes.

Problem 7 Consider the following simple panel data model y_it=x_itβ+α^∗_i +ε_it, i= 1, ..., n t= 1, ..., T where β is one dimensional and where it is assumed that

α^∗_i = ¯x_iλ+α_i, with α_i ∼N ID(0, σ_α²) ε_it∼N ID(0, σ_ε²) mutually independent and independent of allx_its, where x¯_i =P_T

t=1x_it. The parameterβ can be estimated by the fixed effects (or within) estimator given by

βˆ_{F E} = P_n

i=1

P_T

t=1(x_it−x¯_i)(y_it−y¯_i) P_n

i=1

P_T

t=1(x_it−x¯_i)²

As an alternative, the correlation between the error term α_i^∗+ε_it and x_it can be handled by instrumental variables.

1. Give an expression for theβˆ_IV usingx_it−x¯_i as an instrument for x_it. Show that βˆ_IV and βˆ_{F E} are identical.

2. Another way to eliminate the individual effects α^∗_i from the model is doing the following transformation:

y_it−y¯_i = (x_it−x¯_i)β+ (ε_it−ε¯_i)

Which is the OLS estimator (β) based on this model? In which condi-ˆ tions is (βˆ) a consistent estimator of β?

3. Consider the between estimator βˆ_B for β. Give an expression for βˆ_B

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1. Give an expression for the probability that r_it = 1 given z_it and ψ_i. 2. Compute the likelihood for this model.

3. Explain why it is not possible to treatψs as fixed unknown parameters and estimateδconsistently (for fixed T) from this fixed effects probit?

Problem Set # 2

Problem Set # 2

Master in Business and Quantitative Methods

Contents

0.1 Exercises on limited dependent variables

0.2 Exercises on panel data