Alastair Hall ECON61001: Semester 1, 2020-21 Econometric Methods
Problem Set for Tutorial 4
In this question you explore further the different scalings of the sample mean in the WLLN and CLT. In the Lecture Notes Section 3.1 Example 3.2, it is remarked that the CLT is an exact result for Case (i) in which {vt}Tt=1 are independently and identically distributed normal random variables. In this question you verify this statement and consider its implications for the large sample behaviour of the sample mean scaled by different functions of the sample size.
1. Let {vt}Tt=1 be a sequence of independently and identically distributed standard normal ran- dom variables (often written using the mathematical shorthand vt ∼ I N (0, 1), t = 1, 2, . . . T ) a n d s e t v ̄ T = T − 1 Tt = 1 v t .
(a) Show that T1/2v ̄T ∼ N(0,1). Hint: Write Tt=1vt = ι′Tv where v = (v1,v2,…,vT)′ and ιT is a T × 1 vector of ones, and use Lemma 2.1 in the Lecture Notes, noting that vt ∼ IN(0,1), t = 1,2,…T implies v ∼ N(0,IT).
(b) Let n be a finite positive constant and z ∼ N (0, 1). Using part (a), show that P (|v ̄T | < n) = P (|z| < T 1/2n) and use this result to deduce limT →∞P (|v ̄T | < n). Relate this limiting behaviour to the WLLN.
(c) Now consider Tv ̄T. Show that P(|Tv ̄T| < n) = P(|z| < T−1/2n) and so deduce l i m T → ∞ P ( | T v ̄ T | < n ) .
In this question, you consider the probability distribution of the errors in a type of regression model known as the linear probability model (LPM).
2. Consider the linear regression model
yi = x′iβ0 + ui
in which yi is an indicator variable that takes the value one if an event occurs (such as an individual is employed) and zero otherwise. Assess: (i) whether xi and ui are independent; (ii) whether conditional on xi, ui can have a normal distribution.
In lectures, we stated that the the OLS estimator of the error variance is consistent. In this question, you establish this result.
3. Consider the linear regression model
yi = x′iβ0 + ui
where x′i = (1, x′2,i), the data are cross-sectional and the model satisfies Assumptions CS1 - CS5 so that: { (ui,x′i), i = 1,2,...N} forms an independent and identically distributed sequence; E[xix′i] = Q, finite, p.d.; E[ui|xi] = 0; V ar[ui|xi] = σ02, a positive, finite constant. Let σˆN2 denote the OLS estimator of σ02. Show that σˆN2 →p σ02.
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In Tutorial 2, Question 2, you derived the bias of the OLS estimator when relevant regressors have been omitted from the model. In this question, you establish the large sample analogue to this result.
4. Consider the linear regression model
yi = x′i,1β0,1 + x′i,2β0,2 + ui
where x′i = (x′i,1,x′i,2), xi,l is kl ×1 for l = 1,2, k = k1 +k2, and Assumptions CS1 - CS5 are satisfied so that: { (ui,x′i), i = 1,2,...N} forms an independent and identically distributed sequence; E[xix′i] = Q, finite, p.d.; E[ui|xi] = 0; V ar[ui|xi] = σ02, a positive, finite constant. Suppose that a researcher estimates the following model by OLS,
yi = x′i,1γ∗ + error.
Let γˆN be the OLS estimator of γ∗. Show that γˆN →p β0,1 +Q−1Q1,2β0,2, where E[xi,jx′ ] =
Qj,l forj,l=1,2.
In Lecture 4, we discussed methods for testing nonlinear restrictions on β0. In this question, you
must use these methods to propose an appropriate statistic for testing a particular hypothesis.
5. Consider the linear regression model
yi = x′iβ0 + ui
where: { (ui, x′2,i), i = 1, 2, . . . N } forms an independent and identically distributed sequence; E[xix′i] = Q, finite, p.d.; E[ui|xi] = 0; V ar[ui|xi] = σ02, a positive, finite constant. Assume k=5thatis,β0 is5×1andthatβ0,i ̸=0fori=2,3. SupposeitisdesiredtoH0 : β0,2β0,3−1=0versusH1 : β0,2β0,3−1̸=0. Proposeateststatisticandstatetheappropriate decision rule.
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