Alastair Hall ECON61001: Autumn 2020 Econometric Methods
Problem Set for Tutorial 1
In the first lecture, we saw how the linear regression model subsumes models in which the dependent variable and regressors may be functions of certain underlying variables. In this question we explore how the functional form of the variables affects the interpretation of the parameters.
1. Let x be a continuous random variable and assume u is independent of x. Derive ∂y/∂x in the following models.
(a) y=β0,1 + β0,2x + u.
(b) y = β0,1 + β0,2ln(x) + u. (c) ln(y) = β0,1 + β0,2x + u.
(d) ln(y) = β0,1 + β0,2ln(x) + u. (e) y=β0,1 + β0,2x + β0,3×2 + u.
For models (a)-(e), deduce the interpretation of β0,2.
The next two questions consider certain linear algebra results that are used in our derivation of the
OLS estimator.
2. Leth(θ)=a′θandg(θ)=θ′Aθwhereθandaarep×1vectorsandAisap×pmatrix. Show that:
(a) ∂h(θ)/∂θ = a.
(b) ∂g(θ)/∂θ = (A + A′)θ.
(c) ∂2g(θ)/∂θ∂θ′ = A + A′.
(d) If A is symmetric, then show how the results in parts (b) and (c) can be simplified.
Hint: by definition, ∂h(θ)/∂θ is p×1 vector whose ith element is ∂h(θ)/∂θi where θi is the ith element of θ. So the results can be shown by deriving the form of ∂h(θ)/∂θi and then using this to deduce the appropriate result for ∂h(θ)/∂θ. Similarly, ∂2g(θ)/∂θi∂θj is the i − jth element of ∂2g(θ)/∂θ∂θ′.
3.(i) Recall that a k × k matrix M is positive definite if c′Mc > 0 for any non-null k × 1 vector c. Define X to be a T × k matrix with rank(X) = k. Show that X′X is a positive definite matrix. Hint: Show c′X′Xc = b′b for a certain choice of b and then deduce the result by considering the properties of b.
3.(ii) Suppose now that rank(X) < k what can be said about the sign of c′X′Xc? 1
This question considers a property of OLS estimators for the linear model,
y = Xβ0 + u
where all definitions and dimensions are the same as our discussion in the lectures. It is assumed
̄
tion, show that: (i) e ̄ = 0; (ii) y ̄ = yˆ. Hint: part (ii) follows from part (i).
that the model includes an intercept and so X = [ιT , X2] where ιT is a T × 1 vector of ones.
ˆ th th −1T ̄
4.Letyˆ=XβT witht elementyˆt,e=y−yˆwitht elementet,y ̄=T t=1yt,yˆ= T−1 Tt=1 yˆt, and e ̄ = T−1 Tt=1 et. By considering the first order conditions of OLS estima-
In this question, you establish the partitioned matrix inversion result to which we appealed in our discussionof the Frisch-Waugh-Lovell Theorem.
5. Consider the partitioned matrix
A=A1,1 A1,2
A2,1 A2,2
where |Aii| ≠ 0. Prove that A−1 = B where B is a similarly partitioned matrix with
B1,1 = (A1,1 − A1,2A−1A2,1)−1, 2,2
B2,2 = (A2,2 − A2,1A−1A1,2)−1, 1,1
B1,2 = B2,1 =
−A−1A1,2B2,2, 1,1
−A−1A2,1B1,1. 2,2
2