Problem 1: 50 Marks
Evaluate whether each of statements (a)-(d) below is True, False or Uncertain. Explain the rea- soning behind your answers. Your mark will depend on the quality and clarity of your explanation. Each is worth 12.5 marks.
(a) Consider the following model: yi = β0 + β1xi + εi. Let yi∗ = yi + vi.
• Statement (a) : β1 may be consistently estimated with an OLS regression of
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yi∗ on xi.
(b) Consider the impact of a binary treatment on some outcome yi. Let Di be a treatment indicator equal to one if unit i receives treatment and 0 otherwise. Let yi0,yi1 be potential outcomes for i. Define the selection effect as E[yi0|Di = 1] − E[yi0|Di = 0].
• Statement (b): If the selection effect is equal to 0, then E[yi|Di = 1] − E[yi|Di = 0]
is equal to the average treatment effect.
(c) Let xt denote a mean 0 column vector of excess returns for m assets. Define Σx = Cov(xt). Let Γ be a square matrix with the eigenvectors of Σx as columns (ordered by the size of the corresponding eigenvalue). Let pt = Γ′xt. Note that we may write
xt = Γkpkt + εt
where Γk and pkt represent the first k < m columns of Γ and elements of pt, respectively.
• Statement (c): Cov(εt) is a diagonal matrix.
(d) Let xt be a vector of returns on m assets. Define Σx = cov(xt), and let P denote the number of unique variance-covariance terms contained within the general version of this m × m matrix.
Now suppose we assume that xt is driven by a two factor model:1 xt = α + Bft + εt.
Our model covariance matrix is given by
Σ ̃ x = B Ω f B ′ + Ψ .
Let Q be the number of unique parameters in this formulation (i.e. the total number of parameters in B, Ωf and Ψ).
• Statement (d): Q < P.
1 Here α is a vector of length m, B is an m × 2 matrix of factor loadings, ft is a 2 × 1 vector of factor realizations with cov(ft) = Ωf , and Cov(εt) = Ψ is diagonal.
Author: CJH
©Imperial College London 2021
Problem 2: 15 Marks
Consider the following difference-in-difference model for individual i in period t ∈ {1, 2}: yit = β0 + β1Di × Aftert + β2Di + β3Aftert + εit.
Here Di is an indicator variable denoting treated individuals and Aftert is an indicator variable equal to 1 in the 2nd period. Please compute OLS estimates βˆols,βˆols,βˆols and βˆols using the data
i t yi Di Aftert 11300 21500 31110 41310 12101 22901 32211 42211
Author: CJH
©Imperial College London 2021
Problem 3: 15 Marks
Suppose we are interested in estimating the coefficients β0, β1, and β2 in the following linear model:
yi∗ =β0 +β1x1i +β2x2i +vi.
While we observe x1i and x2i, we are unable to observe yi∗ entirely. Instead, we see yi, where yi is given by:
k if yi∗2 > c yi= yi∗ifyi∗2≤c
for some known constants c > 1 and k > c. Let vi ∼ N(0,1) be a standard normal random variable with probability density function f(z|x1i,x2i) = φ(z) and cumulative distribution function F (z|x1i, x2i) = Φ(z).
(a) What is the probability distribution function of yi given x1i, x2i and the parameters β0, β1, β2: g(yi|x1i,x2i;β0,β1,β2)? (15 marks)
Author: CJH
©Imperial College London 2021
Problem 4: 20 Marks
Consider the following model for yi:
y i = β 0 + X i′ β + ε i .
Here Xi = . and β = . . You may assume that yi and all elements of Xi have mean 0.
. . xik βk
The objective function for RIDGE is given by:
NK βˆRIDGE=argmin(yi−Xi′β)2 subjectto βk2≤c
xi1 β1 xi2 β2
for some c > 0. Alternatively:
βˆRIDGE = arg min (yi − Xi′β)2 + λ βk2.
(a) Derive βˆRIDGE in terms of Xi, yi, and λ.2 (10 marks)
(b) Discuss the bias-variance tradeoff in prediction exercises. Why does this tradeoff arise? Are there problems in which we might we prefer a biased estimator to an unbiased one? How does this relate to βˆRIDGE? (10 marks)
FeelfreetouseX= . andY =.ifyouprefermatrixnotation.
X 1′ y 1 ′
. . Xk′ yk
Author: CJH
©Imperial College London 2021
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