# CS计算机代考程序代写 Problem 1: 15 Marks

Problem 1: 15 Marks

Suppose we see excess returns xit on m assets (i = 1,2,··· ,m) over T time periods (t = 1,2,··· ,T). We may write these together as a vector at time t:

x1t x2t

xt= . . .

xmt

Suppose these returns are driven by the following two factor model:

xit = αi + β1if1t + β2if2t + εit. The covariance matrix of f1t and f2t is given by:

σ f2 σ f ρ f Ωf=σfρf 1.

Suppose the covariance matrix of εit for all i is given by:

σ12 0 ··· 0

0 σ2 ··· 0 Ψ=. .. . .

.···.. 0 0 · · · σ m2

You may assume that Cov(fkt,εit′) = 0 for any k, i, t and t′, and that both fkt and εit are uncorrelated over time.

(a) Let Σ = Cov(xt) be the covariance of asset returns. Express Σ in terms of Ωf , Ψ and anything else you need (please clearly define any notation you use). (10 marks)

(b) How many unique parameters are contained in this formulation of Σ. (5 marks)

Author: CJH

©Imperial College London 2018/2019

Problem 2: 25 Marks

Suppose we see excess returns xit on m assets (i = 1,2,··· ,m) over T time periods (t = 1,2,··· ,T). We may write these together as a vector at time t:

x1t x2t

xt= . . .

xmt

You may assume that each excess return has mean 0 and that the empirical covariance matrix

ˆˆ

is given by Cov(xt) = Σx. Suppose further that we perform an eigendecomposition to recover

Σˆx = ΓΛΓ′. Here Λ is a diagonal matrix of eigenvalues of Σˆx (with the eigenvalues on the diagonal ordered from largest to smallest), and Γ is a matrix with the eigenvectors of Σˆx as columns.

(a) Define pt, the set of principle components of xt (10 marks) (b) State two desirable properties of pt (5 marks)

(c) Suppose m = 3 and the eigenvalues of Σˆx are 5, 2 and 1. What fraction of the total variance of xt can be explained by the first two principle components? (10 marks)

Author: CJH

©Imperial College London 2018/2019

Problem 3: 25 Marks

Suppose we are interested in predicting some outcome variable yi with a vector of p+1 explanatory variables Xi, where Xi is given by:

1 x1i

x

Xi = 2i. .

xpi

The matrix containing these Xi for all i is can be written as:

1 x11 x21 ··· xp1 . . . … .

X = 1 x1i x2i · · · xpi .

. . . .. . . . . . .

1 x1n x2n ··· xpn

You may assume that yi and all xki have been standardized to have mean 0 and variance 1. Consider

the following two minimization problems:1 np

min (yi −Xi′β)2 +λβj2 (1)

β

i=1 j=1

np

min (yi −Xi′β)2 +λ|βj| . (2)

β

i=1 j=1

(a) Suppose we set λ = 0 in each of the above. Please provide βˆ that solves the minimization problems (1) and (2). (5 marks)

(b) Describe generally in a sentence or two why we might we be interested in the solution to either (1) or (2) with λ > 0. Additionally, describe at least one advantage of (2) over (1). (10 marks)

(c) Suppose a researcher sets λ = 0 and estimates parameters for p = 1500 different explanatory variables. The researcher finds that this explains 99.1% of the variation in the data used to estimate the parameters. As a result, the researcher claims that they will be able to almost perfectly predict yi out of sample. Discuss this claim (a few sentences or short paragraph should be sufficient). (10 marks)

β0

β1

1Here Xi′ denotes the transpose of Xi and β = . . Note that the summations λ pj=1 βj2 and λ pj=1 |βj | in

. βp

equations (1) and (2) begin at j=1 and hence do not include the term β0.

Author: CJH

©Imperial College London 2018/2019

Problem 4: 25 Points

Suppose we are interested in estimating the coefficient β1 in the following linear model: Yi = β0 + β1Xi + vi.

Suppose that corr(Xi, vi) ̸= 0, but that we have access to an instrumental variable Zi. (a) List two necessary conditions for Zi to be a valid instrument (5 marks)

(b) Suppose we see 5 observations of Xi, Zi, and Yi, shown in the table below:

Yi Xi Zi 3 -3 0 821 461 0 -2 1 310

Calculate βˆiv, the instrumental variables estimate of β 11

of your choice. (20 marks)

given the data above using an estimator

Author: CJH

©Imperial College London 2018/2019

Problem 5: 10 Marks

Consider again the data given in Problem 4:

Yi Xi Zi 3 -3 0 821 461 0 -2 1 310

Suppose we were interested in non-parametrically estimating h(X) = E[Y |X] using the nearest neighbors approach. Use the above to do so at the point X = 0 using the 3 nearest neighbors. (10 marks)

Author: CJH

©Imperial College London 2018/2019