mathematical model
mathematical model | math – 该题目是一个常规的mathematical model的练习题目代写, 涉及了mathematical model等代写方面
Evaluate whether each of statements (a)-(d) below isTrue, FalseorUncertain. 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= 0 + 1 xi+i.Letyi=yi+vi.
- Statement (a) : 1 may be consistently estimated with an OLS regression of yionxi.
(b) Consider the impact of a binary treatment on some outcomeyi. LetDibe a treatment indicator equal to one if unitireceives treatment and 0 otherwise. Letyi 0 ,yi 1 be potential outcomes for i.Define the selection effect asE[yi 0 |Di= 1]E[yi 0 |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) Letxtdenote a mean 0 column vector of excess returns formassets. Define x=Cov(xt).
Let be a square matrix with the eigenvectors of xas columns (ordered by the size of the
corresponding eigenvalue). Letpt= xt. Note that we may write
xt= kpkt+t
where kandpktrepresent the firstk < mcolumns of and elements ofpt, respectively.
- Statement (c): Cov(t)is a diagonal matrix.
(d) Letxtbe a vector of returns onmassets. Define x=cov(xt),and letP denote the number of unique variance-covariance terms contained within the general version of thismmmatrix. Now suppose we assume thatxtis driven by a two factor model:^1 xt=+Bft+t. Our model covariance matrix is given by x=BfB+ . LetQbe the number of unique parameters in this formulation (i.e. the total number of parameters inB, fand ).
- Statement (d): Q<P.
(^1) Hereis a vector of lengthm,Bis anm2 matrix of factor loadings,ftis a 21 vector of factor realizations withcov(ft) = f, and Cov(t) = is diagonal. Author: CJH
Consider the following difference-in-difference model for individualiin periodt{ 1 , 2 }:
yit= 0 + 1 DiAftert+ 2 Di+ 3 Aftert+it.
HereDiis an indicator variable denoting treated individuals and Aftertis an indicator variable equal to 1 in the 2nd period. Please compute OLS estimates 0 ols, 1 ols,ols 2 andols 3 using the data below:
i t yi Di Aftert
1 1 3 0 0
2 1 5 0 0
3 1 1 1 0
4 1 3 1 0
1 2 1 0 1
2 2 9 0 1
3 2 2 1 1
4 2 2 1 1
Author: CJH
Suppose we are interested in estimating the coefficients 0 , 1 , and 2 in the following linear model:
yi= 0 + 1 x 1 i+ 2 x 2 i+vi.
While we observex 1 iandx 2 i, we are unable to observeyientirely. Instead, we seeyi, whereyiis given by:
yi=
{
kifyi^2 > c
yiifyi^2 c
for some known constantsc >1 andk > c. LetviN(0,1) be a standard normal random variable with probability density functionf(z|x 1 i,x 2 i) =(z) and cumulative distribution function F(z|x 1 i,x 2 i) = (z).
(a) What is the probability distribution function ofyigivenx 1 i,x 2 iand the parameters 0 , 1 , 2 : g(yi|x 1 i,x 2 i; 0 , 1 , 2 )?(15 marks)
Author: CJH
Consider the following model foryi:
yi= 0 +Xi+i.
HereXi=
xi 1
xi 2
..
.
xik
and=
1
2
..
.
k
.You may assume thatyiand all elements ofXihave mean 0.
The Objective function for RIDGE is given by:
RIDGE= arg min
N
i=
(yiXi)^2 subject to
K
k=
k^2 c
for somec >0. Alternatively:
RIDGE= arg min
N
i=
(yiXi)^2 +
K
k=
^2 k.
(a) DeriveRIDGEin terms ofXi,yi, and.^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 toRIDGE?(10 marks)
(^2) Feel free to useX= X 1 X 2 .. . Xk andY= y 1 y 2 .. . yk if you prefer matrix notation. Author: CJH