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日期：2024-10-07 12:51

Homework 21

Due date: October 9, 2024 (Wednesday).

Please submit your answer by 11:59pm.

There are total of 6 questions.

Q1 (Reflection): Read the solution to HW 1.

• Are your own answers in line with the solutions? If not, list the questions you missed. • Discuss what you could have done better.

• OnascaleofA,B,C,D,howwouldyougradeyourHW1?

Q2 (Conditional expectation): Let the random vector (y,x)0 have a normal distribution with mean vector μ = (μy,μx)0 and covariance matrix

Σ= σy2 σyσxρ, σ x σ y ρ σ x2

where σy and σx are the standard deviations and ρ is the correlation between y and x. The joint density is

fY,X(y,x)= 1 exp −1(w−μ)0Σ−1(w−μ) , −∞<y,x<∞. 2π|Σ|1/2 2

The determinant of the covariance matrix is

|Σ| = σy2σx2(1 − ρ2),

and the inverse of the covariance matrix is

Σ−1=1 σy2 −σyσx.

fY,X(y,x)=

1Last compiled: September 27, 2024; STAT5200, Fall 2023

−2ρ + .

exp −

2(1 − ρ2) σy

σy σx

σx

1ρ 1−ρ2−ρ 1

  Thus, the joint density can be written as

1 2πσyσx(1 − ρ2)1/2

1 y−μy2 y−μy x−μx x−μx2

σ y σ x σ x2

The marginal density of x is

fX(x) =

that is, normal with mean μx and variance σx2.

fY,X(y,x)dy = √ 1. Derive conditional distribution of y given x.

exp

Z∞ −∞

1 2πσx

1 x−μx 2 −2 σ ,

 2. Compute linear projection of y on x = (1,x). That is, derive and express L(y|1,x) as a function of μx, μy, ρ, σx, σy.

3. Define u = y − L(y|1, x). What is the distribution of u?

Q3 (Linear projection): The textbook (Wooldridge)’s definition of the linear projection is slightly different from that was introduced in the lecture notes (Notes 01). Wooldridge defines the linear projection in the following way,

Define x = (x1,...,xK) as a 1 × K vector, and make the assumption that the K × K variance matrix of x is nonsingular (positive definite). Then the linear projection of y on1,x1,x2,...,xK alwaysexistsandisunique:

L(y|1,x1,...,xK)=L(y|1,x)=β0 +β1x1 +...+βKxK =β0 +xβ, where, by definition,

β = [V ar(x)]−1Cov(x, y)

β0 =E[y]−E[x]β=E[y]−β1E[x1]−∙∙∙−βKE[xK].

Explain why this definition coincides with the definition that is introduced in the lecture notes (Notes 01). Provide a formal derivation as well. Hint: Answer is in Notes 01.

Q4 (Asymptotics, asymptotic normality): Let yi, i = 1, 2, ... be an independent, identically distributed sequence with E[yi2] < ∞. Let μ = E[yi] and σ2 = V ar(yi).

1. Let yN denote the sample average based on a sample size of N . Find V ar(√N (yN − μ)). 2. What is the asymptotic variance of √N (yN − μ)?

3. What is the asymptotic variance of yN ? Compare this with V ar(yN ).

4. What is the asymptotic standard deviation of yN ?

2

x

Q5 (Asymptotics, delta method): Let θˆ be a √N-asymptotically normal estimator for the scalar θ > 0. Let γˆ = log(θˆ) be an estimator of γ = log(θ).

1. Why is γˆ a consistent estimator of γ?

2. Find the asymptotic variance of √N(γˆ−γ) in terms of the asymptotic variance of √N(θˆ−θ).

3. Suppose that, for a sample of data, θˆ = 4 and se(θˆ) = 2. What is γˆ and its (asymptotic) standard error?

4. Consider the null hypothesis H0 : θ = 1. What is the asymptotic t statistic for testing H0, given the numbers from part 3?

5. Now state H0 from part 4 equivalently in terms of γ, and use γˆ and se(γˆ) to test H0. What do you conclude?

Q6 (Paper question): Find the paper that uses a delta method in your field. If you can’t find it, then find such paper from “American Economic Review”, which is one of the premier journal in economics.

1. Find an academic paper2 that (1) was published in one of those journals from your field of interest, AND (2) contains the word delta method, AND (3) the term delta method in the paper refers to the method that we learnt from the class, AND (4) applies the delta method.

One way to find such a paper is to use Google Scholar. Type the following in the search box

                    source:"[name of the journal]" "delta method"

2. Properly cite the paper you found (name of the author, the title of the article, year of publication, the name of the journal, etc.)

3. Read the paper and explain what is the main research question of the paper in one paragraph.

4. What is the parameter of interest in their empirical model, and why do the authors use the delta method?

5. (Optional reading; will not be graded). Read the following paper

Ver Hoef, J.M., 2012. Who invented the delta method?. The American Statistician, 66(2), pp.124-127.

(I put the copy of the paper in the HW section).

2If you can’t find such paper from the field of your interest, then find it from the “American Economic Review”, which is one of the premier journals in economics.

3