Reflection Conditional expectation

Homework 21Due 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.
• On a scale of A, B, C, D, how would you grade your HW 1?

Q2 (Conditional expectation): Let the random vector (y, x)0 have a normal distribution withmean vector μ = (μy, μx)0 and covariance matrixwhere σy and σx are the standard deviations and ρ is the correlation between y and x. The jointThe determinant of the covariance matrix is|Σ| = σy 2σx 2(1 − ρ2),and the inverse of the covariance matrix isLast compiled: September 27, 2024; STAT5200, Fall 2023The marginal density of x isthat is, normal with mean μx and variance σx 2.

1. Derive conditional distribution of y given x.
2. Compute linear projection of y on x = (1, x). That is, derive and express L(y|1, x) as afunction 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 isslightly different from that was introduced in the lecture notes (Notes 01). Wooldridge definesthe linear projection in the following way,

Define x = (x1, ..., xK) as a 1 × K vector, and 代 寫Reflection Conditional expectation make the assumption that the K × Kvariance matrix of x is nonsingular (positive definite). Then the linear projection of yon1, x1, x2, ..., xK always exists and is unique: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, identicallydistributed sequence with E[yi 2] < ∞. 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 ?2Q5 (Asymptotics, delta method): Let θ ˆ be a √ N-asymptotically normal estimator for thescalar θ > 0. Let ˆγ = log(θ ˆ) be an estimator of γ = log(θ).

1. Why is ˆγ a consistent estimator of γ?
2. Find the asymptotic variance of √ N(ˆγ −γ) in terms of the asymptotic variance of √ N(θ ˆ−θ).
3. Suppose that, for a sample of data, θ ˆ = 4 and se(θ ˆ) = 2. What is ˆγ 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 ˆγ and se(ˆγ) to test H0. Whatdo you conclude?

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

1. Find an academic paper2 that (1) was published in one of those journals from your field ofinterest, AND (2) contains the word delta method, AND (3) the term delta method in thepaper 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 boxsource:"[name of the journal]" "delta method"

2. Properly cite the paper you found (name of the author, the title of the article, year ofpublication, 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 thedelta method?

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