Linear regression equations

(a) The OLS estimator for β minimizes the Sum of Squared Residuals: n βˆ=argmin ô°…ô°„(y −βx)2ô°† βii i=1 Take the first-order condition to show that (b) Show that ˆ ô°ƒni=1 xiyi β=ô°ƒn x2. i=1 i ˆ ô°ƒni=1 xiεi β=β+ ô°ƒni=1x2i What is E[βˆ | β] and Var(βˆ | β)? Use this to show that conditional on β βˆ has the following distribution: ˆ ô° σ2 ô°‚ β|β∼Nβô°ƒn x2. i=1 i 1 (c) Suppose we believe that β is distributed normally with mean 0 and variance σ2 ; that is λ β ∼ N(0 σ2 ). Additionally assume that β is independent of εi. Compute the mean and λ variance of βˆ. That is what is E[βˆ] and Var(βˆ)?(Hint you might find useful: E[w1] = E[E[w1 | w2]] and Var(w1) = E[Var(w1 | w2)] + Var(E[w1 | w2]) for any random variables w1 and w2.) Question 2 Let us consider the linear regression model yi = β0 + β1xi + ui (i = 1 … n) which satisfies Assumptions MLR.1 through MLR.5 (see Slide 7 in “Linear_regression_review” under “Modules” on Canvas)1. The xis (i = 1 … n) and β0 and β1 are nonrandom. The randomness comes from uis (i = 1 … n) where var (ui) = σ2. Let βˆ0 and βˆ1 be the usual OLS estimators (which are unbiased for  y1  1  y2  1β0 and β1 respectively) obtained from running a regression of  .  on  .  (the intercept column) and  .   y n − 1  on   .  only x1  x2  .  .   . Suppose you also run a regression of    y1  x1   x2   x n − 1 xn yn xn a) Give the expression of β ̃1 as a function of yis and xis (i = 1 … n). (excluding the intercept column) to obtain another estimator β ̃1 of β1. ô° ̃ô°‚ ̃

b) Derive E β1 in terms of β0 β1 and xis. Show that β1 is unbiased for β1 when β0 = 0. If β0 ̸= 0 when will β ̃1 be unbiased for β1? c) Derive Varô°Î² ̃ ô°‚ the variance of β ̃ in terms of σ2 and x s (i = 1…n). 11i 1The model is a simple special case of the general multiple regression model in “Linear_regression_review”. Solving this question does not require knowledge about matrix operations.   y n − 1   1  yn 1  y2   .   . x n − 1  2 d) Show that Varô°Î² ̃ ô°‚ is no greater than Varô°Î²Ë† ô°‚; that is Var ô°Î² ̃ ô°‚ ≤ Var ô°Î²Ë† ô°‚. When do 1111 you have Var ô°Î² ̃ ô°‚ = Var ô°Î²Ë† ô°‚? (Hint you might find useful: use ô°ƒn x2 ≥ ô°ƒn (x − x Ì„)2 where 11 i=1ii=1i x Ì„ = n1 ô°ƒni=1 xi.)e) Choosing between βˆ1 and β ̃1 leads to a tradeoff between the bias and variance. Comment on this tradeoff.

Question 3 Let vˆ be an estimator of the truth v. Show that E (vˆ − v)2 = Var (vˆ) + [Bias (vˆ)]2 where Bias (vˆ) = E (vˆ) − v. (Hint: The randomness comes from vˆ only and v is nonrandom). Applied questions (with the use of R) For this question you will be asked to use tools from R for coding. Installation

Requirements: answer the question | .doc file

Question 1 Supposethatwehaveamodelyi =βxi+εi (i=1…n)wherey= n1 ô°ƒni=1yi =0x=ô°ƒni=1xi =0 and εi is distributed normally with mean 0 and variance σ2; that is εi ∼ N(0σ2).

(d) Since everything is normally distributed it turns out that E[β | βˆ] = E[β] + Cov(β βˆ) · (βˆ − E[βˆ]). Var(βˆ) Let βˆRR = E[β | βˆ]. Compute Cov(ββˆ) and use the value of E[β] along with the values of E[βˆ] Cov(ββˆ) and Var(βˆ) you have computed to show that ˆRR ˆ 􏰃ni=1 x2i ˆ β = E[β | β] = 􏰃ni=1 x2i + λ · β (Hint: Cov(w1 w2) = E[(w1 − E[w1])(w2 − E[w2])] and E[w1w2] = E[w1E[w2 | w1]] for any random variables w1 and w2)

(e) Does βˆRR increase or decrease as λ increases? How does this relate to β being distributed N(0 σ2 )? λ To install R please see

https://www.r-project.org/.

Once you install R please install also R Studio https://rstudio.com/products/rstudio/ download/.

You will need to use R Studio to solve the problem set. Question 1 Supposethatwehaveamodelyi =βxi+εi (i=1…n)wherey= n1 ô°ƒni=1yi =0x=ô°ƒni=1xi =0 and εi is distributed normally with mean 0 and variance σ2; that is εi ∼ N(0σ2).

(a) The OLS estimator for β minimizes the Sum of Squared Residuals: nβˆ=argmin 􏰅􏰄(y −βx)2􏰆 βiii=1 Take the first-order condition to show that

(d) Since everything is normally distributed it turns out that E[β | βˆ] = E[β] + Cov(β βˆ) · (βˆ − E[βˆ]). Var(βˆ) Let βˆRR = E[β | βˆ]. Compute Cov(ββˆ) and use the value of E[β] along with the values of E[βˆ] Cov(ββˆ) and Var(βˆ) you have computed to show that ˆRR ˆ 􏰃ni=1 x2i ˆ β = E[β | β] = 􏰃ni=1 x2i + λ · β (Hint: Cov(w1 w2) = E[(w1 − E[w1])(w2 − E[w2])] and E[w1w2] = E[w1E[w2 | w1]] for any random variables w1 and w2)

(e) Does βˆRR increase or decrease as λ increases? How does this relate to β being distributed N(0 σ2 )? λ

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