Homework Answers
P![ras because by least square cilerian 2 = arg minst yo- a)- a & lyi- d)² = mx 2a - 28 yi da ial of ý E[Q] 2ELU C . ! Var ( 2 )](http://img.homeworklib.com/questions/a3341590-ac60-11ea-9d7a-b9f899f70a71.png?x-oss-process=image/resize,w_560)
![varl & ) = Van (yi- j) ܛ ܛܛܠܝܐܫܕܬܝܝܢܬܐ ܝܣܥܘܠ - ܙ * * ( 7- . . . 7- . . . /hܬܘܢ ܐܟܣ ܪܨ ܢܝܡ ]= ܡܢ ܝ ܐܠܢܒܡܓܟ݁ܠ ܫܐܢܝܡܠܡ ܀ ܒܡ .](http://img.homeworklib.com/questions/a3d3cc00-ac60-11ea-91af-c7c629a430b0.png?x-oss-process=image/resize,w_560)
![true model is )e܀ ܛܰܬ݁ ܪ ܀ ܕܐ z 3 and i a z not fit Y = Bo Xiter B. = { Yi Xi - [ 2 ]: EL 5Y: Xi L EX 2 - hao Bo Exo² Narl 6](http://img.homeworklib.com/questions/a4efb270-ac60-11ea-9f19-17d9a2346ed6.png?x-oss-process=image/resize,w_560)
![A : LJAL -8_(ܘܗ 5X12 - ܐ ܠ ܐ \ ܙ - Etb1 ܟܐ ܐ ܙ - ܙ ܕܐܐ ܐܰܠܐܝܰܬ݁ܺܝܩܳܛܟܦܣܵܐ ܣ ]. ; z SK .. ܘ8+ ;X ܐ:x£ .ܐ 2 _ 2380 ܬX- ܠܓ-:J] 3](http://img.homeworklib.com/questions/a5a4c1c0-ac60-11ea-9432-bfeb3ccce06a.png?x-oss-process=image/resize,w_560)
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For observations {Y, X;}=1, recall that for the model Y = 0 + Box: +e the...
2. Consider the simple linear regression model: where e1, .. . , es, are i.i.d. N (0, o2), for i= 1,2,... , n. Suppose...
2. Consider the simple linear regression model: where e1, .. . , es, are i.i.d. N (0, o2), for i= 1,2,... , n. Suppose that we would like to estimate the mean response at x = x*, that is we want to estimate lyx=* = Bo + B1 x*. The least squares estimator for /uyx* is = bo bi x*, where bo, b1 are the least squares estimators for Bo, Bi. ayx= (a) Show that the least squares estimator for...
3. (20 pts) Suppose that we have 4 observations for 3 variables y , x\, X2...
3. (20 pts) Suppose that we have 4 observations for 3 variables y , x\, X2 and consider a problem of regressing y on two (qualitative) variables x\, xz. Data y (Income) x (Gender) X2 (Management Status) obs no. Female None 2 Male None 3 Female Yes 4 Male Yes Y4 To handle the qualitative variables x\, x2, we define dummy variables z1, 22 as Male for for 1, 1, T2= Yes Z1= Z2= -1 for for 1 1 =...
X and Y are random variables (a) Show that E(X)=E(B(X|Y)). (b) If P((X x, Y )...
X and Y are random variables (a) Show that E(X)=E(B(X|Y)). (b) If P((X x, Y ) P((X x})P({Y y)) then show that E(XY) = E(X)E(Y), i.e. if two random variables are independent, then show that they are uncorrelated. Is the reverse true? Prove or disprove (c) The moment generating function of a random variable Z is defined as ΨΖφ : Eez) Now if X and Y are independent random variables then show that Also, if ΨΧ(t)-(λ- (d) Show the conditional...
4. Recall that the covariance of random variables X, and Y is defined by Cov(X,Y) =...
4. Recall that the covariance of random variables X, and Y is defined by Cov(X,Y) = E(X - Ex)(Y - EY) (a) (2pt) TRUE or FALSE (circle one). E(XY) 0 implies Cov(X, Y) = 0. (b) (4 pt) a, b, c, d are constants. Mark each correct statement ( ) Cov(aX, cY) = ac Cov(X, Y) ( ) Cor(aX + b, cY + d) = ac Cov(X, Y) + bc Cov(X, Y) + da Cov(X, Y) + bd ( )...
Consider the simple linear regression model y - e, where the errors €1, ,en are iid....
Consider the simple linear regression model y - e, where the errors €1, ,en are iid. random variables with Eki-0, var(G)-σ2, i-1, .. . ,n. Solve either one of the questions below. 1. Let Bi be the least squares estimator for B. Show that B is the best linear unbiased estimator for B1. (Note: you can read the proof in wikipedia, but you cannot use the matrix notation in this proof.) 2. Consider a new loss function Lx(A,%) 71 where...
1. A simple regression model is given by Y81B2X+ e for t 1, (1) ,n errors e with Var (e) a follow AR(1) model where th...
1. A simple regression model is given by Y81B2X+ e for t 1, (1) ,n errors e with Var (e) a follow AR(1) model where the regression et pet-1 + , t=1...n where 's are uncorrelated random variables with constant variance, that is, E()0, Var (v) = , Cov (, ,) 0 for t Now given that Var (e) = Var (e1-1)= , and Cov (e-1, v)0 (a) Show that (b) Show that E (ee-1)= p. (c) What problem(s) will...
Simple linear regression model Assumptions: AI E[u] 0 for all i, i1, .., n On average,...
Simple linear regression model Assumptions: AI E[u] 0 for all i, i1, .., n On average, random component is zero Model runs through expected values of Yand Y A2 E[uaij]-0 for all i and j where i /j COV(IIİlh)- Unobserved component not related across observations E[14"]= for all i All observations have random component dravn from a distribution with the same variance σ2 , f(0,02) A3 var(11i)-σ (Homoskedasticitv) A4 E[Alli] = 0 for all i Random component and covariate not...
3. (20 pts) Suppose that we have 4 observations for 3 variables y,I, 2 and consider...
3. (20 pts) Suppose that we have 4 observations for 3 variables y,I, 2 and consider a problem of regressing y on two (qualitative) variables r, 2. Data: 22 obs no. y (Income) 2 (Management Status) I (Gender) 1 None Female 2 None Male Yes Female Yes Male 4 To handle the qualitative variables r, 12, we define dummy variables 1, 22 as for 1, 22= Yes Male for 1, 219 22 -1. for 22= None for 1= Female -1,...
S y, and that yi-μ +Ei. You can assume that Ele]-0 for all i, Ele: -σ2...
S y, and that yi-μ +Ei. You can assume that Ele]-0 for all i, Ele: -σ2 for all i, and Ele#3-0 for all i j You want to estimate a sample mean, and your friend tells you to use the following estimator: uppose that vou have collected n observations on where w is a known sample weight for observation i (this means w; is non-random) (a) Find E( (b) Under what conditions, if any, is p an unbiased estimator? Under...
6. This problem considers the simple linear regression model, that is, a model with a single...
6. This problem considers the simple linear regression model, that is, a model with a single covariate r that has a linear relationship with a response y. This simple linear regression model is y = Bo + Bix +, where Bo and Bi are unknown constants, and a random error has normal distribution with mean 0 and unknown variance o' The covariate a is often controlled by data analyst and measured with negligible error, while y is a random variable....
2. Consider the simple linear regression model: where e1, .. . , es, are i.i.d. N (0, o2), for i= 1,2,... , n. Suppose that we would like to estimate the mean response at x = x*, that is we want to estimate lyx=* = Bo + B1 x*. The least squares estimator for /uyx* is = bo bi x*, where bo, b1 are the least squares estimators for Bo, Bi. ayx= (a) Show that the least squares estimator for...
3. (20 pts) Suppose that we have 4 observations for 3 variables y , x\, X2 and consider a problem of regressing y on two (qualitative) variables x\, xz. Data y (Income) x (Gender) X2 (Management Status) obs no. Female None 2 Male None 3 Female Yes 4 Male Yes Y4 To handle the qualitative variables x\, x2, we define dummy variables z1, 22 as Male for for 1, 1, T2= Yes Z1= Z2= -1 for for 1 1 =...
X and Y are random variables (a) Show that E(X)=E(B(X|Y)). (b) If P((X x, Y ) P((X x})P({Y y)) then show that E(XY) = E(X)E(Y), i.e. if two random variables are independent, then show that they are uncorrelated. Is the reverse true? Prove or disprove (c) The moment generating function of a random variable Z is defined as ΨΖφ : Eez) Now if X and Y are independent random variables then show that Also, if ΨΧ(t)-(λ- (d) Show the conditional...
4. Recall that the covariance of random variables X, and Y is defined by Cov(X,Y) = E(X - Ex)(Y - EY) (a) (2pt) TRUE or FALSE (circle one). E(XY) 0 implies Cov(X, Y) = 0. (b) (4 pt) a, b, c, d are constants. Mark each correct statement ( ) Cov(aX, cY) = ac Cov(X, Y) ( ) Cor(aX + b, cY + d) = ac Cov(X, Y) + bc Cov(X, Y) + da Cov(X, Y) + bd ( )...
Consider the simple linear regression model y - e, where the errors €1, ,en are iid. random variables with Eki-0, var(G)-σ2, i-1, .. . ,n. Solve either one of the questions below. 1. Let Bi be the least squares estimator for B. Show that B is the best linear unbiased estimator for B1. (Note: you can read the proof in wikipedia, but you cannot use the matrix notation in this proof.) 2. Consider a new loss function Lx(A,%) 71 where...
1. A simple regression model is given by Y81B2X+ e for t 1, (1) ,n errors e with Var (e) a follow AR(1) model where the regression et pet-1 + , t=1...n where 's are uncorrelated random variables with constant variance, that is, E()0, Var (v) = , Cov (, ,) 0 for t Now given that Var (e) = Var (e1-1)= , and Cov (e-1, v)0 (a) Show that (b) Show that E (ee-1)= p. (c) What problem(s) will...
Simple linear regression model Assumptions: AI E[u] 0 for all i, i1, .., n On average, random component is zero Model runs through expected values of Yand Y A2 E[uaij]-0 for all i and j where i /j COV(IIİlh)- Unobserved component not related across observations E[14"]= for all i All observations have random component dravn from a distribution with the same variance σ2 , f(0,02) A3 var(11i)-σ (Homoskedasticitv) A4 E[Alli] = 0 for all i Random component and covariate not...
3. (20 pts) Suppose that we have 4 observations for 3 variables y,I, 2 and consider a problem of regressing y on two (qualitative) variables r, 2. Data: 22 obs no. y (Income) 2 (Management Status) I (Gender) 1 None Female 2 None Male Yes Female Yes Male 4 To handle the qualitative variables r, 12, we define dummy variables 1, 22 as for 1, 22= Yes Male for 1, 219 22 -1. for 22= None for 1= Female -1,...
S y, and that yi-μ +Ei. You can assume that Ele]-0 for all i, Ele: -σ2 for all i, and Ele#3-0 for all i j You want to estimate a sample mean, and your friend tells you to use the following estimator: uppose that vou have collected n observations on where w is a known sample weight for observation i (this means w; is non-random) (a) Find E( (b) Under what conditions, if any, is p an unbiased estimator? Under...
6. This problem considers the simple linear regression model, that is, a model with a single covariate r that has a linear relationship with a response y. This simple linear regression model is y = Bo + Bix +, where Bo and Bi are unknown constants, and a random error has normal distribution with mean 0 and unknown variance o' The covariate a is often controlled by data analyst and measured with negligible error, while y is a random variable....
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