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...
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...
4. Consider the regression model, y1B22+ BKiK+ei -.. where errors may be heteroskedastic. Choose the most incorrect statement (a) The OLS estimators are consistent and unbiased (b) We should report the OLS estimates with the robust standard errors (c) The Gauss-Markov theorem may not apply (d) The GLS cannot be used because we do not know the error variances in practice (e) We should take care of heteroskedasticity only if homoskedasticity is rejected Consider the regression model, +BKIK+et e pet-1+...
1.Given the Multiple Linear regression model as Y-Po + β.X1 + β2X2 + β3Xs + which in matrix notation is written asy-xß +ε where -έ has a N(0,a21) distribution + + ßpXo +ε A. Show that the OLS estimator of the parameter vector B is given by B. Show that the OLS in A above is an unbiased estimator of β Hint: E(β)-β C. Show that the variance of the estimator is Var(B)-o(Xx)-1 D. What is the distribution o the...
Consider the following simple regression model: where the e, are independent errors with E(ed-0 and var(et)-Ơ2X? a. In this case, would an ordinary least squares regression provide you with the best b. c. linear unbiased estimates? Why or why not? What is the transformed model that would give you constant error variance? Given the following data: y = (4,3,1,0,2) and x = (1,2,1,3,4) Find the generalized least squares estimates of β1 and β2 (Do this by hand! Not with excel)
Q3. [10 points [Serial Correlation Consider a simple linear regression model with time series data: Suppose the error ut is strictly exogenous. That is Moreover, the error term follows an AR(1) serial correlation model. That where et are uncorrelated, and have a zero mean and constant variance a. 2 points Will the OLS estimator of P be unbiased? Why or why not? b. [3 points Will the conventional estimator of the variance of the OLS estimator be unbiased? Why or...
2.25 Consider the simple linear regression model y = Bo + B x + E, with E(E) = 0, Var(e) = , and e uncorrelated. a. Show that Cov(Bo, B.) =-TOP/Sr. b. Show that Cov(5, B2)=0. in very short simple way
Q. 1 Consider the multiple linear regression model Y = x3 + €, where e indep MV N(0,0²V) and V +In is a diagonal matrix. a) Derive the weighted least squares estimator for B, i.e., Owls. b) Show Bwis is an unbiased estimator for B. c) Derive the variances of w ls and the OLS estimator of 8. Is the OLS estimator of still the BLUE? In one sentence, explain why or why not.
1. Consider the simple linear regression model: Ү, — Во + B а; + Ei, where 1, . . , En are i.i.d. N(0,02), for i1,2,... ,n. Let b1 = s^y/8r and bo = Y - b1 t be the least squared estimators of B1 and Bo, respectively. We showed in class, that N(B; 02/) Y~N(BoB1 T;o2/n) and bi ~ are uncorrelated, i.e. o{Y;b} We also showed in class that bi and Y 0. = (a) Show that bo is...
5) Consider the simple linear regression model N(0, o2) i = 1,...,n Let g be the mean of the yi, and let â and ß be the MLES of a and B, respectively. Let yi = â-+ Bxi be the fitted values, and let e; = yi -yi be the residuals a) What is Cov(j, B) b) What is Cov(â, ß) c) Show that 1 ei = 0 d) Show that _1 x;e; = 0 e) Show that 1iei =...
1. Consider the following regression model with a single endogenous variable, ya : and given the reduced from for x: where z and are exogenous variables in the sense that cov(u,21) = cov(11,2 ) = 0 and cov(v,21) = cov(ng) = 0 (i.e., both zi and z2 are uncorrelated with u and v, and u is uncorrelated with v) (a) By substituting x into the equation for y, we obtain the reduce formed equation for y: Find the a in...