Consider the following slope estimator: b=2i=1 Yi Suppose the true model is ki + Bo +...
3. Consider simple linear regression model yi = Bo + B12; + &; and B. parameter estimate of the slope coefficient Bi: Find the expectation and variance of 31. Is parameter estimate B1 a) unbiased? b) linear on y? c) effective optimal in terms of variance)? What will be your answers if you know that there is no intercept coefficient in your model?
Assume that the variable Y is actually determined by the following equation Y; = Bo + B1X1,i+ B2X2,i + Uj additionally assume that corr(X1, X2) = p. The usual assumptions for a linear model hold in this case. You are interested in estimating B1. To accomplish this you collect a sample of the variables Y and X1 and estimate the following model Y; = Yo + 91X1,i+ vi (3) Answer the following questions 6. If p= 0 and B2 >...
Question 1 Consider the following model Yi = B.z; + u (a) Derive the OLS estimator of B, B. (6 marks] (b) Show that is unbiased. [9 marks] (c) Find the variance of B. [7 marks]
Question 1 Consider the following model Yi = Bx; +ui (a) Derive the OLS estimator of B, ß. (6 marks] (b) Show that B is unbiased. (9 marks] (c) Find the variance of ß. [7 marks] -r.pdf
Question 1 Consider the simple regression model (only one covariate): y= BoB1 u Let B1 be the OLS estimator of B1. a) What are the six assumptions needed for B1 to be unbiased, have a simple expression for its variance, and have normal distribution? (3 points) b) Under Assumptions 1-6, derive the distribution of B1 conditional on x\,..., xn. (3 points) In lecture we described how to test the null hypothesis B1 bo against the alternative hypothesis B1 bo, where...
6. Consider the following regression model without an intercept: Y = B,X, +U, One possible estimator for this model is given by: BE ANXJ Assume that you can make all of the usual ordinary least squares assumptions about the model, including the assumption that the true model does not include an intercept. Is B, an unbiased estimator? Please prove your conclusion, being sure to state the assumptions you use. [5 points]
1. Suppose the data is generated by model yi = B2.+ Ej. Suppose further that E( X) = 0, var(EX) = o2 and ( yi) is iid with finite fourth moment and and are jointly normal. But you mistakenly estimate it using the following model: y = a1 + 02.1; +e, and obtain the estimated coefficient parameters. Without looking at the analysis report, determine whether the following statement is true or false. please briefly explain. (a) lê = 0 (b)...
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...
Consider the model, Yi = Bo + B1 Xi + Uj, where you suspect Xi is endogenous. You have an exogenous instrument and you estimate the first stage to recover the residuals, Vhati. You want to test for endogeneity so you estimate the following model using OLS: Y; = Bo + B1 Xì + B2 Vhat; + Uj. The estimation results from 100 observations are in the table: Coefficient Standard Errors constant 2.96 0.47 X 0.75 0.85 Vhat 0.37 0.15...
Exercise 7.9 The model is where x; R. Consider the two estimators 1-yi (a) Under the stated assumptions, are both estimators consistent for β? (b) Are there conditions under which either estimator is efficient?