2. Changing Units Suppose we estimate a standard OLS regression equation on data X and Y...
Hint: For (C) we proved that 2. Changing Units Suppose we estimate a standard OLS regression equation on data X and Y and have the standard formulas: Now suppose that X,-1 + 1.6Zi for some Zi-Le., suppose that Xi were generated by trans- orming some Zi.
2. Suppose we have the simple regression model Y =a+8X:+E, and their OLS coefficient estimators a and b. Answer the following questions. (a) Suppose we multiply X, by 1/2 for all i and do the OLS estimation again using X as the regressor (the independent variable). What will be your new estimators, denoted by ă (intercept) and b (slope)? Compare them with the original OLS estimators a and b, respectively (b) Compare Var[b] and Var[b]. Are they the same or...
f and g 1. Consider the standard bivariate regression: Y = Bo + B,X, + a) What is the above function called? b) What are Y, X, X, Bo, and B, called? Graph an example of the function along with some fake data points (X,Y) and label each of the parts. c) Suppose that we estimate the above function using a sample of data and the ordinary least squares method (OLS). Write down the sample regression function. d) What is...
Suppose we have a regression model (x, > 0) with n samples of i.i.d. data 0, Varuir] 2, and (a) Obtain the OLS estimator β0Ls for β (b) Obtain the optimal WLS estimator ws for B Suppose we have a regression model (x, > 0) with n samples of i.i.d. data 0, Varuir] 2, and (a) Obtain the OLS estimator β0Ls for β (b) Obtain the optimal WLS estimator ws for B
Problem 2. (Regression without intercept, 50 pts) Suppose you are given the model: Y; = BX; + Ui, E[u;|Xį] = 0. A) Derive the OLS estimator ß. B) After you estimate B, you can obtain the residual û; = Y; – ĢXį. Does 21-1 Ûi = 0? Explain why and show your derivation.
Consider the following simple regression model: a. Suppose that OLS assumptions 1 to 4 hold true. We know that homoskedasticity assumption is statedas: Var[UjIx] = σ2 for all i Now, suppose that homoskedasticity does not hold. Mathematically, this is expressed as In other words, the subscript i in σ12 means that the conditional variance of errors for each individual i is different. Under heteroskedasticity, we can derive the expression for the variance of Var(B) as SST Where SSTx is the...
2. Consider the following model: y = XB + u where y is a (nx1) vector containing observations on the dependent variable, B = Bi , B X is a (n x 3) matrix. The first column of X is a column of ones whilst the second and third columns contain observations on two explanatory variables (x and x2 respectively). u is (n x 1) vector of error terms. The following are obtained: 1234.7181 1682.376 7345.581 192.0 259.6 1153.1) X'X...
(a) Suppose you are given the following (x, y) data pairs. x 1 2 5 y 2 1 8 Find the least-squares equation for these data (rounded to three digits after the decimal). ŷ = + x (b) Now suppose you are given these (x, y) data pairs. x 2 1 8 y 1 2 5 Find the least-squares equation for these data (rounded to three digits after the decimal). ŷ = + x (c) In the data for parts (a) and (b), did...
Question 4 3 pts Consider the estimated multiple regression model using OLS, with the standard errors in parentheses below each estimated coefficient. There are 1,576 observations in the sample: Y = 10 + 2X2i - 5Xzi (3) (1.5) (2) Suppose that the sample mean of Y is 30. For the 18th observation (i=18) in the sample, the value of X2 is 50, the value of X3 is 16, and the value of Y is 20. The residual associated with the...
(a) Suppose you are given the following (x, y) data pairs. x 1 2 6 y 4 3 7 Find the least-squares equation for these data (rounded to three digits after the decimal). ŷ = + x (b) Now suppose you are given these (x, y) data pairs. x 4 3 7 y 1 2 6 Find the least-squares equation for these data (rounded to three digits after the decimal). ŷ = + x (c) In the data for parts...