1. For the general multivariate regression model, the least squares estimator is given by Show that for the slope estimator in the simple (bivariate) regression case, this is equivalent to ja! įs] 2. In the general multivariate regression model, the variance of the least squares estimator, Va( is σ2(XX)". Show that for the simple regression case, this is equivalent to a. Var(B- b. Var(B)o i, Σ (Xi-X) 2 C. What is the covariance between β° and β,?
For a multiple linear regression model with four predictors, show that: For a multiple linear regression model with four predictors, show that:
3. Consider the multiple linear regression model iid where Xi, . . . ,Xp-1 ,i are observed covariate values for observation i, and Ei ~N(0,ơ2) (a) What is the interpretation of B1 in this model? (b) Write the matrix form of the model. Label the response vector, design matrix, coefficient vector, and error vector, and specify the dimensions and elements for each. (c) Write the likelihood, log-likelihood, and in matrix form. aB (d) Solve : 0 for β, the MLE...
Consider the regression model where the εi are i.i.d. N(0,σ2) random variables, for i = 1, 2, . . . , n. (a) (4 points) Show βˆ is normally distributed with mean β and variance σ2 . 1 1SXX Question 6 Consider the regression model y = Bo + B12 + 8 where the €, are i.i.d. N(0,0%) random variables, for i = 1,2, ..., n. (a) (4 points) Show B1 is normally distributed with mean B1 and variances
1. Consider a regression model Yi = x;ß +ei, i = 1,...,n. You estimate this model using the OLS estimator. (a) Present and discuss assumptions for the OLS estimation.
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....
derive the general formula for leverage in a regression model using a single binary covariate with
Consider the simple linear regression model: Yi = Bo + Bilitei, i = 1,...,n. with the least squares estimates ỘT = (Bo ß1). We observe a new value of the predictor: x] = (1 xo). Show that the expression for the 100(1 - a)% prediction interval reduces to the following: . (xo – x2 Ēo + @130 Etap 11+ntan (x; – 7)2
e. Consider the multiple regression model Y = X1?1 + X2?2 + . The Gauss-Markov conditions hold. Show that Y0 (I ? H)Y = Y0 (I ? H1)Y ? ?ˆ0 2X0 2 (I ? H1)Y. e. Consider the multiple regression model Ý = XiA + X2ß2 + E. The Gauss-Markov conditions hold. show that Y'(l-H)Y-Y'(1-HJY-? (1-H,)Y
1.) What is the difference between a simple regression model and a multiple regression model? a.) There isn’t one. The two terms are equivalent b.) A simple regression model has a single predictor whereas a multiple regression model has potentially many c.) A simple regression model can handle only limited amounts of data whereas a multiple regression model can handle large data sets d.) A simple regression is appropriate for a dichotomous outcome variable, whereas a multiple regression model should...