1. GRA 6020 Multivariate Statistics Regression examples Ulf H. Olsson Professor of Statistics

2. Ulf H. Olsson Regression Analysis

3. Ulf H. Olsson Classical Assumptions Y is stochastic, x1, x2,….,xk are not Linearity in the parameters The error term has const.variance The error term is norm. Distributed with expectation equal to zero The error terms are independent The x-variables are linearly independent

4. Ulf H. Olsson GAUSS-MARKOV OLS is BLUE given the Classical Assumptions B = Best L=Linear U=Unbiased E=Estimator

5. Ulf H. Olsson Regression Analysis The error term has constant variance The error term follows a normal distribution with expectation equal to zero The x-variables are independent of the error term The x-variables are linearly independent

6. Ulf H. Olsson TSLS If the error term is correlated with one or more of the independent variables; The OLS estimate is not consitent, i.e., it is biased even in large samples. If there are instrument variables, that are not correlated with the error term; The TSLS estimator can be used to estimate the Betas consistently. The TSLS is a two-step OLS procedure For every x-variable which is correlated with the error term there must be at least one instrument variable outside the set of x-variables

7. Ulf H. Olsson TSLS and Heteroscedasticity Heteroscedastic Errors There are many forms of heteroscedasticity The most common is on the form TSLS might be a better alternative if the error term is heteroscedastic (asymptotically)

8. Ulf H. Olsson Klein’s equations

9. Ulf H. Olsson Factor Analysis Data Reduction Common Factor Latent variable Factor Loading How many factors