Gra6036- Multivartate Statistics with Econometrics (Psychometrics) Distributions Estimators

1. Gra6036- Multivartate Statistics withEconometrics (Psychometrics)DistributionsEstimators Ulf H. Olsson Professor of Statistics

2. Two Courses in Multivariate Statistics • Gra 6020 Multivariate Statistics • Applied with focus on data analysis • Non-technical • Gra 6036 Multivariate Statistics with Econometrics • Technical – focus on both application and understanding “basics” • Mathematical notation and Matrix Algebra Ulf H. Olsson

3. Course outline Gra 6036 • Basic Theoretical (Multivariate) Statistics mixed with econometric (psychometric) theory • Matrix Algebra • Distribution theory (Asymptotical) • Application with focus on regression type models • Logit Regression • Analyzing panel data • Factor Models • Simultaneous Equation Systems and SEM • Using statistics as a good researcher should • Research oriented Ulf H. Olsson

4. Evaluation • Term paper (up to three students) 75% • 1 – 2 weeks • Multipple choice exam (individual) 25% • 2 – 3 hours Ulf H. Olsson

5. Teaching and communication • Lecturer 2 – 3 weeks: 3 hours per week (UHO) • Theory and demonstrations • Exercises 1 week: 2 hours (DK) • Assignments and Software applications (SPSS/EVIEWS/LISREL) • Blackboard and Homepage • Assistance: David Kreiberg (Dep.of economics) Ulf H. Olsson

6. Week hours Read 2 Basic Multivariate Statistical Analysis. Asymptotic Theory 3 Lecture notes 3 Logit and Probit Regression 3 Compendium: Logistic Regression 4 Logit and Probit Regression 3 Compendium: Logistic Regression 5 Exercises 2 6 Panel Models 3 Book chapter (14): Analyzing Panel Data: Fixed – and Random-Effects Models 7 Panel Models 3 Book chapter (14): Analyzing Panel Data: Fixed – and Random-Effects Models 8 Exercises 2 Ulf H. Olsson

7. 9 Factor Analysis/ Exploratory Factor Analysis 3 Structural Equation Modeling. David Kaplan, 2000 10 Confirmatory Factor Analysis 3 Structural Equation Modeling. David Kaplan, 2000 11 Confirmatory Factor Analysis 3 Structural Equation Modeling. David Kaplan, 2000 12 Exercises 2 13 Simultaneous Equations 3 Structural Equation Modeling. David Kaplan, 2000 15 Structural Equations Models 3 Structural Equation Modeling. David Kaplan, 2000 16 Structural Equations Models 3 Structural Equation Modeling. David Kaplan, 2000 17 Exercises 2 Ulf H. Olsson

8. Any Questions ? Ulf H. Olsson

9. Univariate Normal Distribution Ulf H. Olsson

10. Cumulative Normal Distribution Ulf H. Olsson

11. Normal density functions Ulf H. Olsson

12. The Chi-squared distributions Ulf H. Olsson

13. The Chi-squared distributions Ulf H. Olsson

14. Bivariate normal distribution Ulf H. Olsson

15. Standard Normal density functions Ulf H. Olsson

16. Estimator • An estimator is a rule or strategy for using the data to estimate the parameter. It is defined before the data are drawn. • The search for good estimators constitutes much of econometrics (psychometrics) • Finite/Small sample properties • Large sample or asymptotic properties • An estimator is a function of the observations, an estimator is thus a sample statistic- since the x’s are random so is the estimator Ulf H. Olsson

17. Small sample properties Ulf H. Olsson

18. Large-sample properties Ulf H. Olsson

19. Introduction to the ML-estimator Ulf H. Olsson

20. Introduction to the ML-estimator • The value of the parameters that maximizes this function are the maximum likelihood estimates • Since the logarithm is a monotonic function, the values that maximizes L are the same as those that minimizes ln L Ulf H. Olsson

21. Introduction to the ML-estimator • In sampling from a normal (univariate) distribution with mean  and variance 2 it is easy to verify that: MLs are consistent but not necessarily unbiased Ulf H. Olsson

22. Two asymptotically Equivalent Tests Likelihood ration test Wald test

23. The Likelihood Ratio Test Ulf H. Olsson

24. The Wald Test Ulf H. Olsson

25. Example of the Wald test • Consider a simpel regression model Ulf H. Olsson

26. Likelihood- and Wald. Example from Simultaneous Equations Systems • N=218; # Vars.=9; # free parameters = 21; • Df = 24; • Likelihood based chi-square = 164.48 • Wald Based chi-square = 157.96 Ulf H. Olsson

27. Assessing Normality and Multivariate Normality (Continuous variables) Skewness Kurtosis Mardias test

28. Bivariate normal distribution Ulf H. Olsson

29. Positive vs. Negative SkewnessExhibit 1 These graphs illustrate the notion of skewness. Both PDFs have the same expectation and variance. The one on the left is positively skewed. The one on the right is negatively skewed. Ulf H. Olsson

30. Low vs. High KurtosisExhibit 1 These graphs illustrate the notion of kurtosis. The PDF on the right has higher kurtosis than the PDF on the left. It is more peaked at the center, and it has fatter tails. Ulf H. Olsson

31. J-te order Moments • Skewness • Kurtosis Ulf H. Olsson

32. Skewness and Kurtosis Ulf H. Olsson

33. To Next week • Down load LISREL 8.8. Adr.: http://www.ssicentral.com/ • Read: David Kaplan: Ch.3 (Factor Analysis) • Read: Lecture Notes Ulf H. Olsson