Gra6036- Multivartate Statistics with Econometrics (Psychometrics) Distributions Estimators Ulf H. Olsson Professor of Statistics

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

Ulf H. Olsson 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 Week hoursRead 2Basic Multivariate Statistical Analysis. Asymptotic Theory 3Lecture notes 3Logit and Probit Regression3Compendium: Logistic Regression 4Logit and Probit Regression3Compendium: Logistic Regression 5Exercises2 6Panel Models3Book chapter (14): Analyzing Panel Data: Fixed – and Random-Effects Models 7Panel Models3Book chapter (14): Analyzing Panel Data: Fixed – and Random-Effects Models 8Exercises2

Ulf H. Olsson 9Factor Analysis/ Exploratory Factor Analysis 3Structural Equation Modeling. David Kaplan, Confirmatory Factor Analysis3Structural Equation Modeling. David Kaplan, Confirmatory Factor Analysis3Structural Equation Modeling. David Kaplan, Exercises2 13Simultaneous Equations3Structural Equation Modeling. David Kaplan, Structural Equations Models3Structural Equation Modeling. David Kaplan, Structural Equations Models3Structural Equation Modeling. David Kaplan, Exercises2

Ulf H. Olsson Any Questions ?

Ulf H. Olsson Univariate Normal Distribution

Ulf H. Olsson Cumulative Normal Distribution

Ulf H. Olsson Normal density functions

Ulf H. Olsson The Chi-squared distributions

Ulf H. Olsson The Chi-squared distributions

Ulf H. Olsson Bivariate normal distribution

Ulf H. Olsson Standard Normal density functions

Ulf H. Olsson 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 Small sample properties

Ulf H. Olsson Large-sample properties

Ulf H. Olsson Introduction to the ML-estimator

Ulf H. Olsson 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 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

Two asymptotically Equivalent Tests Likelihood ration test Wald test

Ulf H. Olsson The Likelihood Ratio Test

Ulf H. Olsson The Wald Test

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

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

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

Ulf H. Olsson Bivariate normal distribution

Ulf H. Olsson Positive vs. Negative Skewness Exhibit 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 Low vs. High Kurtosis Exhibit 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 J-te order Moments Skewness Kurtosis

Ulf H. Olsson Skewness and Kurtosis

Ulf H. Olsson To Next week Down load LISREL 8.8. Adr.: Read: David Kaplan: Ch.3 (Factor Analysis) Read: Lecture Notes