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Gra6036 Multivartate Statistics withEconometrics Psychometrics DistributionsEstimatorsUlf H Olsson.

Professor of Statistics 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 AlgebraUlf 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 orientedUlf H Olsson Evaluation Term paper up to three students 75 1 2 weeks.

Multipple choice exam individual 25 2 3 hoursUlf 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 hours Read2 Basic Multivariate Statistical 3 Lecture notesAnalysis Asymptotic Theory3 Logit and Probit Regression 3 Compendium Logistic.

Regression4 Logit and Probit Regression 3 Compendium LogisticRegression5 Exercises 26 Panel Models 3 Book chapter 14 Analyzing.

Panel Data Fixed andRandom Effects Models7 Panel Models 3 Book chapter 14 AnalyzingPanel Data Fixed andRandom Effects Models.

8 Exercises 2Ulf H Olsson 9 Factor Analysis Exploratory 3 Structural Equation Modeling Factor Analysis David Kaplan 200010 Confirmatory Factor Analysis 3 Structural Equation Modeling .

David Kaplan 200011 Confirmatory Factor Analysis 3 Structural Equation Modeling David Kaplan 200012 Exercises 213 Simultaneous Equations 3 Structural Equation Modeling .

David Kaplan 200015 Structural Equations Models 3 Structural Equation Modeling David Kaplan 200016 Structural Equations Models 3 Structural Equation Modeling David Kaplan 2000.

17 Exercises 2Ulf H Olsson Any Questions Ulf H Olsson Univariate Normal Distribution.

Ulf H Olsson Cumulative Normal DistributionUlf H Olsson Normal density functions2 1 1 2 x 2 2 .

x e1 x u e 2u N 0 1 Ulf H Olsson.

The Chi squared distributionsIf u N 0 1 then z u 1 If z1 z2 zn are n independent 2 1 var iablesthen zi 2 n E n n.

Var 2 n 2nUlf H Olsson The Chi squared distributionsIf u1 u2 un are n independent N 0 2 var iablesthen u 2i i 2 n .

If u1 u2 un are n independent N 2 var iablesthen u 2i i 2 n E n n Var 2 n 2n 4 Ulf H Olsson.

Bivariate normal distributionUlf H Olsson Standard Normal density functions u e 2 u 2 2 uv v 2 .

2 1 u v e2 1 1 u 1u u1 u2 un n 2 1 2.

2 u u1 u2 un 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 thusa sample statistic since the x s are random so is the estimatorUlf H Olsson Small sample propertiesUnbiased E .

Biased E 1 is more efficient Var 1 Var 2 Ulf H Olsson Large sample properties.

Consistency lim n P n 1for all Asymptotic unbiased lim n E n Var 1 1 is Asymptotic Efficent lim n .

Var for all Ulf H Olsson Introduction to the ML estimatorLet be the data matrix.

x1 x2 xk where xi are vectorsThe Likelihood function is as a function of the unknownparameter vector f x1 x2 xk f xi L X Ulf H Olsson.

Introduction to the ML estimator The value of the parameters that maximizes this function are the maximum likelihood Since the logarithm is a monotonic function the values that maximizes L are the same asthose that minimizes ln LThe necessary conditions for max imiz in g L is.

ln L We denote the ML estimator ML L L is the Likelihood function evaluated at Ulf H Olsson.

Introduction to the ML estimator In sampling from a normal univariate distribution withmean and variance 2 it is easy to verify that 1 n ML ML xi and.

2 1 n ML xi x 2 MLs are consistent but not necessarily unbiasedUlf H Olsson Two asymptotically Equivalent Tests.

Likelihood ration test The Likelihood Ratio TestLet be a vector of parameters to be estimated Two ML estimates U and RThe likelihood ratio is .

The l arg e sample distribution of 2 ln is 2 d Ulf H Olsson The Wald TestIf x N then x x is d .

H 0 c q then under H 0W c q U c q is d Ulf H Olsson.

Example of the Wald test Consider a simpel regression modely x H 0 0 0 .

we know z or t W 0 Var 0 0 zis 2 1 .

Ulf H Olsson Likelihood and Wald Example fromSimultaneous Equations Systems N 218 Vars 9 free parameters 21 Df 24 .

Likelihood based chi square 164 48 Wald Based chi square 157 96Ulf H Olsson Assessing Normality and MultivariateNormality Continuous variables .

Mardias test Bivariate normal distributionUlf H Olsson Positive vs Negative SkewnessThese graphs illustrate the notion of skewness Both.

PDFs have the same expectation and variance The oneon the left is positively skewed The one on the right isnegatively skewed Ulf H Olsson Low vs High Kurtosis.

These graphs illustrate the notion of kurtosis The PDF onthe right has higher kurtosis than the PDF on the left It ismore peaked at the center and it has fatter tails Ulf H Olsson J te order Moments.

Skewness KurtosisPopulation central moments j E X j j 1 E X X is continuous and random.

Skewness 1 2 3 2Kurtosis 2 2 3Ulf H Olsson Skewness and Kurtosis.

1 and 2 can be estimated from a sample We can test H 0 Skewnes 0 and H 0 Kurtosis 0by z and 2 testsWe can even estimate and test for multi var iate kurtosis Multi var iate kurtosis 2 p E X 1 X 2.

Ulf H Olsson 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 OlssonAsymptotic 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 â€“ and Random-Effects Models 7 Panel Models 3 Book chapter (14): Analyzing Panel Data: Fixed â€“ and ...

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