Professor William Greene Stern School of Business IOMS Department Department of Economics Statistical Inference and Regression Analysis: Stat-GB , Stat-UB

Part 4 – Statistical Inference

4.1 – The Normal Family of Distributions

4/34 Part 4 – Statistical Inference Normal

5/34 Part 4 – Statistical Inference Standard Normal

6/34 Part 4 – Statistical Inference Chi Squared 1 = Square of N(0,1) 6

7/34 Part 4 – Statistical Inference Limit Result for Square of N(0,1) 7

8/34 Part 4 – Statistical Inference Sum of Two Independent Chi Squared(1) Variables 8

9/34 Part 4 – Statistical Inference Sum of N Independent Chi Squareds 9

10/34 Part 4 – Statistical Inference Limit Result for Square of Normal 10

11/34 Part 4 – Statistical Inference Noncentral Chi Squared 11

12/34 Part 4 – Statistical Inference t distribution 12 If v=1, t=N[0,1]/N[0,1] = Cauchy. No finite moments.

13/34 Part 4 – Statistical Inference Limiting Form of t 13

14/34 Part 4 – Statistical Inference F Distribution 14

15/34 Part 4 – Statistical Inference Limiting Form of F 15

16/34 Part 4 – Statistical Inference 16 Multiply value in last row by degrees of freedom. Equals value for chi-squared. 95% critical values for chi squared 95% critical values for limiting F distribution

17/34 Part 4 – Statistical Inference Special Case of F 17

18/34 Part 4 – Statistical Inference Independence of Sample Mean and Variance in Normal Sampling 18

19/34 Part 4 – Statistical Inference Useful Result 19

20/34 Part 4 – Statistical Inference Distribution of the t statistic 20

4.2 – Interval Estimation

22/34 Part 4 – Statistical Inference Estimation Point Estimator: Provides a single estimate of the feature in question based on prior and sample information. Interval Estimator: Provides a range of values that incorporates both the point estimator and the uncertainty about the ability of the point estimator to find the population feature exactly. 22

23/34 Part 4 – Statistical Inference Obtaining a Confidence Interval Pivotal quantity f(estimator, parameters) that has a known distribution free of parameters and data Probability statement can be made about the pivotal quantity Manipulate the interval to describe the parameter. 23

24/34 Part 4 – Statistical Inference Example – Normal Mean 24

25/34 Part 4 – Statistical Inference t distribution – values of t* 25

26/34 Part 4 – Statistical Inference Normal Variance 26

27/34 Part 4 – Statistical Inference 27

28/34 Part 4 – Statistical Inference GSOEP Income Data 28 Descriptive Statistics for 1 variables Variable| Mean Std.Dev. Minimum Maximum Cases Missing HHNINC| For the mean, t* for 24-1 = 23 degrees of freedom = Confidence interval for mean is / * (.15708/sqr(24)) = / Confidence interval for variance: Critical values from chi squared 23 are and Confidence interval for  2 is (24-1) /38.08 to (24-1) /11.69 = to Confidence interval for  is to Notice, not symmetric around s 2 or s.

29/34 Part 4 – Statistical Inference Large Sample Results There are almost no other cases in which there exists an exact pivotal quantity Most estimators rely on large sample results based on central limit theorems (estimator – parameter)  N(0,1) standard error of estimator 29

30/34 Part 4 – Statistical Inference Confidence Intervals 30

31/34 Part 4 – Statistical Inference Interpretation of The Interval Not a statement about probabilities that  will lie in specific intervals. (1-  ) percent of the time, the interval will contain the true parameter 31

32/34 Part 4 – Statistical Inference Application: Credit Modeling 1992 American Express analysis of Application process: Acceptance or rejection; X = 0 (reject) or 1 (accept). Cardholder behavior Loan default (D = 0 or 1). Average monthly expenditure (E = $/month) General credit usage/behavior (Y = number of charges) 13,444 applications in November, 1992

33/34 Part 4 – Statistical Inference is the true proportion in the population of 13,444 we are sampling from.

34/34 Part 4 – Statistical Inference Estimates plus and minus 1 and 2 standard errors