Introd, Policies, Regression

1. Introd, Policies, Regression Aim: Get you ready to read journal articles in Economics and Etrics. 1) I will merely whet your appetite! 2) Lot of self-teaching! No spoon-feeding! Homeworks Knowledge grows Giving/Sharing 3)Review of What is Etrics and Regression methods

2. Ancient Prayer for Learning • Written between 3000 BC Vedas 600 BC Budha (The Martin Luther like figure for Hinduism who protested against Yajna rituals with animal sacrifices. May our studies be brilliant and may there be no animosity between the teacher and the student. I want tRemember knowledge cannot be stolen and grows by giving

3. Scientific Method, Econ Data • Nothing on faith, question everything, Socrates gave his life. • Ancient Indian sages (Vedas 3000BC). Always permit questioning of fundamental beliefs. Patanjali promoted Yoga. • Quiet mind easier during Yoga. In 1967 Herb Benson showed: Meditators use 17% less oxygen, increase theta brain waves, are calmer & happier, they deactivate frontal lobe. (Time Magazine, Aug 4, 2003). Happiness depends on Inner Self (humans are designed to tolerate temp range 50 to 100 degrees, Learn not to complain)

4. Econometrics I, ECGA 7910 • Prof H. D. Vinod, Ph. D. (Harvard). • Title: Professor of Economics. • Fellow: Journal of Econometrics • www.fordham.edu/economics/vinod • Click Graduate Class Resources. or • www.fordham.edu/economics/vinod/grad.htm • Office location: Dealy Hall 5th floor East side, Room beyond Economics Dept. Room E526. Office hours: Thur, 1:15 to 2:15 and by appointment

5. What is Econometrics • Bag of tricks? Econ+Stats+Mtx Alg+Math+PC • Partial –incomplete info yet want models. • How to serach for Quantitative economic knowledge? How to inference/reasoning. Probability-econometrics models, parametric, semi and nonparametric • How to specify? Role of data. Observable, unobservables, source of errors, measurement/approximation/specification

6. Software. Learn several packs • Gauss comes with Mittelhammer text CD • LIMDEP comes with Greeene text CD • R-language is free downloadable software instructions on my web page, sophisticated and powerful. A good free stats book for regression using R is available at • www.stat.lsa.umich.edu/~faraway/book • Excel is everywhere, specific instruction on my web page

7. Davidson, Russell and J. G. KacKinnon (2004) Econometric Theory and Methods. New York, Oxford Univ. Press, ISBN 0-19-512372-7 (required text) • Mittlehammer, Ron C. G.G. Judge and D. J. Miller (2000), Econometric Foundations, Cambridge University Press, NY, Gauss software included on a CD in the book. (req’red) • Greene, 2003 William H. Econometric Analysis, MacMillan Publishing, 5th ed, NY with LIMDEP CD. (Optional text).

8. Supplementary texts: • Gujarati, Damodar, “Basic Econometrics” McGraw Hill (latest edition) • Salvatore D. and D. Reagle, Schaum series Econometrics (these are undergrad texts with spoon feeding details included) • Vinod-Ullah “Recent Advances in Regression Methods" 1982, Marcel Dekker, New York. (out of print has the SVD)

9. Homework in the form of a Diary assignment (* exercises of the text) and quizzes (if any) 50% • Final Exam: 40 % • Class attendance and participation: 10 % Hardest and most demanding course? Stats, Maths, Matrix Algebra,Lots of software use are all needed.

10. Regression Line • The regression line is given by the familiar equation y = b 0 + b 1 x. • The regression line is used as a descriptive tool to summarize the relationship between the two variables x and y.

11. Population Regression Line • Y=dependent variable • X= regressor • To discuss inference, a distinction must be made between a regression line estimated from a sample and one estimated from a population. The population regression line is written as y = b0 + b1 x. • b0 is the Y intercept for the population regression model, and b1 is the slope.

12. Sample Regression Line • The sample regression line is written as y = b 0 + b 1 x. • b0 is an estimate of b0. • b1 is an estimate of b1.

13. y x Is a sample exactly representative? Even if a random sample from the population is drawn, there can be no guarantee that the sample will be exactly representative of the population. Sample Regression Line Population Regression Line (unknown)

14. Confidence Intervals and Hypothesis Testing • How accurate will the sample estimates b0 and b1 be? • Two familiar inferential techniques, confidence intervals and hypothesis testing, will be used to analyze the model’s estimated coefficients and the predicted values.

15. Assumptions of the Simple Linear Model

16. Simple Linear Regression Model • To make inferences about the linear model, a number of assumptions are necessary. • Recall that an error term was incorporated in the model, because virtually no real set of bivariate data is exactly linear. • Incorporating the error term in the population regression line produces the simple linear regression model, y i = b0 + b1 x i + ei .

17. The Error Term • The error term represents the variation in y not accounted for by the linear regr model. In order to perform inference on the model, some assumptions about the nature of the error terms are required. 1. The ei are presumed to be normally distributed with a mean of 0 and a variance of se2. 2. The ei are presumed to be independent.

18. Estimation of the Parameters b0, b1, and se2

19. Inference--How good is the estimate of b1?

20. The Confidence Interval for b1 The confidence interval will serve two purposes, to place bounds on the location of b1 as well as to provide information about the quality of the point estimate b1. Form of the Confidence Interval sample estimate of parameter (a certain number of standard deviation units depending on the desired confidence) (the standard deviation of the sample estimate)

21. The Confidence Interval for b1 • The sample estimate of b1 is b1. The variance of b1 is given by The sample est of the variance of b1 is The standard deviation of the sample estimate is given by

22. The Confidence Interval for b1 Definition : The (1 - a) confidence interval for b1 is given by where is the critical value for a t-distribution with n - 2 degrees of freedom which captures an area of a/2 in the right tail of the distribution.

23. Homework in the form of a Diary assignment and quizzes (if any) 50% • Final Exam: 40 % • Class attendance and participation: 10 % Hardest and most demanding course? Stats, Maths, Matrix Algebra all needed.

24. Regression in Matrix Notation • We will review elementary Regression analysis without matrix algebra and then see how y = X  +  is so simple and better. y is n by 1 vector, X is n by p matrix and  is p by 1vector. Transpose=  • Minimizing error sum of squares is simply minimizing  w.r.t   = (y  X) (y  X) = y y  Xy  yX + XX

25. Derivation of Least Sq • So we need to Minimize  2yX + XX w.r.t , Let us change notation and write A=yX and C= XX, now minimand is •  2A + C to which we apply following two mtx algebra rules for derivatives • (1) derivative of A w.r.t  is A • (2) derivative of C w.r.t  is 2C

26. Derivation of Least Sq 2 • So we need to Minimize  2yX + XX w.r.t , Let us change notation and write A=yX and C= XX, now minimand is •  2A + C to which we apply following two mtx algebra rules for derivatives • (1) derivative of A w.r.t  is A • (2) derivative of C w.r.t  is 2C

27. Derivation of Least Sq 3 • A=yX and C= XX, now minimand is •  2A + C and its derivative is set equal to zero. Then we have: •  2 A + 2C =0 Now substitute A = Xy and C= XX  2 Xy + 2 XX  =0 Now move a term 2 XX  = 2 Xy, cancel 2 to give XX  = Xy

28. Derivation of Least Sq 4 Solution to the minimization problem is XX  = Xy Pre-multiply both sides by (XX)1 We have  = (XX)1 Xy as the solution to the minimization problem by using usual calculus methods for minimizing anything. Much of econometrics deals with b = (XX)1 Xy as the OLS estimator and its extensions different assumptions for 

29. Testing a Hypothesis Concerning b1

30. Relationship Between y and x • One of the fundamental questions concerning the construction of a model is whether a linear relationship exists between y and x. • The linear model connects x to y through the slope parameter b1. yi = b0 + b1 xi + ei , i = 1,..., n

31. [ ] b1 The Confidence Interval for b1 The expression, , creates the following interval. The expression relates the width of the interval to the amount of confidence required.

32. What if b1 = 0? • If b1 = 0, then there is no linear relationship between x and y since the term b1xi = 0. • Regardless of the value of x, the model becomes yi = b0 + ei , • which says that y is equal to a constant b0 plus a random error.

33. Using P-values to Test Hypothesis Concerning b1 • The P-value measures the level of rareness of the test statistic assuming the null hypothesis is true. • Specifically, a P-value is the probability of observing a test statistic as large or larger (in absolute value) than what has been observed given that the null hypothesis is true. • Reject the null hypothesis if the P-value is less than a.