Econ 5410 (former 641) Econometrics I. Prof. J. Huston McCulloch

Transcription

1 Econ 5410 (former 641) Econometrics I Prof. J. Huston McCulloch

2 Lecture 1 Introduction Chapter 1 Elementary Statistics Random Variables, distributions Mean, variance Appendi B.1 B.3.

3 Levels of Econometrics

4 Levels of Econometrics Gourmet 2 nd year grad sequence Invent new recipes

5 Levels of Econometrics Gourmet 2 nd year grad sequence Invent new recipes Cookbook 1 st year grad sequence Assemble Ingredients from scratch, following recipe

6 Levels of Econometrics Gourmet 2 nd year grad sequence Invent new recipes Cookbook 1 st year grad sequence Assemble ingredients from scratch, following recipe Microwave Econ 5410, 5420 Select nutritious entree Punch in # of minutes

7 Levels of Econometrics Gourmet 2 nd year grad sequence Invent new recipes Cookbook 1 st year grad sequence Assemble Ingredients from scratch, following recipe Microwave Econ 5410, 5420 Select nutritious entree Punch in # of minutes Junk Food Unhealthy diet Darrell Huff, How to Lie with Statistics, 1954 Data Mining Cherry picking, Lemon dropping Duplication fallacy Newey-West HAC standard errors

8 Types of Data Cross Sectional Several individuals, one point in time Econ 5410 Continuous vs discrete dependent variables Logit, Probit Econ 5410 Time Series Several points in time, one individual Mostly Econ 5420 Pooled Cross Sections Several (different) individuals, several points in time Panel Data same individuals across time Econ 5420 Systems of simultaneous equations, Instrumental Variables e.g. Supply and Demand Econ 5410, 5420

9 Concerns Determining precision of estimates, forecasts standard errors, confidence intervals Holding other things constant Ceteris Paribus multiple regression Joint tests of significance F test, confidence ellipses Serial Correlation Reduces effective sample size

10 Problems Nonlinearity 5410 Data Mining 5410 Heteroskedasticity (non-const. variance) WLS, HCC 5410 Non-Gaussian errors Tests for normality 5410 Robust regression 5410 Censored dependent variable Tobit Econ 5410 Serial correlation Econ 5410, 5420 Fied, Random effects in Panel data 5420 Endogenous regressors Instrumental variables, 2SLS 5410, 5420

11 Elem. Statistics I Wooldridge App. B Random Variables (RVs) Appendi B follows convention: Upper case (e.g. X) indicates RV itself Lower case indicates possible values RV may take e.g. if continuous, 1,, 2, etc if discrete Tilde (~) also often used to indicate RV ~ e.g. X

12 Probability Distributions Characterized by Cumulative Distribution Function (CDF): F() = P(X ) CDF must be non-decreasing, bounded by 0 and 1: F() 1

13 Distributions may be discrete or continuous Discrete Distributions 1, 2,... n = possible values Probability Mass Function (PMF) gives probability of each outcome: n 1 P(X = i ) = p i CDF is a step function Eg fair die 1 = 1,... 6 = 6 p i = 1/6, i = 1,... 6 F() 1 5/6 2/3 ½ 1/3 1/6 i1 p i F() i p i

14 Continuous Distributions CDF is a continuous function P(X=) = 0, all, so PMF uninformative Probability Density Function (pdf) f() = F() F()/ = local prob. per unit of f() f() 0

15 Area under pdf between a and b gives P(a X b) = F(b) F(a) b a f ( ) d f() a b

16 Area under pdf to left of gives CDF: P(X ) = F() f ( ) d ( = Greek i) f()

17 Area under pdf to right of gives complemented CDF: P(X > ) = F c () = 1- F() f() f () d

18 Total area under pdf must be 1: f ( ) d F( ) 1 f()

19 Epected Value or population mean E(X) or ( = mu) Discrete Distribution: E( X ) E.g. fair die: E(X) = (1/6)(1) + (1/6)(2) +... (1/6)(6) = 3.5 i1 Continuous Dist: n i p i E( X ) f ( ) d

20 Linearity of Epectation Operator for any constants a and b, E(aX + b) = ae(x) + b, for any 2 RVs X, Y, E(X + Y) = E(X) + E(Y)

21 Variance Var(X) or 2 = E(X- ) 2 or n i1 i p 2 f ( ) d 2 i ( = sigma) has units of X 2 e.g. X~[$] Var(X)~[$ 2 ] (!)

22 Standard deviation sd( X ) Var( X ) has same units as X X~[$] sd(x)~[$]

23 Nonlinearity of Variance Operator Var(aX + b) = a 2 Var(X) but s.d.(ax + b) = a s.d.(x)

24 Standardized RVs Z-scores Suppose that for some RV X, E(X) =, Var(X) = 2 so s.d.(x) =. Then for Z = (X - ) /, E(Z) = 0, Var(Z) = s.d.(z) = 1. Also, F X () = P(X < ) = P(Z < ( - )/ ) = F Z (z) for z = ( - )/ so that F Z (z) tells us F X ().

25 Net Class Covariance Estimation of mean, variance Appendices B, C HW1 Due Friday 5 PM Bo of Shin-Wu Yu outside Arps 410