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# Econ 103, Week 1 Lecture slides Econ 103

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This 45 page Class Notes was uploaded by kodasbigmove on Wednesday April 20, 2016. The Class Notes belongs to Econ 103 at University of California - Los Angeles taught by Rojas in Spring 2015. Since its upload, it has received 46 views. For similar materials see Econometrics in Economcs at University of California - Los Angeles.

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Date Created: 04/20/16

Lecture 1: Introduction and Review of Probability and Statistics Rodrigo Pinto UCLA Spring, 2016 Course Logistics Teaching Assistants YUN FENG (yfengpetervon@ucla.edu) WENSHU GUO (guowenshu2008@ucla.edu) LU LIU (luliu@ucla.edu) MATIAS VIEYRA (mativieyra@ucla.edu) LINA ZHANG (lnzhang@ucla.edu) Time and location Lectures, Tue. & Ths. 2:00-3:15PM, Humants A51 Lab lectures, Friday 1:00-1:50PM, Dodd 147 O▯ce hours: Ths. 4:00{5:00PM, Bunche Hall 8385 Midterm Exam Tuesday, May 10, 2016, 2:00PM{3:15PM Final Exam Thursday, June 9, 2016, 11:30AM{2:30PM One has to register to both Econ 103 AND Econ 103L. You cannot take one without the other Tentative Course Schedule: Course Schedule Date Book Lecture Week Lecture Month Day Chapter Content Week 1 Lec. 1 March 29 Ch. 1 Review of Probability and Statistics Week 1 Lec. 2 March 31 Ch. 1 A Short Introduction to Econometrics Week 2 Lec. 1 April 5 Ch. 2 Linear Regression Week 2 Lec. 2 April 7 Ch. 2 Linear Regression Week 3 Lec. 1 April 12 Ch. 3 Interval Estimation and Hypothesis Testing Week 3 Lec. 2 April 14 Ch.s 2{3 Exercises on Linear Regression and Hypothesis Testing Week 4 Lec. 1 April 19 Ch. 4 Prediction, Goodness-of-Fit, and Modeling Issues Week 4 Lec. 2 April 21 Ch. 4 Prediction, Goodness-of-Fit, and Modeling Issues Week 5 Lec. 1 April 26 Ch. 5 Multiple Regression Model Week 5 Lec. 2 April 28 Ch. 5 Multiple Regression Model Week 6 Lec. 1 May 3 Ch. 6 Inference in the Multiple Regression Model Week 6 Lec. 2 May 5 Ch. 6 Inference in the Multiple Regression Model Week 7 Lec. 1 May 10 Ch.s 1{6 Midterm Exam Week 7 Lec. 2 May 12 Ch. 7 Indicator Variables Week 8 Lec. 1 May 17 Ch. 7 Indicator Variables Week 8 Lec. 2 May 19 Ch. 8 Heteroskedasticity Week 9 Lec. 1 May 24 Ch. 8 Heteroskedasticity Week 9 Lec. 2 May 26 Ch. 10 Random Regressors and Moment-Based Estimation Week 10 Lec. 1 May 31 Ch. 10 Random Regressors and Moment-Based Estimation Week 10 Lec. 2 June 2 Ch. 10 Wrap-up and review Textbooks and Other Material 1 Principles of Econometrics. Hill, R. C., Gri▯ths, W. E. and G. C. Lim, 4th Edition, 2011. Wiley and Sons. (Required!) 2 Using Stata for Principles of Econometrics. Lee C. Adkins & R. Carter Hill, 4th Edition. Wiley and Sons. (Strongly Recommended) 3 Other: Basic Econometrics. Gujarati, D. N., 5th Revised Edition, 2010, McGraw-Hill. Introductory Econometrics. Wooldridge, J., 5th Edition,2013, South Western. Probability and Statistical Inference. Hogg, R. V. and E. A. Tanis, 8th Edition, 2009, MacMillan. 4 Lecture notes: Slides will be posted on the class website before class. I will not bring copies! 5 Course outline - see Syllabus Requirements 1. 6 problem sets, for 20% of the ▯nal grade. Mix of multiple choice questions, analytic exercises, and STATA work. Grading of problem sets: 5 points for completing homework. 5 points for correct answer to question chosen randomly by TAs. Schedule: See tentative schedule in the syllabus. Due at end of class on due date. No exceptions! Must hand in hardcopy. Note! You are encouraged to work together, but everyone must hand in their own hardcopy, and do the computer work independently. Requirements 2. Midterm { 30% (or 0%) In class on Tuesday, May 10, 2016, 2:00PM{3:15PM. Very much like the problem sets. Optional { I will compute the ▯nal grade both with and without the midterm and use the better grade. No makeup exam! Requirements 3) Final { 50% (or 80%) Thursday, June 9, 2016, 11:30AM{2:30PM. Cumulative. Make sure you don’t have any con icts on that date. The is no makeup exam! Questions/Problems/.... Questions on course material and problem sets: Come to my o▯ce hours or your TA’s o▯ce hours. Post questions to the class website discussion board. TA’s and myself will be monitoring the discussion board daily to answer such questions. Emails to me or TA’s on such questions will not be answered. Regrading You can email your TA to request a regrade. The entire exam or homework will be regraded. Please check the online syllabus for updates, corrections, changes, etc. Course Overview Econometrics... uses statistical methods to analyze economic data. aims to answer quantitative questions. Main tool: regression analysis We want to determine the causal e▯ect of one variable (X) on another variable (Y). Econ 41: statistical analysis of one variable Econ 103: analysis of the relationship between two (or more) variables Examples of questions of interest: How does demand change with the price of the good? What is the e▯ect of a new marketing campaign on sales? How does class size a▯ect education outcomes (e.g. test scores)? How much an additional year of schooling increases wages? What is the relationship between credit scores and loan default rates? How much does output grow if the Fed cuts interest rates by 1%? Do more policemen reduce crime? In this course, you will learn... Theory: How to interpret results in articles, or the results your computer gives. To understand the caveats of regression analysis. Why it is essential to do good empirical work. How to do empirical analysis yourself: You will work with real datasets. You will use statistical software, namely STATA. Review of Probability & Statistics (Econ 41) 1. De▯nitions Random Variable (X): takes on di▯erent outcomes with certain probabilities. Outcomes (x ):imutually exclusive potential results of a random process. Probability of an outcome: proportion of times that the outcome occurs in the long run. Types of Random Variables: Discrete Random Variable: takes on a ▯nite number of values. Example: coin toss; the number of times a computer crashes. Continuous Random Variable: takes on any value in a real interval, each speci▯c value has zero probability. Example: height of an individual; time it takes to commute to school. 1. De▯nitions The function f that summarizes the information relating the possible outcomes of a random variable X and the corresponding probabilities is called the PDF, which stands for: the Probability Distribution Function, for discrete variables; or the Probability Density Function for continuous functions. PDF’s must satisfy: 1 f(xi) ▯ 0;and R 2 å if(x i = 1 (if discrete) or f(x )dx = 1 (if continuous) 1. De▯nitions Examples: Coin toss: x =heads, x =tails,( )x= f( ) = 1 1 2 1 2 2 Height of adult men in inches. PDF’s of discrete (left) and continuous (right) random variables 1. De▯nitions Cumulative Distribution Function (CDF): describes the probability that the random variable is less than or equal to a particular value. CDF’s of discrete (left) and continuous (right) random variables 1. De▯nitions Moments: summary statistics of the probability distribution: Expected Value or Expectation or Mean 8 < å ixif(xi) if X is discrete mX = E(X ) = : R xf(x )dx if X is continuous Measure of central tendency Weighted average of the possible values of X with the probability (x) serving as weights 1. De▯nitions Moments (continued): Variance 2 ▯ 2▯ sX = Var(X ) = E X ▯ mX) 8 < å ixi▯ mX)2f(xi) if X discrete = : R 2 x ▯ mX )f x dx if X cont. Measure of dispersion How the values of X are spread around its mean Standard deviation q sX = Var(X ) 2. De▯nitions - Two Random Variables Joint Probability Density Function of X and Y: describes the probability that the random variables simultaneously take on certain values, x and y. It is the function X,Y x,y ) = Pr(X = x,Y = y). Marginal Probability Distribution: Y (y) = å Pr(X = xi,Y = y) = å fX,Y(x,y) i x 2. De▯nitions - Two Random Variables Conditional Probability Distribution of Y given X: is the probability that Y takes on the value y when X takes on the value x. fYjX(y jx) = PrY = y jX = x) = Pr(X = x,Y = y) Pr(X = x) = X,Y x,y ) X (x) 2. De▯nitions - Two Random Variables Example: \Men say they will vote for the Republican candidate rather than the Democratic candidate in their districts by a margin of 45 percent to 32 percent. The numbers are nearly reversed for women, with 36 percent saying they will vote Republican and 43 percent saying they will vote Democratic." New York Times, September 20, 2010 Assume that there are 50% men and 50% women. 8 ▯ < 1 Democrat 1 male X = 2 female Y = : 2 Republican 3 other 2. De▯nitions - Two Random Variables The joint PDF of X and Y is: x1= 1 x2 = 2 y1= 1 .16 .215 y2= 2 .225 .18 y3= 3 .115 .105 2. De▯nitions - Two Random Variables The marginal distributions of X and Y are in red: x1= 1 x2 = 2 å x i y1 = 1 .16 .215 .375 y2 = 2 .225 .18 .405 y3 = 3 .115 .105 .22 å yj .5 .5 1 2. De▯nitions - Two Random Variables The conditional distribution of being a Democrat, given being female: P(Democrat jFemale)=Pr(Y = 1 jX = 2) PrY = 1,X = 2) .215 = PrX = 2) = .5 ▯ .43 The conditional distribution of being a Republican, given being a female: Pr(RepublicjFemale)= Pr(Y = 2 jX = 2) = Pr(Y = 2,X = 2) = .18▯ .36 Pr(X = 2) .5 2. De▯nitions - Two Random Variables Covariance: is a measure of linear association between two random variables, and is de▯ned as sXY = Cov(X,Y ) = E((X ▯ mX)(Y ▯ mY)) Correlation coe▯cient: is a unit-free measure of linear association between two random variables, and is de▯ned as Cov(X,Y ) rXY = s s X Y It can be shown that ▯1 ▯ r▯ 1 Independence: fX,Y (x,y ) = fX (x)fY (y ) If two random variables are independent, then the conditional distribution of each variable coincides with its marginal distribution, that is fYjX (y j x) = fY(y ), and fXjY (x j y) = fX(x ). Properties of Expectations, Variance and Covariance E(a) = a E(aX + b) = aE(X ) +b E(X + Y ) =E (X) + E(Y ) Var(a) =0 Var(aX + b) =a Var(X ) Cov(X,X ) = Var(X) Cov(X,Y ) = E(XY ) ▯ E(X)E (Y) Var(aX + bY) = a Var(X ) +b Var(Y ) +2abCov(X,Y ) Properties of Expectations, Variance and Covariance If X and Y are independent, then E(XY ) = E X )E(Y ) Var(X + Y ) =Var X ) +Var Y ) Cov(X,Y ) =0, but the reverse is not true! Zero covariance does not imply independence. Special Probability Distributions 2 Normal distribution: (m,s ) Standard normal distribution: 0,1 ) A variable that follows a standard normal distribution is often denoted by Z. Its CDF is often denoted by F, tha(Zi▯, c) =F (c)for any c. 2 X ▯m Note that if X▯ N (m,s ), then Z= s ▯ N (0,1). Note also that if X and Y are normally distributed, so is X + Y (and any other linear combination). Special Probability Distributions Standard Normal PDF Special Probability Distributions Chi-Squared distribution: assume tha▯ N(0,1),8 i. Then i m 2 2 W = å Zi ▯ c m i=1 where m is the degrees of freedom. \c distribution with m degrees of freedom" Special Probability Distributions Chi-Squared PDF Special Probability Distributions Student t distribution: assume▯tNa(0,), W ▯ cm,and Z and W are independent. Then p Z ▯ t W/m m \t distribution with m degrees of freedom" Note: ¥ = N(0,1) Special Probability Distributions Student t PDF Special Probability Distributions Standard Normal vs. Student t distributions Special Probability Distributions Chi-Squared PDF 2 2 F distribution: assume that▯Wcm, V ▯ cn, and W and V are independent. ThenW/m ▯ F m,n V/n \F distribution with m and n degrees of freedom" 2 2 Note: mFm,¥ = cm and 1,n= tn Special Probability Distributions F distribution PDF Computing Probabilities Assume that adult male heights are normally distributed with mean 70 inches (178 cm) and standard deviation 4 inches (7.6 cm). If you are 65 inches tall (165 cm), what percentage of men are shorter than you? Let X denote the height of men▯ N 70,16). Let us standardize X: Z = X ▯ 70 ▯ N 0,1) 4 We have to do the same manipulation for 65. 65 ▯ 70 4 = ▯ 1.25 Computing Probabilities What is the probability that▯ ▯ 1.25? This is given by its CDF! F(▯ 1.25) = 10.56% You can compute this value in Stata using command di normal(-1.25). Alternatively, see Appendix Table 1 in textbook. Computing Probabilities Computing Probabilities Computing Probabilities Computing Probabilities Computing F (▯1.25) from Standard Normal table: 1▯ 0.8944 = 0.1056!

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