Introduction to Econometrics

1. Introduction to Econometrics • What do I expect of you before you come to class? • Print out the slides. • Read the chapter, and as you read, write questions down on the slides. • Therefore, when I am lecturing, I do not expect it to be the first time you are hearing about a concept. • If you don’t do this, it will seem like I am going really, really fast. • If this approach to my teaching/your learning, which places high demand on your pre-class preparation, doesn’t suit you, I won’t be offended if you take Eco205 from someone else.

2. Brief Overview of the Course • Economic theory often suggests the sign of important relationships, often with policy implications, but rarely suggests quantitative magnitudes of causal effects. • What is the quantitative effect of reducing class size on student achievement? Expected sign is ? • How does another year of education change earnings? • What is the effect on output growth of a 1 percentage point decrease in interest rates by the Fed? • What is the effect on housing prices of environmental improvements?

3. This course is about using data to measure causal effects. • Typically only have observational (nonexperimental) data • level of education vs. wages • cigarette price vs. quantity demanded • selectivity of a college vs. wages • class size vs. test scores • democracy measure vs. GDP per capita (income) • Difficulties arise from using observational data to estimate causal effects • confounding effects (omitted factors) • simultaneous causality • Remember, correlation does not imply causation ! • Randomized experiments often not feasible

4. Source: Acemoglu, Johnson, Robinson, and Yared (AER 2008)

5. Source: Ruhm, Christopher (J Health Economics, 1996)

6. Review of Probability and Statistics(SW Chapters 2, 3) • Empirical problem: Class size and educational output • Policy question: What is the effect on test scores (or some other outcome measure) of reducing class size by one student per class? By 8 students/class?

7. The California Test Score Data Set

8. Initial look at the data(You should already know how to interpret this table) • What do we learn about the relationship between test scores and the STR?

9. Do districts with smaller classes have higher test scores? STR

10. Numerical Evidence

11. Compare districts with “small” (STR < 20) and “large” (STR ≥ 20) class sizes 1. Estimation of  = population difference between group means 2. Test the hypothesis that  = 0 3. Construct a confidence interval for 

12. 1. Estimation • Is this a large difference in a real-world sense? • Standard deviation across districts = 19.1 • Difference between 60th and 75th percentiles of test score distribution is 667.6 – 659.4 = 8.2 • Is this a big enough difference to be important for school reform discussions, for parents, for a school committee?

13. 2. Hypothesis testing

14. Two sample Difference-of-means t-test

15. 3. 95% Confidence interval

16. Review of Statistical Theory

17. Review of Statistical Theory

18. (a) Population, random variable, and distribution • Population • The group or collection of all possible entities of interest (school districts) • We will think of populations as infinitely large • Random variable Y • Numerical summary of a random outcome (district average test score, district STR) • Population distribution • Gives the probabilities of different values of Y • when Y is discrete, Pr[Y = 650] • when Y is continuous, Pr[640 ≤ Y ≤ 660]

19. (b) Moments of a population distribution

20. (b) Moments of a population distribution

22. Two Random Variables • Two random variables have a joint distribution • cov(X,Z) = E[(X – X)(Z – Z)] = XZ • Linear association • Units? • If X and Z are independently distributed, then cov(X,Z) = 0 (but not vice versa!!) • cov(X,X) = E[(X – X)(X – X)] = E[(X – X)2]

23. Covariance is negative so is the correlation…

24. Population correlation coefficient

26. (c) Conditional distributions • Conditional distributions • distribution of test scores, given that STR < 20 • Conditional moments • conditional mean is written E(Y|X = x) • E(Test scores|STR < 20) • note that the prob here = (1/ns) for the test scores, yielding the average test score among small districts • conditional variance is written Var(Y|X=x) • Var(Test scores|STR < 20)

27. Examples of Conditional Mean • Wages of all female workers (Y = wages, X = gender) • Mortality rate of patients given an experimental treatment (Y = live/die; X = treated/not treated) • The difference in means from the t-test •  = E(Test scores|STR < 20) – E(Test scores|STR ≥ 20)

28. Properties of Conditional Mean • Law of Iterated Expectations E[Y] = E[ E[Y|X] ] • Recall that • And expected value of E[Y|X] is • Note that y takes on k outcomes, x takes on l outcomes

29. L.I.E. example • Consider the following joint probability distribution table for two random variables, the number of children a household has (C) and the location of the household (L). • Number of Children (C) • Location (L) 0 1 2 3 • West (L = 0) 0.10 0.05 0.10 0.05 • Central (L = 1) 0.10 0.02 0.10 0.02 • East (L = 2) 0.15 0.18 0.10 0.03 • Show that L.I.E. holds

30. Properties of Conditional Mean • If E(X|Z) = X, then corr(X,Z) = 0 (not necessarily vice versa) • Proof: Assume X = 0 and Z = 0 for simplicity • First, note that corr(X, Z) = 0 implies cov(X,Z) = 0. Why? • Start with definition of cov(X,Z) …

31. (d) Distribution of a sample of data drawn randomly from a population: Y1,…, Yn • The data set is (Y1, Y2, … , Yn), where Yi = value of Y for the ith individual (district, entity) sampled • Yiare said to be i.i.d. “independent and identically distributed”

33. (a) Sampling distribution of when Y ~ Bernoulli (p = .78):

34. Things we want to know about the sampling distribution:

35. Mathematics of Expectations • Read Appendix 2.1 carefully • Let’s prove this one, for practice

36. General sampling distribution of

37. The sampling distribution of when n is large

38. The Law of Large Numbers

39. The Central Limit Theorem (CLT)

40. Sampling distribution of when Y is Bernoulli, p = 0.78:

41. Same example: sampling distribution of :

42. (b) Why Use To Estimate Y?

43. 3. Hypothesis Testing • H0: Y= Y,0 vs. H1: Y> Y,0 , < Y,0 , ≠ Y,0 • p-value= probability of drawing a statistic at least as adverse to H0 as the value actually computed with your data, assuming that H0 is true. • “lowest significance level at which you can reject H0” • The significance level of a test is a pre-specified probability of incorrectly rejecting H0, when H0is true.

44. At this point, you might be wondering,...

45. Comments on the Student t-distribution • Astounding result really … if Yi are i.i.d. normal, then you can know the exact, finite-sample distribution of the t-statistic … it’s the Student’s t-distribution. • tn-1 approaches z“quickly” as n increases • t30,.05=2.042, t60,.05=2.000, t100,.05=1.983 • Requires the impractical assumption that population distribution of X is normal

47. Comments on Student t distribution 4. Consider the statistic to test difference in means between 2 groups (s,l): It does not have an exact t-distribution in small samples, even if Y is normally distributed. This statistic does though (when Y normal), but only if Bottom line: That’s not likely, so pooled std error formula usually inappropriate. So use different-variance formula with large-sample z critical values.

48. Confidence Intervals A 95% confidence intervalfor Y is an interval that is expected to contain the true value of Y in 95% of repeated samples of size n. Note: What is random here?

49. Confidence intervals

50. Summary