1. Introduction to Econometrics The Statistical Analysis of Economic (and related) Data

2. 2 Brief Overview of the Course

3. 3 This course is about using data to measure causal effects.

4. 4 In this course you will:

5. 5 Review of Probability and Statistics (SW Chapters 2, 3)

6. 6 The California Test Score Data Set

7. 7 Initial look at the data: (You should already know how to interpret this table)  This table doesn’t tell us anything about the relationship between test scores and the STR.

8. 8 Do districts with smaller classes have higher test scores? Scatterplot of test score v. student-teacher ratio What does this figure show?

9. 9 We need to get some numerical evidence on whether districts with low STRs have higher test scores – but how?

10. 10 Initial data analysis: Compare districts with “small” (STR < 20) and “large” (STR ≥ 20) class sizes: 1.Estimation of  = difference between group means 2.Test the hypothesis that  = 0 3.Construct a confidence interval for  Class SizeAverage score ( ) Standard deviation (s B Y B ) n Small657.419.4238 Large650.017.9182

11. 11 1. Estimation

12. 12 2. Hypothesis testing

13. 13 Compute the difference-of-means t-statistic:

14. 14 3. Confidence interval

15. 15 What comes next…

16. 16 Review of Statistical Theory

17. 17 (a) Population, random variable, and distribution

18. 18 Population distribution of Y

19. 19 (b) Moments of a population distribution: mean, variance, standard deviation, covariance, correlation

20. 20 Moments, ctd.

21. 21

22. 22 2 random variables: joint distributions and covariance

23. 23 so is the correlation… The covariance between Test Score and STR is negative:

24. 24 The correlation coefficient is defined in terms of the covariance:

25. 25 The correlation coefficient measures linear association

26. 26 (c) Conditional distributions and conditional means

27. 27 Conditional mean, ctd.

28. 28 (d) Distribution of a sample of data drawn randomly from a population: Y 1,…, Y n

29. 29 Distribution of Y 1,…, Y n under simple random sampling

30. 30

31. 31 (a) The sampling distribution of

32. 32 The sampling distribution of, ctd.

33. 33 The sampling distribution of when Y is Bernoulli (p =.78):

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

35. 35 The mean and variance of the sampling distribution of

36. 36

37. 37 Mean and variance of sampling distribution of, ctd.

38. 38 The sampling distribution of when n is large

39. 39 The Law of Large Numbers:

40. 40 The Central Limit Theorem (CLT):

41. 41 Sampling distribution of when Y is Bernoulli, p = 0.78:

42. 42 Same example: sampling distribution of :

43. 43 Summary: The Sampling Distribution of

44. 44 (b) Why Use To Estimate  Y ?

45. 45 Why Use To Estimate  Y ?, ctd.

46. 46

47. 47

48. 48 Calculating the p-value, ctd.

49. 49 Calculating the p-value with  Y known:

50. 50 Estimator of the variance of Y :

51. 51 Computing the p-value with estimated:

52. 52 What is the link between the p-value and the significance level?

53. 53 At this point, you might be wondering,...

54. 54 Comments on this recipe and the Student t-distribution

55. 55 Comments on Student t distribution, ctd.

56. 56 Comments on Student t distribution, ctd.

57. 57

58. 58 Comments on Student t distribution, ctd.

59. 59 The Student-t distribution – summary

60. 60

61. 61 Confidence intervals, ctd.

62. 62 Summary:

63. 63 Let’s go back to the original policy question: