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

2 Brief Overview of the Course

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

4 In this course you will:

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

6 The California Test Score Data Set

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

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

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 Small Large

11 1. Estimation

12 2. Hypothesis testing

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

14 3. Confidence interval

15 What comes next…

16 Review of Statistical Theory

17 (a) Population, random variable, and distribution

18 Population distribution of Y

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

20 Moments, ctd.

21

22 2 random variables: joint distributions and covariance

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

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

25 The correlation coefficient measures linear association

26 (c) Conditional distributions and conditional means

27 Conditional mean, ctd.

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

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

30

31 (a) The sampling distribution of

32 The sampling distribution of, ctd.

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

34 Things we want to know about the sampling distribution:

35 The mean and variance of the sampling distribution of

36

37 Mean and variance of sampling distribution of, ctd.

38 The sampling distribution of when n is large

39 The Law of Large Numbers:

40 The Central Limit Theorem (CLT):

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

42 Same example: sampling distribution of :

43 Summary: The Sampling Distribution of

44 (b) Why Use To Estimate  Y ?

45 Why Use To Estimate  Y ?, ctd.

46

47

48 Calculating the p-value, ctd.

49 Calculating the p-value with  Y known:

50 Estimator of the variance of Y :

51 Computing the p-value with estimated:

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

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

54 Comments on this recipe and the Student t-distribution

55 Comments on Student t distribution, ctd.

56 Comments on Student t distribution, ctd.

57

58 Comments on Student t distribution, ctd.

59 The Student-t distribution – summary

60

61 Confidence intervals, ctd.

62 Summary:

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