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

2 What do economists study?

3 How do we answer these questions?

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

5 The California Test Score Data Set

6 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.

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

8 How do we answer this question with data?

9 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 Y ) n Small Large

10 1. Estimation

11 2. Hypothesis testing

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

13 3. Confidence interval

14 Review of Statistical Theory

15 (a) Population, random variable, and distribution

16 Population distribution of Y

(b) Characteristics (a.k.a. moments) of a population distribution 17

Flip coin to see how many heads result from 2 flips  E(Y) = 0*(0.25) + 1*(0.50) + 2*(0.25) = = 1  var(Y) = (0.25)*(0 - 1)² + (0.50)*(1 – 1)² + (0.25)*(2 – 1)² = =.50  stdev(Y) = √.50 =

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21 2 random variables: joint distributions and covariance

Joint Probability  Example: The relationship between commute time and rain Pr(X=x, Y=y) is the joint probability, where X = 0 if raining = 1 otherwise Y = 1 if commute time is short (<20 minutes) = 0 if commute time is long (>= 20 minutes) Positive or negative relationship? 22

Conditional Probability  Conditional probability is used to determine the probability of one event given the occurrence of another related event.  Conditional probabilities are written as P(X | Y). They are read as “the probability of X given Y” and are calculated as: 23

Joint Independence  Two random variables, X and Y, are independently distributed if for all X and Y Pr(X = x,Y = y) = Pr(X = x)*Pr(Y = y) or Pr(Y = y | X = x) = Pr(Y = y) 1. Do these hold in the rain and commute example? 2. Pr (X = 1, Y=1) = ? 3. E (X | Y=1) = ? 4. Pr (X=0 | Y=0) = ? 24

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

26 The correlation coefficient measures linear association

27 (c) Conditional distributions and conditional means

28 Conditional mean, ctd.

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

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

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32 (a) The sampling distribution of

33 The sampling distribution of, ctd.

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

35 Things we want to know about the sampling distribution:

36 The mean and variance of the sampling distribution of

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38 Mean and variance of sampling distribution of, ctd.

39 The sampling distribution of when n is large

40 The Law of Large Numbers:

41 The Central Limit Theorem (CLT):

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

43 Same example: sampling distribution of :

44 Summary: The Sampling Distribution of

45 (b) Why Use To Estimate  Y ?

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 Test statistic = t-statistic:  Significance level: Specified probability of Type I error  Significance level = α  Critical Value: Value of test statistic for which the test just rejects the null at a given significance level Language of Hypothesis Testing 47

Language of Hypothesis Testing, ctd.  p-value  Probability of drawing a statistic (e.g. Y) at least as adverse to the null hypothesis as the value computed with your data, assuming the null hypothesis is true  The smallest significance level at which you can reject the null hypothesis |Test statistic| > |critical value| → reject null hypothesis |Test statistic| < |critical value| → fail to reject null hypothesis 48

49 Calculating the p-value with  Y known:

50 Estimator of the variance of Y :

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

Common Critical Values One-Tail TestTwo-Tail Test 1-ααCritical Value 1-αα/2Critical Value

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