Introduction to Econometrics

Introduction to Econometrics • What do I expect of you before you come to class?

1. Print out the slides.

2. 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.

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?

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

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

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

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?

The California Test Score Data Set

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?

Do districts with smaller classes have higher test scores?

STR

Numerical Evidence

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

Class Size Average score ( )

Standard deviation (s)

n

Small 657.4 19.4 238

Large 650.0 17.9 182

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?

2. Hypothesis testing

Two sample Difference-of-means t-test

3. 95% Confidence interval

Review of Statistical Theory

Review of Statistical Theory

(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]

(b) Moments of a population distribution

(b) Moments of a population distribution

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]

so is the correlation…

Covariance is negative

Population correlation coefficient

(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)

•

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)

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

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 3West (L = 0) 0.10 0.05 0.10 0.05Central (L = 1) 0.10 0.02 0.10 0.02East (L = 2) 0.15 0.18 0.10 0.03

• Show that L.I.E. holds

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) …

(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

• Yi are said to be i.i.d. “independent and identically distributed”

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

Things we want to know about the sampling distribution:

Mathematics of Expectations • Read Appendix 2.1 carefully

• Let’s prove this one, for practice

General sampling distribution of

The sampling distribution of when n is large

The Law of Large Numbers

The Central Limit Theorem (CLT)

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

Same example: sampling distribution of :

(b) Why Use To Estimate Y?

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 H0 is true.

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

Comments on the Student t-distribution

1. 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.

2. tn-1 approaches z “quickly” as n increases

• t30,.05=2.042, t60,.05=2.000, t100,.05=1.983

3. Requires the impractical assumption that population distribution of X is normal

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.

Confidence IntervalsA 95% confidence interval for 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?

Confidence intervals

Summary

Let’s go back to the original policy question: