Econometrics Session 2 – Introduction: Inference Amine Ouazad, Asst. Prof. of Economics Outline of the course 1. Introduction: Identification 2. Introduction: Inference 3. Linear Regression 4. Identification Issues in Linear Regressions 5. Inference Issues in Linear Regressions Previous session: Identification • Golden Benchmark: Randomization – D = E(Y(1)|D=1) – E(Y(0)|D=0) • We do not in fact observe E(Y(d)|D=d)… • But we observe: 1 (Y(d)|d)= , =1 This session Introduction: Inference • What problems appear because of the limited number of observations? • Hands-on problem #1: – At the dinner table, your brother-in-law suggests playing heads or tails using a coin. You suspect he is cheating. How do you prove that the coin is unbalanced? This session Introduction: Inference • Hands-on problem #2: – Using a survey of 1,248 subjects in Singapore, you determine that the average income is $29,041 per year. How close is this mean to the true average income of Singaporeans? Do we have enough data? This session Introduction: Inference • • • • • Convergence The Law of Large Numbers The Central Limit Theorem Hypothesis Testing Inference for the estimation of treatment effects. Session 2 - Inference 1. CONVERGENCE Warning (you can ignore this) • Proofs of the LLN and the CLT are omitted since most of their details are irrelevant to daily econometric practice. • There are multiple flavors of the LLN and the CLT. I only introduce one flavor per theorem. I will introduce more versions as needed in the following sessions, but do not put too much emphasis on the distinctions (Appendix D of the Greene). Notations • An estimator of a quantity is a function of the observations in the sample. • Examples: – Estimator of the fraction of women in Singapore. – Estimator of the average salary of Chinese CEOs. – Estimator of the effect of a medication of patients’ health. • An estimator is typically noted with a hat. • An estimator sometimes has an index n for the number of observations in the sample. Convergence • Convergence in probability. – An estimator qn of q is converging in probability to q if for all epsilon, P(|qn-q|>e) -> 0 as n->∞. – We write plim qn = q • An estimator of q is consistent if it converges in probability to q. Session 2 - Inference 2. LAW OF LARGE NUMBERS Law of Large Numbers • Let X1, …, Xn be an independent sequence of random variables, with finite expected value mu = E(Xj), and finite variance sigma^2 = V(Xj). Let Sn = X1+…+Xn. Then, for any epsilon>0, Sn P( - m > e ) ® 0 n • As n->infinity Law of large numbers • The empirical mean of a series of random variables X1, …, Xn converges in probability to the actual expectancy of the sequence of random variables. • Application: What is the fraction of women in Singapore? – Xi = 1 if an individual is a woman. – EXi is the fraction of women in the population. – Empirical mean is arbitrarily close to the true fraction of women in Singapore. – Subtlety? Another application • Load the micro census data. • Take 100, 5% samples of the dataset. Calculate the fraction of women in the dataset, for each dataset. • Consider the approximation that the fraction of women is 51% exactly. • Illustrate that for epsilon = 0.5%, the number of samples with a mean above 51+-0.5 is shrinking as the size of the sample increases. Session 2 - Inference 2. CENTRAL LIMIT THEOREM Central Limit Theorem Lindeberg-Levy Central Limit Theorem: • If x1,…,xn are an independent random sample from a probability distribution with finite mean m and finite variance s2, and 1 n x n = å xi then n i=1 n ( x n ) d N (0, s 2 ) • Proof: Rao (1973, p.127) using characteristic functions Applications: Central Limit Theorem • Exercise #1: – You observe heads,tails,tails,heads,tails,heads. – Give an estimate of the probability of heads, with a 95% confidence interval. • Exercise #2: – Solve the hands-on problem #2 at the beginning of these slides. – Discuss the assumptions of the CLT. Session 2 - Inference 3. HYPOTHESIS TESTING Hypothesis testing • Null hypothesis H0. • Alternative hypothesis Ha. • Unknown parameter q. • Typical null hypothesis: – Is q = 0 ? – Is q > 3 ? – Is q = f ? (if f is another unknown parameter). – Is q = 4 f ? Hypothesis Testing: Applications • Application #1 (Coin toss): is the coin balanced? – Write the null hypothesis. – Given the information presented before, can we reject the null hypothesis at 95%? • Application #2 (Average income): is the average income greater than $29,000 ? – Write the null hypothesis. – Given the information presented before, can we reject the null hypothesis at 90%? t-test • From the Central Limit Theorem, if the standard deviation were known, under the null hypothesis: − 0 → (0,1) / • But the s.d. is estimated, and, under the null hypothesis: − 0 → ( − 1) / Critical region • Region for which the null hypothesis is rejected. • If the null hypothesis is true, then the null is rejected in 5% of cases if the critical region is: = −∞, +∞ − [ − 2.5 ; + 97.5 ] • Where cq is the qth quantile of the student distribution with n-1 degrees of freedom. Flavors of t-tests • One-sample, two-sided. – See previous slides. • One-sample, one-sided. • Two-sample, two-sided. – Equal and unequal variances. • Two-sample, one-sided. – Equal and unequal variances. Errors Null hypothesis is not rejected Null hypothesis is rejected Null hypothesis is true Cool, no worries Type I error Probability a Null hypothesis is wrong Type II error Probability b Cool, no worries • E.g. in judicial trials, medical tests, security checks. • Power of a test 1-b: probability of rejecting the null when the null is false. • Size of a test a: proba of type I error. Quirky • Many papers run a large number of tests on the same data. • Many papers report only significant tests… – What is wrong with this approach? • Many papers run “robustness checks”, i.e. tests where the null hypothesis should not to be rejected. – What is wrong with this approach? • Conclusion: – This is wrong, but common practice. For more , see January 2012 of Strategic Management Journal. Session 2 - Inference 4. INFERENCE FOR TREATMENT EFFECTS Treatment effects: Inference (inspired by Lazear) • There are two groups, a treatment and a control group. • 128 employees are randomly allocated to the treatment and to the control. • Treatment employees: piece rate payoff. • Control employees: fixed pay. • Treatment workers is 38.3 pieces per hour in the treatment group, and is 23.1 in the control group. Questions 1. Why do we perform a randomized experiment? 2. Do we have enough information to get an estimator of the treatment effect? 3. Is the estimator consistent? 4. Is the estimator asymptotically normal? 5. Do we have enough information to get a 95% confidence interval around the estimator of the treatment effect? 6. Test the hypothesis that the medication is effective at raising the health index.