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**Session 2 – Introduction: Inference**

Econometrics Session 2 – Introduction: Inference Amine Ouazad, Asst. Prof. of Economics

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**Outline of the course Introduction: Identification**

Introduction: Inference Linear Regression Identification Issues in Linear Regressions Inference Issues in Linear Regressions

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**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: 𝐸 (Y(d)|d)= 1 𝑁 𝑑 𝑘=1 𝑁 𝑑 𝑌 𝑑,𝑘

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**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?

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**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?

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**This session Introduction: Inference**

Convergence The Law of Large Numbers The Central Limit Theorem Hypothesis Testing Inference for the estimation of treatment effects.

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Session 2 - Inference 1. convergence

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**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).

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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.

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**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.

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Session 2 - Inference 2. Law of large numbers

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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, As n->infinity

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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?

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**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 is shrinking as the size of the sample increases.

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Session 2 - Inference 2. Central limit theorem

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**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 then Proof: Rao (1973, p.127) using characteristic functions

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**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.

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Session 2 - Inference 3. Hypothesis testing

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**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 ?

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**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 ? Given the information presented before, can we reject the null hypothesis at 90%?

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t-test From the Central Limit Theorem, if the standard deviation were known, under the null hypothesis: But the s.d. is estimated, and, under the null hypothesis: 𝑋 − 𝑚 0 𝜎/ 𝑛 →𝑁(0,1) 𝑋 − 𝑚 0 𝜎/ 𝑛 →𝑆(𝑛−1)

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**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: Where cq is the qth quantile of the student distribution with n-1 degrees of freedom. 𝐶= −∞,+∞ −[𝑚− 𝑐 2.5 𝑠;𝑚+ 𝑐 97.5 𝑠]

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**Flavors of t-tests One-sample, two-sided. One-sample, one-sided.**

See previous slides. One-sample, one-sided. Two-sample, two-sided. Equal and unequal variances. Two-sample, one-sided.

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**Errors E.g. in judicial trials, medical tests, security checks.**

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 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.

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**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. Conclusion: This is wrong, but common practice. For more , see January 2012 of Strategic Management Journal.

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**4. Inference for treatment effects**

Session 2 - Inference 4. Inference for treatment effects

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**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.

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**Questions Why do we perform a randomized experiment?**

Do we have enough information to get an estimator of the treatment effect? Is the estimator consistent? Is the estimator asymptotically normal? Do we have enough information to get a 95% confidence interval around the estimator of the treatment effect? Test the hypothesis that the medication is effective at raising the health index.

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