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
𝐸(Y(d)|d)=
𝑁𝑑
𝑁𝑑
𝑌𝑑,𝑘
𝑘=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
•
•
•
•
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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:
𝑋 − 𝑚0
→ 𝑁(0,1)
𝜎/ 𝑛
• But the s.d. is estimated, and, under the null
hypothesis:
𝑋 − 𝑚0
→ 𝑆(𝑛 − 1)
𝜎/ 𝑛
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:
𝐶 = −∞, +∞ − [𝑚 − 𝑐2.5 𝑠; 𝑚 + 𝑐97.5 𝑠]
• 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.