Research Method

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Research Method

• Lecture 11-1 (Ch15)
• Instrumental Variables Estimation and Two Stage
  Least Square

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MotivationOne explanatory variable case

• Consider the following regression.

• Since ability is not observed, we can only run
  the following regression.

• Since ability is correlated with educ, educ is
  endogenous (i.e, correlated with u). Thus,
  will be biased.

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• We learned two methods to eliminate the bias.
• Plug in the proxy variable for ability, such as
  IQ.
• Use panel data method (either the fixed effect or
  the first differenced model).
• Instrumental variable method is another method to
  eliminate the bias.

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Instrumental variable method One explanatory
variable case.

• Consider the following model.
• Suppose that x is endogenous, that is cov(x,
  u)?0.
• Further, suppose that you have another variable,
  z, which satisfies the following conditions.
• Cov(z,u)0 (instrument exogeneity)
  ....(1)
• Cov(z,x)?0 (instrument relevance)(2)
• If the above conditions are satisfied, we call z
  an instrumental variable.

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• There are two ways to intuitively understand
  these conditions.
• Instrumental variable is a variable that is not
  correlated with the omitted variable, but is
  correlated with the endogenous explanatory
  variable.
• Instrumental variable is a variable that affects
  y only through x.

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• The condition Cov(z,u)0 involves unobserved u.
  Therefore, we cannot test this condition. (When
  you have extra instrumental variables, you can
  test this. This will be discussed later).
• The condition Cov(z,x)?0 is easy to test. Just
  runt the following OLS,
• xp0p1zv
• then test
• H0p10

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Instrumental variable estimation One explanatory
variable-one instrument case

• Now, consider
• Then we have
• Cov(z,y)Cov(z,ß0ß1xu)
• So we have,
• Cov(z,y) ß1Cov(z,x)Cov(z,u)
• Since Cov(z,u)0, we have

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• By replacing Cov(z,y) and Cov(z,x) with their
  sample covariances, we have the instrumental
  variable estimator of ß1 which is given by
• You can easily show that is a consistent
  estimator of ß1.

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Statistical inference with IV Homoskedasticity
case

• Homoskedasticity assumption in the case of IV
  regression is stated in terms of z.
• E(u2z)s2
• It can be shown that the asymptotic variance of
  is given by
• where is the variance of x, and is
  the correlation between x and z.

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• Now, the estimator of var( ) is obtained by
  replacing s2, , and with their sample
  estimates.
• Sample estimator of s2 is obtained in the
  following way. First, obtain the IV estimates for
  ß0 and ß1, then compute
• The estimator for s2 is then computed as

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• The sample estimator for is given as
• Finally, sample estimator for can be most
  easily obtained in the following way. First,
  regress x on z. Then the R-squared from this
  regression equals the square of the sample
  correlation. Let call this R2x,z. (Off course,
  you can compute the sample correlation and raise
  it by power 2. You will get the same result).

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• Then, the estimator for the variance of is
  given by
• You can show that this is a consistent estimator
  of the asymptotic variance given by (5).

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Note R-squared in IV regression

• The R-squared for IV regression is computed as
• R21-SSR/SST
• Where SSR is the sum of the squared IV residuals.
  (The IV residual is given by (6)).
• Unlike in the case of OLS, SSR can be greater
  than SST. Thus, R2 can be negative. In IV
  regression, R2 does not have a natural
  interpretation.

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Finding the instrumental variable

• The most difficult part of the instrumental
  variable estimation is to find suitable
  instrumental variables.
• Consider the following regression
• Then, you have to find z that is correlated with
  educ, but not correlated with abil. What can be z?

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• Consider the fathers education. Perhaps a person
  whose father is highly educated tends to take
  more education as well. So the fathers education
  is likely correlated with educ.
• But, for fathers education to be an instrument,
  this should not be correlated with the unobserved
  ability. A highly educated father may nurture his
  child better, so fathers education may be
  correlated with the unobserved ability. If this
  is the case, fathers education is not a good
  instrument.
• Nonetheless, many studies have used fathers and
  mothers education as instruments.

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Exercises

• 1. Run the following regression using OLS, using
  MROZ.dat
• 2. Using the fathers education as an instrument
  for edu, estimate the same model using IV
  regression. Also check if fathers education is
  correlated with educ.

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OLS
IV regression
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Check if fathers education is correlated with
educ.
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An application

• Angrist and Krueger (1991), Does Compulsory
  School Attendance Affect Schooling and Earning?
• They used the quarter of the birth dummy as an
  instrument for education to estimate the effect
  of education on wage.

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• In the US, the compulsory schooling law requires
  students to remain in school until their 16th
  birthday.
• At the same time, schools usually requires
  Children to be 6 years old on January 1st to be
  admitted to school. Therefore, children who were
  born in the first quarter were older than
  children who were born in the last quarter when
  they were first admitted to schools (6.45 v.s.
  6.07 years).

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• This also means that children who were born in
  the first quarter of the year has shorter
  schooling when they reach the legal drop out age.
  So, children who were born in the first quarter
  can legally drop out of school with less
  education than children who were born in other
  quarters.
• If some people want to take as little education
  as possible but are constrained by the compulsory
  schooling law, the quarter of birth should affect
  the education attainment.

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• At the same time, the quarter of birth is
  unlikely to be correlated with the unobserved
  ability.
• Therefore, the dummy variable indicating if a
  person was born in the first quarter of the year
  is a good instrument for education.

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Those born in the first quarter of the year tend
to have lower education attainment
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This is the IV regression