Research Method

1
Research Method

• Lecture 9 (Ch9)
• More on specification and Data issues

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Using Proxy Variables for Unobserved Explanatory
Variables

• Suppose you are interested in estimating the
  return to Education. So you consider the
  following model.
• Log(Wage)ß0ß1Educ ß2Exp (ß3Abilityu) (1)
• Ability is unobserved, so it is included in the
  composite error term. If Ability is correlated
  with the year of education, ß1 will be biased.
• Question if ability is correlated with Educ,
  what is the direction of the bias?

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• One way to eliminate the bias is to use a Panel
  data then apply the fixed effect or the first
  differencing method.
• Another method is to use a proxy variable for
  ability. This is the topic of this section.
• Suppose that IQ is a proxy variable for ability,
  and that IQ is available in your data.

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• Then, the basic idea is to estimate the
  following.
• Regress Log(Wage) on Educ, Exp, and IQ (2)
• This is called the plug-in solution to the
  omitted variables problem.
• The question is under what conditions (2)
  produces consistent estimates for the original
  regression (1). I will explain these conditions
  using the above example (though the arguments can
  be easily generalized).
• It turns out, the following two conditions ensure
  that you get consistent estimates by using the
  plug-in solution.

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• Condition 1 u is uncorrelated with IQ. In
  addition, the original equation should satisfy
  the usual conditions (i.e, u is also uncorrelated
  with Educ, Exp, and Ability).
• Condition 2 E(AbilityEduc, Exp,
  IQ)E(AbilityIQ)
• Condition 2 means that, once IQ is conditioned,
  Educ and Exp does not explain Ability. More
  simple way to express condition 2 is that the
  ability can be written as
• Abilityd0d3IQv3
  (3)
• where, v3 is a random error which is uncorrelated
  with either IQ, Educ or Exp. What it means is
  that Ability is a function of IQ only.

Omitted variable
The initial explanatory variables
The proxy variable
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• Then, it is clear why these two conditions
  guarantee that the plug-in condition produces
  consistent estimates. Just plug (3) into (1).
  Then you have
• Log(Wage)(ß0d0)ß1Educ ß2Exp ß3d3IQ
  (uß3v3 ) (4)
• Where
• Since u and v3 are uncorrelated with any of the
  explanatory variables under condition1 and
  condition 2, the slope parameters are consistent.
  The intercept has changed, but usually you are
  not interested in the intercept. Importantly, you
  get consistent estimates for the slope
  parameters.

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• It is also obvious that, if condition 2 is
  violated, then the plug in solution will not
  work. If the condition 2 is violated, then
  ability will be a function of not only IQ, but
  also Educ and Exp. So you will have
• Abilityd0 d1Educd2Expd3IQv3 (5)
• If you plug (5) into (1), you have
• Log(Wage)(ß0d0)(ß1ß3d1)Educ (ß2ß3d2)Exp
  ß3d3IQ (uß3v3 ) (4)
• Thus, the coefficient for Educ is no longer ß1,
  but it is ß1ß3d1. Thus, the plug-in solution
  produces inconsistent estimates when condition 2
  is violated.

If condition 2 is violated then, ability is a
function of all the variables.
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Exercise

• Ex.1 Use Wage2.dta to estimate a log wage
  equation to examine the return to education.
  Include in the equation exper, tenure, married,
  south, urban, black. Do you think that the return
  to education is unbiased? What do you think is
  the direction of the bias
• Ex.2 Now, use IQ as a proxy for unobserved
  ability. Did the result change? Was your
  prediction of the direction of the bias correct?

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Answer OLS without IQ
10
Answer OLS with IQ
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Using lagged dependent variable as proxy variables

• Often the lag of the dependent variable is used
  as a proxy for the unobserved variables.
• First consider the following model.
• (Crime rate) ß0ß1(unemp)
  ß2(expenditure) u
• If there are omitted factors that directly affect
  crime rate and at the same time correlated with
  unemployment rate, ß1 will be biased. The omitted
  factors may be some pre-existing conditions, like
  demographic features (age, race etc). Crime rate
  could be different among cities for historical
  factors.

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• The idea is that, the lag of the dependent
  variable may summarize such pre-existing
  conditions.
• So, estimate the following equation
• (Crime rate)it ß0ß1(unemp)it
  ß2(expenditure)it
• ß3(Crime rate)it-1
  uit
• The following slides estimate the model using
  CRIME2.dta

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Example

• We estimate Crime2.dta to estimate the
  regressions. Results are the following.

Without the lag of dependent varriable.
With the lag of dependent variable.
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Measurement error

• The existence of important omitted variables
  causes endogeneity problem.
• Another source of endogeneity is the measurement
  error.
• This section explains under what circumstance
  the measurement error causes endogeneity, and
  under what circumstance it does not.

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Measurement error in explanatory variable.

• When the explanatory variables are measured with
  errors, this causes the endogeneity problem.
• This is a common problem. For example, in a
  typical survey, the respondents may report their
  annual incomes with a lot of errors. Variables
  such as GPA or IQ may be reported with errors as
  well.

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• Now, let us understand the nature of the problem.
• Suppose that you want to estimate the following
  simple regression.
• yß0ß1x1 u .(1)
• where x1 is the measurement-error free variable.
  Suppose that this regression satisfies MLR.1
  through MLR.4.
• Now, suppose that you only observe the
  error-ridden variable x1. That is
• x1x1e1
• where e1 is a random error uncorrelated with x1.

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• To be more precise, the measurement error is such
  that
• x1x1e1 .(2)
• and
• Cov(x1, e1)0 .(3)
• (2) and (3) is called the classical
  errors-in-variables (CEV) assumption.
• Note that the above assumption has nothing to do
  with u. We maintain the assumption that u is
  uncorrelated with both x1 and x1. This also
  means that u is uncorrelated with e1.

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• Because we only observe the error-ridden variable
  x1, we can only estimate the following model.
• yß0ß1x1v.(4)
• Under the CEV assumption, the observed
  (error-ridden) variable in regression (4) is
  endogenous.
• To see this, plug x1x1-e1 into the original
  regression (1) to get
• yß0ß1x1(u- ß1e1).(5)

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• So, we have vu- ß1e1
• Now, notice that
• Cov(x1, v)Cov(x1, u- ß1e1) ?0
• See the front board for the proof.
• Therefore, x1 is correlated with the error term.
  Therefore, x1 is endogenous. Thus, OLS will be
  biased.

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• Under the CEV assumption, we can predict the
  direction of the inconsistency (characterization
  of the bias is difficult). Let be the
  estimated coefficient from the error-ridden
  variable regression (4). Then, we have

• Proof see the front board
• Since the term inside the parenthesis is always
  smaller than 1, there is a bias towards zero.
  This is called the attenuation bias.

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Error in variable (more general case)

• Suppose you want to estimate the following model.
• yß0ß1x1ß2x2.ßkxku
• where x1 is measurement free variable.
• However, you only observe error-ridden variable
  x1. So you can only estimate the above regression
  by replacing x1 with x1.

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• Assume that other variables are measurement error
  free.
• Then the probability limit of is given by

where is the population error from the
following regression. x1d0d1x2
dk-1xk r1
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Measurement error in the dependent variable

• When the measurement error is in the dependent
  variable, but explanatory variables have no
  measurement-errors, there will be no bias in OLS.
• Consider the following model.
• yß0ß1x1 u .(1)
• where y is the measurement free variable.
• But, you only observe the error-ridden variable y.

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• Assume the following
• yye ..(2)
• and
• Cov(y, e)0 ...(3)
• Again, we maintain the assumption that u is
  uncorrelated with both x1 and x1. This also
  means that u is uncorrelated with e1.
• By plugging yy-e into (1), we have the
  following OLS.
• yß0ß1x1 (ue) (5)
• Since e and u are not correlated with the
  explanatory variables, (5) causes no bias in the
  estimation.

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Non random sampling1 Exogenous sampling

• Consider the following regression
• Savingß0ß1(income)ß2(age)u
• Suppose that the survey is conducted for people
  over 35 years old. This is non-random sampling,
  but the sampling criteria is based on the
  independent variable. This is called the sample
  selection based on the independent variables, and
  is an example of exogenous sample selection.
• In this case, OLS regression of the above model
  has no bias.

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Non random sampling2 Enogenous sampling

• Consider the following regression.
• Wealthß0ß1(Educ)ß2(Exper)u
• However, suppose that only people with wealth
  below 250,000 are included in the sample. Then
  the sample selection criteria is based on the
  dependent variable. This is called the sample
  selection based on dependent variable, and is an
  example of endogenous sample selection.
• In this case, OLS estimate of the above
  regression are always biased.

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Stratified sampling

• This is a common survey method, in which the
  population is divided into non-overlapping
  groups, or strata. The sampling is random within
  each group.
• However, some groups are often oversampled in
  order to increase observations for that group.
  Whether this causes the bias depends on whether
  the selection is exogenous or endogenous.

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• If females are oversampled, and you are
  interested in the gender differences in savings,
  then this is the exogenous sample selection.
  Thus, this causes no bias.
• If people with low wealth are oversampled, and if
  you are interested in the wealth regression, then
  this is endogenous sample selection. This causes
  a bias in the regression.

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More subtle form of sample selection.

• Suppose that you are interested in estimating the
  wage offer regression.
• Low(wage offer) ß0ß1(Educ)ß2(Exper)u
• When the wage offer is too low for a particular
  person, the person may decide not to work. Thus,
  this person will not be included in the sample.
  This is the case where sample selection is caused
  by the persons decision to work or not.

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• When the decision is based on unobserved factors,
  then the OLS regression causes a bias. This is
  called the sample selection bias.
• This is typically a problem for the study of the
  wage offer for women.
• This course does not cover the method to correct
  for this type of bias. In the fall semester, I
  will cover this type of issues in a new course
  the Cross Section and Panel Data Analysis.