5' Endogenous right hand side variables

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5. Endogenous right hand side variables

• 5.1 The problem of endogeneity bias
• 5.2 The basic idea underlying the use of
  instrumental variables
• 5.3 When the endogenous right hand side variable
  is continuous
• 5.4 When the endogenous right hand side variable
  is binary

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5.1 Endogeneity bias

• Consider a simple OLS regression
• Yit a0 a1 X1it uit
• Recall that our estimate of a1 will be unbiased
  only if we can assume that X1it is uncorrelated
  with the error term (uit)
• We have discussed two ways to help ensure that
  this assumption is true
• First, we should control for any observable
  variables that affect Yit and which are
  correlated with X1it. For example, we should
  control for X2it if X2it affects Yit and X2it is
  correlated with X1it (see Chapter 2)
• Yit a0 a1 X1it a2 X2it uit

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5.1 Endogeneity bias

• Second, if we have panel data, we can control for
  any unobservable firm-specific characteristics
  (ui) that affect Yit and which are correlated
  with the X variables.
• From Chapter 4
• Yit a0 a1 X1it a2 X2it ui eit
• We control for the correlations between ui and
  the X variables by estimating fixed effects
  models.
• Our estimates of a1 and a2 are unbiased if the X
  variables are uncorrelated with eit. In this
  case, we say that the X variables are exogenous.

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5.1 Endogeneity bias

• Unfortunately, multiple regression and fixed
  effects models do not always ensure that the X
  variables are uncorrelated with the error term
• if we do not observe all the variables that
  affect Y and that are correlated with X, multiple
  regression will not solve the problem.
• if we do not have panel data, the fixed effects
  models cannot be estimated.
• even if we have panel data, the Y and X variables
  may display little variation over time in which
  case the fixed effects models can be unreliable
  (Zhou, 2001).
• even if we have panel data and the Y and X
  variables display sufficient variation over time,
  the unobservable variables that are correlated
  with X may not be constant over time in which
  case the fixed effects models will not solve the
  problem.

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• A variable is more likely to be correlated with
  the error term if it is endogenous
• Endogenous means that the variable is
  determined within the economic model that we are
  trying to estimate.
• For example, suppose that Y2it is an endogenous
  explanatory variable
• Y1it a0 a1 Y2it a2 Xit uit (1)
• Y2it b0 b1 Xit b2 Zit vit
  (2)
• Equations (1) and (2) have a triangular
  structure since Y2it is assumed to affect Y1it,
  but Y1it is assumed not to affect Y2it
• Given this triangular structure, the OLS estimate
  of a1 in equation (1) is unbiased only if vit is
  uncorrelated with uit
• If vit is correlated with uit, then Y2it is
  correlated with uit which means that the OLS
  estimate of a1 would be biased
• To avoid this bias, we must estimate equation (1)
  instrumental variables (IV) regression rather
  than OLS.

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• Equations (1) and (2) are called structural
  equations because they describe the economic
  relationship between Y1it and Y2it
• We can obtain a reduced-form equation by
  substituting eq. (2) into eq. (1)
• Y1it a0 a1 (b0 b1 Xit b2 Zit vit) a2
  Xit uit
• In this reduced-form equation, all the
  explanatory variables (Xit and Zit) are exogenous
• The basic idea underlying IV regression is to
  remove vit from the Y1it model so that our
  estimate of a1 is unbiased.

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5.2 The basic idea underlying the use of
instrumental variables

• Note that vit is removed from the Y1it model if
  we use the predicted rather than the actual
  values of Y2it on the right hand side.
• We predict Y2it using all the exogenous variables
  in the system (in our example, we use the two
  exogenous variables Xit and Zit)

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5.2 The basic idea

• We then use the predicted rather than the actual
  values of Y2it when estimating the Y1it model
• The a1 estimate is biased in eq. (3) but it is
  unbiased in eq. (4) because the vit term has been
  removed.

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• In eq. (4) the estimated coefficient for the Zit
  variable is
• We already know the value of from eq.
  (2)
• Therefore
• It is important to note that the
  coefficient can be estimated only if there is at
  least one exogenous variable in the structural
  model for Y2it that is excluded from the
  structural model for Y1it
• This is the Zit variable in eq. (2)

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• In eq. (4) the coefficient is just
  identified because there is only one exogenous
  variable (Zit) that is in the Y2it model and that
  is excluded from the Y1it model

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• Suppose we had included Zit in both models
• In this case, the coefficient cannot be
  identified because we estimate and
  
• In other words, we cannot determine whether the
  effect of Zit on Y1it is a main effect (a3) or an
  indirect effect through Y2it (a1b2)
• Here we say that the system of equations is
  under-identified

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• Suppose we had included two exogenous variables
  in the Y2it model and we excluded both these
  variables from the Y1it model
• Now we have estimates of , ,
  , and .
• Therefore
• Here we say that the system of equations is
  over-identified

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5.3 When the endogenous right hand side variable
is continuous

• When the models have a triangular structure, the
  models can be estimated using the ivreg command
  (NB In STATA 10.0 this command has changed to
  ivregress)
• In our example, the system is triangular because
  there are two equations and one endogenous
  right-hand side variable

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5.3.1 Estimating triangular models using 2SLS
(ivreg)

• Go to http//ihome.ust.hk/accl/Phd_teaching.htm
• Open up the housing.dta file which provides data
  from 50 U.S. states (1980 Census)
• use "C\phd\housing.dta", clear
• pct_population_urban the of the population
  that lives in urban areas
• family_income median annual family income
• housing_value median value of private housing
• rent median monthly housing rental payments
• region1 region 4 dummy variables for four
  regions in the U.S.

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• Suppose we want to estimate the following
• rent a0 a1 pct_population_urban
  a2 housing_value u
• housing_value b0 b1 family_income
  b2 region2 b3 region3 b4 region4 v
• This is a triangular system because there are two
  equations and one endogenous right hand side
  variable (housing_value)
• If u and v are correlated, the OLS estimate of a2
  will be biased in the rent model

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• If we ignore the endogeneity problem and estimate
  the rent model using simple OLS
• reg rent housing_value pct_population_urban
• To take account of the potential endogeneity
  problem we use the ivreg command
• ivreg depvar1 varlist1 (depvar2 varlistiv)
• depvar1 is the dependent variable for the model
  which has an endogenous regressor
• varlist1 are the exogenous variables in the model
  that has the endogenous regressor
• depvar2 is the endogenous regressor
• varlistiv are the exogenous variables that are
  believed to affect the endogenous regressor

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• The models that we want to estimate are
• rent a0 a1 pct_population_urban
  a2 housing_value u
• housing_value b0 b1 family_income
  b2 region2 b3 region3 b4 region4 v
• Therefore
• ivreg rent pct_population_urban (housing_value
  family_income region2 region3 region4)
• The housing_value model can be estimated using
  OLS as there are no endogenous regressors.

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• STATA tells us that
• the endogenous regressor is housing_value
• the pct_population_urban, family_income region2 -
  region4 variables are assumed to be exogenous
  (i.e., they are instruments)

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• We would get exactly the same coefficients if we
  first estimated the housing_value model and then
  included the predicted values of housing_value in
  the rent model
• However, the standard errors are biased under OLS
  so, in practice, you should use the ivreg command
  
• reg housing_value family_income region2 region3
  region4 pct_population_urban
• NB the housing_value model must be estimated on
  all the exogenous variables (including
  pct_population_urban)
• predict housing_value_hat
• reg rent housing_value_hat pct_population_urban

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• These OLS coefficients are the same as we
  obtained using ivreg.
• However, the standard errors are different.

• The OLS coefficients from the second-stage model
  using the predicted housing_value variable are

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• We should test whether
• our chosen instruments are exogenous (i.e., they
  should be uncorrelated with the error term) and
• it is valid to exclude some of them from the
  model that has the endogenous regressor.
• If they are not exogenous or they should not be
  excluded, they are not valid instruments.

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• The Sargan and Basmann tests are used to test for
  instrument validity
• they are tests of over-identifying restrictions
  because the tests can only be performed if the
  model with the endogenous regressor is
  overidentified
• the tests assume that at least one of the chosen
  instruments is valid (unfortunately this
  assumption cannot be tested)
• In our example, the instrumented housing_value
  variable is overidentified because four of the
  exogenous variables (family_income region2
  region3 region4) are excluded from the rent
  model.
• If we had excluded only one of these variables,
  the instrumented housing_value variable would
  have been just identified in which case the
  Sargan and Basmann tests would have been
  unavailable.

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• We obtain the Sargan and Basmann tests by typing
  overid after we run ivreg
• ivreg rent pct_population_urban (housing_value
  family_income region2 region3 region4)
• overid
• If overid is not installed on your computer you
  can install it from the STATA Technical bulleting
  by typing findit overid
• findit commandname is a very useful way of
  downloading commands that have been written but
  not installed on your version of STATA (e.g.,
  your version is out of date or the command has
  not yet been included in the latest version)

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• We obtain the Sargan and Basmann tests by typing
  overid after we run ivreg
• ivreg rent pct_population_urban (housing_value
  family_income region2 region3 region4)
• overid
• These tests are statistically significant, which
  means the chosen instruments are not valid.
• This is not surprising because we did not have
  good reason to assume that they are exogenous and
  validly excluded from the rent model. For
  example
• family incomes may depend on housing values and
  rents (e.g., families may own housing for
  investment purposes), so family_income is
  endogenous
• rents may be different across the four regions,
  so the region dummies should not be excluded from
  the rent model

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• We can also test whether the coefficient of the
  endogenous regressor is biased under OLS.
• The Hausman tests for endogeneity bias are only
  reliable if the chosen instruments are valid.
• We obtain two Hausman tests for endogeneity bias
  by typing ivendog after we run ivreg
• Given these results, we can strongly reject the
  hypothesis that housing_value is exogenous
• Therefore, we have reason to be concerned about
  endogeneity bias (however, this test is not
  reliable as our chosen instruments are not
  valid).

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• It is easy to correct for heteroscedasticity and
  time-series dependence because we can use the
  robust cluster() option with ivreg
• However, we can only run the overid and ivendog
  commands after running ivreg not after running
  ivreg, robust cluster()
• Therefore, you can get the correct standard
  errors using robust cluster() and then test for
  endogeneity bias and instrument validity by
  running ivreg without the robust cluster() option

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Class exercise 5a

• Using the fees.dta file, estimate the following
  models for audit fees and company size
• lnaf a0 a1 lnta a2 big6 u
• lnta b0 b1 ln_age b2 listed v
• where lnaf is the log of audit fees, lnta is the
  log of total assets, ln_age is the log of the
  companys age in years, listed is a dummy
  variable indicating whether the companys shares
  are publicly traded on a market.
• Is the instrumented lnta variable
  over-identified, just-identified, or
  under-identified? Explain.
• Estimate the audit fee model using IV regression,
  controlling for heteroscedasticity and
  time-series dependence.
• Check that the coefficients are the same if you
  instead use the OLS two-step approach.
• Test the validity of the chosen instrumental
  variables.
• Test whether the lnta variable is affected by
  endogeneity bias.
• Verify that the test for instrument validity is
  not available if you change the model so that it
  is just-identified.

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Class exercise 5a

• The instrumented lnta variable is over-identified
  because two exogenous variables (ln_age and
  listed) are excluded from the lnaf model.
• Generating the variables and dropping
  observations with missing data
• use "C\phd\Fees.dta", clear
• gen fyedate(yearend, "mdy")
• format fye d
• gen yearyear(fye)
• gen age year-incorporationyear
• gen ln_ageln(age)
• gen listed0
• replace listed1 if companytype2
  companytype3 companytype5
• gen lnafln(auditfees)
• gen lntaln(totalassets)
• egen missrmiss(ln_age listed lnaf lnta big6)
• drop if miss!0
• ivreg lnaf big6 (lntaln_age listed), robust
  cluster(companyid)

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Class exercise 5a
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Class exercise 5a

• Checking against the two-step OLS results
• reg lnta ln_age listed big6
• predict lnta_hat
• reg lnaf lnta_hat big6

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Class exercise 5a

• Testing for instrument validity and endogeneity
  bias
• Remember to drop the robust cluster() option
• ivreg lnaf big6 (lnta ln_age listed)
• overid
• ivendog

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Class exercise 5a

• Checking that the test for instrument validity
  requires the model to be over-identified, we can
  include ln_age or listed in the audit fee model
  so that it becomes just-identified. For example
• ivreg lnaf big6 ln_age (lnta ln_age listed)
• overid
• Or
• ivreg lnaf big6 listed (lnta ln_age listed)
• overid
• If we include both ln_age and listed in the audit
  fee model, it is under-identified and we cannot
  estimate the effect of company size on fees
• ivreg lnaf big6 ln_age listed (lnta ln_age
  listed)

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• The key to estimating IV models is to find one or
  more exogenous variables that explains the
  endogenous regressor and that can be safely
  excluded from the main equation.
• Unfortunately, most accounting studies that use
  IV regression do not attempt to justify why their
  chosen instruments are exogenous or why they can
  be excluded from the structural model.
• As a result, Larcker and Rusticus (2007)
  criticize the way in which accounting studies
  have applied IV regression
• A key problem is that the IV results can be very
  sensitive to the researchers choice of which
  variables to exclude from the structural model
  and, in many studies, these variables have been
  chosen in a very arbitrary way

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• When testing instrument validity (overid) and
  endogeneity bias (ivendog), it is important to
  consider your sample size
• in large samples, the tests may reject a null
  hypothesis that is nearly true.
• in small samples, the tests may fail to reject a
  null hypothesis that is very false.
• Larcker and Rusticus (2007) recommend that formal
  tests for instrument validity and endogeneity
  should be supplemented with sensitivity analyses.
  For example, researchers should
• report both the OLS and IV results
• examine whether the results are sensitive to
  using different instrumental variables

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• It is important to note that the overid test for
  instrument validity relies on the assumption that
  at least one of the chosen instrumental variables
  is valid
• Larcker and Rusticus (2007) recommend that the
  overid test should be used to check the validity
  of instruments that are justified using theory
  or some basic economic intuition
• the overid test should not be used to select
  instruments on purely statistical grounds

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5.3.2 Estimating simultaneous equations using
3SLS (reg3)

• So far we have been examining a triangular
  system. For example, Y2it affects Y1it but Y1it
  does not affect Y2it
• Y1it a0 a1 Y2it a2 Xit a3 Z2it uit
  
• Y2it b0 b2 Xit b3 Z1it vit
  
• In a simultaneous system, both dependent
  variables affect each other
• Y1it a0 a1 Y2it a2 Xit a3 Z2it uit
  
• Y2it b0 b1 Y1it b2 Xit b3 Z1it vit

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• Y1it a0 a1 Y2it a2 Xit a3 Z2it uit
  
• Y2it b0 b1 Y1it b2 Xit b3 Z1it vit
• In this case, the OLS estimates are biased
  because
• Eq. (1) shows that uit affects Y1it while eq. (2)
  shows that Y1it affects Y2it. As a result, it
  must be true that uit is correlated with Y2it in
  eq. (1). Therefore, the OLS estimate of a1 would
  be biased in eq. (1).
• Eq. (2) shows that vit affects Y2it while eq. (1)
  shows that Y2it affects Y1it. As a result, it
  must be true that vit is correlated with Y1it in
  eq. (2). Therefore, the OLS estimate of b1 would
  be biased in eq. (2).

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• For example, it seems reasonable to argue that
  housing values depend on rents as well as rents
  depending on housing values
• rent a0 a1 housing_value a2
  pct_population_urban u
• housing_value b0 b1 rent b2 family_income
  b3 region2 b4 region3 b5 region4 v
• Note that for identification, each equation must
  contain at least one exogenous variable that is
  not included in the other equation. These are
• pct_population_urban in the rent model
• family_income, region2 - region4 in the
  housing_value model

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• We estimate this kind of model using the reg3
  command
• reg3 (depvar1 varlist1) (depvar2 varlist2)
• use "C\phd\housing.dta", clear
• reg3 (rent housing_value pct_population_urban)
  (housing_value rent family_income region2
  region3 region4)
• Note that the robust cluster() option and the
  overid and ivendog commands are not available
  with reg3

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5.4 When the endogenous right hand side variable
is binary

• So far we have been dealing with the case where
  the endogenous regressor is continuous.
• We may want to estimate a model in which the
  endogenous regressor is binary.
• This brings us to a special class of models which
  are known as self-selection or Heckman
  models. Selectivity Endogeneity where the
  endogenous regressor is binary
• The basic idea is similar to the instrumental
  variable techniques that we have already
  discussed.
• Unfortunately, many accounting researchers have
  been misusing the Heckman model (Lennox and
  Francis, 2009).

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• Examples of endogenous binary variables in
  accounting
• Companies decide whether to use hedge contracts
  (Barton, 2001 Pincus and Rajgopal, 2002).
• Companies decide whether to grant stock options
  (Core and Guay, 1999).
• Companies decide whether to hire Big 5 or non-Big
  5 auditors (e.g., Chaney et al., 2004).
• Governments decide whether to fully or partially
  privatize (Guedhami and Pittman, 2006).
• Companies decide whether to follow international
  financial reporting strategy (Leuz and
  Verrecchia, 2000).
• Companies decide whether to recognize financial
  instruments at fair value or disclose (Ahmed et
  al., 2006).
• Companies decide whether or not to go private
  (Engel et al., 2002).

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Selection model

• Concerns about selectivity arise when the RHS
  dummy variable (D) is endogenous
• Endogeneity results in bias if E(u D) ? 0. The
  intuition underlying Heckman is to estimate and
  then control for E(u D). First model the choice
  of D
• Z is a vector of exogenous variables that affect
  D but have no direct effect on Y.

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Selection model
D
Z
Y
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Selection model

• The error terms in the two equations (u and v)
  are assumed to have a bivariate normal
  distribution with mean zero and
  variance-covariance matrix
• If u and v are correlated (? ? 0), then
  E(u D) ? 0, in which case the OLS
  estimate of the effect of D on Y would be biased.

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Selection model

• The intuition underlying Heckman is to estimate
  E(u D) and include it as a control variable on
  the RHS of the Y model
• E(u D) ?? IMR where

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Selection model

• The IMR variable is added as a control for
  selectivity in the Y model
• The OLS estimate of the effect of D on Y is now
  unbiased because E(e D) ? 0.
• The D and Y models can be estimated in two-steps
  or estimated jointly using maximum likelihood
  (ML)
• ML yields separate estimates of ? and ?.
• The two-step yields an estimate of ??.
• Under the null of no selectivity bias, ? 0 and
  ?? 0.

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Selection model

• In the Y model it has been assumed that the slope
  coefficients on the X variables are equal for the
  cases where D 0, 1
• This assumption can be relaxed by estimating the
  model separately on the two sub-samples

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Selection model

• Again, selectivity is controlled for by including
  IMR variables on the RHS

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• For example, Chaney, Jeter and Shivakumar (2004)
  examine the case where Y audit fees and D a
  dummy for Big 6 (Non-Big 6) audits.
• They argue that an OLS regression of eq. (1)
  gives biased estimates of the Big 6 fee premium
  (?).
• They also argue that the slope coefficients (?)
  may differ between Big 6 and Non-Big 6 audit
  clients

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Class exercise 5b

• As an example, we are going to look at a
  fictional dataset on 2,000 women.
• use "C\phd\heckman.dta", clear
• sum age education married children wage
• Suppose we believe that older and more highly
  educated women earn higher wages. Why would it be
  wrong to estimate the following model?
• reg wage age education
• Estimate a probit model to test whether women are
  more likely to be employed if they are married,
  have children, are older and more highly educated.

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Class exercise 5b

• Of the 2,000 women in our dataset, only 1,343 are
  in a paid job.
• This raises a selection (endogeneity) problem
  because the sub-sample of 1,343 women is probably
  not representative of the population (which
  includes women who are not earning wages).
• Put another way, we do not observe the wages that
  would have been earned by the 657 women if they
  had been in employment.

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Class exercise 5b

• wage a0 a1 age a2 education u
• If older and more highly educated women earn
  higher wages, we expect a1 gt 0 and a2 gt 0.
• However, the dependent variable (wage) is only
  observed for women who are in employment.
• To overcome this problem, we need to think about
  what determines the likelihood of female
  employment. For example, we may argue that women
  are more likely to be employed if they are
  married, have children, are older and more highly
  educated
• emp b0 b1 married b2 children b3 age
  b4 education v
• gen emp0 if wage.
• replace emp1 if wage!.
• probit emp married children age education

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5.4 When the endogenous right hand side variable
is binary (heckman)

• It is easy to estimate the two-step Heckman model
  in STATA
• heckman depvar1 varlist1, select (depvar2
  varlist1), twostep
• where depvar1 is the dependent variable in the
  main equation and depvar2 is the dependent
  variable in the selection model
• Going back to our dataset on female wages
• heckman wage education age, select(emp married
  children education age) twostep

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• The 657 censored observations are the women who
  are not in employment.
• The Wald chi2 tests the overall significance of
  the model.

• Womens wages are higher if they are older and
  more highly educated

• The probit model of employment is exactly the
  same as what we had before
• Women are more likely to be in employment if they
  are married, have children, are more highly
  educated or older.

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• Recall that we are trying to estimate the error
  in the wage equation which is truncated because
  we only observe wages of the women who are in
  employment

• The lamba variable is simply the IMR that was
  estimated from the emp model ( ).
• The IMR coefficient (4.00) is
• Since the IMR coefficient is statistically
  significant, it may be concluded that there is
  statistically significant evidence of a selection
  effect.

• The IMR coefficient can also be written as the
  product of rho and sigma ( )
• rho ( ) is the correlation between u and v
• sigma ( ) is the standard deviation of u
• Thus, 4.00 0.67 5.95

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• The selection model can also be estimated using
  maximum likelihood (ML) rather than the two-step
  approach.
• This can be useful if we want to test the
  statistical significance of rho.
• If rho 0, the IMR coefficient must also be zero
  in which case there is no need to control for
  selectivity.
• There is an unresolved debate in the econometrics
  literature as to whether the two-step or ML
  approach is best.
• STATA automatically gives us the ML results if we
  do not specify twostep as an option
• heckman wage education age, select(emp married
  children education age)

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• Here, the results for the wage and employment
  models are similar using either ML or the
  two-step.
• NB Sometimes the results are different between
  ML and the two-step. Also you may find that the
  ML model does not converge if the likelihood
  function is not concave.

• /athroh the inverse hyperbolic tangent of
• /lnsigma is the log of the standard deviation of
  u ( )
• STATA estimates athrho and lnsigma rather than
  rho and sigma directly in order to increase the
  numerical stability of the maximization routine
  for the likelihood function.
• STATA also reports the untransformed values of
  rho and sigma.

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• The Likelihood-ratio statistic allows us to
  reject the hypothesis that rho 0, which means
  that there is significant evidence of
  selectivity.

• When rho 0, it is also true that
  equals zero.
• The statistical significance of athrho implies
  that there is significant evidence of selectivity.

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Class exercise 5c

• Estimate the following audit fee models
  separately for Big 6 and Non-Big 6 audit clients
• lnaf a0 a1 lnta u (1)
• lnaf a0 a1 lnsales u (2)
• where lnaf log of audit fees, lnta log of
  total assets, lnsales log of sales
• Use the heckman command to control for
  endogeneity with respect to the companys
  selected auditor. Your auditor choice models are
  as follows
• big6 b0 b1 lnsales b2 lnta v
• nbig6 c0 c1 lnsales c2 lnta w
• where big6 1 (big6 0) if the company chooses
  a Big 6 (Non-Big 6) auditor and nbig6 1 (nbig6
  0) if the company chooses a Non-Big 6 (Big 6)
  auditor.

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Class exercise 5c

• What exclusion restrictions are you imposing in
  equations (1) and (2)?
• Is there statistically significant evidence of
  selectivity?
• For the two different specifications of the audit
  fee model
• what are the signs of the IMR coefficients?
• what are the signs of rho?

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Class exercise 5c

• In equation (1) we impose the restriction that
  lnsales does not affect lnaf. In equation (2) we
  impose the restriction that lnta does not affect
  lnaf.
• use "C\phd\Fees.dta", clear
• gen lnsalesln(sales)
• gen lnafln(auditfees)
• gen lntaln(totalassets)
• egen missrmiss(lnaf lnta lnsales)
• drop if miss!0
• gen nbig60
• replace nbig61 if big60
• heckman lnaf lnta, select (big6 lnta lnsales)
  twostep
• heckman lnaf lnsales, select (big6 lnta
  lnsales) twostep
• heckman lnaf lnta, select (nbig6 lnta lnsales)
  twostep
• heckman lnaf lnsales, select (nbig6 lnta
  lnsales) twostep

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Class exercise 5c

• The coefficients on the IMRs and rho are positive
  in equations (1) and (2) when the fee models are
  estimated for Non-Big 6 clients.
• The coefficients on the IMRs and rho are positive
  in equation (1) but they are negative in equation
  (2) when the fee models are estimated for Big 6
  clients.
• Therefore, the estimated effects of selectivity
  are sensitive to which exclusion restrictions are
  imposed on the audit fee model for Big 6 clients.
• The problem is that we have chosen arbitrary
  exclusion restrictions that lack any intuitive or
  theoretical justification.

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Treatment effects model

• In exercise 5c, we estimated the audit fee models
  separately for the Big 6 and non-Big 6 audit
  clients
• To do this, we use the heckman command

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Treatment effects model

• Suppose that we want to estimate one audit fee
  model with Big 6 on the right hand side of the
  equation (i.e., we assume that the X coefficients
  have the same slope in the two equations)

70
Treatment effects model

• We can estimate this model using the treatreg
  command
• treatreg lnaf lnta, treat (big6 lnta lnsales)
  twostep
• treatreg lnaf lnsales, treat (big6 lnta
  lnsales) twostep
• If we dont specify the twostep option we will
  get the ML estimates (sometimes the ML model will
  not converge due to a nonconcave likelihood
  function)
• treatreg lnaf lnta, treat (big6 lnta lnsales)
  
• treatreg lnaf lnsales, treat (big6 lnta
  lnsales)

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Treatment effects model

• The results for both the treatment effects and
  Heckman models can be very sensitive to the model
  specification.
• For example, the Big 6 fee premium can easily
  flip signs from positive to negative
• treatreg lnaf lnta, treat (big6 lnta lnsales)
  twostep
• treatreg lnaf lnta lnsales, treat (big6 lnta
  lnsales) twostep
• Note that there are no exclusion restrictions (Z
  variables) in the second specification since lnta
  and lnsales appear in both the first stage and
  second stage models

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Exclusion restrictions

• Francis and Lennox (2009) argue that many
  accounting studies have estimated the Heckman and
  treatment effects models incorrectly
• It is well recognized (in economics) that
  exogenous Z variables from the first stage choice
  model need to be validly excluded from the second
  stage outcome regression (Little, 1985 Little
  and Rubin, 1987 Manning et al., 1987).
• Accounting studies have generally failed to (a)
  impose exclusion restrictions, or (b) provide
  compelling grounds for the validity of the
  exclusion restrictions.

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Exclusion restrictions

• Of the 38 accounting studies in our survey
• 4 studies explicitly fail to nominate any Z
  variable (5 studies estimate specifications both
  with and without Z variables 4 studies fail to
  disclose whether they include a Z variable).
• Only 2 studies provide a rationale for including
  the Z variable in the first stage model and
  excluding it from the second stage.
• Only 2 studies report robustness tests using
  alternative exclusion restrictions (i.e.,
  alternative Z variables).

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Exclusion restrictions

• Economists recognize that it is important to
  justify why the Zs can be validly excluded from
  the Y model.
• For example, Angrist (1990) examines how military
  service affects the earnings of veteran soldiers
  after they are discharged from the army.
• This involves a selection issue because
  individuals join the military if they have poor
  wage offers in other types of job.
• Angrist (1990) tackles the selectivity issue
  using data from the Vietnam era, when military
  service was partly determined by a draft lottery.

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Exclusion restrictions
D military service
Z Random lottery
Y civilian earnings
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Exclusion restrictions

• Other examples from economics
• Levitt (1997) tests whether additional policing
  results in less crime
• Selectivity is an issue because more police are
  hired if crime increases (or if it is expected
  that crime will increase)
• Uses the electoral cycle as an instrument for
  policing.
• Angrist and Evans (1998) test whether child
  bearing reduces female participation in the labor
  market
• Selectivity is an issue because women are more
  likely to have children rather than enter the
  labor market if their wage offers would be low
  (i.e., lower opportunity cost).
• Use the gender of the second child as instrument
  for the decision to have a third child.

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Levitt (1997) Exclusion restriction
D policing
Z electoral cycle
Y crime
78
Angrist and Evans (1998) Exclusion restriction
D decision to have a third child
Z Sex composition of first two children
Y female participation in labor market
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Exclusion restrictions

• Of the 38 accounting studies in our survey, only
  two attempt to justify why Z has no direct impact
  on Y.
• Many studies do not report results for the D
  model, so the reader cannot evaluate the power of
  the Z variables for identifying selectivity.
• At least 9 studies (possibly 13) estimate models
  in which there are no nominated Z variables.

80
Exclusion restrictions

• When there are no exclusion restrictions,
  identification of the IMR coefficients relies on
  the assumed non-linearity
• The IMRs would capture any misspecification of
  the functional relation between X and Y (e.g.,
  non-linearity) in addition to any selectivity
  bias.

81
Exclusion restrictions

• Little (1985) Relying on nonlinearities to
  identify selectivity bias is unappealing
  because it is very difficult to distinguish
  empirically between selectivity and
  misspecification of the models functional form.
• STATA manual Theoretically, one does not need
  such identifying variables, but without them, one
  is depending on functional form to identify the
  model. It would be difficult to take such results
  seriously since the functional-form assumptions
  have no firm basis in theory.
• A failure to nominate any Z variables can lead to
  serious problems of multicollinearity (Manning et
  al., 1987 Puhani, 2000 Leung and Yu, 2000).

82
Re-examine Chaney et al. (2004)

• In some respects, their study is fairly typical
  of those in our survey
• 26 out of 38 papers attempt to control for
  selectivity in a treatment variable.
• 15 studies rely on the selection model for their
  primary results (even if those results contradict
  the OLS findings).
• The Chaney et al. study does not include any Z
  variables.

83
Chaney, Jeter and Shivakumar (2004)
D BIG5 (company hires a Big 5 or non-Big 5
auditor)
Y Audit fees
Z null set
84

• OLS models of audit fees

85

• CJS argue that it is important to allow the slope
  coefficients to differ between Big 5 and Non-Big
  5 clients
• Without controlling for selectivity, the mean fee
  premiums of Big 5 auditors are
• Without controlling for selectivity, the mean fee
  premiums of non-Big 5 auditors are
• What do these results mean?

86

• The results are very different when the IMRs are
  added as RHS variables
• After controlling for selectivity, the mean fee
  premiums of Big 5 auditors are
• After controlling for selectivity, the mean fee
  premiums of non-Big 5 auditors are
• What do these results mean?

87
Chaney, Jeter and Shivakumar (2004)

• We want to test whether these results are robust.
• Company size is the most important determinant of
  both auditor choice and audit fees.
• We find evidence of multicollinearity problems
  due to the high correlations between company size
  (LTA) and the IMRs.
• We try alternative specifications of the company
  size variable in the auditor choice and audit fee
  models.

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Chaney, Jeter and Shivakumar (2004)

• To ensure a level playing field, we estimate the
  same specifications of the fee models without
  controlling for selectivity
• LTA alone
• LTS alone
• LTA and LTS
• The results consistently indicate that Big 5
  clients pay significant fee premiums.

92
Matched propensity scores (MPS)

• Given the problems with using selection models,
  it would be good to find an alternative or
  complementary approach.
• The major advantage of MPS is that exclusion
  restrictions and assumptions about functional
  form are unnecessary because the Y model does not
  include the IMRs.
• Selection is assumed to take place on the
  independent variables in the D model so MPS does
  not control for any selectivity on
  unobservables.
• MPS is used by only 3 of the 38 accounting
  studies in our survey.

93
Matched propensity scores

• Steps
• Estimate the D model.
• Obtain the predicted probability that D 1 for
  each observation in the sample.
• Match each D 1 observation to a D 0
  observation that has the closest predicted
  probability.
• The above three steps can be done using the
  psmatch2 command in STATA
• Estimate the Y model on the matched sample.
• Compare results to the unmatched sample to
  determine if there is selectivity bias.

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Conclusions

• The conclusions of prior studies may be fragile
  especially when
• they attempt to control for selectivity in a
  treatment variable
• results for the selection model are not
  corroborated by single equation estimates
• researchers fail to nominate or justify the
  chosen exclusion restrictions.
• This is true of nearly all studies in our survey!

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99
Example Leuz and Verrecchia (2000)
D IR97 (international reporting)
Z ROA, Capital intensity, UK/US listing.
Y Cost of capital
100
Leuz and Verrecchia (2000)

• Is it valid to assume that ROA, Capital
  intensity, and UK/US listing have no direct
  effect on the cost of capital?
• Are these Z variables really exogenous?

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Leuz and Verrecchia (2000)

• Are the tests for selectivity bias powerful?
• Are the results sensitive to functional form?
  (see the free float variable).
• LV do not report results using OLS
• LV do not report whether their results are
  sensitive to alternative model specifications.
• LV do not report tests for multicollinearity, nor
  do they try the MPS approach.

103
Going forward

• Researchers need to be aware that Heckman and
  treatment effects models can provide results that
  are extremely fragile. Sensitivity primarily
  affects the RHS variable that is assumed to be
  endogenous (D) and the IMRs.
• Studies need to discuss
• why the Zs are exogenous
• why the Zs have no direct effect on Y
• whether the Zs are powerful predictors of D
• The signs and significance of the IMRs alone do
  not provide compelling evidence as to the
  direction or existence of selectivity bias.
• Selection studies should routinely report tests
  for multicollinearity problems.
• Researchers can consider using the MPS
  methodology to determine whether there is
  evidence of selection on observables.

104
Summary

• When the endogenous regressor is continuous, you
  can control for endogeneity using the ivreg or
  reg3 commands.
• When the endogenous regressor is binary, you can
  control for endogeneity using the heckman or
  treatreg commands.
• If you want to control for endogeneity, it is
  vitally important that you have a good
  justification for your chosen exclusion
  restrictions.
• Choosing arbitrary exclusion restrictions will
  very likely give you garbage results.

105
Concluding comments

• When writing a paper, you normally follow three
  steps
• Find a research idea (either before or after you
  get the data)
• Perform the empirical analysis
• Write up the results
• This course has focused on step 2 but it has also
  touched on step 1
• There are opportunities to improve on what prior
  accounting studies have done (e.g., Rock et al.,
  2001 Larcker and Rusticus, 2007 Francis and
  Lennox, 2008).
• Step 3 is also very important
• You should spend lots of time learning how to
  write
• practice is very important, just as it is with
  data analysis and programming.
• Having a well written paper is crucial for
  publication
• badly written papers are sometimes rejected even
  if the idea is good and the data analysis is well
  done.
• well written papers are sometimes accepted even
  if the empirical analysis is poor or the results
  are misinterpreted.