5. Endogenous right hand side variables

1
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

2
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

3
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.

4
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.

5

• 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.

6

• 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.

7
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)

8
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.

9

• 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)

10

• 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

11

• 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

12

• 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
• In this example, the system is triangular
  because there are two equations and one
  endogenous right-hand side variable

13
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 ivregress
  command
• The models can be estimated using 2SLS or LIML or
  GMM
• 2SLS is more commonly used in practice

14
5.3.1 Estimating triangular models using 2SLS
(ivregress)

• Go to MySite
• Open up the housing.dta file which provides data
  from 50 U.S. states (1980 Census)
• use "J\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.

15

• 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

16

• 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 ivregress command
• ivregress estimator depvar1 varlist1 (depvar2
  varlistiv)
• estimator is either 2sls or liml or gmm
• 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

17

• 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
• The rent model has an endogenous regressor
• ivregress 2sls rent pct_population_urban
  (housing_value family_income region2 region3
  region4)
• ivregress liml rent pct_population_urban
  (housing_value family_income region2 region3
  region4)
• ivregress gmm 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
• reg housing_value family_income region2 region3
  region4

18

• 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.

19

• The tests for instrument validity are also known
  as 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 it
  would not be possible to test for instrument
  validity.

20

• We obtain the tests for instrument validity by
  typing estat overid after we run ivregress
• ivregress 2sls rent pct_population_urban
  (housing_value family_income region2 region3
  region4)
• estat overid
• These tests are statistically significant, which
  means the chosen instruments are not valid.

21

• 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_income is endogenous if family incomes
  depend on housing values and rents
• Why would this be true?
• rents may be different across the four regions,
  so the region dummies should not be excluded from
  the rent model

22

• We obtain different statistics for the tests of
  instrument validity if the models are estimated
  using LIML or GMM
• However, the conclusions are the same as in our
  previous example
• ivregress liml rent pct_population_urban
  (housing_value family_income region2 region3
  region4)
• estat overid
• ivregress gmm rent pct_population_urban
  (housing_value family_income region2 region3
  region4)
• estat overid

23

• Note that we cannot test for instrument validity
  when the endogenous regressor is just identified
• This is because the test statistics are obtained
  under the assumption that at least one of the
  instruments is valid
• For example
• ivregress 2sls rent pct_population_urban
  (housing_value family_income)
• estat overid
• ivregress liml rent pct_population_urban
  (housing_value family_income)
• estat overid
• ivregress gmm rent pct_population_urban
  (housing_value family_income)
• estat overid

24

• We can also test whether the coefficient of the
  endogenous regressor is biased under OLS.
• We obtain two Hausman tests for endogeneity bias
  by typing estat endogenous after we run ivregress
• ivregress 2sls rent pct_population_urban
  (housing_value family_income region2 region3
  region4)
• estat endogenous
• (The Durbin statistic uses an estimate of the
  error terms variance assuming that the variable
  being tested is exogenous whereas the Wu-Hausman
  statistic assumes that the variable being tested
  is endogenous)
• Given these results, we may be tempted to reject
  the hypothesis that housing_value is exogenous
• However, the Hausman tests for endogeneity bias
  are only reliable if the chosen instruments are
  valid. In our example they are not, and so we
  cannot draw conclusions about the potential for
  endogeneity bias.

25
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 2SLS.
• 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.

26

• 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 (2010)
  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

27
(No Transcript)
28
(No Transcript)
29
(No Transcript)
30

• Larcker and Rusticus (2010) recommend that
  researchers justify their chosen instruments
  using theory or economic intuition
• the estat overid test should not be used to
  select instruments on purely statistical grounds
  particularly as the test is invalid if all of the
  chosen instruments are also invalid
• When testing instrument validity (estat overid)
  and endogeneity bias (estat endog), it is also
  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.

31
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

32

• 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).

33

• 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

34

• We estimate this kind of model using the reg3
  command
• reg3 (depvar1 varlist1) (depvar2 varlist2)
• use "J\phd\housing.dta", clear
• reg3 (rent housing_value pct_population_urban)
  (housing_value rent family_income region2
  region3 region4)
• Unfortunately, the overid and endog commands are
  not currently available with reg3

35
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.

36

• 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).

37
Selection model

• Concerns about selectivity arise when the RHS
  dummy variable (D) is endogenous
• Endogeneity results in bias if E(u D) ? 0.
• If u and v are correlated, then E(u D) ? 0, in
  which case the OLS estimate of the effect of D on
  Y would be biased.

38
Selection model

• 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.

39
Selection model
D
Z
Y
40
Selection model

• Estimate E(u D) and include it as a control
  variable on the RHS of the Y model
• E(u D) ?? IMR where ? captures the
  correlation between u and v while ? is the
  standard deviation of u and

41
Selection model

• The MILLS 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.

42
Class exercise 5b

• We are going to look at a fictional dataset on
  2,000 women.
• use "J\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.

43
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

44
(No Transcript)
45

• 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.

46

• The lamba variable is simply the IMR that was
  estimated from the emp model
• The IMR coefficient is 4.00 and statistically
  significant
• there is statistically significant evidence of a
  selection effect.

• The IMR coefficient is the product of rho and
  sigma (??)
• Thus, 4.00 0.67 5.95

47
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.

48
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 MILLS coefficients?
• what are the signs of rho?

49
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
• 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)

50
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)

51
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

52
Exclusion restrictions

• Francis, Lennox, Francis Wang (2012) 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.

53
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.

54
Exclusion restrictions
D military service
Z Random lottery
Y civilian earnings
55
Exclusion restrictions

• 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.

56
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
57
Exclusion restrictions

• In accounting, many studies fail 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.
• Some studies estimate models in which there are
  no nominated Z variables.

58
Exclusion restrictions

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

59
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 worsen
  the problems of multicollinearity (Manning et
  al., 1987 Puhani, 2000 Leung and Yu, 2000).

60
Example Chaney, Jeter and Shivakumar (2004)
D BIG5 (company hires a Big 5 or non-Big 5
auditor)
Y Audit fees
Z null set
61
Example Leuz and Verrecchia (2000)
D IR97 (international reporting)
Z ROA, Capital intensity, UK/US listing.
Y Cost of capital
62
(No Transcript)
63
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?

64
(No Transcript)
65
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.

66
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.

67
Summary

• When the endogenous regressor is continuous, you
  can control for endogeneity using the ivregress
  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
  probably give you garbage results.