4. Using panel data

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4. Using panel data

• 4.1 The basic idea
• 4.2 Linear regression
• 4.3 Logit and probit models
• 4.4 Other models

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4.1 The basic idea

• Panel data data that are pooled for the same
  companies across time.
• In panel data, there are likely to be unobserved
  company-specific characteristics that are
  relatively constant over time.
• I have already explained that it is necessary to
  control for this time-series dependence in order
  to obtain unbiased standard errors.
• In STATA we can do this using the robust cluster
  () option

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4.1 The basic idea

• The first advantage of panel data is that we are
  using a larger sample compared to the case where
  we have only one observation per company.
• The larger sample permits greater estimation
  power, so the coefficients can be estimated more
  precisely.
• Since the standard errors are lower (even when
  they are adjusted for time-series dependence), we
  are more likely to find statistically significant
  coefficients.
• use "C\phd\Fees.dta", clear
• gen fyedate(yearend, "mdy")
• format fye d
• gen yearyear(fye)
• sort year
• gen lnafln(auditfees)
• gen lntaln(totalassets)
• by year reg lnaf lnta, robust cluster(companyid)
• reg lnaf lnta, robust cluster(companyid)

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4.1 The basic idea

• The second advantage of panel data is that we can
  estimate dynamic models.
• For example, suppose we believe that audit fees
  depend not only on the companys size but also
  its rate of growth
• sort companyid fye
• gen growth lnta- lnta_n-1 if companyid
  companyid_n-1
• reg lnaf lnta growth, robust cluster( companyid)
• We find that audit firms offer lower fees to
  companies that are growing more quickly
• If we had had only one year of data, we would not
  have been able to estimate this model.

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4.1 The basic idea

• The third and most important advantage of
  panel data is that we are able to control for
  unobservable company-specific effects that are
  correlated with the observed explanatory
  variables
• Lets start with a simple regression model
• Lets assume that the error term has an
  unobserved company-specific component that does
  not vary over time and an idiosyncratic component
  that is unique to each company-year observation

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4.1 The basic idea

• Putting the two together
• Recall that the standard error of ? will be
  biased if we do not adjust for time-series
  dependence
• this adjustment is easy using the robust cluster
  () option
• The OLS estimate of the ? coefficient will be
  unbiased as long as the unobservable
  company-specific component (ui) is uncorrelated
  with Xit

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4.1 The basic idea

• Unfortunately, the assumption that ui is
  uncorrelated with Xit is unlikely to hold in
  practice.
• If ui is correlated with Xit then ?it is also
  correlated with Xit
• The OLS estimate of ? will be biased if ?it is
  correlated with Xit (recall our previous
  discussion and notes on omitted variable bias)

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4.1 The basic idea

• An example can illustrate this bias.
• Go to http//ihome.ust.hk/accl/Phd_teaching.htm
• use "C\phd\beatles.dta", clear
• list
• This dataset is a panel of four individuals
  observed over three years (1968-70)
• In each year they were asked how satisfied they
  are with their lives
• this is the lsat variable which takes larger
  values for increasing satisfaction
• You want to test how age affects life
  satisfaction
• reg lsat age
• It appears that they became slightly more
  satisfied as they got older.

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4.1 The basic idea

• Suppose you now include dummy variables for each
  individual
• tab persnr, gen(dum_)
• Recall that you must omit one dummy variable or
  the intercept in order to avoid perfect
  collinearity (see the previous notes about
  multicollinearity)
• reg lsat age dum_1 dum_2 dum_4
• reg lsat age dum_1 dum_2 dum_3 dum_4, nocons
• There now appears to be a highly significant
  negative impact of age on life satisfaction
• Whats going on here?

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4.1 The basic idea

• Recall that fitting a simple OLS model (lsat on
  age) is equivalent to plotting a line of best fit
  through the data
• twoway (lfit lsat age) (scatter lsat age)

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4.1 The basic idea

• I am now going to introduce a new command,
  separate , by()
• separate lsat, by(persnr)
• This creates four separate life satisfaction
  variables for each of the four individuals
• Now graph the relationship between life
  satisfaction and age for each of the four people
• twoway (lfit lsat1 age) (scatter lsat1 age)
• twoway (lfit lsat2 age) (scatter lsat2 age)
• twoway (lfit lsat3 age) (scatter lsat3 age)
• twoway (lfit lsat4 age) (scatter lsat4 age)

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• It is clear that each of the four individuals
  became less satisfied as they got older.
• The simple OLS regression was biased because John
  and Ringo (who happened to be older) were
  generally more satisfied than Paul and George
  (who happened to be younger)
• The multiple OLS regression controlled for these
  idiosyncratic differences by including dummy
  variables for each person
• We can see this by plotting the simple OLS
  results and the multiple OLS results
• reg lsat age dum_1 dum_2 dum_3 dum_4, nocons
• predict lsat_hat
• separate lsat_hat, by(persnr)
• twoway (line lsat_hat1-lsat_hat4 age) (lfit lsat
  age) (scatter lsat1-lsat4 age)

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4.1 The basic idea

• What does all this have to do with panel data
  being advantageous?
• Without panel data we would not have been able to
  control for the idiosyncracies of the four
  individuals.
• If we had had data for only one year, we would
  not have known that the age coefficient was
  biased in the simple regression.
• We can demonstrate this by running a regression
  of lsat on age for each year in the sample
• sort time
• by time reg lsat age
• Without panel data, we would have incorrectly
  concluded that people get happier as they get
  older

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4.1 The basic idea

• In the multiple regression, we include dummy
  variables (dum_1 dum_2 dum_3 dum_4) which control
  for the individual-specific effects (ui)
• Without including the person dummies, our
  estimate of ? would be biased because the dummies
  are correlated with age.
• The person dummies explain all the
  cross-sectional variation in life satisfaction
  across the four individuals.
• The only variation that is left is the change in
  satisfaction within each person as he gets older.
  
• Therefore, the model with dummies is sometimes
  called the within estimator or the
  fixed-effects model.

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4.1 The basic idea

• In small datasets like this, it is easy to create
  dummy variables for each person (or each
  company).
• In large datasets, we may have thousands of
  individuals or companies.
• The number of variables in STATA is restricted
  due to memory limits.
• Also it is not very inconvenient to have results
  for thousands of dummy variables (just imagine
  how long your log file would be!).

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4.1 The basic idea

• Instead of including dummy variables, we can
  control for idiosyncratic effects by transforming
  the Y and X variables.
• Taking averages of eq. (1) over time gives
• Subtracting eq. (2) from eq. (1) gives
• The key thing to note here is that the
  individual-specific effects (ui) have been
  differenced out so they will not bias our
  estimate of ?.

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4.1 The basic idea

• Another transformation that will do the same
  trick is to take differences rather than subtract
  means
• Lagging by one period
• Subtracting eq. (2) from eq. (1) gives
• Again the individual-specific effects (ui) have
  been differenced out so they will not bias our
  estimate of ?.

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Class exercise 4a

• Estimate the following models, where Y life
  satisfaction and X age.
• Compare the age coefficients in these models to
  the age coefficient in the untransformed model
  with person dummies (ignore the standard errors
  of the age coefficients because they are biased)

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Class exercise 4a

• You should find that the age coefficients are
  exactly the same.
• First, we create the variables
• sort persnr time
• gen chlsatlsat-lsat_n-1 if persnrpersnr_n-1
  
• gen chageage-age_n-1 if persnrpersnr_n-1
• (NB the chage variable is just a constant
  because each person gets older by one from one
  year to the next list persnr time chage)
• by persnr egen avlsatmean(lsat)
• by persnr egen avagemean(age)
• gen difflsatlsat-avlsat
• gen diffageage-avage
• Next, we run the three regressions without
  constant terms (recall that the chage variable is
  a constant)
• reg chlsat chage, nocons
• reg difflsat diffage, nocons
• reg lsat age dum_1 dum_2 dum_3 dum_4, nocons

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4.2 Linear regression using panel data (xtreg, fe
i())

• Fortunately, STATA has a command that
• allows us to avoid creating dummy variables for
  each person
• corrects the standard errors
• xt is a prefix that tells STATA we want to
  estimate a panel data model
• The fe option tells STATA we want to estimate a
  fixed effects model
• in OLS this is equivalent to including dummy
  variables to control for person-specific effects
• The i() term tells STATA the variable that
  identifies each unique person
• xtreg lsat age , fe i(persnr)

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• Note that the age coefficient and t-statistic are
  exactly the same as in the OLS model that
  includes person dummies
• reg lsat age dum_1 dum_2 dum_3 dum_4, nocons
• There are 12 person-years, 3 persons, and the
  minimum, average and maximum number of
  observations per person is 4.

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• Since we are estimating a within-effects model,
  it is the within R2 that is directly relevant
  (93.2).
• If we used the same independent variables to
  estimate a between-effects model, we would have
  an R2 of 88.4 (I will explain later what we mean
  by the between-effects model).
• If we used the same independent variables to
  estimate a simple OLS model, we would get an R2
  of 16.5. (reg lsat age)
• The F-statistic is a test that the coefficient(s)
  on the X variable(s) (i.e., age) are all zero.

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• sigma_u is the standard deviation of the
  estimates of the fixed effects, ui (?u)
• sigma_e is the standard deviation of the
  estimates of the residuals, eit (?e)
• rho ?u2 / (?u2 ?e2)
• 4.932 / (4.932 0.472) 0.99

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• The correlation between uit and Xit is -0.83.
• This correlation appears to be high confirming
  our prior finding that the fixed effects are
  correlated with age.
• The F-test allows us to reject the hypothesis
  that there are no fixed effects.
• If we had not rejected this hypothesis, we could
  estimate a simple OLS instead of the
  fixed-effects model.

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4.2 Linear regression (predict)

• After running the fixed-effects model, we can
  obtain various predicted statistics using the
  predict command
• predict , xb
• predict , u
• predict , e
• predict , ue

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4.2 Linear regression (predict)

• For example
• xtreg lsat age , fe i(persnr)
• drop lsat_hat
• predict lsat_hat, xb
• predict lsat_u, u
• predict lsat_e, e
• predict lsat_ue, ue
• Checking that lsat_ue lsat_u lsat_e
• list lsat_u lsat_e lsat_ue
• Checking that the correlation between uit and Xit
  is -0.83
• corr lsat_hat lsat_u

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4.2 Linear regression

• I have explained that there are three main
  advantages of panel data
• The larger sample increases power, so the
  coefficients are estimated more precisely
• We can estimate models that incorporate dynamic
  variables (e.g., the effect of growth on audit
  fees)
• We can control for unobservable fixed effects
  (e.g., company-specific or person-specific
  characteristics) by estimating fixed-effects
  models.

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4.2 Linear regression

• Are there any disadvantages?
• Yes, unfortunately we cannot investigate the
  effect of explanatory variables that are held
  constant over time.
• From a technical point of view, this is because
  the time-invariant variable would be perfectly
  collinear with the person dummies.
• From an economic point of view, this is because
  fixed-effect models are designed to study what
  causes the dependent variable to change within a
  given person. A time-invariant characteristic
  cannot cause such a change.

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4.2 Linear regression

• For example, suppose that the height of the four
  persons is constant over the three years.
• Lets create a height variable and test the
  effect of height on life satisfaction
• gen height185 if dum_11
• replace height180 if dum_21
• replace height175 if dum_31
• replace height170 if dum_41
• list persnr height
• Note that the height variable is a constant for
  each person.
• We can estimate the effect of height as long as
  we do not control for unobservable
  person-specific effects
• reg lsat age height

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4.2 Linear regression

• If we try to control for person-specific effects
  by including dummy variables
• reg lsat age height dum_1 dum_2 dum_3 dum_4,
  nocons
• Note that STATA has to throw away either a dummy
  variable or the height variable.
• The reason is that the height variable is
  collinear with the four dummy variables.
• The only way we can include dummies for each
  person is if we do not include the height
  variable.
• reg lsat age dum_1 dum_2 dum_3 dum_4, nocons
• If we try to estimate the effect of height using
  the xtreg, fe i() command, STATA will
  inform us that there is a problem of perfect
  collinearity
• xtreg lsat age height, fe i( persnr)

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4.2 Linear regression

• Note that the height coefficient can be estimated
  if there is some variation over time for one or
  more persons.
• The fixed-effects estimator can exploit this time
  variation to estimate the effect of height on
  life satisfaction.
• For example, suppose that each person became 1cm
  taller in 1970.
• replace height height1 if time1970
• xtreg lsat age height, fe i( persnr)

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• The xtreg, fe i() command estimates the following
  fixed-effects model
• Recall that we derived this model by taking
  averages
• The averages model is sometimes called the
  between estimator because the comparison is
  cross-sectional between persons rather than over
  time.
• Like OLS, the between estimator provides unbiased
  estimates of ? only if the unobservable
  company-specific component (ui) is uncorrelated
  with Xit
• If we wanted to estimate the between effects
  model, the command in STATA is xtreg , be i()
• xtreg lsat age, be i( persnr)

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• Note that the age coefficient is positive
• the reason is that we are not controlling for
  person-specific effects, which are correlated
  with age.
• therefore, the between-effects estimate of the
  age coefficient is biased.
• Since we are estimating a between-effects model,
  it is the between R2 that is relevant (88.4).
• Note that this is also the between-effects R2
  that was previously reported using the
  fixed-effects model.
• Note that the R2 for the between-effects model is
  high despite that the age coefficient is severely
  biased. Again, this reinforces the fact that a
  high R2 does not imply that the model is well
  specified.

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• The between estimator is also less efficient than
  simple OLS because it throws away all the
  variation over time in the dependent and
  independent variables.
• In fact the between estimator is equivalent to
  estimating an OLS model on the averages for just
  one year
• Recall that we have already created averages for
  the lsat and age variables (avlsat avage)
• reg avlsat avage if time1968
• reg avlsat avage if time1969
• reg avlsat avage if time1970
• xtreg lsat age height, be i( persnr)
• Since we actually have three years of data, it
  seems silly (and it is silly) to throw data away

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4.2 Linear regression (xtreg)

• Normally, then, we would never be interested in
  estimating a between-effects model
• The estimates are biased if the person-specific
  effects are correlated with the X variables
• The estimates are inefficient because we are
  ignoring any time-series variation in the data
• The fixed effects estimator is attractive because
  it controls for any correlation between ui and
  Xit
• An unattractive feature is that it is forced to
  estimate a fixed parameter for each person or
  company in the data
• you can think of these parameters as being the
  coefficients on the person dummy variables

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4.2 Linear regression (xtreg)

• An alternative is the random effects model in
  which the ui are assumed to be randomly
  distributed with a mean of zero and a constant
  variance (ui IID(0, ?2u) rather than fixed.
• Intuitively, the random effects model is like
  having an OLS model where the constant term
  varies randomly across individuals i.
• Like simple OLS, the random effects model assumes
  that there is zero correlation between ui and Xit
• If ui and Xit are correlated, the random-effects
  estimates are biased.

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4.2 Linear regression (xtreg)

• The random-effects model can be thought of as an
  intermediate case of OLS and the fixed-effects
  model

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4.2 Linear regression (xtreg)

• The OLS model corresponds to ? 0.
• The fixed-effects model corresponds to ? 1.
• The random-effects model (0 ? ? ? 1) is also
  known as the generalized least squares model
  (i.e., it is more general than OLS or the
  fixed-effects model).

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4.2 Linear regression (xtreg)

• If we want to estimate a random effects model,
  the command is xtreg , re i()
• For example
• xtreg lsat age, re i( persnr)
• Note that because we have controlled for (random)
  unobserved person effects, the age coefficient is
  estimated with the correct negative sign.

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• The rest of the output is similar to the
  fixed-effects model except
• We use a Wald statistic instead of an F statistic
  to test the significance of the independent
  variables. Here we can reject the hypothesis that
  age is insignificant.
• The Wald statistic is used because only the
  asymptotic properties of the random-effects
  estimator are known.
• The output explicitly tells us that we have
  imposed the assumption that ui and Xit are
  uncorrelated.
• This is the key difference between the
  random-effects and fixed-effects models.

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• We can test whether ui and Xit are correlated.
• If they are correlated, we should use the
  fixed-effects model rather than OLS or the
  random-effects model (otherwise the coefficients
  are biased).
• If they are not correlated, it is better to use
  the random-effects model (because it is more
  efficient).
• The test was devised by Hausman
• if ui and Xit are correlated, the random-effects
  estimates are biased (inconsistent) while the
  fixed-effects coefficients are unbiased
  (consistent)
• In this case, there will be a large difference
  between the random-effects and fixed-effects
  coefficient estimates
• if ui and Xit are uncorrelated, the
  random-effects and fixed-effects coefficients are
  both unbiased (consistent) the fixed-effects
  coefficients are inefficient while the
  random-effects coefficients are efficient.
• In this case, there will not be a large
  difference between the random-effects and
  fixed-effects coefficient estimates
• The Hausman test indicates whether the two sets
  of coefficient estimates are significantly
  different

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• Null hypothesis (H0) ui and Xit are uncorrelated
• The Hausman statistic is distributed as chi2 and
  is computed as
• If the chi2 statistic is positive and
  statistically significant, we can reject the null
  hypothesis. This would mean that the
  fixed-effects model is preferable because the
  coefficients are consistent.
• If the chi2 statistic is not positive and
  statistically significant, we cannot reject the
  null hypothesis. This would mean that the
  random-effects model is preferable because the
  coefficients are consistent and efficient.
• NB The (Vc-Ve)-1 matrix is guaranteed to be
  positive only asymptotically. In small samples,
  this asymptotic result may not hold in which case
  the computed chi2 statistic will be negative.

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4.2 Linear regression (estimates store, hausman)

• The procedure for executing a Hausman test is as
  follows
• Save the coefficients that are consistent even if
  the null is not true
• xtreg lsat age, fe i( persnr)
• estimates store fixed_effects
• Save the coefficients that are inconsistent if
  the null is not true
• xtreg lsat age, re i( persnr)
• estimates store random_effects
• The command for the Hausman test is
• hausman name_consistent name_efficient
• hausman fixed_effects random_effects

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• b is the fixed-effects coefficient while B is the
  random-effects coefficient.
• The (Vc-Ve)-1 matrix has a negative value on the
  leading diagonal and, as a result, the square
  root of the leading diagonal is undefined. This
  is why the Chi2 statistic is negative.
• Since the Chi2 statistic is not significantly
  positive, we might decide that we cannot reject
  the null hypothesis (see p. 57 of the STATA
  reference manual for the Hausman test).
• On the other hand, this result is not very
  reliable because the asymptotic assumption fails
  to hold in this small sample.

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• If we reject the null hypothesis that ui and Xit
  are uncorrelated, the fixed-effects model is
  preferable to the OLS and random-effects models.
• If we cannot reject the null hypothesis that ui
  and Xit are uncorrelated, we need to determine
  whether the ui are distributed randomly across
  individuals.
• Recall that the random-effects model is like
  having an OLS model where the constant term
  varies randomly across individuals i.
• Therefore, we need to test whether there is
  significant variation in ui across individuals.

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• rho ?u2 / (?u2 ?e2)
• 1.032 / (1.032 0.472) 0.83
• ?u2 captures the variation in ui across
  individuals.
• If ?u2 is significantly positive, the
  random-effects model is preferable to the OLS
  model.
• The Breusch and Pagan (1980) Lagrange multiplier
  test is used to investigate whether ?u2 is
  significantly positive.

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• We perform the Breusch-Pagan test by typing
  xttest0 after xtreg, re
• Our estimate of ?u2 is 1.067 (note that ?u is
  estimated to be 1.032 which is the same as
  sigma_u on the previous slide).
• We are unable to reject the hypothesis that ?u2
  0. Therefore, we cannot conclude that the
  random-effects model is preferable to the OLS
  model.
• NB Our Hausman and LM tests lack power because
  the sample consists of only 12 observations. In
  larger samples, we are more likely to reject the
  hypothesis that ?u2 0 and we are more likely to
  reject the hypothesis that ui and Xit are
  uncorrelated.

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Class exercise 4b

• Estimate models in which the dependent variable
  is the log of audit fees.
• Estimate the models using
• OLS without controlling for ui
• Fixed-effects models
• Random-effects models
• How do the coefficient estimates vary across the
  different models?
• Which of these models is preferable?

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Class exercise 4b

• The lnta coefficients are largest in the OLS
  model that does not control for ui
• The lnta coefficients are smallest in the
  fixed-effects model
• The Hausman test rejects the hypothesis that ui
  and Xit are uncorrelated. Therefore, the
  fixed-effects model is preferable.
• The LM test rejects the hypothesis that ?u2 0
  (given that ui and Xit are significantly
  correlated, we would not actually need to carry
  out this test).

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Class exercise 4b

• use "C\phd\Fees.dta", clear
• gen fyedate(yearend, "mdy")
• format fye d
• gen yearyear(fye)
• sort year
• gen lnafln(auditfees)
• gen lntaln(totalassets)
• reg lnaf lnta
• xtreg lnaf lnta, fe i(companyid)
• estimates store fixed_effects
• xtreg lnaf lnta, re i(companyid)
• estimates store random_effects
• hausman fixed_effects random_effects
• xttest0

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4.2 Linear regression

• Compared to economics and finance, there are not
  many accounting studies that exploit panel data
  in order to control for unobserved
  company-specific effects (ui).
• Most studies simply report OLS estimates on the
  pooled data.
• Some studies even fail to adjust the OLS standard
  errors for time-series dependence
• this can be a very serious mistake especially
  when the panels are long (e.g., the sample period
  covers many years).
• If you use the xtreg command, STATA automatically
  recognizes that you are using panel data and it
  will give you the correct standard errors.
• Therefore, there is no need to use the robust
  cluster() option and, in fact, there is no robust
  cluster() option with xtreg
• xtreg lnaf lnta, fe i(companyid) robust
  cluster(companyid)

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4.2 Linear regression

• Ke and Petroni (2004) is an example of an
  accounting study that estimates fixed-effects
  regressions to control for unobservable
  company-specific effects.
• Their dependent variable is the change in the
  ownership of institutional investors in
  companies.
• They test whether there are significant changes
  in institutional ownership prior to a break in a
  string of consecutive quarterly earnings
  increases.
• Bhattacharya et al. (2003) is an example of an
  accounting study that estimates fixed-effects
  regressions to control for unobservable
  country-specific effects.
• Their dependent variable is the cost of equity
  for 34 countries between 1984-1998 (they are
  using a cross-country panel)
• They test how earnings opacity at the country
  level affects the cost of equity
• They acknowledge that there is a potentially
  serious problem of omitted variable bias

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• Bhattacharya et al. (2003) argue that they
  largely avoid this problem because they control
  for fixed country-specific effects

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4.2 Linear regression

• It is important to recognize that the fixed
  effects estimator relies only on the time-series
  variation in Y and X within a given company
• If the extent of time-series variation is small,
  either or will be close
  to zero.
• In this case, the fixed effects estimator is not
  reliable because there is insufficient variation
  in either the dependent or treatment variable.

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4.2 Linear regression

• As in any model, we require a reasonable amount
  of variation in the Y and X variables.
• If either variable displays little variation, the
  results may be unreliable.

• We saw an example of this previously.
• Except for one observation, the independent
  variable is a constant.
• As a result the fitted regression line is
  unreliable.

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4.2 Linear regression

• This point was made by Zhou (JFE, 2001) who
  criticized the use of fixed effects models when
  the treatment variable is management ownership.
• Because management ownership usually remains
  constant from one year to the next, the
  term is typically equal to zero (or very
  small).

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4.3 Logit and probit models

• When the dependent variable is continuous, it is
  easy to transform the model such that unobserved
  firm-specific effects are washed away
• When the dependent variable is binary, the
  required transformation is different and more
  complicated
• if you are interested in the derivation, see the
  Baltagi textbook (pages 178-180).
• in the fixed-effects logit, the fixed effects
  (ui) are not actually estimated, instead they are
  conditioned out of the model.
• the fixed-effects logit model is not equivalent
  to logit dummy variables.

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4.3 Logit models (xtlogit)

• We can estimate a fixed-effects logit model using
  the command xtlogit , fe i()
• NB Your version of STATA 9.0 may have a problem
  with estimating the fixed effects logit model.
  You can instead use version 8.0 or 10.0.
• version 8.0
• Before we estimate the fixed-effects logit model,
  we need to understand a complication that arises
  because the dependent variable is binary.

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• Suppose we have five annual observations on two
  companies.
• For company 1, there is no variation in the
  dependent variable over time (Y 0 in every
  year).
• A fixed effect for this company will perfectly
  predict the outcome (Y 0)
• Consequently, the first company will be dropped
  from the estimation sample.
• In fact, the fixed-effects logit model will drop
  all companies that exhibit no variation in the
  dependent variable over time.

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4.3 Logit models (xtlogit)

• use "C\phd\xtlogit.dta", clear
• list
• The sample consists of three companies.
• Company 1 exhibits no variation in the dependent
  variable over time while companies 2 3 do
  exhibit time-series variation.
• There is no problem estimating this model on the
  full sample if we do not control for fixed
  effects
• logit y x
• Running a fixed effects logit model results in
  the first company being thrown away
• xtlogit y x, fe i(id)

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4.3 Logit models (xtlogit)

• In many empirical settings, we are likely to find
  a large number of companies that exhibit no
  variation in the binary dependent variable during
  the sample period.
• Example 1
• Yit 1 if company i is engaged in fraud in year
  t Yit 0 otherwise.
• The vast majority of companies do not engage in
  fraud at any point in time (Yit 0 for all t).
• All such non-fraud companies would be dropped
  from the estimation sample.
• The estimation sample would include only the
  companies that commit fraud at some point during
  the sample period.

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4.3 Logit models (xtlogit)

• Example 2
• Yit 1 if company i hires a Big 6 auditor in
  year t Yit 0 if company i hires a non-Big 6
  auditor in year t.
• The vast majority of companies keep the same
  auditor in the following year and switches
  between Big 6 and non-Big 6 auditors are
  especially rare.
• All companies that do not switch between Big 6
  and Non-Big 6 auditors would be dropped from the
  sample.
• The estimation sample would include only the
  companies that switch between Big 6 and Non-Big 6
  auditors at some point during the sample period.

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4.3 Logit models (xtlogit)

• Alternatively, we can estimate a random-effects
  logit model using the command xtlogit , re i()
• The company effects (ui) are now assumed to be
  random rather than fixed.
• Consequently, the random effects model does not
  throw away companies that lack time-series
  variation in the dependent variable.
• For example
• xtlogit y x, re i(id)

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• The estimation sample is now 15 rather than 10
  (i.e., all 3 companies are included in the
  sample).
• lnsig2u ln(?u2) -1.625
• sigma_u ?u 0.444 exp(-1.625)0.5
• rho ?u2 / (?u2 ?e2) 0.056

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• If rho ?u2 / (?u2 ?e2) 0, there would be no
  variation in the ui across companies (i.e., each
  company would have the same ui).
• In this case, there would be no need to control
  for company-specific effects, i.e., we could rely
  on logit instead of estimating xtlogit , re i()
• The likelihood-ratio statistic tests the null
  hypothesis that rho equals zero.
• If we reject this hypothesis, the random effects
  model is preferable to ordinary logit.
• In our data, we are unable to reject, so we could
  use an ordinary logit model instead of the random
  effects logit model. This would be a good idea
  because the ordinary logit is more efficient
  (fewer parameters need to be estimated).

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4.3 Logit models (xtlogit)

• Recall that we previously used a Hausman test to
  determine whether the xtreg, fe i() or xtreg, re
  i() model is preferable.
• Fortunately, we can do the same test in STATA for
  deciding whether the fixed-effects or
  random-effects logit models are preferable.
• The only difference is that we have to use the
  equations() option with the Hausman test
• actually, this point is not explained in the
  STATA manual but a question and answer were
  posted about this topic on the statalist
  (www.stata.com/statalist/archive/2004-01/msg00669.
  html)
• the equations() option specifies, by number, the
  pairs of equations that are to be compared.
• usually, we are estimating just one equation in
  each model, in which case the option is
  equations(11)

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4.3 Logit models (xtlogit)

• For example
• xtlogit y x, fe i(id)
• estimates store fixed_effects
• xtlogit y x, re i(id)
• estimates store random_effects
• hausman fixed_effects random_effects
• STATA is telling us there is an error (we need to
  specify the equation numbers)
• hausman fixed_effects random_effects, eq(11)
• The Chi2 statistic is negative (again there is a
  small sample problem which causes the asymptotic
  assumption to fail).

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Class exercise 4c

• Open the fee.dta data set.
• Estimate models in which big6 is the dummy
  dependent variable using
• ordinary logit
• fixed-effects logit
• random-effects logit
• Why is the estimation sample much smaller in the
  fixed effects model?
• Which of the three models is most preferable?

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Class exercise 4c

• use "C\phd\Fees.dta", clear
• gen lntaln(totalassets)
• logit big6 lnta, robust cluster(companyid)
• xtlogit big6 lnta, fe i(companyid)
• estimates store fixed_effects
• xtlogit big6 lnta, re i(companyid)
• estimates store random_effects
• hausman fixed_effects random_effects, eq(11)
• The estimation sample is much smaller in the
  fixed effects model because the majority of
  companies do not switch between Big 6 and Non-Big
  6 auditors during the sample period.
• The likelihood ratio test of rho 0 indicates
  that the random-effects model is preferable to
  the ordinary logit.
• The Hausman test indicates that the fixed-effects
  model is preferable to the random-effects logit.

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4.3 Probit models (xtprobit)

• Recall that there are two commands available when
  the dependent variable is binary (ordinary
  logit and probit).
• There is no command for a fixed-effects probit
  model because no-one has yet found a
  transformation that will allow the fixed effects
  to be washed out.
• If you type xtprobit big6 lnta, fe i(companyid)
  you will get an error message.
• A random-effects probit model is available,
  however
• xtprobit big6 lnta, re i(companyid)
• Just as with the random-effects logit model,
  there is a likelihood ratio test that helps us to
  choose between the random-effects probit and the
  ordinary probit models.
• In our data, we can reject the hypothesis that
  rho 0, so we may decide not to use an ordinary
  probit model.

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4.4 Other models
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4.4 Other models

• If you look at the STATA manual for panel data
  (Cross-Sectional Time-series), you will find
• Fixed-effects and random-effects models are
  available for count data (xtpoisson and xtnbreg)
• We can test which model is preferable using a
  Hausman
• Random-effects models are available for censored
  data (xttobit and xtintreg)
• fixed-effects models are not available
• therefore there is no need for a Hausman test

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4.4 Other models

• Duration data is, by its very nature, in the form
  of panel data.
• What about the multinomial and ordered models
  that we previously looked at (mlogit, mprobit,
  ologit, oprobit)? It appears that STATA does not
  have random- or fixed-effects versions of these
  models.

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4.4 Other models

• You can use the search command in STATA to find
  out if a command is available.
• The search command looks through official STATA
  commands, frequently asked questions (on the
  STATA website), the STATA journal (SJ) and the
  STATA technical bulletins (STBs)
• The SJ and STBs are where you can sometimes find
  commands that will appear in future versions of
  STATA
• search multinomial logit
• We can find the multinomial logit command but
  there does not appear to be any command
  specifically for the multinomial model with panel
  data

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4.4 Other models

• Even if the command you want is not available
  from STATA, you may be able to find a STATA user
  who has already written the program that you
  need.
• Statalist (www.stata.com/statalist/) is an email
  listserver where over 2,500 Stata users discuss
  all things statistical and Stata.
• Click on Archives provided by Statacorp and
  search the archives

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4.4 Other models

• For example, suppose you want to estimate a
  random-effects ordered probit
• Typing this into the statalist archive I found
  that someone has written a program with this
  command (reoprob) www.stata.com/statalist/archive/
  2006-02/msg00509.html
• The message tells us we can download it to STATA
  by typing
• findit reoprob

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4.4 Other models

• If you cannot find someone who has already
  written the program and if it is a command that
  you really do need, you will either have to write
  the program yourself or wait for someone else to
  do it.
• In fact, it is not too difficult to learn how to
  write new programs in STATA
• you would need to take a STATA programming course
• www.stata.com/netcourse/
• net courses 151 152

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Summary

• There are three advantages to using panel data
• We can control for unobservable fixed effects
  that might otherwise bias the coefficient
  estimates.
• these unobservable fixed effects can be
  company-specific, country-specific, or
  person-specific.
• The larger sample means that the coefficients are
  estimated more precisely.
• We can include lagged or change variables in our
  models.

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Summary

• The xtreg command is used to estimate
  fixed-effects and random-effects models (where
  the dependent variable is continuous).
• We can test whether the fixed-effects or
  random-effects model is preferable using the
  hausman test.
• If there is a significant correlation between ui
  and Xit, the fixed effects model is preferable to
  the OLS and random effects models.
• If there is no significant correlation between ui
  and Xit, we can test whether the OLS or
  random-effects model is preferable using a LM
  test.

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Summary

• When the dependent variable is binary we can
  estimate fixed-effects or random-effects logit
  models.
• Again, we can test which model is preferable
  using a Hausman test.
• Only the random-effects model is available in the
  case of the probit model.