Managers' perceptions of product

Document Sample
Managers' perceptions of product Powered By Docstoc
					        4. Using panel data
 4.1 The basic idea
 4.2 Linear regression
 4.3 Logit and probit models
 4.4 Other models




                                1
                4.1 The basic idea
 Panel data = data that are pooled for     Company Year
  the same companies across time.
 In panel data, there are likely to be
                                            A       1996
  unobserved company-specific
  characteristics that are relatively
  constant over time.                       A       1997
 I have already explained that it is
  necessary to control for this time-
  series dependence in order to obtain      B       1996
  unbiased standard errors.
       In STATA we can do this using the   B       1997
        robust cluster () option


                                                           2
                   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 fye=date(yearend, "mdy")
        format fye %d
        gen year=year(fye)
        sort year
        gen lnaf=ln(auditfees)
        gen lnta=ln(totalassets)
        by year: reg lnaf lnta, robust cluster(companyid)
        reg lnaf lnta, robust cluster(companyid)

                                                                            3
                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 company’s 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.


                                                                      4
              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
   Let’s start with a simple regression model:


   Let’s 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:



                                                            5
               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

                                                              6
               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)

                                                                    7
                 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.

                                                                             8
              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
 What’s going on here?


                                                       9
                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)




                                                                 10
               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)

                                                                 11
12
 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)
                                                                13
14
                 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


                                                         15
                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.
                                                                16
           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!).

                                                      17
                 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 .
                                                                     18
                  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 .


                                                                         19
              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)




                                                          20
                   Class exercise 4a
   You should find that the age coefficients are exactly the same.
   First, we create the variables:
        sort persnr time
        gen chlsat=lsat-lsat[_n-1] if persnr==persnr[_n-1]
        gen chage=age-age[_n-1] if persnr==persnr[_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 avlsat=mean(lsat)
        by persnr: egen avage=mean(age)
        gen difflsat=lsat-avlsat
        gen diffage=age-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

                                                                        21
    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)
                                                            22
23
   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.




                                                                24
   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.
                                                           25
 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




                                              26
 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.

                                                   27
 4.2 Linear regression (predict)
      running the fixed-effects model, we
 After
 can obtain various predicted statistics
 using the predict command


     predict   , xb
     predict   ,u
     predict   ,e
     predict   , ue

                                            28
    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



                                                             29
             4.2 Linear regression
Ihave 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.

                                                     30
            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.

                                                               31
               4.2 Linear regression
   For example, suppose that the height of the four persons
    is constant over the three years.
   Let’s create a height variable and test the effect of height
    on life satisfaction
       gen height=185 if dum_1==1
       replace height=180 if dum_2==1
       replace height=175 if dum_3==1
       replace height=170 if dum_4==1
       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

                                                              32
              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)

                                                               33
            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= height+1 if time==1970
       xtreg lsat age height, fe i( persnr)


                                                   34
   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)


                                                                 35
36
   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.

                                                                            37
   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 time==1968
       reg avlsat avage if time==1969
       reg avlsat avage if time==1970
       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



                                                             38
     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


                                                                  39
    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.

                                                     40
     4.2 Linear regression (xtreg)
   The random-effects model can be thought of as an
    intermediate case of OLS and the fixed-effects model:




                                                            41
     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).


                                                              42
    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.

                                                            43
   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.



                                                                       44
   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
                                                                                45
   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.
                                                                         46
               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
                                                                47
   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.
                                                                              48
 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.
                                                    49
   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.

                                50
 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.

                                              51
             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?

                                            52
            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).
                                                    53
            Class exercise 4b
   use "C:\phd\Fees.dta", clear
   gen fye=date(yearend, "mdy")
   format fye %d
   gen year=year(fye)
   sort year
   gen lnaf=ln(auditfees)
   gen lnta=ln(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


                                           54
             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)

                                                                        55
              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
                                                                           56
   Bhattacharya et al. (2003) argue that they largely avoid
    this problem because they control for fixed country-
    specific effects




                                                               57
           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.

                                                        58
               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.
                                                                      59
           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).




                                                           60
        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.


                                                                              61
         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.


                                                                 62
   Suppose we have five annual               Id   Year   Y
    observations on two companies.            1    2000   0
                                              1    2001   0
   For company 1, there is no variation
    in the dependent variable over time       1    2002   0
    (Y = 0 in every year).                    1    2003   0

   A fixed effect for this company will      1    2004   0
    perfectly predict the outcome (Y = 0)     2    2000   1
                                              2    2001   1
   Consequently, the first company will
    be dropped from the estimation            2    2002   0
    sample.                                   2    2003   0

   In fact, the fixed-effects logit model    2    2004   1
    will drop all companies that exhibit no
    variation in the dependent variable
    over time.

                                                              63
            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)

                                                     64
          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.



                                                                       65
         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.

                                                                 66
          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)


                                               67
   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

                                                                  68
   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).


                                                                               69
          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(1:1)

                                                                       70
         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(1:1)
   The Chi2 statistic is negative (again there is a
    small sample problem which causes the
    asymptotic assumption to fail).
                                                        71
               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?


                                                           72
                 Class exercise 4c
       use "C:\phd\Fees.dta", clear
       gen lnta=ln(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(1:1)
   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.
                                                            73
        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.

                                                                        74
                         4.4 Other models
Dependent              Examples                               Estimation            STATA
variable (Y)                                                  method(s)
Discrete and           Method of transport
unordered              (train, bus, car, bicycle)             Multinomial logit     mlogit
(Y = 0, 1, 2,..)       Type of company                        Multinomial probit    mprobit
                       (private, public unquoted, quoted)
Discrete and ordered   Type of peer review report (adverse,   Ordered probit        oprobit
(Y = 0, 1, 2,..)       modified, unmodified)                  Ordered logit         ologit

Discrete count data    Number of weaknesses disclosed in      Poisson               poisson
(Y = 0, 1, 2, …)       peer review report                     Negative binomial     nbreg


Continuous and
censored               Non-audit fees                         Tobit                 tobit
(kL  Y < kH)          Football attendance                    Interval regression   intreg

Duration data          Duration of unemployment
(often censored)       CEO tenure                             Cox proportional      stcox
kL  Y < kH            Company survival                       hazards
                                                                                         75
                 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



                                                                   76
           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.

                                                  77
                 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
                                                              78
            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

                                                     79
                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


                                                        80
                 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
                                                       81
                         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.



                                                                       82
                        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.

                                                                83
                  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.



                                                     84

				
DOCUMENT INFO