Objectives of this class

About This Presentation

Title:

Objectives of this class

Description:

bicycle, car, bus, train, walk (here there are five categories ) ... Credit rating agencies assess the credit worthiness of companies ... – PowerPoint PPT presentation

Number of Views:51

Avg rating:3.0/5.0

Slides: 158

Provided by: accl5

Category:

more less

Transcript and Presenter's Notes

Title: Objectives of this class

1
Objectives of this class

By the end of the class you should be able to
Explain why OLS should not be used when the
dependent variable is discrete
Understand and use logit and probit models
Use multinomial, ordered and count-data models in
the correct situations
Implement tobit and interval regression models,
explaining why OLS should not be used when the
dependent variable is truncated
Handle duration data and estimate the Cox
proportional hazards model

2
3. When the dependent variable is not continuous
and unbounded

3.1 Why not OLS?
3.2 The basic idea underlying logit models
3.3 Estimating logit models
3.4 Multinomial models
3.5 Ordinal dependent variables
3.6 Count data models
3.7 Tobit models and interval regression
3.8 Duration models

3
3.1 Why not OLS?

A variable is categorical if it takes discrete
values.
For example, a dummy variable is categorical
because it takes two possible values (one, or
zero).
In some situations, we may want to estimate a
model in which the dependent variable is
categorical.
For example, we may want to know why some
companies choose large audit firms while other
companies choose small audit firms
The dependent variable (big6) is categorical as
it takes two discrete values, zero or one.
Why should we not use OLS to estimate the model
when the dependent variable is categorical?

4
3.1 Why not OLS?

We believe that company size is an important
determinant of the choice of auditor.
Suppose we try to graph the relationship between
big6 and company size.
The observations lie only on two horizontal lines
(where big60 and big61)
If larger companies are more likely to choose
big6 auditors, the number of observations on the
1-line should be further to the right than the
number on the 0-line
use "C\phd\Fees.dta", clear
gen lntaln(totalassets)
scatter big6 lnta, msize(tiny)

5
3.1 Why not OLS?

This graph is not very informative because the
observations lie directly on top of each other,
hiding the number of observations.
You can use the jitter() option to provide a more
informative graph.
jitter() adds a small random number to each
observation, thus showing observations that were
previously hidden under other data points. The
number in brackets is from 1 to 30 and controls
the size of the random number
scatter big6 lnta, msize(tiny) jitter(30)
graph twoway (scatter big6 lnta, msize(tiny)
jitter(30)) (lfit big6 lnta)

6
3.1 Why not OLS?

The graph shows one major problem with using
linear regression for dichotomous dependent
variables
the predicted values of big6 can be lt 0 or gt 1,
yet according to the mathematical definition of a
probability, the big6 probability should lie
between 0 and 1,
given sufficiently small or large values of X, a
model that uses a straight line to represent
probabilities will inevitably produce values that
are negative or greater than one.

7
3.1 Why not OLS?

A second major problem is that linear regression
provides errors that do not have a constant
variance
This violates the assumption of OLS that the
errors are homoscedastic. For example
The residuals are

8
3.1 Why not OLS?

The variance of the residuals is
The variance of the residuals is larger as the
predicted values approach 0.5.
Since the variance of the residuals is a function
of the predicted values, the residuals do not
have a constant variance.
Because of this heteroscedasticity, the standard
errors of the coefficients are biased.

9
Class exercise 3a

Using the regress command, estimate an OLS model
where the dependent variable is big6 and the
independent variable is lnta.
Using the predict command, obtain the predicted
big6 values and the predicted residuals.
Draw a scatterplot of the residuals against the
predicted values of big6.
Do you notice any pattern between the residuals
and the fitted values?
Why does this pattern exist?

10
3.1 Why not OLS?

To summarize, we would have two statistical
problems if we use OLS when the dependent
variable is categorical
Not all the predicted values can be interpreted
(i.e., they can be negative or greater than one)
The standard errors are biased because the
residuals are heteroscedastic.
Instead of OLS, we can use a logit model

11
3.2 The basic idea underlying logit models

The values calculated with linear regression are
not subject to any restrictions, so any values
between -? and ? may emerge.
We need to create a variable that
has an infinite range,
reflects the likelihood of choosing a big6
auditor versus a non-big6 auditor.
This variable is known as the log odds ratio.

The odds ratio is as follows
The log odds ratio is obtained by taking
natural logs of the odds ratio.

13
(No Transcript)
14
3.2 The basic idea

Col. 1 shows the probability of a company
choosing a big6 auditor
Note that the probabilities lie between 0 and 1.
Cols. 2 3 shows the odds ratios
Note that the odds ratios lie between 0 and ?.
Using the odds ratio solves the problem that the
linear predicted values may exceed 1.
However, we still have the problem that the
linear predicted values may be negative
We solve this problem by using the natural log of
the odds ratio (which are also called logits)
Col. 4 shows the logits
Note that the logits lie between -? and ?.
As the big6 probability approaches zero, the
logits approach -?.
As the big6 probability approaches one, the
logits approach ?.
Note also that the logits are symmetric (e.g.,
when the big6 probability is 0.5, the logit
zero when the big6 probability is 0.4, the logit
-0.41 when the big6 probability is 0.6, the
logit 0.41).

15
3.2 The basic idea

The logit can take any value between -? and ?,
and it is symmetric.
The logit is therefore suitable for use as a
dependent variable.
Writing the logit (i.e., the log of the odds
ratio) as L, it is easy to transform the logits
back into probabilities

16
3.2 The basic idea

The logit model uses a linear combination of
independent variables to predict the values of
the logit, L.
L a0 a1 X1 a2 X2 e
There is a one-to-one mapping between values of
the continuous L variable and values of the dummy
variable
big6 1 if 0 lt L lt ?
big6 0 if -? lt L ? 0

17
3.2 The basic idea

The interpretation of the coefficients is the
same as in OLS
L a0 a1 X1 a2 X2 e
For example when X1 increases by one unit, the
predicted values of the logit (L) increase by a1
The coefficients for the logit model are
estimated by maximizing the likelihood function.

The likelihood function is as follows
The coefficients (a0 a1 X1 a2 X2) are
estimated such that they maximize the value of
the likelihood function.
Unlike OLS, there is no analytical solution to
characterize formulas for the estimated
coefficients.
Instead, the likelihood function is maximized
using iterative algorithms (this is known as
maximum likelihood estimation).

19
3.3 Estimating logit models and interpreting the
results

There are two commands in STATA for estimating
the logit model where the dependent variable is
binary
logit reports the values of the estimated
coefficients
logistic reports the odds ratios
Typically, accounting researchers report the
coefficient estimates rather than the odds
ratios, so we will be using the logit command.

20
Example

Suppose we wish to test the effect of the
companys age and size on its choice between a
big6 or non-big6 auditor
gen fyedate(yearend, "mdy")
format fye d
gen yearyear(fye)
gen age year-incorporationyear
sum age, detail
replace age0 if agelt0
logit big6 lnta age
In many respects, the output from the logit model
looks similar to what we obtain from OLS
regression.

21
3.3 Estimating logit models
22

The coefficient on lnta tells us that the log
odds of hiring a big6 auditor increase by 0.58 if
lnta increases by one unit.
Usually, we are mainly interested in the signs
and statistical significance of the coefficients
We find that larger companies and younger
companies are significantly more likely to hire
big6 auditors.

When using maximum likelihood estimation, there
is typically no closed-form mathematical solution
to obtain the coefficient estimates.
Instead, an iterative procedure must be used that
tries a sequence of different coefficient values.
As the algorithm gets closer to the solution, the
value of the likelihood function increases by
smaller and smaller increments.

The first and last values of the likelihood
function are of most interest. The larger the
difference between these two values, the greater
the explanatory power of the independent
variables in explaining the dependent variable.
The pseudo-R2 is similar to the R-squared in the
OLS model as it tells you how high is the models
explanatory power.
pseudo-R2 (ln(L0) - ln(LN)) / ln(L0)
(-175224146215) / -175224

Besides the pseudo-R2, the likelihood-ratio is
another indicator of the models explanatory
power
Chi2 -2(ln(L0) - ln(LN)) -2(-175224146215)
58018
As with the F value in linear regression, you can
use the likelihood-ratio statistic to test the
hypothesis that the independent variables have no
explanatory power (i.e., all coefficients except
the intercept are zero).
The probability that this hypothesis is true is
reported in the line Prob gt chi2. In our
example we can reject this hypothesis because the
probability is virtually zero.

26
3.3 Estimating logit models

Just as with OLS, we can use the robust option to
correct for any heteroscedasticity and we can use
the cluster() option to control for correlated
errors
logit big6 lnta age
logit big6 lnta age, robust
logit big6 lnta age, robust cluster(companyid)

27
3.3 Estimating logit models

The predict command generates a new variable that
contains the predicted probability of choosing a
big6 auditor for every observation in the sample
logit big6 lnta age, robust cluster(companyid)
drop big6hat
predict big6hat
sum big6hat, detail
Note that these predicted probabilities lie
within the range 0, 1

28
3.3 Estimating logit models

Note that the predicted probabilities are not the
same as the predicted logit values
gen big6hat1 _b_cons_blntalnta
_bageage
sum big6hat1, detail
The predicted logit values lie in the range -? to
?.
We can easily obtain the predicted probabilities
using the logit values
replace big6hat1exp(big6hat1)/(1exp(big6hat1))
sum big6hat big6hat1

29
3.3 Estimating logit models

Alternatively, we can predict the logit values
using the ,xb option
drop big6hat big6hat1
logit big6 lnta age, robust cluster(companyid)
predict big6hat
predict big6hat1, xb
sum big6hat1, detail
replace big6hat1exp(big6hat1)/(1exp(big6hat1))
sum big6hat big6hat1

30
3.3 Estimating logit models

Just as with OLS models, we can report the
economic significance of the coefficients.
For example, we can calculate the change in the
predicted probability of hiring a big6 auditor as
the companys age increases from 10 to 20 years
old
logit big6 lnta age, robust cluster(companyid)
gen big10 exp(_b_cons_blntalnta
_bage10) / (1(exp(_b_cons_blntalnta
_bage10)))
gen big20 exp(_b_cons_blntalnta
_bage20) / (1(exp(_b_cons_blntalnta
_bage20)))

31
Class exercise 3b

Calculate the change in the predicted probability
of hiring a big6 auditor as the companys age
increases by one standard deviation around the
mean
Hint remember that you can obtain the mean and
standard deviations using the sum command and the
return codes r()
To see the list of return codes, type return list

32
3.3 Estimating logit models

We have been assuming that assets has a monotonic
log-linear relationship with the log of the odds
ratio
We can check the validity of assuming a
log-linear relation by creating dummy variables
for each size decile
If the correlation between assets and auditor
choice is log-linear and monotonic, the lnta
coefficients should increase continuously
xtile lnta_categlnta, nquantiles(10)
tabulate lnta_categ, gen (lnta_)
logit big6 lnta_2- lnta_10 age, robust
cluster(companyid)
Notice that the lnta coefficients are
montonically increasing
therefore an increase in size increases the
probability of hiring a big 6 auditor for each
size decile

33
3.3 An alternative to logit

In the logit model, we predict P(Y 1) using a
linear combination of X variables
To ensure the predicted P(Y 1) values lie
between 0 and 1, we used a logit transformation
An alternative is the probit transformation,
which is used in probit models
The main difference is that the logit uses a
logarithmic distribution whereas the probit uses
a normal distribution

34
3.3 An alternative to logit

In the logit model, the likelihood function is
where
In the probit model, the likelihood is
where ? is the cumulative normal distribution
function

35
3.3 An alternative to logit

The coefficients of the probit model are also
estimated using maximum likelihood
Usually, the predicted probabilities of probit
models are very close to those of logit models
The coefficients tend to be larger in probit
models but the levels of statistical significance
are often similar
capture drop big6hat big6hat1
logit big6 lnta age, robust cluster(companyid)
predict big6hat
probit big6 lnta age, robust cluster(companyid)
predict big6hat1
pwcorr big6hat big6hat1

36
3.4 Multinomial models

Multinomial models are used when
the dependent variable takes on three or more
categories and
the categories are not ranked
For example, the dependent variable might be your
method of transport to university
bicycle, car, bus, train, walk (here there are
five categories )
there is no particular ranking from best to
worst (bicycle and walking may be cheaper and
more healthy but car may be quicker it may not
be obvious whether the car or train is quicker
and cheaper so these choices cannot be ranked)

37
3.4 Multinomial models

In our dataset the companytype variable has six
different categories
Suppose we are interested in three categories
private, public, and publicly traded
companytype 1, 6 if private company,
companytype 4 if public but not traded on a
market,
companytype 2, 3, 5 if company is publicly
traded on a market.
gen cotype10 if companytype1 companytype6
replace cotype11 if companytype4
replace cotype12 if companytype2
companytype3 companytype5
Our dependent variable (cotype1) now has three
possible values
In a multinomial model, we predict the
probability of each of these three outcomes

38
3.4 Multinomial models

Alternatively, you could create a binary variable
for each of the three categories
gen private0
replace private1 if cotype10
gen public_nontraded0
replace public_nontraded1 if cotype11
gen public_traded0
replace public_traded1 if cotype12
And then estimate logit models using each binary
variable as the dependent variable
logit private lnta, robust cluster(companyid)
predict private_hat
logit public_nontraded lnta, robust
cluster(companyid)
predict public_nontraded_hat
logit public_traded lnta, robust
cluster(companyid)
predict public_traded_hat

39
3.4 Multinomial models

A problem with this approach is that the
predicted probabilities from the three logit
models do not sum to one
They should sum to one because there are only
three categories (private, public non-traded,
public traded)
gen sum_prob private_hat public_nontraded_hat
public_traded_hat
sum sum_prob, detail
This problem arises because the three logit
models are estimated in an unconnected way
Instead we need to estimate the models jointly
such that the predicted probabilities sum to one

40
3.4 Multinomial models

Recall that when the dependent variable has two
possible outcomes (e.g., big6 1, 0), there is
one equation estimated
The observations where big6 0 are used as a
benchmark for evaluating why companies choose
big6 auditors

41
3.4 Multinomial models

Similarly, when the dependent variable has three
possible outcomes (cotype1 0, 1, 2), there are
two equations estimated.
One of the three outcomes is used as a benchmark
for evaluating what determines the other two
outcomes.
More generally, if the dependent variable has N
possible outcomes (cotype1 0, 1, 2, , N),
there are N-1 equations estimated.
It does not matter which outcome we choose to be
the benchmark.
By default, STATA chooses the most frequent
outcome as the benchmark, but you can override
this if you wish.

42
3.4 Multinomial models (mlogit)

The STATA command for the multinomial logit model
is mlogit
In early versions of STATA (e.g., STATA 8) there
was no option to estimate a multinomial probit
model. A multinomial probit model is now
available in STATA 9 10 (mprobit).
The maximum likelihood algorithms for the
multinomial probit model are complicated. As a
result, the multinomial probit can be
time-consuming to estimate especially when the
dependent variable has several categories or the
sample is large.
mprobit cotype1 lnta, robust cluster(companyid)
Because mprobit is so time-consuming, I am going
to stick with mlogit for the sake of
demonstration.
mlogit cotype1 lnta, robust cluster(companyid)

43
3.4 Multinomial models (mlogit)

There are now two sets of coefficient estimates
The first set contains the coefficients of the
equation for public non-traded companies
(cotype11)
The second set contains the coefficients of the
equation for public traded companies (cotype12)
The coefficients of the equation for private
companies (cotype10) are set at zero, because
STATA chose this to be the benchmark group.

44
3.4 Multinomial models (mlogit)

The coefficients need to be interpreted
carefully because private companies comprise the
benchmark group, .
The results show that larger companies are
significantly more likely to be in the public
non-traded category than in the private category
(i.e., 1 vs. 0).
Also, larger companies are significantly more
likely to be in the public traded category than
in the private category (i.e., 2 vs. 0).
Suppose we wish to test whether larger companies
are significantly more likely to be in category 2
(public traded) versus category 1 (public
non-traded)

45
3.4 Multinomial models (test, basecategory())

After running the mlogit command, we can test
whether the coefficients in the two equations are
equal
test Equation no. Equation no. Variable name
mlogit cotype1 lnta, robust cluster(companyid)
test 12 lnta
test 12 _cons
The results indicate that larger companies are
significantly more likely to be in category 2
(public traded) than in category 1 (public
non-traded).
Having performed this test it is now valid to
conclude that larger companies are significantly
more likely to be in the public traded category
(i.e., we have compared 2 vs. 1 and 2 vs. 0)
We can easily change the benchmark comparison
group using the , basecategory() option
mlogit cotype1 lnta, basecategory(1) robust
cluster(companyid)

46
Class exercise 3c

Estimate the multinomial model using outcome 2
(public traded) as the base category.
Test whether the lnta coefficients are the same
for private versus public non-traded companies.
Why are the signs of the lnta coefficients
negative whereas they were positive when outcome
0 is the base category?
Does the negative lnta coefficient for outcome 1
imply that larger companies are less likely to be
public non-traded?

47
3.5 Ordinal dependent variables

Multinomial models are used when the values of
the dependent variable do not have an ordinal
ranking
For example, it does not make economic sense to
rank public traded companies higher or lower than
private companies.
The values of cotype1 are simply used to identify
different types of company.
Therefore, we use the multinomial logit model
when the dependent variable is cotype1.

48
3.5 Ordinal dependent variables

In other cases, it may make sense for the
dependent variable to have an ordinal ranking
For example, a professor marks an exam taken by
five students and ranks the students in order of
their marks
1 top, 2 second place, , 5 bottom of the
class
Ordinal dependent variables are common when
researchers are using survey data about peoples
perceptions
How concerned are you about crime in Guangzhou?
1 very concerned, 2 quite concerned, 3 not
concerned.
Credit rating agencies assess the credit
worthiness of companies
Moodys rating scales AAA, Aa1, Aa2, Aa3, A1,
A2, A3, Baa1, Baa2, Baa3, Ba1, Ba2, Ba3, B1, B2,
B3, Caa1, Caa2, Caa3, Ca, C
Blume et al. (1998) use an ordered probit model
to examine whether rating agencies are using more
stringent standards in assigning ratings.

49
3.5 Ordinal dependent variables

Recall that in the binary logit model there is a
one-to-one mapping between values of the
continuous L variable and values of the dummy
variable (big6)
L a0 a1 X1 a2 X2 e
big6 1 if 0 lt L lt ?
big6 0 if -? lt L ? 0
The zero is basically a cut-off point for mapping
the transformation of the observed dummy variable
(big6) to the unobserved latent variable (L)
The underlying logit score (L) is estimated as a
linear function of the X variables using the
cut-off of zero for the two values of big6.

50
3.5 Ordinal dependent variables

Similarly, in the ordered logit model, there is a
one-to-one mapping between values of the
continuous L variable and values of the ordinal
dependent variable.
Unlike the binary logit model, there are at least
three values for the dependent variable.
For example, when the dependent variable takes
three possible values (Y 1, 2, 3), we have two
cut-off values (k1 and k2)
L a0 a1 X1 a2 X2 e
Y 3 if k2 lt L lt ?
Y 2 if k1 lt L ? k2
Y 1 if -? lt L ? k1
The probability of observing outcome Y 1, 2 or
3 corresponds to the probability that the
estimated linear function plus random error (a0
a1 X1 a2 X2 e) lies within the range of the
cut-off points estimated.

51
3.5 Ordinal dependent variables

More generally, the ordered dependent variable
may take N possible values (Y 1, 2, , N) in
which case there are N-1 cut-off points
L a0 a1 X1 a2 X2 e
Y N if kN-1 lt L lt ?
Y N-1 if kN-2 lt L ? kN-1
....etc.
Y 2 if k1 lt L ? k2
Y 1 if -? lt L ? k1

52
3.5 Ordinal dependent variables

I have a paper that uses data on the opinions
that peer reviewers issue to audit firms (Hilary
and Lennox, 2005).
Audit firms in the U.S. are now regulated
independently by the Public Company Accounting
Oversight Board whereas they are previously
self-regulated by the AICPA under a system of
peer review.
Peer review an auditors assessment of the
quality of auditing services provided by another
audit firm.
The motivation for the paper is to examine
whether the system of self-regulated peer reviews
was perceived to be credible.

53
3.5 Ordinal dependent variables

Reviewers issue one of three types of peer review
opinion
unmodified (no serious weaknesses),
modified (serious weaknesses),
adverse (very serious weaknesses),
Reviewers also disclose how many significant
weaknesses they found at the audit firm (where
significant is not as bad as serious).

54
(No Transcript)
55
(No Transcript)
56
(No Transcript)
57
(No Transcript)
58
3.5 Ordinal dependent variables

Go to http//ihome.ust.hk/accl/Phd_teaching.htm
Download and open peer_review.dta
use "C\phd\peer_review.dta", clear
sum opinion, detail
The opinion variable is coded as follows
0 if unmodified 0 weaknesses
1 if unmodified 1 weakness
2 if unmodified 2 weaknesses
3 if unmodified 3 weaknesses
..
11 if modified 1 weakness
12 if modified 2 weaknesses
13 if modified 3 weaknesses
..
24 if adverse 4 weaknesses
25 if adverse 5 weaknesses
26 if adverse 6 weaknesses
..

59
3.5 Ordinal dependent variables

The peer review opinion variable is ordered and
discrete
adverse opinions (20-29) are worse than modified
(10-19)
modified opinions (10-19) are worse than
unmodified (0-9)
within each type of opinion (i.e., unmodified,
modified, adverse) the opinion is worse if it
discloses more weaknesses
We need to account for the fact that the opinion
variable is ordered and discrete when we estimate
a model that explains the determinants of peer
review opinions.
Note that the numbers assigned to the opinion
variable have no cardinal meaning
For example, a value of 12 for the opinion
variable does not mean that the opinion is twice
as bad as an opinion with a value of 6.
The assigned numbers are simply used to provide
an ordering from best (opinion 0) to worst
(opinion 29)

60
3.5 Ordinal dependent variables

For example, the opinion variable could have been
coded as follows
0 if unmodified 0 weaknesses
1 if unmodified 1 weakness
2 if unmodified 2 weaknesses
3 if unmodified 3 weaknesses
..
101 if modified 1 weakness
102 if modified 2 weaknesses
103 if modified 3 weaknesses
..
2004 if adverse 4 weaknesses
2005 if adverse 5 weaknesses
2006 if adverse 6 weaknesses
..
The numbers are different but the ordering is the
same as before

61
3.5 Ordinal dependent variables

We want an estimation procedure that relies only
on the ordering of the dependent variable, not
the actual numbers assigned.
We cannot use OLS because OLS assigns a cardinal
(i.e., a literal) meaning to the numbers.
Assigning different numbers to the dependent
variable would result in different coefficient
estimates if we used OLS.

62
3.5 Ordinal dependent variables

Lets create a second opinion variable that
preserves the same order but assigns different
numerical values
gen opinion1opinion
replace opinion1101 if opinion11
replace opinion1102 if opinion12
replace opinion1103 if opinion13
replace opinion1104 if opinion14
replace opinion1105 if opinion15
replace opinion12004 if opinion24
replace opinion12005 if opinion25
replace opinion12006 if opinion26
replace opinion12007 if opinion27
replace opinion12009 if opinion29

63
3.5 Ordinal dependent variables

OLS will provide different estimation results
according to whether the dependent variable is
opinion or opinion1
reg opinion reviewed_firm_also_reviewer
litigation_dummy, robust
reg opinion1 reviewed_firm_also_reviewer
litigation_dummy, robust
In contrast, the ordered logit model relies only
on the ordering of the dependent variable, not
the actual numerical values.
ologit opinion reviewed_firm_also_reviewer
litigation_dummy, robust
ologit opinion1 reviewed_firm_also_reviewer
litigation_dummy, robust

64
3.5 Ordinal dependent variables

The output of the ologit model is the same as
with logit except that there are _cut1 _cut2 etc.
values shown beneath the coefficient estimates.
These are the cut-off values kN-1, kN-2, . . . ,
k2, k1
Y N if kN-1 lt L lt ?
Y N-1 if kN-2 lt L ? kN-1
....etc.
Y 2 if k1 lt L ? k2
Y 1 if -? lt L ? k1
Usually, we are not particularly interested in
the cut-off values so we would not bother
reporting them in our paper.
Another difference is that there is no intercept
term in the ordered logit and ordered probit
models.

65
3.5 Ordinal dependent variables

We could alternatively estimate the model using
ordered probit (oprobit)
oprobit opinion reviewed_firm_also_reviewer
litigation_dummy, robust
Notice that the ologit and oprobit results are
quite close to each other
usually it doesnt make much difference whether
you use ordered logit or ordered probit.

66
Choosing between multinomial and ordered models

The key differences between multinomial and
ordered models are that
You should only use the ordered model if the
dependent variable has an ordering.
The multinomial model requires that you estimate
coefficients for N-1 equations (where N is the
number of outcomes for the dependent variable)
whereas the ordered model requires that you
estimate coefficients for just one equation.

67
Choosing between multinomial and ordered models

Lau (1987) was one of the first researchers in
accounting to use a multinomial model
A five-state financial distress prediction
model
Most prior bankruptcy studies had used a dummy
dependent variable equal to
one if the company goes bankrupt in the following
year,
zero if the company survives in the following
year.
These bankruptcy models are generally estimated
using either logit or probit.

68
Choosing between multinomial and ordered models

In Laus paper, the dependent variable takes five
possible values

69
Choosing between multinomial and ordered models

Francis and Krishnan (1999) use an ordered probit
model when examining how auditors reporting
choices are affected by accruals

70
Choosing between multinomial and ordered models

In their study, the dependent variable takes
three possible values

71
Class exercise 3d

Do you agree with Laus decision to use a
multinomial model?
Do you agree with Francis and Krishnans decision
to use an ordered model?

72
3.6 Count data models

Count data models are used where the dependent
variable takes discrete non-negative values (Y
0, 1, 2, 3, ) that represent a count.
In contrast, to the ordered logit and ordered
probit models, the numerical values assigned to
the dependent variable do have a literal meaning.
For example, consider the number of financial
analysts that follow a given company
if the company is not followed by any analysts, Y
0
if the company is followed by one analyst, Y 1
if the company is followed by two analysts, Y 2
if the company is followed by two analysts, Y 3
Etc
The values are non-negative because a company
cannot have negative analyst following.
The values are discrete because a company cannot
have a half or a quarter of an analyst.
The values have real meaning (i.e., they are not
arbitrarily assigned to denote a ranking or to
identify different categories).

73
3.6 Count data models

Bhushan (JAE, 1989) uses OLS models to explain
the determinants of analyst following.
OLS should not be used if the dependent variable
is a count data variable, such as analyst
following.
Using OLS is wrong because
Linear regression assumes that the dependent
variable is distributed between -? and ?. In
contrast, analyst following is truncated at
zero (negative values of analyst following are
impossible).
Linear regression assumes that the dependent
variable is continuous. In contrast, analyst
following takes discrete integer values 0, 1, 2,
3, etc.

74
3.6 Count data models

Two distributions that fulfill the criteria of
having non-negative discrete integer values are
the Poisson and the negative binomial.
Rock et al. (2001) published a paper in JAE in
which they re-examine the OLS results reported in
Bhushan (1989) for the determinants of analyst
following.
Rock et al. (2001) compare the results of OLS
models (the wrong method) with the results from
Poisson and negative binomial models.
some of the results in Bhushan (1989) are found
to be unreliable
for example the number of institutional investors
is inversely related with analyst following.
The Rock et al. paper illustrates that you can
publish in the top journals by showing that prior
researchers have used faulty econometric
methodology.

75
3.6 Count data models

There are lots of other examples of count data
variables
The number of RD patents awarded
The number of airline accidents
The number of murders
The number of times that Chinese people have
visited Guangzhou
The number of weaknesses found by peer reviewers
at audit firms

76
3.6 Count data models

Recall that the peer review opinion variable that
we used previously was coded using
The type of opinion issued (adverse, modified or
unmodified)
The number of significant weaknesses disclosed in
the peer review report
I assigned numerical values to the opinion
variable and I made two assumptions when ordering
the opinions
Adverse is worse than modified and modified is
worse than unmodified
For each type, the opinions are worse when they
disclose a greater number of weaknesses.
Because the numerical values had no literal
meaning, we used ordered logit and ordered probit
models.

77
Class exercise 3e

Suppose that we are only interested in the number
of weaknesses disclosed and not the type of peer
review opinion (i.e., we are not interested in
whether it is unmodified, modified or adverse).
Using the opinion variable, create a variable
that equals the number of weaknesses disclosed in
the peer review opinion.
Examine the distribution of the weaknesses
variable and explain why it is a count date
variable.

78
3.6 Count data models

STATA estimates two types of count data model
the negative binomial (nbreg)
the Poisson (poisson)
As I will explain later, the Poisson model can be
regarded as a special case of the negative
binomial model.
The Poisson distribution is most often used to
determine the probability of x occurrences per
unit of time
E.g., the number of murders per year
Count data models are sometimes used to determine
the number of occurrences not involving time. For
example
the number of analysts per company
the number of weaknesses per peer review report

79
3.6 Count data models

The basic assumptions of the Poisson distribution
are as follows
The time interval can be divided into small
subintervals such that the probability of an
occurrence in each subinterval is very small
Note that this assumption is unlikely to hold for
analyst following or the number of weaknesses in
peer review reports. The smallest subinterval is
one company (one audit firm) but the probability
of an occurrence is not very small.
The probability of an occurrence in each
subinterval remains constant over time
The probability of two or more occurrences in
each subinterval must be small enough to be
ignored
An occurrence or nonoccurrence in one subinterval
must not affect the occurrence or nonoccurrence
in any other subinterval (this is the
independence assumption).

80
3.6 Count data models

For example, consider the number of murders per
year. Assume that
The probability of a murder occurring during any
given minute is small
The probability of a murder occurring during any
given minute remains constant during the year
The probability of more than one person being
murdered during any given minute is very small
The number of murders in any given time period is
independent of the number of murders in any other
time period.
If these assumptions hold, we can use a Poisson
model to estimate a model that explains the
number of murders.

81
3.6 Count data models

The only parameter needed to characterize the
Poisson distribution is the mean rate at which
events occur
This is known as the incidence rate and is
usually represented using ?
For example, ? can be the average number of
murders per month or the average number of
analysts per company
The probability function for the Poisson
distribution is used to determine the probability
that N occurrences take place.
The value of ? must be positive and N can be any
non-negative integer value

82
3.6 Count data models

The probability function for the Poisson
distribution is
For example, suppose that on average there are 2
murders per month (? 2) and we want to know the
probability that there will be three murders
during the month (N 3).

83
3.6 Count data models

An attractive feature of the Poisson model is
that the distribution is defined using only one
parameter (?)
In fact the mean and variance of the Poisson
distribution are exactly the same, they both
equal ?
In the Poisson model, the incidence rate is
written as an exponential function of the
explanatory variables and their coefficients.

84
3.6 Count data models (poisson)

The Poisson model is estimated using the poisson
command
As usual you can control for heteroscedasticity
using the robust option
If this were a panel dataset (it isnt) you would
also need to control for time-series dependence
using the cluster() option
poisson weaknesses reviewed_firm_also_reviewer
litigation_dummy , robust

85
3.6 Count data models

The Poisson model imposes the assumption that the
mean and variance of the distribution are equal
(?)
This is sometimes known as the equidispersion
feature of the Poisson distribution
Unobserved heterogeneity in the data (e.g.,
omitted variables) will often cause the variance
to exceed the mean (a phenomenon known as
overdispersion).
As noted by Rock et al. (p. 357)

86
3.6 Count data models

It is therefore important that we test whether
the data are consistent with the assumed
distribution for the Poisson
In STATA we can do this using the poisgof command
after we run the Poisson model
If this test is significant, the Poisson model is
inappropriate and we should instead use the
negative binomial model, which is a more general
version of the Poisson model
The negative binomial does not assume that the
mean and variance of the distribution are the same

87
3.6 Count data models (poisgof)

poisson weaknesses reviewed_firm_also_reviewer
litigation_dummy , robust
poisgof
The goodness-of-fit statistic is highly
significant which means we can strongly reject
the assumption that the data are Poisson
distributed
Given this finding, it is necessarily to relax
the assumption that the data are Poisson
distributed and we need to estimate a more
general model (the negative binomial)

88
3.6 Count data models

One derivation of the negative binomial model is
that there is an omitted variable (Z) such that
expZ follows a gamma distribution with mean 1 and
variance ?
? is referred to as the overdispersion
parameter. The larger is ?, the greater the
overdispersion.
The Poisson model corresponds to the special case
where ? 0 (i.e., any omitted variables are just
constants and are captured in the a0 intercept)

89
3.6 Count data models

The negative binomial model is estimated using
the nbreg command (again we can use the robust
and cluster() options)
nbreg weaknesses reviewed_firm_also_reviewer
litigation_dummy , robust
In our data, the coefficients and t-statistics
from the negative binomial are similar to the
poisson even though the data do not fit the
poisson distribution
Nevertheless, it would be better to report the
results for the more general nbreg model

90
3.6 Count data models

Instead of estimating ? directly, STATA estimates
the natural log of alpha which is reported in the
output as lnalpha
In our model ln(?) -0.427, implying that ?
0.652
nbreg does the anti-log transformation for us at
the bottom of the output
0.652 exp(-0.427).
Note that ? is significantly different from zero,
which is what we would expect given that the
poisgof test rejected the validity of the Poisson
model for our data

91
3.7 Tobit and interval regression models

A common problem in applied research is censoring
(or truncation) of the dependent variable.
This occurs when values of the dependent variable
are reported as (or transformed to) a single
value.
For example suppose we are interested in
explaining
The demand for attending football matches
we do not observe the demand, we only observe the
football attendance
the attendance cannot exceed the capacity of the
stadium
in this case, the data are right-censored at the
capacity
The demand for cigarettes
we do not observe the demand, we only the number
of cigarettes purchased
the amount purchased will be zero for all
non-smokers
in this case, the data are left-censored at
zero

The variable of interest is the demand for
attending football matches.
The dependent variable that we actually measure
is the crowd attendance at the match.
If the match is sold out, we know that the demand
was greater than the capacity but we do not
observe what the attendance would have been if
there had been no limit to the stadiums capacity
The data are right-censored

The variable of interest is the demand for
cigarettes.
The dependent variable that we actually measure
is the number of cigarettes purchased.
If the person is a non-smoker, his/her demand may
be less than zero
the non-smoker may not smoke even if cigarettes
were given away for free
The data are left-censored at zero.

Suppose that in our data, the price of cigarettes
varies between P0 and P1 (we do not observe how
the purchase of cigarettes would change outside
of this range).
We expect that smokers would smoke more if the
price is lower.

The marginal non-smoker is indifferent between
smoking and not smoking at price P0.
If the price is lowered towards zero, some
non-smokers might be persuaded to smoke.
Other non-smokers may not smoke even if the price
is zero (i.e., even if cigarettes were given away
free of charge)
Even these non-smokers might be induced to smoke
at negative prices (i.e., they might be willing
to smoke if they were paid to do so).
Therefore, there is a negatively sloped demand
curve even for non-smokers.

The problem is that, in our data, we would
observe only zero values for the purchases of
non-smokers (i.e., the price does not fall below
P0)
If we fit a line through the data points for
non-smokers and smokers, the slope coefficient
would be biased because the purchases of
cigarettes by non-smokers are censored at zero.

97
3.7 Tobit model

The censoring problem can be solved by estimating
a tobit model
the name tobit refers to an economist, James
Tobin, who first proposed the model (Tobin 1958).
it is assumed that the uncensored distribution of
the dependent variable is normal (see earlier
slides)

98
3.7 Tobit model

Recall that in the probit model, we assume a
one-to-one mapping between an unobserved
continuous variable (Y) and the observed dummy
variable (Y)
Y a0 a1 X e
Y 0 if -? lt Y ? 0
Y 1 if 0 lt Y lt ?
The tobit model is somewhat similar
Y a0 a1 X e
Y 0 if -? lt Y ? 0
Y Y if 0 lt Y lt ?
The Y and Y variables are both observed when
they are greater than zero (Y is unobserved when
Y 0)
Both the probit and tobit models assume that the
errors (e) are normally distributed.

99
3.7 Tobit model (tobit)

Recall that in our fee dataset, the nonauditfees
variable is left-censored at zero because many
companies choose not to purchase any non-audit
services
This phenomenon is like some individuals choosing
not to purchase any cigarettes when the price
exceeds P0
use "C\phd\Fees.dta", clear
sum nonauditfees, detail
taking logs to reduce the influence of outliers
with large positive values
gen lnnafln(1 nonauditfees)
sum lnnaf, detail

100
3.7 Tobit model (tobit)

The STATA command for estimating a tobit model is
tobit
The option ll() specifies the lower limit at
which the dependent variable is left-censored
For example, non-audit fees are left-censored at
zero, so we type ll(0)
The option ul() specifies the upper limit at
which the dependent variable is right-censored
For example, the capacity at Manchester Uniteds
stadium is 75000. So attendances at their home
games are right-censored at 75000 (we would type
ul(75000) if our dependent variable is
attendances at Manchester Uniteds home games)

101
3.7 Tobit model (tobit)

Suppose we wish to test whether larger companies
purchase more non-audit services
gen lntaln(totalassets)
egen missrmiss(lnnaf lnta)
tobit lnnaf lnta if miss0, ll(0)
We do not even have to tell STATA that the
censoring point is at zero
As long as we tell STATA that the data are
left-censored, STATA will find the minimum value
for the dependent variable and assume that the
minimum is the censoring point
In other words, we get exactly the same results
by typing
tobit lnnaf lnta if miss0, ll
STATA tells us how many observations are
left-censored and how many are uncensored . We
can easily check this
count if miss0 lnnaf0
count if miss0 lnnafgt0

102
Class exercise 3f

Estimate the same model using OLS instead of
tobit
Is the coefficient on company size (lnta) larger
in the tobit model or the OLS model?
Is the intercept larger in the tobit model or the
OLS model?
Explain why the OLS coefficients are biased.

103
3.7 Tobit model (tobit)

Tobit models can also be used when the data are
both left-censored and right-censored.
For example, lets pretend that we only observe
the true values of lnnaf if they lie between 0
and 5
values less than 0 are recorded as 0 (i.e., the
data are left-censored as before)
values in excess of 5 are recorded as 5 (i.e.,
the data are now right-censored as well)
sum lnnaf, detail
gen lnnaf1lnnaf
replace lnnaf15 if lnnafgt5 lnnaf!.
tobit lnnaf1 lnta if miss0, ll(0) ul(5)

104
3.7 Tobit model (tobit)

In fact, we dont need to tell STATA that the
censoring points are at 0 and 5 because STATA
will automatically choose these as they are the
minimum and maximum values of lnnaf1
tobit lnnaf1 lnta if miss0, ll ul
Note that we get exactly the same output if we
use the lnnaf variable (which is not
right-censored) but we tell STATA to censor this
variable at 5
tobit lnnaf lnta if miss0, ll(0) ul(5)
Note also that we dont even need to tell STATA
that there is left-censoring at zero
tobit lnnaf lnta if miss0, ll ul(5)
However, the results will be different if we use
lnnaf instead of lnnaf1 and we dont tell STATA
to censor at 5 (STATA will choose the maximum
value of lnnaf as the censoring point)
tobit lnnaf lnta if miss0, ll ul

105
3.7 Interval regression

Tobit is simply a special case of a more general
type of model known as interval regression.
A particular advantage of interval regression is
that we can adjust the standard errors for
heteroscedasticity and for time-series dependence
using the robust cluster () option
The robust cluster () option is unavailable for
the tobit command

106
3.7 Interval regression (intreg)

The interval regression command is intreg and
this time we have to specify two dependent
variables
the first dependent variable
takes a missing value (.) for the left-censored
data,
takes the actual value for the uncensored data,
takes a value equal to the upper censoring point
for the right-censored data
the second dependent variable
takes a value equal to the lower censoring point
for the left-censored data,
takes the actual value for the uncensored data,
takes a missing value (.) for the right-censored
data

107
3.7 Interval regression (intreg)

For example, lets use the intreg command to
estimate the same model as before (where lnnaf is
left-censored at zero and there is no
right-censoring).
The first dependent variable
takes a missing value (.) for the left-censored
data (i.e., at lnnaf 0),
takes the actual value for the uncensored data
(there is no right-censoring)
drop lnnaf1
gen lnnaf1lnnaf
replace lnnaf1. if lnnaf0
The second dependent variable
takes a value equal to the lower censoring point
for the left-censored data (i.e., equals 0 when
lnnaf 0),
takes the actual value for the uncensored data
(there is no right-censoring)
this second dependent variable is simply lnnaf

108
3.7 Interval regression (intreg)

For example
We can now run intreg
intreg lnnaf1 lnnaf lnta
Notice that the output is exactly the same as
before when we ran tobit with left-censoring at
zero
tobit lnnaf lnta, ll(0)

109

Just like before, we can also use interval
regression when the dependent variable is both
left-censored and right-censored.
For example, lets estimate a model in which
lnnaf is left-censored at 0 and right-censored at
5.
The first dependent variable
takes a missing value (.) for the left-censored
data (i.e., missing when lnnaf 0),
takes the actual value for the uncensored data,
takes a value equal to the upper censoring point
for the right-censored data
drop lnnaf1
gen lnnaf1lnnaf
replace lnnaf1. if lnnaf0
replace lnnaf15 if lnnafgt5
The second dependent variable
takes a value equal to the lower censoring point
for the left-censored data (i.e., 0 at lnnaf
0),
takes the actual value for the uncensored data,
takes a missing value (.) for the right-censored
data
gen lnnaf2lnnaf
replace lnnaf2. if lnnafgt5 lnnaf!.

110
3.7 Interval regression (intreg)

For example
We can now run intreg
intreg lnnaf1 lnnaf2 lnta if miss0
Notice that the output is exactly the same as
when we run tobit with left-censoring at 0 and
right-censoring at 5
tobit lnnaf lnta if miss0, ll(0) ul(5)

111
3.7 Interval regression (intreg)

The main advantage of using the intreg command
instead of tobit is that we can use the robust
and cluster() options to control for the effect
of heteroscedasticity and time-series dependence
on the standard errors.
intreg lnnaf1 lnnaf2 lnta if miss0, robust
cluster(companyid)
Notice that the estimated standard errors are
biased downwards (and the t-statistics are biased
upwards) if we do not adjust for
heteroscedasticity and time-series dependence.

112
3.7 Interval regression

Caramanis and Lennox (JAE, 2008) is an example of
a study that uses interval regression rather than
OLS because the dependent variables are
truncated.
The study examines how audit effort affects
earnings management.

113
3.7 Interval regression