程序代做CS代考 Excel Basics LPM Logit models Probit models

Basics LPM Logit models Probit models

CORPFIN 2503 – Business Data Analytics:
Applications of logit and probit models

čius

Week 5: August 23rd, 2021

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Basics LPM Logit models Probit models

Outline

Basics

LPM

Logit models

Probit models

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Introduction

We use simple and multiple linear regressions if the dependent
variable is continuous.

For example, the dependent variable is the car price.

What if the dependent variable is a dummy variable, such as
SEDAN=1 if a car type is sedan and 0 otherwise?

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Introduction II

Other interesting issues:
• dividend payers vs non-payers
• firms going bankrupt and not
• firms issuing equity vs debt securities
• credit rating downgrades and upgrades
• will a firm become an M&A target or not
• etc.

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Introduction III

If we use OLS regressions in these cases, we would estimate linear
probability models (LPMs).

It is better to estimate logit or probit models.

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LPM example

Let’s estimate an LPM where the dependent variable equals 1 if a
firm pays dividends using ASX data from Workshop 3:

data work.asx;
set work.asx;
label dividend_payer=”Dividend payer”;
ln_assets=log(assets);
re_equity_d=0;
if re_equity>0 & re_equity ne . then re_equity_d=1;
run;

PROC REG DATA=work.asx;
MODEL dividend_payer=re_equity_d ln_assets;
RUN;

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LPM example II

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LPM example II

The results suggest that:
• firms with positive RE/equity ratio have 25.6% higher
probability of paying dividends

• the increase in ln(assets) by 1 leads to 9.4% higher probability
of paying dividends.

If RE/equity > 0 and ln(assets) = 15 then a firm has 18%
probability of paying dividends (–1.48192 + 0.25610 + 15 ×
0.09350 = 0.17668).

Let’s look at the residuals.

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LPM example III

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Slide #61 of Lecture 4

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LPM example IV

Residuals are not normally distributed and are subject to
heteroscedasticity.

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Introduction II

LPMs are subject to:
1. heteroskedasticity (standard errors will be wrong, and

hypothesis tests will be incorrect)
2. residuals will not be normally distributed (as the residuals can

take on two possible values)
3. the predicted values for the dependent variable might be

greater than 1 or lower than 0 (as LPM assumes the linear
impact of the independent variable):

• if the dependent variable equals 1 if a firm pays dividends and
0 otherwise, values outside the region [0, 1] are not logical

• probability of paying dividends cannot be negative.

The solution is to use logit (logistic) or probit models.

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Theory
Suppose p is the probability that Y = 1:

p = P (Y = 1).

Functional form of LPM with 2 independent variables is:

p = β0 + β1×1 + β2×2.

The LHS of the LPM can range from 0 to 1, but the RHS can vary
from −∞ to ∞.

=⇒ We need to transform the dependent variable to eliminate the
0 to 1 constraint.

We can eliminate the upper bound (p = 1) by using the ratio p
1−p .

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Theory II

p
1−p is the odds of an event occurring.

Let’s assume that the probability of success of some event is 0.75.

Then the probability of failure is 1 – 0.75 = 0.25.

The odds of success are defined as the ratio of the probability of
success over the probability of failure.

=⇒ The odds of success are 0.75/0.25 = 3.

=⇒ The odds of success are 3 to 1.

Similarly, if the probability of success is 0.5, i.e., 50-50 percent
chance, then the odds of success are 1 to 1.

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Theory III

p
1−p is the odds of an event occurring.

p Odds

0.0001 0.0001
0.01 0.0101
0.1 0.1111
0.25 0.3333
0.5 1
0.75 3
0.9 9
0.99 99

0.9999 9999

• When probability is either very
small or very big, changes in odds
hardly impact probability.

• When probability is between 0.1
and 0.9, changes in odds
substantially impact probability.

• The range of odds is between 0 and
∞.

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Theory IV
We can eliminate the lower bound of 0 by taking the natural
logarithm of the odds ratio.

-10

-8

-6

-4

-2

0

2

0 1 2 3 4 5 6

y

Odds

y=ln(Odds)

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Theory V
The log odds of the event occurring is lnOdds = ln

[
p

1−p

]
.

p Odds ln(Odds)

0.0001 0.0001 –9.210
0.01 0.0101 –4.595
0.1 0.1111 –2.197
0.25 0.3333 –1.099
0.5 1 0
0.75 3 1.099
0.9 9 2.197
0.99 99 4.595

0.9999 9999 9.210

• When probability is either very
small or very big, changes in log
odds hardly impact probability.

• When probability is between 0.1
and 0.9, changes in log odds
substantially impact probability.

• If p < 0.5, then odds < 1 and log odds < 0. • If p = 0.5, then odds = 1 and log odds = 0. • If p > 0.5, then odds > 1 and
log odds > 0.

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Theory VI

Functional form of logit (logistic) model with 2 independent
variables:

lnOdds = ln
[

p

1− p

]
= β0 + β1×1 + β2×2.

The dependent and independent variables can vary between −∞
and ∞.

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Theory VII

Let’s derive the predicted value of p:

ln

[
p

1− p

]
= β0 + β1×1 + β2×2,

p

1− p
= eβ0+β1×1+β2×2 ,

p = e
β0+β1×1+β2×2

1+eβ0+β1×1+β2×2
,

where e ≈ 2.71828.

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Theory VIII

The impact of Q on the predicted probability depends on Q:
• if Q is low, the impact is small
• if Q is neither low or high, the impact is big
• if Q is high, the impact is small.

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Theory IX

In case of logit models, predicted probability is never < 0 or > 1,
and the line is not straight.

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Theory X
Functional form of logit (logistic) model:

p =
eβ0+β1×1+β2×2

1 + eβ0+β1×1+β2×2
.

Suppose that eβ0+β1×1+β2×2 = 1, 000, 000 then:

p =
1, 000, 000

1 + 1, 000, 000
< 1. Suppose that eβ0+β1x1+β2x2 = 0.0000001 then: p = 0.0000001 1 + 0.0000001 > 0.

=⇒ p is always between 0 and 1.
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Logit models

In SAS, there are several procedures to estimate logit models:
• LOGISTIC (our main choice)
• QLIM
• GENMOD
• PROBIT
• MDC
• PHREG and
• CATMOD.

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Logit models II

Now let’s estimate logit model using the same dataset:

PROC LOGISTIC DATA=work.asx;
MODEL dividend_payer (EVENT=’1’) = re_equity_d ln_assets;
RUN;

Option (EVENT=’1’) makes SAS estimate the probability of paying
dividends.

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Logit models III

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Logit models IV

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Logit models V

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Logit models VI

Coefficient estimates for firm size and for positive RE/equity ratio
dummy are significantly positive.

The results suggest that larger firms as well as firms with positive
RE/equity ratio are more likely to pay dividends:

• a 1 unit increase in RE/equity ratio dummy will result in a
1.6120 increase in the log odds to pay dividends (if there are 2
firms with identical ln(assets), the log odds for the one with
positive RE/equity ratio would be 1.6120 greater than the log
odds for the firm with negative RE/equity ratio)

• a 1 unit increase in ln(assets) will result in a 1.1059 increase in
the log odds to pay dividends.

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Logit models VII

To compute the probability of paying dividends for a particular firm,
we simply need to plug in its ln(assets) and RE/equity dummy
value in the equation below:

p =
e−22.4746+1.6120×RE/equity dummy+1.1059×ln(assets)

1 + e−22.4746+1.6120×RE/equity dummy+1.1059×ln(assets)
.

Suppose ln(assets)=20 and RE/equity dummy=1, then p=0.778.

Suppose ln(assets)=17 and RE/equity dummy=0, then p=0.025.

Suppose ln(assets)=20 and RE/equity dummy=0, then p=0.412.

Suppose ln(assets)=17 and RE/equity dummy=1, then p=0.113.

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Descriptive statistics

Let’s look at the properties of ln_assets and re_equity_d:

proc univariate data=work.asx plots;
var ln_assets;
run;

proc freq data=work.asx;
tables dividend_payer * re_equity_d /
norow nocol nopercent;
run;

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Descriptive statistics II: ln_assets

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Descriptive statistics III: ln_assets

ln(assets)=20: assets=e20= 485,165,195. Very large firms!

ln(assets)=17: assets=e17= 24,154,952. Average firms.

The former firms are 20 times larger than the latter firms.

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Descriptive statistics IV: re_equity_d

Two-way table:

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Odds ratios

Odds ratio estimates are used to see the exact impact of each
individual variable on the odds of the positive outcome of the
model.

E.g., the odds ratio estimate for RE/equity dummy indicates the
impact of RE/equity dummy on the odds of paying dividends:

• What is the change in the odds when there is a unit change in
the independent variable?

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Odds ratios II

Odds ratio for RE/equity dummy is 5.013.

Suppose ln(assets)=20 and RE/equity dummy=1:
• p(div. payer)=0.778 & p(div. non-payer)=1 – 0.778=0.222
• odds of paying over not paying dividends = 0.778

0.222
= 3.51.

Suppose ln(assets)=20 and RE/equity dummy=0:
• p(div. payer)=0.412 & p(div. non-payer)=1 – 0.412=0.588
• odds of paying over not paying dividends = 0.412

0.588
= 0.70.

Change in odds with unit change in RE/equity dummy is
3.51
0.70

= 5.013.

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Odds ratios III

For 2 otherwise identical firms, the odds to pay dividends:
• for a firm with the positive RE/equity would be exp(1.6120) =
5.013 times greater

• for a firm with ln(assets) greater by 1 unit would be
exp(1.1059) = 3.022 times greater.

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Marginal effects

Interpreting the impacts on log odds and odds might be tricky.

Why not look at the impact of a variable on the probability to pay
dividends, holding all other variables in the model constant?

Yes, we can. This is known as a marginal effect.

However, it depends on the the variable values.

Thus, we compute marginal effect at each observation and then
calculate the sample average of individual marginal effects to obtain
the overall marginal effect.

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Marginal effects II

SAS procedure LOGISTIC does not compute marginal effects but a
procedure QLIM does:

PROC QLIM DATA=work.asx;
MODEL dividend_payer = re_equity_d ln_assets
/ discrete(d=logistic);
OUTPUT OUT=work.marginal_effects MARGINAL;
RUN;

PROC MEANS DATA=work.marginal_effects mean min max maxdec=3;
VAR Meff_P2_re_equity_d Meff_P2_ln_assets;
title ’Average of the Individual Marginal Effects
(Logit Model)’;
RUN;

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Marginal effects III
The results:

On average:
• having positive RE/equity ratio increases the probability of
paying dividends by 0.106

• ln(assets) greater by 1 unit increases the probability of paying
dividends by 0.073.

These values are smaller than those from the LPM (0.25610 &
0.09350).

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Model fit statistics

If we estimate several logit models, how do we know which one is
the best?

AIC (Akaike Information Criterion) and SC ( ) are
used to compare two or more models and pick the best one.

A model with minimum AIC and SC values are preferred:
• such model would have fewer independent variables and
• better fit to the data.

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Model fit statistics II
Let’s estimate 4 logit models and compare their fit statistics:

PROC LOGISTIC DATA=work.asx;
MODEL dividend_payer (EVENT=’1’) = re_equity_d ln_assets;
RUN;

PROC LOGISTIC DATA=work.asx;
MODEL dividend_payer (EVENT=’1’) = re_equity_d assets;
RUN;

PROC LOGISTIC DATA=work.asx;
MODEL dividend_payer (EVENT=’1’) = re_equity_d;
RUN;

PROC LOGISTIC DATA=work.asx;
MODEL dividend_payer (EVENT=’1’) = ln_assets;
RUN;

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Model fit statistics III

Summary of results:
• Model 1 is the best (its indep. var.:
RE/equity dummy and ln(assets)).

• Model 3 is the worst (its indep.
var.: RE/equity dummy).

• Models with covariates are better
than the model without any
covariate.

In general, AIC and SC lead to the same
model being selected.

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Probit models

The key difference between probit and logit models is their
functional form.

For logit (logistic) models, it is the cumulative standard logistic
distribution function:

p =
eβ0+β1×1+β2×2

1 + eβ0+β1×1+β2×2
.

For probit models, it is the cumulative standard normal probability
distribution function:

p = Φ (β0 + β1×1 + β2×2) .

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Probit models II

Now let’s estimate probit model using the same dataset:

PROC PROBIT DATA=work.asx;
MODEL dividend_payer (EVENT=’1’) = re_equity_d ln_assets;
RUN;

or

PROC LOGISTIC DATA=work.asx;
MODEL dividend_payer (EVENT=’1’) = re_equity_d ln_assets
/ LINK=PROBIT;
RUN;

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Probit models III

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Probit models IV

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Probit models V

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Probit models VI

Coefficient estimates for firm size and for positive RE/equity ratio
dummy are significantly positive.

The results suggest that larger firms as well as firms with positive
RE/equity ratio are more likely to pay dividends.

Interpretation of the coefficients in probit regression is not as
straightforward as the interpretations of coefficients in LPM or logit
models.

If LOGISTIC procedure is used, then one gets AIC and SC values
that can be used to identify the best model (which is the one with
the lowest AIC and SC values).

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Probit models VII
To compute the predicted probability of paying dividends for a
particular firm, we simply need to plug in its ln(assets) and
RE/equity dummy value in the equation below:

p = Φ(−11.7732 + 0.8823× RE/equity dummy
+ 0.5767× ln(assets))

where Φ is the cumulative standard normal distribution function
(Excel function: NORM.S.DIST).

Suppose ln(assets)=20 and RE/equity dummy=1, then p=0.740.

Suppose ln(assets)=17 and RE/equity dummy=0, then p=0.024.

Suppose ln(assets)=20 and RE/equity dummy=0, then p=0.405.

Suppose ln(assets)=17 and RE/equity dummy=1, then p=0.139.

The results are very similar to those of logit model.
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Marginal effects
SAS procedure PROBIT also does not compute marginal effects;
thus, we can use the procedure QLIM:

PROC QLIM DATA=work.asx;
MODEL dividend_payer = re_equity_d ln_assets
/ discrete(d=probit);
OUTPUT OUT=work.marginal_effects2 MARGINAL;
RUN;

PROC MEANS DATA=work.marginal_effects2 mean min max
maxdec=3;
VAR Meff_P2_re_equity_d Meff_P2_ln_assets;
title ’Average of the Individual Marginal Effects
(Probit Model)’;
RUN;

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Marginal effects II
The results:

On average:
• having positive RE/equity ratio increases the probability of
paying dividends by 0.109

• ln(assets) greater by 1 unit increases the probability of paying
dividends by 0.071.

These values are almost identical to those from logit model.

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Probit models VIII

The results are very similar to those of logit model.

Logit and probit models lead to essentially the same results in most
cases.

Logit models tend to converge a little bit faster.

=⇒ Thus, use either of them.

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Required reading

Konasani, V. R. and Kadre, S. (2015). “Practical Business
Analytics Using SAS: A Hands-on Guide”: chapter 11.

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Basics
LPM
Logit models
Probit models