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CORPFIN 2503 – Business Data Analytics: Applications of multinomial logit models
Week 6: August 30th, 2021
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Introduction Data
Predictions
Other issues
Introduction Data
Model Predictions Other issues
CORPFIN 2503, Week 6
Introduction Data Model Predictions Other issues
Introduction
The dependent variable in logit regressions have 2 possible outcomes (e.g., to pay dividends or not).
The dependent variable in multinomial logit regressions have more than 2 possible unordered outcomes.
E.g., for corporate bond issues:
• currency: AUD, USD, EUR, GBP
• coupon: 0, fixed, variable
• credit rating agency: Moody’s, Fitch, S&P.
E.g., buying a car:
• type: sedan, truck, SUV
• brand: BMW, Tesla, Toyota • colour: blue, white, red.
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Introduction Data Model Predictions
Multinomial logit regressions
Suppose we have n possible outcomes.
1 outcome will be chosen as “base” outcome.
The other n − 1 outcomes are separately regressed against the “base” outcome.
n − 1 logit models will be estimated.
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Multinomial logit regressions II
Let’s assume that we have 3 possible outcomes (Y = 0, 1, 2). Base outcome is Y = 0.
There are 2 independent variables: X1 and X2. Then the following models will be estimated:
lnPr(Y =1) =α1 +β1 ×X1 +γ1 ×X2, Pr(Y = 0)
lnPr(Y =2) =α2 +β2 ×X1 +γ2 ×X2. Pr(Y = 0)
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Other issues
US corporate bond data from Workshop 4.
Variables of interest: • Currency
• Maturity
• Credit rating.
To predict bond’s currency using its maturity and credit rating.
In other words:
Does bond’s currency depend on its maturity and credit rating?
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Introduction Data Model Predictions
Let’s consider 3 currencies: USD, EUR, and other.
data work.bonds;
set work.bonds;
currency2=currency;
if currency in (“Australia” “British P” “Canadian”
“Swiss Fra”) then currency2=”Other”;
currency2 will be our dependent variable.
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Introduction Data Model Predictions
Let’s generate frequency distribution table for currency2.
proc freq data=work.bonds;
tables currency2;
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Frequency distribution table for currency2:
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The independent variables include:
1. bond maturity ln_maturity2
2. credit rating dummy cr_rating_d.
data work.bonds;
set work.bonds;
maturity2=(maturity-today())/365;
ln_maturity2=log(maturity2);
cr_rating_d=0;
if s_p in (“AAA” “AA+” “AA” “AA-” “A+” “A” “A-“)
then cr_rating_d=1;
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Introduction Data Model Predictions
Let’s get descriptive statistics of bond maturity:
PROC MEANS DATA=work.bonds mean std min p25 median
p75 max maxdec=3;
VAR maturity2 ln_maturity2;
title ’Descriptive statistics of bond maturity’;
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Let’s generate a two-way table for currency and credit rating dummy:
proc freq data=work.bonds;
tables currency2*inv_grade_d / norow nocol nopercent;
title ’Two-way table for currency and credit rating
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We assume that currency2 = f (ln_maturity2, cr_rating_d). SAS code for multinomial logit model:
proc logistic data = work.bonds;
model currency2 = ln_maturity2 cr_rating_d
/ link = glogit;
output out=work.bonds_pred predprobs=(individual);
Alternatively, we could use SAS procedure CATMOD.
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The results
SAS produces a few tables:
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The results II
More tables:
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The results III
This is the test whether none of the predictors in either of the
models have non-zero coefficients.
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Not over yet …
The results IV
Null hypothesis:
There is no relation between the predictor variable and the outcome (i.e., the estimates of the predictor in both of the fitted models are 0).
If the p-value is less than the specified α (e.g., 0.1), then this null hypothesis can be rejected.
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Introduction Data
Main results:
The results V
Predictions Other issues
ln_maturity2 = −1.7201:
• bonds with shorter maturities are more likely to be in EUR than in USD
• a 1-unit increase in ln_maturity2 is associated with a 1.7201 decrease in the relative log odds of making bond issue in EUR vs. USD.
CORPFIN 2503, Week 6
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The results VI
Main results:
ln_maturity2 = −1.0010:
• bonds with shorter maturities are more likely to be in other currency than in USD
• a 1-unit increase in ln_maturity2 is associated with a 1.0010 decrease in the relative log odds of making bond issue in other currency vs. USD.
CORPFIN 2503, Week 6
Introduction Data
Model Predictions Other issues
Main results:
cr_rating_d = 1 if a firm has a good credit rating.
cr_rating_d = 0.6414:
• bonds with good credit ratings are more likely to be in EUR than in USD
• a good credit rating increases the relative log odds of making bond issue in EUR vs. USD by 0.6414.
The results VII
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Introduction Data
Main results:
cr_rating_d = 1 if a firm has a good credit rating.
Model Predictions Other issues
The results VIII
cr_rating_d = −0.5957:
• bonds with good credit ratings are less likely to be in other currency than in USD. The impact is insignificant
• a good credit rating decreases the relative log odds of making bond issue in other currency vs. USD by 0.5957.
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Introduction Data
Last table:
• 0.367; if ln_maturity2 increases by 1-unit then odds of making bond issue in other currency vs. USD would be expected to decrease by a factor of 0.367 (e−1.0010=0.367).
Model Predictions Other issues
The results IX
ln_maturity2:
• 0.179; if ln_maturity2 increases by 1-unit then odds of making bond issue in EUR vs. USD would be expected to decrease by a factor of 0.179 (e−1.7201 =0.179)
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The results X
Last table:
• 0.551; a good credit rating is expected to decrease the odds of making bond issue in other currency vs. USD by a factor of 0.551 (e−0.5979=0.551). The impact is insignificant.
cr_rating_d:
• 1.899; a good credit rating is expected to increase the odds of making bond issue in EUR vs. USD by a factor of 1.899 (e0.6414=1.899)
CORPFIN 2503, Week 6
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Predictions
SAS generated work.Bonds_pred file which includes the following new variables:
_INTO_: IP_Euro: IP_Other:
the actual value of the dependent variable (currency2)
the predicted value of the dependent variable
the predicted probability that currency2 = ”EURO” the predicted probability that currency2 = ”Other”
IP_US_Dollar: the predicted probability that currency2 = ”US Dollar”.
IP_Euro + IP_Other + IP_US_Dollar = 1.
_INTO_ is equal to the value which predicted probability is the highest.
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work.Bonds_pred
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Predictions II
Model with 2 independent variables and 3 possible outcomes:
lnPr(Y =1) =α1 +β1 ×X1 +γ1 ×X2, Pr(Y = 0)
lnPr(Y =2) =α2 +β2 ×X1 +γ2 ×X2. Pr(Y = 0)
Predicted probabilities are:
eα1 +β1 ×X1 +γ1 ×X2
Pr(Y = 1) = 1 + eα1+β1×X1+γ1×X2 + eα2+β2×X1+γ2×X2 ,
eα2 +β2 ×X1 +γ2 ×X2
Pr(Y = 2) = 1 + eα1+β1×X1+γ1×X2 + eα2+β2×X1+γ2×X2 ,
Pr(Y =0)=1−Pr(Y =1)−Pr(Y =2).
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Predictions III
Let’s predict the probabilities for the first observation:
ln_maturity2 = 2.0808104673 ≈ 2.081 cr_rating_d = 0.
Predicted probabilities are:
Pr(Y = ”Euro”) =
1 + e1.8417−1.7201×2.081+0.6414×0 + e−0.7925−1.0010×2.081−0.5957×0
Pr(Y = ”Other”) = e−0.7925−1.0010×2.081−0.5957×0
1 + e1.8417−1.7201×2.081+0.6414×0 + e−0.7925−1.0010×2.081−0.5957×0 Pr(Y = ”U S Dollar”) = 1 − 0.1427 − 0.0458 = 0.8115.
e1.8417−1.7201×2.081+0.6414×0
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Predictions IV
Pr(Y = ”US Dollar”) is the highest:
Thus, the predicted currency is US_Dollar.
Our calculations are consistent with the results in work.Bonds_pred file.
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Predictions V
Let’s predict the probabilities for the bond with:
ln_maturity2 = 1.4 cr_rating_d = 1.
Predicted probabilities are:
Pr(Y = ”Euro”) =
1 + e1.8417−1.7201×1.4+0.6414×1 + e−0.7925−1.0010×1.4−0.5957×1
Pr(Y = ”Other”) = e−0.7925−1.0010×1.4−0.5957×1
1 + e1.8417−1.7201×1.4+0.6414×1 + e−0.7925−1.0010×1.4−0.5957×1 Pr(Y = ”U S Dollar”) = 1 − 0.5038 − 0.0287 = 0.4674.
e1.8417−1.7201×1.4+0.6414×1
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Predictions VI
Pr(Y = ”Euro”) is the highest:
Thus, the predicted currency is Euro.
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Marginal effects
What about marginal effects?
It is possible to compute, but there is no built-in command in SAS. =⇒ We do not need to know how.
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Multinomial probit model
Multinomial probit model:
• is similar to multinomial logit model
• is superior to multinomial logit model
• does not allow to make predictions easily.
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Ordered logit/probit models
If the values of the dependent variable can be ordered (e.g., low, medium, high) then one should use:
• ordered logit/probit models
• OLS if there are 5 or more categories (not ideal method)
• multinomial logit models (but one would fail to use some of the information available).
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