dummy x -variables dummy y-variables - wordpress.com · 2011-05-09 · dummy y -variables discrete...

3/3/2014

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CDS M Phil Econometrics Vijayamohan

CDS M Phil Econometrics

Vijayamohanan Pillai N

13-Mar-14 CDS Mphil Econometrics Vijayamohan

Dummy variable ModelsDummy variable Models

CDS Mphil Econometrics Vijayamohan

Dummy XDummy X--variablesvariables

Dummy Dummy YY--variablesvariables




53-Mar-14


Dummy variable: Dummy variable:

variable assuming values 0 and 1 to indicate variable assuming values 0 and 1 to indicate

some attributes some attributes

To classify data into mutually exclusive To classify data into mutually exclusive

categoriescategories

Also called: Also called:

indicator variable, binary variable, indicator variable, binary variable,

dichotomous variable, categorical variable, dichotomous variable, categorical variable,

qualitative variablequalitative variable

3-Mar-14 CDS M Phil Econometrics Vijayamohan

6


YYii = = αα + + ββDDii + + uuii

YYii = Wage rate of an agricultural = Wage rate of an agricultural labourerlabourer

DDii = 1, if male worker= 1, if male worker0, otherwise.0, otherwise.

Mean wage of a male agri. worker?Mean wage of a male agri. worker?

E(YE(Yii | D| Dii = 1) = = 1) = αα + + ββ

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7





Mean wage of a female agri. worker?Mean wage of a female agri. worker?

E(YE(Yii | D| Dii = 0) = = 0) = αα


8





HH00: no sex discrimination : no sex discrimination ⇒⇒

HH00: : ββ= 0.= 0.


9





Analysis of Variance (ANOVA) Model:Analysis of Variance (ANOVA) Model:

Mean difference testMean difference test


103-Mar-14

In eco applications, In eco applications, control for other sociocontrol for other socio--eco factors:eco factors:caste, nature of work, experience, caste, nature of work, experience, …………Both quantitative and qualitative Both quantitative and qualitative variables:variables:Analysis of Covariance (ANCOVA) Analysis of Covariance (ANCOVA) ModelModel



113-Mar-14

YYii = = αα00 ++ αα11DDii ++ββXXii ++ uuii


E(YE(Yii | D| Dii = 0) = = 0) = αα00 + + ββXXii



E(YE(Yii | D| Dii = 1) = (= 1) = (αα00 ++ αα11) + ) + ββXXii

DDii = 1, if male worker= 1, if male worker= 0, otherwise.= 0, otherwise.


123-Mar-14


Ag

ricu

ltu

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ate

Ag

ricu

ltu

ral

wag

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ate

XX

αα00

αα11

((αα 00

++αα 11

))

Differential intercept: Differential intercept: αα11

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Ag

ricu

ltu

ral

wag

e r

ate

Ag

ricu

ltu

ral

wag

e r

ate

XX

αα00

αα11

((αα 00

++αα 11

)) ββ

ββ


143-Mar-14


Ag

ricu

ltu

ral

wag

e r

ate

Ag

ricu

ltu

ral

wag

e r

ate

XX

αα00

αα11

((αα00++αα11))

ffmm

3 March 2014 Vijayamohan CDS 15

YYii = = αα00 ++ αα11DDii ++ββ11XXii ++ββ22DDiiXXii ++ uuii


E(YE(Yii | D| Dii = 0) = = 0) = αα00 + + ββ11XXii


E(YE(Yii | D| Dii = 1) = (= 1) = (αα00++αα11) + () + (ββ11++ββ22)X)Xii

DDii = 1, if male worker= 1, if male worker= 0, otherwise.= 0, otherwise.Interaction termInteraction term


163-Mar-14

Dummy XDummy X--variablesvariablesA

gri

cu

ltu

ral

wag

e r

ate

Ag

ricu

ltu

ral

wag

e r

ate

XX

αα00

ββ11ββ11+ + ββ22

αα11

((αα00++αα11))


Dummy YDummy Y--variablesvariables

Discrete Choice ModelsDiscrete Choice Models Many situations Many situations in in which which the the dependent variable dependent variable is is not a continuous variablenot a continuous variable..

Discrete Discrete or, or, qualitativequalitative


Dummy YDummy Y--variablevariable

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In general 2 types:In general 2 types:

1.1. dependent variables which take one dependent variables which take one of two values of two values (binary/ dichotomous choice), and(binary/ dichotomous choice), and

2. dependent variables which can 2. dependent variables which can take more than two values but are take more than two values but are finite (finite (polychotomouspolychotomous; multiple ; multiple choice).choice).

Dummy YDummy Y--variablevariable

Choose A Don’t Choose A

Individual i

To be Not to be

Binary Choice Model

Binary Choice Model

Choose A Don’t Choose A

Individual i

By car Not

Individual i

Alternatives J

2cycle

3car …

1walk bus train

Multinomial Choice Model

ExamplesExamples

•• Labour Force Labour Force Participation:Participation:

-- occupational choice (multiple choice)occupational choice (multiple choice)-- employed or unemployed (binary choice)employed or unemployed (binary choice)-- to be employed fullto be employed full--time, parttime, part--time or time or

unemployed unemployed (multiple choice(multiple choice))


Binary Choice Model Binary Choice Model

•• Voting Voting Behaviour:Behaviour:

-- to vote or not to vote (binary choice)to vote or not to vote (binary choice)-- to vote to vote Congress, BJP, Communists, Congress, BJP, Communists, or or

abstain abstain (multiple (multiple choicechoice))


ExamplesExamplesBinary Choice Model Binary Choice Model

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Censored Data:Censored Data:Limited dependent variable:Limited dependent variable:

Housing expenditure Housing expenditure ––some may not have purchased: some may not have purchased: so zeroso zero

ExamplesExamplesBinary Choice Model Binary Choice Model

Two questions Two questions ––

(i)(i) Can we still use OLS to estimate such Can we still use OLS to estimate such outcomesoutcomes??

(ii)(ii) If If not, how do we model such outcomes?not, how do we model such outcomes?



Binary Choice Binary Choice ––

(i)(i) Linear Probability Model (LPM)Linear Probability Model (LPM)

(ii)(ii) LogitLogit/ / ProbitProbit ModelModel

Censored/ Limited Dependent Variable Censored/ Limited Dependent Variable Regression ModelRegression Model

(i)(i) TobitTobit ModelModel




283-Mar-14

Linear Probability ModelLinear Probability Model

We We focus on single equation binary outcomes:focus on single equation binary outcomes:

A fundamental difference between a quantitative A fundamental difference between a quantitative response model response model and and a qualitative response a qualitative response model:model:

The latter The latter is is a a probability model. probability model.

{ }10 ,∈iy


29


In general:In general:

ProbProb(event (event j occursj occurs) ) = P(Y = j= P(Y = j) ) = f (relevant variables; = f (relevant variables; parametersparameters))= f(x= f(x ii, , ββ))

where [ ]ik1ii x,...,xx =

are the variables and β is a vector of parameters.

;),x(fP)x1y(P iiii β===

;),x(f1P1)x0y(P iiii β−=−==


Given Given

yi follows Bernoulli probability distribution

{ }10 ,∈iy


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313-Mar-14

How do we How do we specify f(xspecify f(xii,,ββ)?)?

•• Linear Probability ModelLinear Probability Model

An obvious choice is the familiar least squares procedure:

β=β ii x),x(f ⇒ iii uxy +β=

This leads to the linear probability model (LPM).

)1y(P1)0y(P0)xy(E iiii =⋅+=⋅=

β=== iiiii x)x1y(P)xy(E

Conditional expectation = conditional probability

The regression eqn describes the probability that yi = 1 given information on xi


Assuming E(u) = 0, it follows that


XXk

1

0

β1 +β2Xk

y, P

β1

The probability of the event occurring, p, is assumed to be a linear function of the variable X.


The case of a single explanatory variable:

yi = β1 +β2 Xi


Now an example….Now an example….CDS Mphil Econometrics Vijayamohan


Changed into binary: 0 = none; 1 = ≥ 1

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0 → 4511 → 150

F → 315M → 286


363-Mar-14

N

Min

Statistic

Max

Statistic

Mean

Statistic

Std.

Dev

Skewness

Kurtosis

Statistic

Std.

Error Statistic

Std.

Error

Have extramarital

affairs 601 0 1 0.250 0.433 1.160 0.100 -0.656 0.199

Sex 601 0 1 0.476 0.500 0.097 0.100 -1.997 0.199

Age group 601 17.5 57 32.488 9.289 0.889 0.100 0.232 0.199

Married years group 601 0.125 15 8.178 5.571 0.078 0.100 -1.571 0.199

Have children 601 0 1 0.715 0.452 -0.958 0.100 -1.087 0.199

Religiocity group 601 1 5 3.116 1.168 -0.089 0.100 -1.008 0.199

Education group 601 9 20 16.166 2.403 -0.250 0.100 -0.302 0.199

Occupation group 601 1 7 4.195 1.819 -0.741 0.100 -0.776 0.199

Marriage rating

group 601 1 5 3.932 1.103 -0.836 0.100 -0.204 0.199

Interpretation ?

LPM LPM –– Ray Fair ModelRay Fair Model

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UnstandardizedStandar-

dized

Beta

t Sig.

B Std. Error

(Constant) 0.736 0.152 4.859 0.000

Sex 0.045 0.040 0.052 1.129 0.259

Age -0.007 0.003 -0.159 -2.463 0.014

Married years 0.016 0.005 0.206 2.911 0.004

Have children 0.054 0.047 0.057 1.168 0.243

Religiocity -0.054 0.015 -0.145 -3.608 0.000

Education 0.003 0.009 0.017 0.360 0.719

Occupation 0.006 0.012 0.025 0.499 0.618

Marriage rating -0.087 0.016 -0.223 -5.472 0.000

Dependent Variable: Extramarital affairs

Interpretation ?



Chap 13-39



sex Ageyears

married childrenreligiou

sEducat-

ionOccup-ation

marriagrating

PredProb

1 57 15 1 1 20 7 1 0.614

Predicted Probabilities


0 57 15 1 1 20 7 1 0.569

1 57 15 1 1 20 7 5 0.2650 57 15 1 1 20 7 5 0.219


sex Ageyears


sEducat-

ionOccup-ation

marriagrating

PredProb

1 27 4 0 5 9 1 5 -0.0270 27 4 0 5 9 1 5 -0.072

Predicted Probabilities

NegativeNegative probability !probability !

LPM LPM –– Ray Fair ModelRay Fair ModelSome Some serious shortcomings.serious shortcomings.

(i)(i) The The distribution of the disturbance is nondistribution of the disturbance is non--normalnormal..

As y As y can can take only one take only one of two of two values, the values, the error term also error term also has a discrete (nonhas a discrete (non--normal) distributionnormal) distribution..

β−= ixyuThe probability distribution of u is:

β−=⇒= ii x1u1y

β−=⇒= ii xu0y

In effect, u follows a Bernoulli distribution


The Linear Probability Model

with Pi

with (1 – Pi)

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Normal PP Plot of Regression Standardized Residuals

Histogram



443-Mar-14


((ii) the ii) the error term is error term is heteroskedasticheteroskedastic

)x1)(x()u(Var ii β−β=

Which clearly varies with the value of xWhich clearly varies with the value of xii

Some serious shortcomings.Some serious shortcomings.


Chap 13-45



463-Mar-14

these these problems problems not insurmountablenot insurmountable::

•• Problem of nonProblem of non--normality can be normality can be

circumvented provided we have a large circumvented provided we have a large

sample size (invoke the central limit theorem)sample size (invoke the central limit theorem)

•• Problem of Problem of heteroskedasticityheteroskedasticity can be can be

removed by using White’s removed by using White’s heteroskedasticheteroskedastic

standard errorsstandard errors



473-Mar-14

The Linear Probability ModelThe Linear Probability Model

(iii) The main problem (iii) The main problem isis

the the NonNon--fulfilment of 0 fulfilment of 0 ≤≤ E(YE(Yii) ) ≤≤ 11

There is no There is no guaranteeguarantee that the predicted values that the predicted values of Y will all lie between 0 and 1. of Y will all lie between 0 and 1.

NegativeNegativeprobability !probability !

sex Ageyears


sEducat-

ionOccup-ation

marriagrating

PredProb

1 27 4 0 5 9 1 5 -0.0270 27 4 0 5 9 1 5 -0.072



What we require therefore is a way of constraining the LPM constraining the LPM so that the predicted probabilities do lie in the [0,1] range.

In general we use alternative estimation models to do this.

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493-Mar-14

The SolutionThe Solution

0.00

0.25

0.50

0.75

1.00

-8 -6 -4 -2 0 2 4 6

The usual way of avoiding this problem is to hypothesize that

the probability is a sigmoid (S-shaped) function of Z, F(Z),

where Z is a function of the explanatory variables.

Several mathematical functions are sigmoid in character.

The SolutionThe Solution

0.00

0.25

0.50

0.75

1.00

-8 -6 -4 -2 0 2 4 6


Alternatives to Alternatives to



523-Mar-14

Alternatives Alternatives

• The distribution

– Normal: PROBIT, natural for behavior

– Logistic: LOGIT, allows “thicker tails”

– Gompertz: asymmetric, underlies the

basic logit model for multiple choice

Underlying Probability Distributions for Binary ChoiceUnderlying Probability Distributions for Binary Choice



The The LogitLogit ModelModel

0.00

0.25

0.50

0.75

1.00

-8 -6 -4 -2 0 2 4 6

Z

Z

Z e1

e

e1

1)Z(F

+=

+= −

β= ixZ where

Several mathematical functions are sigmoid in Several mathematical functions are sigmoid in character. character.

One One is the logistic is the logistic function. function.

)(ZF

Z

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The The LogitLogit ModelModel

0.00

0.25

0.50

0.75

1.00

-8 -6 -4 -2 0 2 4 6

Z

Z

Z e1

e

e1

1)Z(F

+=

+= −

β= ixZ where

As As Z Z →→ ∞∞, e, e-- ZZ →→ 0 and p 0 and p →→ 1 1

(but cannot exceed 1).

As As Z Z →→ –– ∞∞, e, e-- ZZ →→ ∞∞ and and p p →→ 0 0

(but cannot be below 0).

)(ZF

Z


563-Mar-14

Normal distribution vs. Normal distribution vs. Logistic distributionLogistic distribution


573-Mar-14

Logistic distributionLogistic distributionThe Logistic distribution has density function: The Logistic distribution has density function:

wherewhere

aa is the mean of the distributionis the mean of the distribution

bb is the scale parameteris the scale parameter

ee is the base of the natural logarithm, Euler's e is the base of the natural logarithm, Euler's e

(2.71...) (2.71...)

2b/)az(

b/)az(

)e1(

e)b/1()z(f −−

−−

+=

Here a = 0; b = 1, 2, and 3


583-Mar-14

Logistic distributionLogistic distributionWith a = 0 and b = 1, the Logistic distribution has density

function:

Integrating the pdf gives the distribution function:

2z

z

)e1(

e)z(f −

−

+= –∞ < z < ∞

ze1

1)z(F −+

= –∞ < z < ∞

Here a = 0; b = 1, 2, and 3

Z

)(ZF


0.00

0.25

0.50

0.75

1.00

-8 -6 -4 -2 0 2 4 6Z

Z

Z e1

e

e1

1

+=

+= −

β= ixZ where

PPii = E(y = 1|X= E(y = 1|Xii) = F(Z)) = F(Z)

;e1

1P1

zi+

=−

zZ

z

i

i ee1

e1

P1

P=

++=

− −

;e1

1P

zi −+=

Odds ratioOdds ratio

The Logit Model: Odds Ratio


The Logit Model: Odds Ratio

)(ZF

0.00

0.25

0.50

0.75

1.00

-8 -6 -4 -2 0 2 4 6 Z

zZ

z

i

i ee1

e1

P1

P=

++

=− −

Odds ratio

β= ixZ where

Taking log of the odds ratio,Taking log of the odds ratio,β==

−= i

i

i xZP1

PlnL

L is called L is called LogitLogit..

Hence the Hence the LogitLogit modelmodel

Now an example….Now an example….

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0 → 4511 → 150

F → 315M → 286

LogitLogit –– Ray Fair ModelRay Fair Model

Regression variables estimates

BB S.E.S.E. WaldWald dfdf Sig.Sig. Exp(B)Exp(B)

SexSex 0.2800.280 0.2390.239 1.3741.374 11 0.2410.241 1.3241.324

Age Age --0.0440.044 0.0180.018 5.8815.881 11 0.0150.015 0.9570.957

Married years Married years 0.0950.095 0.0320.032 8.6558.655 11 0.0030.003 1.0991.099

Have childrenHave children 0.3980.398 0.2920.292 1.8611.861 11 0.1730.173 1.4881.488

Religiocity Religiocity --0.3250.325 0.0900.090 13.08913.089 11 0.0000.000 0.7230.723

Education Education 0.0210.021 0.0510.051 0.1740.174 11 0.6770.677 1.0211.021

Occupation Occupation 0.0310.031 0.0720.072 0.1860.186 11 0.6670.667 1.0311.031

Marriage Marriage

rating rating --0.4680.468 0.0910.091 26.55526.555 11 0.0000.000 0.6260.626

ConstantConstant 1.3771.377 0.8880.888 2.4072.407 11 0.1210.121 3.9643.964

CDS Mphil Econometrics VijayamohanCDS Mphil Econometrics Vijayamohan

Logit – Ray Fair ModelRegression variables estimates


Logit – Ray Fair ModelOdds ratio estimates

Regression variables estimatesRegression variables estimates

Logi

tLo

git––

Ray

Fai

r Mod

elR

ay F

air M

odel BB S.E.S.E. WaldWald Sig.Sig. Exp(B)Exp(B)

SexSex 0.2800.280 0.2390.239 1.3741.374 0.2410.241 1.3241.324

Age Age --0.0440.044 0.0180.018 5.8815.881 0.0150.015 0.9570.957

Married years Married years 0.0950.095 0.0320.032 8.6558.655 0.0030.003 1.0991.099

Have childrenHave children 0.3980.398 0.2920.292 1.8611.861 0.1730.173 1.4881.488

Religiocity Religiocity --0.3250.325 0.0900.090 13.08913.089 0.0000.000 0.7230.723

Education Education 0.0210.021 0.0510.051 0.1740.174 0.6770.677 1.0211.021

Occupation Occupation 0.0310.031 0.0720.072 0.1860.186 0.6670.667 1.0311.031

Marriage rating Marriage rating --0.4680.468 0.0910.091 26.55526.555 0.0000.000 0.6260.626

ConstantConstant 1.3771.377 0.8880.888 2.4072.407 0.1210.121 3.9643.964

Wald = (B/SE)Wald = (B/SE)22 = t= t22

Only 4 variables significantly different from zero Only 4 variables significantly different from zero at at αα = 0.05 = 0.05

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Logit – Ray Fair ModelB S.E. Wald Sig. Exp(B)

Sex 0.280 0.239 1.374 0.241 1.324

Age -0.044 0.018 5.881 0.015 0.957

Married years 0.095 0.032 8.655 0.003 1.099

Have children 0.398 0.292 1.861 0.173 1.488

Religiocity -0.325 0.090 13.089 0.000 0.723

Education 0.021 0.051 0.174 0.677 1.021

Occupation 0.031 0.072 0.186 0.667 1.031

Marriage rating -0.468 0.091 26.555 0.000 0.626

Constant 1.377 0.888 2.407 0.121 3.964

If B > 0, OR > 1; If B > 0, OR > 1; If B < 0, OR < 1. If B < 0, OR < 1. If B = 0, odds unchanged If B = 0, odds unchanged

z

i

i eP1

P=

−

Exp(Bi) = odds ratio =

Factor by which the odds change when the ithindependent variable ↑ by one unit.

e.g., When No. of years married e.g., When No. of years married ↑↑ by 1 unitby 1 unit, log , log of odds of odds for affairs for affairs ↑↑ by 1.099 or 9.9%, ceteris paribus. by 1.099 or 9.9%, ceteris paribus.


683-Mar-14


Exp(BExp(Bii) = ) = odds ratio =odds ratio =

z

i

i eP1

P=

−

BB S.E.S.E. WaldWald Sig.Sig. Exp(B)Exp(B)

SexSex 0.2800.280 0.2390.239 1.3741.374 0.2410.241 1.3241.324

Age Age --0.0440.044 0.0180.018 5.8815.881 0.0150.015 0.9570.957



ReligiocityReligiocity --0.3250.325 0.0900.090 13.08913.089 0.0000.000 0.7230.723





e.g., e.g., ReligiocityReligiocity significantly reduces significantly reduces incidence of incidence of

extramarital affairs! extramarital affairs!

‘Married years’ significantly contributes to ‘Married years’ significantly contributes to extramarital extramarital

affairs!affairs!

324.1e 28.0 =


693-Mar-14


Exp(BExp(Bii) = ) = odds ratio =odds ratio =

z

i

i eP1

P=

−

BB S.E.S.E. WaldWald Sig.Sig. Exp(B)Exp(B)

SexSex 0.2800.280 0.2390.239 1.3741.374 0.2410.241 1.3241.324

Age Age --0.0440.044 0.0180.018 5.8815.881 0.0150.015 0.9570.957



ReligiocityReligiocity --0.3250.325 0.0900.090 13.08913.089 0.0000.000 0.7230.723





Those who are married longer are about 1.1 times Those who are married longer are about 1.1 times

more likely to have more likely to have extramarital affairs than those extramarital affairs than those

recently married! recently married!


Logit – Ray Fair ModelModel Discrimination: Goodness of FitModel Discrimination: Goodness of Fit

Compares the observed and predicted group memberships.Cases with a cut value of 0.5 or greater are classified as having extramarital affairs.

>≤

= 5.0y if 1

5.0y if 0y

i

i*i


PredictedPredicted ObservedObserved

11

11

00

00

25/150

25/41435/451

435/560

Model DiscriminationModel Discrimination


1. The likelihood (L): probability of the observed results, given the parameter estimates.

As L is a small number, < 1, we use – 2 times the log of L (–2LL)

A good model is one with a high L of the observed results ⇒ small value for – 2LL.

(If a model fits perfectly, L = 1; –2LL = 0.)

Usually compare with the –2LL of a model with only the constant.

Goodness of Fit

Model DiscriminationModel Discrimination

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733-Mar-14

Logit – Ray Fair ModelGoodness of Fit

The likelihood (L):

Usually compare with the –2LL of a model with only the constant.

Model is good, if –2LL(with all variables) < –2LL(with only constant),

Model DiscriminationModel Discrimination3-Mar-14 CDS M Phil Econometrics

Vijayamohan74


Pseudo R2 : similar to R2. to quantify the proportion of explained ‘variation’ in the logistic regression model.

1. Cox & Snell R2 and Nagelkerke R2: (in SPSS)

(a) Cox & Snell R2 : N/2

2CS

)Max(L

)0(L1R

−=

Mod

el D

iscr

imin

atio

n


75


(a) Cox & Snell R2 :

L(0) = L(with only the constant): Constrained

L(Max) = L(with all variables): Unconstrained

N = sample size.

Cannot achieve a maximum value of 1.

N/22CS

)Max(L

)0(L1R

−=

Mod

el D

iscr

imin

atio

n


Logit – Ray Fair Model

(b)(b)NagelkerkeNagelkerke RR22::

where where RR22

CSCS = = Cox & Snell R2

Goodness of Fit

2max

2CS2

NR

RR =

N/22max )]0(L[1R −=

Mod

el D

iscr

imin

atio

n



Mod

el D

iscr

imin

atio

n


78

Logit – Ray Fair Model: Goodness of Fit

No. of correct predictions = 435 + 25 = 460No. of correct predictions = 435 + 25 = 460N (No. of observations) = 601N (No. of observations) = 601

Count RCount R22 = = (No. of correct predictions)/N(No. of correct predictions)/N460/601 460/601 = 0.765= 0.765

Mod

el D

iscr

imin

atio

n

3/3/2014

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Predicted ProbabilitiesPredicted Probabilities

i

i

Z

Z

ie

eyP

+==

11)( oror

;e1

1P

zi −+=

BB

SexSex 0.2800.280

Age Age --0.0440.044

Married years Married years 0.0950.095

Have childrenHave children 0.3980.398

Religious Religious --0.3250.325

Education Education 0.0210.021

Occupation Occupation 0.0310.031

Marriage rating Marriage rating --0.4680.468

ConstantConstant 1.3771.377

sexsex AgeAge

MarriedyMarriedy

earsears childrenchildren religiousreligious

EducatEducat--

ionion

OccupOccup--

ationation

marriagmarriag

ratingrating ZiZi PiPi

11 5757 1515 11 11 2020 77 11 0.7990.799 0.6900.690

00 5757 1515 11 11 2020 77 11 0.5180.518 0.6270.627

11 5757 1515 11 11 2020 77 55 --1.0751.075 0.2540.254

00 5757 1515 11 11 2020 77 55 --1.3561.356 0.2050.205

11 2727 44 00 55 99 11 55 --2.9042.904 0.0520.052

00 2727 44 00 55 99 11 55 --3.1843.184 0.0400.040

Another commonly used distribution: the probit.

PROBIT MODELPROBIT MODEL

Here the sigmoid function is the cumulative standardized normal distribution.

2Z2

1

e2

1)Z()Z(f

−

π=φ=

∫∞−

−

π=Φ=

Z Z2

1 2

e2

1)Z()Z(F

CDS Mphil Econometrics Vijayamohan –∞ < z < ∞


813-Mar-14

COMPARISON OF COMPARISON OF LOGIT LOGIT AND PROBITAND PROBIT

How do logit and probit models compare?

• Results quite similar although the logistic distribution has slightly fatter tails

• Variance of the probit is 1 (standard normal distribution).

For the logit it is 3

π


823-Mar-14

COMPARISON OF COMPARISON OF LOGIT LOGIT AND PROBITAND PROBIT

How do logit and probit models compare?

• Amemiya (1981) : the relationship between probit and logit models :

βprobit= 0.625βlogit and

βlogit= 1.6βprobit

ESTIMATION OF BINARY PROBIT ESTIMATION OF BINARY PROBIT AND LOGIT MODELSAND LOGIT MODELS

The logit and probit are non-linear. The parameters enter the regression model in a non-linear fashion.

We can no longer use OLS.

Hence the method of Maximum Likelihood.


dummy x -variables dummy y-variables - wordpress.com · 2011-05-09 · dummy y -variables discrete...

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