Ch11D3

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Why do some people go to college while others do not?

Why do some women enter the labor force while others

do not?

Why do some people buy houses while others rent?

Why do some people migrate while others stay?

1

BINARY CHOICE MODELS: LINEAR PROBABILITY MODEL

Financial Economists are often interested in the factors behind the decision-making ofindividuals or enterprises, examples being shown above.

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do not?



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The models that have been developed for this purpose are known as qualitative response orbinary choice models, with the outcome, which we will denote Y, being assigned a value of

1 if the event occurs and 0 otherwise.


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do not?



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Models with more than two possible outcomes have also been developed, but we willconfine our attention to binary choice models.


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The simplest binary choice model is the linear probability model where, as the nameimplies, the probability of the event occurring, p, is assumed to be a linear function of a set

of explanatory variables.


iii XYpp 21)1(

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5

XXi

1

0

1 +2Xi

y, p

Graphically, the relationship is as shown, if there is just one explanatory variable.

iii XYpp 21)1(


1

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Of course pis unobservable. One has data on only the outcome, Y. In the linear probabilitymodel this is used like a dummy variable for the dependent variable.


iii XYpp 21)1(

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Why do some people graduate from high school while

others drop out?

7

As an illustration, we will take the question shown above. We will define a variableGRADwhich is equal to 1 if the individual graduated from high school, and 0 otherwise.


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. g GRAD=0

. replace GRAD=1 if S>11

(523 real changes made)

. reg GRAD ASVABC

Source | SS df MS Number of obs = 570

---------+------------------------------ F( 1, 568) = 112.59

Model | 7.13422753 1 7.13422753 Prob > F = 0.0000

Residual | 35.9903339 568 .063363264 R-squared = 0.1654

---------+------------------------------ Adj R-squared = 0.1640Total | 43.1245614 569 .07579009 Root MSE = .25172

------------------------------------------------------------------------------

GRAD | Coef. Std. Err. t P>|t| [95% Conf. Interval]

---------+--------------------------------------------------------------------

ASVABC | .0121518 .0011452 10.611 0.000 .0099024 .0144012

_cons | .3081194 .0583932 5.277 0.000 .1934264 .4228124

------------------------------------------------------------------------------


8

The Stata output above shows the construction of the variable GRAD. It is first set to 0 forall respondents, and then changed to 1 for those who had more than 11 years of schooling.

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. g GRAD=0



. reg GRAD ASVABC


---------+------------------------------ F( 1, 568) = 112.59

Model | 7.13422753 1 7.13422753 Prob > F = 0.0000



------------------------------------------------------------------------------


---------+--------------------------------------------------------------------

ASVABC | .0121518 .0011452 10.611 0.000 .0099024 .0144012

_cons | .3081194 .0583932 5.277 0.000 .1934264 .4228124

------------------------------------------------------------------------------

9

Here is the result of regressing GRADon ASVABC. It suggests that every additional pointon the ASVABCscore increases the probability of graduating by 0.012, that is, 1.2%.


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. g GRAD=0



. reg GRAD ASVABC


---------+------------------------------ F( 1, 568) = 112.59

Model | 7.13422753 1 7.13422753 Prob > F = 0.0000



------------------------------------------------------------------------------


---------+--------------------------------------------------------------------

ASVABC | .0121518 .0011452 10.611 0.000 .0099024 .0144012

_cons | .3081194 .0583932 5.277 0.000 .1934264 .4228124

------------------------------------------------------------------------------


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The intercept has no sensible meaning. Literally it suggests that a respondent with a 0ASVABCscore has a minus 31% probability of graduating. However a score of 0 is not

possible.

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Unfortunately, the linear probability model has some serious shortcomings. First, there areproblems with the disturbance term.


iii XYpp 21)1(

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As usual, the value of the dependent variable Yiin observation ihas a nonstochasticcomponent and a random component. The nonstochastic component depends on Xiand

the parameters. The random component is the disturbance term.


iii XYpp 21)1( iii uYEY )(

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The nonstochastic component in observation iis its expected value in that observation.This is simple to compute, because it can take only two values. It is 1 with probability piand

0 with probability (1 - pi) The expected value in observation iis therefore 1 + 2Xi.



iiiii XpppYE 21)1(01)(

O O O O

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This means that we can rewrite the model as shown.




iii uXY 21

BINARY CHOICE MODELS LINEAR PROBABILITY MODEL

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XXi

1

0

1 +2Xi

Y, p iii XYpp 21)1( BINARY CHOICE MODELS: LINEAR PROBABILITY MODEL

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The probability function is thus also the nonstochastic component of the relationshipbetween Yand X.

1


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iii uXY 21 iii XuY 2111

iii XuY 210

In observation i, for Yi to be 1, uimust be (1 - 1 - 2Xi). For Yi to be 0, uimust be (- 1 - 2Xi).

iii uYEY )(


iii XYpp 21)1(


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XXi

1

0

1 +2Xi

Y, p iii XYpp 21)1( BINARY CHOICE MODELS: LINEAR PROBABILITY MODEL

1

17

The two possible values, which give rise to the observations A and B, are illustrated in thediagram. Since udoes not have a normal distribution, the standard errors and test

statistics are invalid. Its distribution is not even continuous.

A1 - 1 - 2Xi

B

1 + 2Xi


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XXi

1

0

1 +2Xi

Y, p


1

A1 - 1 - 2Xi

B

1 + 2Xi

18

Further, it can be shown that the population variance of the disturbance term in observation

iis given by (1 + 2Xi)(1 - 1 - 2Xi). This changes with Xi, and so the distribution isheteroscedastic.

)1)(( 21212

iiu XXi


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XXi

1

0

1 +2Xi

Y, p


1

A1 - 1 - 2Xi

B

1 + 2Xi

19

Yet another shortcoming of the linear probability model is that it may predict probabilities ofmore than 1, as shown here. It may also predict probabilities less than 0.


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The Stata command for saving the fitted values from a regression is predict, followed by thename that you wish to give to the fitted values. We are calling them PROB.

BINARY CHOICE MODELS: LINEAR PROBABILITY MODEL. g GRAD=0



. reg GRAD ASVABC


---------+------------------------------ F( 1, 568) = 112.59

Model | 7.13422753 1 7.13422753 Prob > F = 0.0000



------------------------------------------------------------------------------


---------+--------------------------------------------------------------------

ASVABC | .0121518 .0011452 10.611 0.000 .0099024 .0144012

_cons | .3081194 .0583932 5.277 0.000 .1934264 .4228124

------------------------------------------------------------------------------

. predict PROB


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. tab PROB if PROB>1

PROB | Freq. Percent Cum.

------------+-----------------------------------

1.000773 | 33 18.75 18.75

1.012925 | 16 9.09 27.84

1.025077 | 21 11.93 39.77

1.037229 | 26 14.77 54.55

1.04938 | 35 19.89 74.43

1.061532 | 20 11.36 85.80

1.073684 | 16 9.09 94.89

1.085836 | 3 1.70 96.591.097988 | 6 3.41 100.00

------------+-----------------------------------

Total | 176 100.00

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tab is the Stata command for tabulating the values of a variable, and for cross-tabulatingtwo or more variables. We see that there are 176 observations where the fitted value is

greater than 1.



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------------+-----------------------------------

1.000773 | 33 18.75 18.75

1.012925 | 16 9.09 27.84

1.025077 | 21 11.93 39.77

1.037229 | 26 14.77 54.55

1.04938 | 35 19.89 74.43

1.061532 | 20 11.36 85.80

1.073684 | 16 9.09 94.89

1.085836 | 3 1.70 96.591.097988 | 6 3.41 100.00

------------+-----------------------------------

Total | 176 100.00

. tab PROB if PROB

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------------+-----------------------------------

1.000773 | 33 18.75 18.75

1.012925 | 16 9.09 27.84

1.025077 | 21 11.93 39.77

1.037229 | 26 14.77 54.55

1.04938 | 35 19.89 74.43

1.061532 | 20 11.36 85.80

1.073684 | 16 9.09 94.89

1.085836 | 3 1.70 96.591.097988 | 6 3.41 100.00

------------+-----------------------------------

Total | 176 100.00

. tab PROB if PROB

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------------+-----------------------------------

1.000773 | 33 18.75 18.75

1.012925 | 16 9.09 27.84

1.025077 | 21 11.93 39.77

1.037229 | 26 14.77 54.55

1.04938 | 35 19.89 74.43

1.061532 | 20 11.36 85.80

1.073684 | 16 9.09 94.89

1.085836 | 3 1.70 96.591.097988 | 6 3.41 100.00

------------+-----------------------------------

Total | 176 100.00

. tab PROB if PROB

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