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BINARY CHOICE MODELS: LOGIT ANALYSIS
The linear probability model may make the nonsense predictions that an event will occur with probability greater than 1 or less than 0.
XXi
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XZ 21
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.
BINARY CHOICE MODELS: LOGIT ANALYSIS
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Several mathematical functions are sigmoid in character. One is the logistic function shown here. As Z goes to infinity, e–Z goes to 0 and p goes to 1 (but cannot exceed 1). As Z goes to minus infinity, e–Z goes to infinity and p goes to 0 (but cannot be below 0).
BINARY CHOICE MODELS: LOGIT ANALYSIS
XZ 21
)(ZFZe
ZFp
11)(
Z
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The model implies that, for values of Z less than –2, the probability of the event occurring is low and insensitive to variations in Z. Likewise, for values greater than 2, the probability is high and insensitive to variations in Z.
BINARY CHOICE MODELS: LOGIT ANALYSIS
XZ 21
)(ZFZe
ZFp
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Z
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To obtain an expression for the sensitivity, we differentiate F(Z) with respect to Z. The box gives the general rule for differentiating a quotient and applies it to F(Z).
BINARY CHOICE MODELS: LOGIT ANALYSIS
VUY
2VdZdVU
dZdUV
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)1(
)1()(10)1(
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ee
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01 dZdUU
Z
Z
edZdV
eV
)1(
ZeZFp
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2)1()( Z
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dZdpZf
BINARY CHOICE MODELS: LOGIT ANALYSIS
The sensitivity, as measured by the slope, is greatest when Z is 0. The marginal function, f(Z), reaches a maximum at this point.
ZeZFp
1
1)()(ZF
Z
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For a nonlinear model of this kind, maximum likelihood estimation is much superior to the use of the least squares principle for estimating the parameters. More details concerning its application are given at the end of this sequence.
BINARY CHOICE MODELS: LOGIT ANALYSIS
ZeZFp
1
1)()(ZF
XZ 21
Z
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We will apply this model to the graduating from high school example described in the linear probability model sequence. We will begin by assuming that ASVABC is the only relevant explanatory variable, so Z is a simple function of it.
BINARY CHOICE MODELS: LOGIT ANALYSIS
ZeZFp
1
1)()(ZF
ASVABCZ 21
Z
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. logit GRAD ASVABC
Iteration 0: Log Likelihood =-162.29468Iteration 1: Log Likelihood =-132.97646Iteration 2: Log Likelihood =-117.99291Iteration 3: Log Likelihood =-117.36084Iteration 4: Log Likelihood =-117.35136Iteration 5: Log Likelihood =-117.35135
Logit Estimates Number of obs = 570 chi2(1) = 89.89 Prob > chi2 = 0.0000Log Likelihood = -117.35135 Pseudo R2 = 0.2769
------------------------------------------------------------------------------ grad | Coef. Std. Err. z P>|z| [95% Conf. Interval]---------+-------------------------------------------------------------------- asvabc | .1666022 .0211265 7.886 0.000 .1251951 .2080094 _cons | -5.003779 .8649213 -5.785 0.000 -6.698993 -3.308564------------------------------------------------------------------------------
BINARY CHOICE MODELS: LOGIT ANALYSIS
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The Stata command is logit, followed by the outcome variable and the explanatory variable(s). Maximum likelihood estimation is an iterative process, so the first part of the output will be like that shown.
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. logit GRAD ASVABC
Iteration 0: log likelihood = -118.67769Iteration 1: log likelihood = -104.45292Iteration 2: log likelihood = -97.135677Iteration 3: log likelihood = -96.887294Iteration 4: log likelihood = -96.886017
Logit estimates Number of obs = 540 LR chi2(1) = 43.58 Prob > chi2 = 0.0000Log likelihood = -96.886017 Pseudo R2 = 0.1836
------------------------------------------------------------------------------ GRAD | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1313626 .022428 5.86 0.000 .0874045 .1753206 _cons | -3.240218 .9444844 -3.43 0.001 -5.091373 -1.389063------------------------------------------------------------------------------
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In this case the coefficients of the Z function are as shown.
BINARY CHOICE MODELS: LOGIT ANALYSIS
ASVABCZ 131.0240.3ˆ
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iASVABCi ep 131.0240.31
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Since there is only one explanatory variable, we can draw the probability function and marginal effect function as functions of ASVABC.
BINARY CHOICE MODELS: LOGIT ANALYSIS
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ASVABCZ 131.0240.3ˆ
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iASVABCi ep 131.0240.31
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BINARY CHOICE MODELS: LOGIT ANALYSIS
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We see that ASVABC has its greatest effect on graduating when it is below 40, that is, in the lower ability range. Any individual with a score above the average (50) is almost certain to graduate.
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ASVABCZ 131.0240.3ˆ
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The t statistic indicates that the effect of variations in ASVABC on the probability of graduating from high school is highly significant.
BINARY CHOICE MODELS: LOGIT ANALYSIS
. logit GRAD ASVABC
Iteration 0: log likelihood = -118.67769Iteration 1: log likelihood = -104.45292Iteration 2: log likelihood = -97.135677Iteration 3: log likelihood = -96.887294Iteration 4: log likelihood = -96.886017
Logit estimates Number of obs = 540 LR chi2(1) = 43.58 Prob > chi2 = 0.0000Log likelihood = -96.886017 Pseudo R2 = 0.1836
------------------------------------------------------------------------------ GRAD | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1313626 .022428 5.86 0.000 .0874045 .1753206 _cons | -3.240218 .9444844 -3.43 0.001 -5.091373 -1.389063------------------------------------------------------------------------------
ASVABCZ 131.0240.3ˆ
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BINARY CHOICE MODELS: LOGIT ANALYSIS
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Strictly speaking, the t statistic is valid only for large samples, so the normal distribution is the reference distribution. For this reason the statistic is denoted z in the Stata output. This z has nothing to do with our Z function.
. logit GRAD ASVABC
Iteration 0: log likelihood = -118.67769Iteration 1: log likelihood = -104.45292Iteration 2: log likelihood = -97.135677Iteration 3: log likelihood = -96.887294Iteration 4: log likelihood = -96.886017
Logit estimates Number of obs = 540 LR chi2(1) = 43.58 Prob > chi2 = 0.0000Log likelihood = -96.886017 Pseudo R2 = 0.1836
------------------------------------------------------------------------------ GRAD | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1313626 .022428 5.86 0.000 .0874045 .1753206 _cons | -3.240218 .9444844 -3.43 0.001 -5.091373 -1.389063------------------------------------------------------------------------------
ASVABCZ 131.0240.3ˆ
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iASVABCi ep 131.0240.31
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BINARY CHOICE MODELS: LOGIT ANALYSIS
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The coefficients of the Z function do not have any direct intuitive interpretation.
ASVABCZ 131.0240.3ˆ
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However, we can use them to quantify the marginal effect of a change in ASVABC on the probability of graduating. We will do this theoretically for the general case where Z is a function of several explanatory variables.
BINARY CHOICE MODELS: LOGIT ANALYSIS
kk XXZ ...221
ZeZFp
1
1)(
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Since p is a function of Z, and Z is a function of the X variables, the marginal effect of Xi on p can be written as the product of the marginal effect of Z on p and the marginal effect of Xi on Z.
BINARY CHOICE MODELS: LOGIT ANALYSIS
iZ
Z
iii e
eZfXZ
dZdp
Xp 2)1(
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kk XXZ ...221
ZeZFp
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We have already derived an expression for dp/dZ. The marginal effect of Xi on Z is given by its coefficient.
BINARY CHOICE MODELS: LOGIT ANALYSIS
iZ
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eZfXZ
dZdp
Xp 2)1(
)(
2)1()( Z
Z
ee
dZdpZf
kk XXZ ...221
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1
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Hence we obtain an expression for the marginal effect of Xi on p.
BINARY CHOICE MODELS: LOGIT ANALYSIS
iZ
Z
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eZfXZ
dZdp
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)(
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2)1()( Z
Z
ee
dZdpZf
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kk XXZ ...221
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iZ
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eZfXZ
dZdp
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)(
2)1()( Z
Z
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dZdpZf
The marginal effect is not constant because it depends on the value of Z, which in turn depends on the values of the explanatory variables. A common procedure is to evaluate it for the sample means of the explanatory variables.
BINARY CHOICE MODELS: LOGIT ANALYSIS
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The sample mean of ASVABC in this sample is 51.36.
BINARY CHOICE MODELS: LOGIT ANALYSIS
. sum GRAD ASVABC
Variable | Obs Mean Std. Dev. Min Max-------------+-------------------------------------------------------- GRAD | 540 .9425926 .2328351 0 1 ASVABC | 540 51.36271 9.567646 25.45931 66.07963
Logit estimates Number of obs = 540 LR chi2(1) = 43.58 Prob > chi2 = 0.0000Log likelihood = -96.886017 Pseudo R2 = 0.1836
------------------------------------------------------------------------------ GRAD | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1313626 .022428 5.86 0.000 .0874045 .1753206 _cons | -3.240218 .9444844 -3.43 0.001 -5.091373 -1.389063------------------------------------------------------------------------------
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When evaluated at the mean, Z is equal to 3.507.
BINARY CHOICE MODELS: LOGIT ANALYSIS
. sum GRAD ASVABC
Variable | Obs Mean Std. Dev. Min Max-------------+-------------------------------------------------------- GRAD | 540 .9425926 .2328351 0 1 ASVABC | 540 51.36271 9.567646 25.45931 66.07963
Logit estimates Number of obs = 540 LR chi2(1) = 43.58 Prob > chi2 = 0.0000Log likelihood = -96.886017 Pseudo R2 = 0.1836
------------------------------------------------------------------------------ GRAD | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1313626 .022428 5.86 0.000 .0874045 .1753206 _cons | -3.240218 .9444844 -3.43 0.001 -5.091373 -1.389063------------------------------------------------------------------------------
507.336.51131.0240.321 XZ
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e–Z is 0.030. Hence f(Z) is 0.028.
BINARY CHOICE MODELS: LOGIT ANALYSIS
. sum GRAD ASVABC
Variable | Obs Mean Std. Dev. Min Max-------------+-------------------------------------------------------- GRAD | 540 .9425926 .2328351 0 1 ASVABC | 540 51.36271 9.567646 25.45931 66.07963
507.336.51131.0240.321 XZ
030.0507.3 ee Z
028.0)030.01(
030.0)1(
)( 22
Z
Z
ee
dZdpZf
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The marginal effect, evaluated at the mean, is therefore 0.004. This implies that a one point increase in ASVABC would increase the probability of graduating from high school by 0.4 percent.
BINARY CHOICE MODELS: LOGIT ANALYSIS
028.0)030.01(
030.0)1(
)( 22
Z
Z
ee
dZdpZf
004.0131.0028.0)(
iii
ZfXZ
dZdp
Xp
030.0507.3 ee Z
. sum GRAD ASVABC
Variable | Obs Mean Std. Dev. Min Max-------------+-------------------------------------------------------- GRAD | 540 .9425926 .2328351 0 1 ASVABC | 540 51.36271 9.567646 25.45931 66.07963
507.336.51131.0240.321 XZ
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In this example, the marginal effect at the mean of ASVABC is very low. The reason is that anyone with an average score is almost certain to graduate anyway. So an increase in the score has little effect.
BINARY CHOICE MODELS: LOGIT ANALYSIS
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To show that the marginal effect varies, we will also calculate it for ASVABC equal to 30. A one point increase in ASVABC then increases the probability by 2.9 percent.
BINARY CHOICE MODELS: LOGIT ANALYSIS
496.0701.0 ee Z
222.0)496.01(
496.0)1(
)( 22
Z
Z
ee
dZdpZf
029.0131.0222.0)(
iii
ZfXZ
dZdp
Xp
. sum GRAD ASVABC
Variable | Obs Mean Std. Dev. Min Max-------------+-------------------------------------------------------- GRAD | 540 .9425926 .2328351 0 1 ASVABC | 540 51.36271 9.567646 25.45931 66.07963
701.030131.0240.321 XZ
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An individual with a score of 30 has only a 67 percent probability of graduating, and an increase in the score has a relatively large impact.
BINARY CHOICE MODELS: LOGIT ANALYSIS
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. logit GRAD ASVABC SM SF MALE
Iteration 0: log likelihood = -118.67769Iteration 1: log likelihood = -104.73493Iteration 2: log likelihood = -97.080528Iteration 3: log likelihood = -96.806623Iteration 4: log likelihood = -96.804845Iteration 5: log likelihood = -96.804844
Logit estimates Number of obs = 540 LR chi2(4) = 43.75 Prob > chi2 = 0.0000Log likelihood = -96.804844 Pseudo R2 = 0.1843
------------------------------------------------------------------------------ GRAD | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1329127 .0245718 5.41 0.000 .0847528 .1810726 SM | -.023178 .0868122 -0.27 0.789 -.1933267 .1469708 SF | .0122663 .0718876 0.17 0.865 -.1286307 .1531634 MALE | .1279654 .3989345 0.32 0.748 -.6539318 .9098627 _cons | -3.252373 1.065524 -3.05 0.002 -5.340761 -1.163985------------------------------------------------------------------------------
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Here is the output for a model with a somewhat better specification.
BINARY CHOICE MODELS: LOGIT ANALYSIS
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. sum GRAD ASVABC SM SF MALE
Variable | Obs Mean Std. Dev. Min Max-------------+-------------------------------------------------------- GRAD | 540 .9425926 .2328351 0 1 ASVABC | 540 51.36271 9.567646 25.45931 66.07963 SM | 540 11.57963 2.816456 0 20 SF | 540 11.83704 3.53715 0 20 MALE | 540 .5 .5004636 0 1
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We will estimate the marginal effects, putting all the explanatory variables equal to their sample means.
BINARY CHOICE MODELS: LOGIT ANALYSIS
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Logit: Marginal Effects
mean b product f(Z) f(Z)b
ASVABC 51.36 0.133 6.826 0.028 0.004
SM 11.58 –0.023 –0.269 0.028 –0.001
SF 11.84 0.012 0.146 0.028 0.000
MALE 0.50 0.128 0.064 0.028 0.004
Constant 1.00 –3.252 –3.252
Total 3.514
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BINARY CHOICE MODELS: LOGIT ANALYSIS
The first step is to calculate Z, when the X variables are equal to their sample means.
514.3
...221
kk XXZ
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Logit: Marginal Effects
mean b product f(Z) f(Z)b
ASVABC 51.36 0.133 6.826 0.028 0.004
SM 11.58 –0.023 –0.269 0.028 –0.001
SF 11.84 0.012 0.146 0.028 0.000
MALE 0.50 0.128 0.064 0.028 0.004
Constant 1.00 –3.252 –3.252
Total 3.514
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BINARY CHOICE MODELS: LOGIT ANALYSIS
We then calculate f(Z).
030.0514.3 ee Z
028.0)1(
)( 2
Z
Z
eeZf
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The estimated marginal effects are f(Z) multiplied by the respective coefficients. We see that the effect of ASVABC is about the same as before. Mother's schooling has negligible effect and father's schooling has no discernible effect at all.
BINARY CHOICE MODELS: LOGIT ANALYSIS
iii
ZfXZ
dZdp
Xp )(
Logit: Marginal Effects
mean b product f(Z) f(Z)b
ASVABC 51.36 0.133 6.826 0.028 0.004
SM 11.58 –0.023 –0.269 0.028 –0.001
SF 11.84 0.012 0.146 0.028 0.000
MALE 0.50 0.128 0.064 0.028 0.004
Constant 1.00 –3.252 –3.252
Total 3.514
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Logit: Marginal Effects
mean b product f(Z) f(Z)b
ASVABC 51.36 0.133 6.826 0.028 0.004
SM 11.58 –0.023 –0.269 0.028 –0.001
SF 11.84 0.012 0.146 0.028 0.000
MALE 0.50 0.128 0.064 0.028 0.004
Constant 1.00 –3.252 –3.252
Total 3.514
33
BINARY CHOICE MODELS: LOGIT ANALYSIS
iii
ZfXZ
dZdp
Xp )(
Males have 0.4 percent higher probability of graduating than females. These effects would all have been larger if they had been evaluated at a lower ASVABC score.
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Individuals who graduated: outcome probability is
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This sequence will conclude with an outline explanation of how the model is fitted using maximum likelihood estimation.
BINARY CHOICE MODELS: LOGIT ANALYSIS
iASVABCe 2111
ASVABCZ 21
ASVABC
Z
e
eZFp
21111
1)(
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In the case of an individual who graduated, the probability of that outcome is F(Z). We will give subscripts 1, ..., s to the individuals who graduated.
BINARY CHOICE MODELS: LOGIT ANALYSIS
iASVABCe 2111
ASVABCZ 21
ASVABC
Z
e
eZFp
21111
1)(
Individuals who graduated: outcome probability is
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In the case of an individual who did not graduate, the probability of that outcome is 1 – F(Z). We will give subscripts s+1, ..., n to these individuals.
BINARY CHOICE MODELS: LOGIT ANALYSIS
iASVABCe 2111
iASVABCe 21111
Maximize F(Z1) x ... x F(Zs) x [1 – F(Zs+1)] x ... x [1 – F(Zn)]
Individuals who graduated: outcome probability is
Individuals did not graduate: outcome probability is
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Maximize F(Z1) x ... x F(Zs) x [1 – F(Zs+1)] x ... x [1 – F(Zn)]
Did graduate Did not graduate
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We choose b1 and b2 so as to maximize the joint probability of the outcomes, that is, F(Z1) x ... x F(Zs) x [1 – F(Zs+1)] x ... x [1 – F(Zn)]. There are no mathematical formulae for b1 and b2. They have to be determined iteratively by a trial-and-error process.
BINARY CHOICE MODELS: LOGIT ANALYSIS
iASVABCe 2111 iASVABCe 211
11
ns
s
ASVABCbbASVABCbb
ASVABCbbASVABCbb
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21121
21121
111...
111
11...
11