- Email: [email protected]
Rules of thumb for comparing multinomial logit and multinomial probit coefficients
235
Economics Letters 31 (1989) 235-238 North-Holland
RULES OF THUMB FOR COMPARING PROBIT COEFFICIENTS
MULTINOMIAL
LOGIT
AND MULTINOMIAL
Steven STERN University
of Virginia, Charlottesville,
VA 22901, USA
Received 23 January 1989 Accepted 12 April 1989
A rule of thumb is suggested for comparing multinomial logit coefficients with multinomial probit coefficients in the special case where the normal errors are distributed N(0, 1). The rule is a generalization of the ‘1.6’ rule for comparing logit and probit coefficients.
1. Introduction Polychotomous choice models have become popular in the economics literature. Unfortunately, the distribution of the errors must be assumed with little guidance from the theory of the economic model being estimated. Most researchers have assumed the errors were distributed either joint normal 1 or extreme value 2. Interpretation of the coeffients from discrete choice models is not as straightforward as from linear regression models, and it is difficult to compare coefficients across different distributional assumptions. For the case where the agent chooses between only two alternatives (i.e., logit and probit), the rule of thumb to compare coefficients is p,_ = 1.6 fir,, where pL are the logit coefficients and jjp are the probit coefficients. 3 Th.is paper proposes a rule of thumb for comparing multinomial logit and multinomial probit coefficients for which the ‘1.6’ rule is a special case. It is a generalization of Amemiya’s rationalization of the ‘1.6’ rule. 4
2. Notation Consider
the model
U,=x,P+e,,
,..., n,
i=1,2
where e, - iid F( .). The probability Pr[i]=Pr[U,> ’ * 3 4
L$]
that choice i is chosen
forall
(2.1) is
j#i.
See, for example, Hausman and Wise (1978). See, for example, Dubin and McFadden (1984). See Amemiya (1981, p. 1488). See Amemiya (1981, p. 1488).
0165-1765/89/$3.50
0 1989, Elsevier Science Publishers B.V. (North-Holland)
(2.2)
S. Stern / Rules of thumb for logit andprobit
236
coefficients
If the errors are normally distributed, then eq. (2.2) has a multinomial probit form. For the special case of independent and identically distributed errors, (2.3) where CDis the standard normal distribution function + is its density. This model has been considered by Heckman (1981) and Butler and Moffitt (1982). If the errors are distributed extreme value, then eq. (2.2) has a mulitnomial logit form:
(2.4)
Pr[il=w{4&)/ i exp{q&}. j=l
The goal of the paper is to derive a rule of thumb for comparing estimates of BP in eq. (2.3) to estimates of /3r in eq. (2.4).
3. Derivation of the rule The rule of thumb relies upon comparing aPr[i]/aX, evaluated at X, = X, for all j for the two cases. This is the method suggested by Amemiya to compare probit and logit. The multinomial derivative is
which, when evaluated at Xl = X, for all j, becomes &Pr[i] I
=ppJm (n-cc
I)@(u;)~-~+(u,)~
dui.
(3.2)
Through integration by parts, eq. (3.2) can be written as &Pr[ I
i] = $
=
Jm nu,@( u~)“-‘+( --m
$E max[u,,
u2 ,...,
ui) du,
~~3.
The value of eq. (3.3) when BP = 1 for different provides the value of eq. (3.3) for n = 2, 3,. . . ,20. 5
See, for example, Owen (1962, pp. 152-154).
(3.3) n can be taken from many sources. 5 Table 1
S. Stern / Rules of thumb for logit andprobit coefficients Table 1 Values of l/n
231
Emax[u,, u2 ,..., un]. iEmax[u,,
n
2 3
u2 ,..., un]
0.2821 0.2821 0.2574 0.2326 0.2112 0.1932 0.1780 0.1650 0.1539
4
5 6 I 8
9 10
n
iEmax[u,,
11 12 13 14 15 16 17 18 19 20
0.1442 0.1358 0.1283 0.1217 0.1157 0.1104 0.1055 0.1011 0.0971 0.0934
u2 ,..., u,]
The multinomial logit derivative is
&Pdij = &exp{ X,P,}/ I
I
= /?Pr[i](l
Ii ew{X,PL}
j=l
- Pr[i]),
(3.4)
which, when evaluated at X, = X, for all j, becomes &Pr[i] I
= PLn-‘(l
- K’).
(3.5)
The rule of thumb follows from comparing eq. (3.3) to eq. (3.5): 1 /3’;Emax[u,,
u,,...,u,]=P,~~l(l-n-‘).
(3.6)
This can be written in the form BP = a,,PL where a, is presented in table 2 for n = 2, 3,. . . ,20.
Table 2 Comparison factors: BP = a,PL. n
a”
n
0”
2 3 4 5 6 7 8 9 10
0.8862 0.7877 0.7284 0.6879 0.6576 0.6338 0.6145 0.5986 0.5848
11 12 13 14 15 16 17 18 19 20
0.5731 0.5625 0.5534 0.5450 0.5378 0.5307 0.5248 0.5190 0.5135 0.5086
238
S.Stern/ Rulesof thumb for logit andprobit coefficients
There are two things to notice in table 2. First, the standard rule of thumb for n = 2 is a special case of this rule. To see this, note that probit and logit are written in terms of the distribution of the difference of errors. Thus, the variance of the difference is two times the variance of the errors for the notation of this paper. If the entry in table 2 for n = 2 is divided by a, the proper normalizing factor, it becomes 0.6266, which is the reciprocal of 1.596. The second thing is that the comparison factors decline with n. Thus, using the ‘1.6’ rule of thumb for polychotomous discrete choice models will result in errors.
4. Conclusions The advantage of multinomial probit estimation over multinomial logit estimation is that it allows for a general covariance matrix. The rule described in this paper provides a way to compare multinomial probit and logit estimates and to decompose the difference into a component due to the general covariance matrix and a component due to normality-extreme value differences. Multinomial logit coefficients can be translated using table 2 and then be directly compared to multinomial probit estimates.
References Amemiya, Takeshi, 1981, Qualitative response models: A survey, Journal of Economic Literature, Dec. Butler, J.S. and Robert Moffitt, 1982, A computationally efficient quadratic procedure for the one-factor multinomial probit model, Econometrica, May. Dubin, Jeffrey and Daniel McFadden, 1984, An econometric analysis of residential electric appliance holdings and consumption, Econometrica, March. Hausman, Jerry and David Wise, 1978, A conditional probit model for qualitative choice: Discrete decisions recognizing interdependence and heterogeneous preferences, Econometrica, March. Heckman, J., 1981, Statistical models for discrete panel data, in: D. McFadden and C. Manski, eds., The econometrics of panel data (MIT Press, Cambridge, MA). Owen, D.B., 1962, Handbook of statistical tables (Addison-Wesley, Reading, MA).
Economics Letters 31 (1989) 235-238 North-Holland
RULES OF THUMB FOR COMPARING PROBIT COEFFICIENTS
MULTINOMIAL
LOGIT
AND MULTINOMIAL
Steven STERN University
of Virginia, Charlottesville,
VA 22901, USA
Received 23 January 1989 Accepted 12 April 1989
A rule of thumb is suggested for comparing multinomial logit coefficients with multinomial probit coefficients in the special case where the normal errors are distributed N(0, 1). The rule is a generalization of the ‘1.6’ rule for comparing logit and probit coefficients.
1. Introduction Polychotomous choice models have become popular in the economics literature. Unfortunately, the distribution of the errors must be assumed with little guidance from the theory of the economic model being estimated. Most researchers have assumed the errors were distributed either joint normal 1 or extreme value 2. Interpretation of the coeffients from discrete choice models is not as straightforward as from linear regression models, and it is difficult to compare coefficients across different distributional assumptions. For the case where the agent chooses between only two alternatives (i.e., logit and probit), the rule of thumb to compare coefficients is p,_ = 1.6 fir,, where pL are the logit coefficients and jjp are the probit coefficients. 3 Th.is paper proposes a rule of thumb for comparing multinomial logit and multinomial probit coefficients for which the ‘1.6’ rule is a special case. It is a generalization of Amemiya’s rationalization of the ‘1.6’ rule. 4
2. Notation Consider
the model
U,=x,P+e,,
,..., n,
i=1,2
where e, - iid F( .). The probability Pr[i]=Pr[U,> ’ * 3 4
L$]
that choice i is chosen
forall
(2.1) is
j#i.
See, for example, Hausman and Wise (1978). See, for example, Dubin and McFadden (1984). See Amemiya (1981, p. 1488). See Amemiya (1981, p. 1488).
0165-1765/89/$3.50
0 1989, Elsevier Science Publishers B.V. (North-Holland)
(2.2)
S. Stern / Rules of thumb for logit andprobit
236
coefficients
If the errors are normally distributed, then eq. (2.2) has a multinomial probit form. For the special case of independent and identically distributed errors, (2.3) where CDis the standard normal distribution function + is its density. This model has been considered by Heckman (1981) and Butler and Moffitt (1982). If the errors are distributed extreme value, then eq. (2.2) has a mulitnomial logit form:
(2.4)
Pr[il=w{4&)/ i exp{q&}. j=l
The goal of the paper is to derive a rule of thumb for comparing estimates of BP in eq. (2.3) to estimates of /3r in eq. (2.4).
3. Derivation of the rule The rule of thumb relies upon comparing aPr[i]/aX, evaluated at X, = X, for all j for the two cases. This is the method suggested by Amemiya to compare probit and logit. The multinomial derivative is
which, when evaluated at Xl = X, for all j, becomes &Pr[i] I
=ppJm (n-cc
I)@(u;)~-~+(u,)~
dui.
(3.2)
Through integration by parts, eq. (3.2) can be written as &Pr[ I
i] = $
=
Jm nu,@( u~)“-‘+( --m
$E max[u,,
u2 ,...,
ui) du,
~~3.
The value of eq. (3.3) when BP = 1 for different provides the value of eq. (3.3) for n = 2, 3,. . . ,20. 5
See, for example, Owen (1962, pp. 152-154).
(3.3) n can be taken from many sources. 5 Table 1
S. Stern / Rules of thumb for logit andprobit coefficients Table 1 Values of l/n
231
Emax[u,, u2 ,..., un]. iEmax[u,,
n
2 3
u2 ,..., un]
0.2821 0.2821 0.2574 0.2326 0.2112 0.1932 0.1780 0.1650 0.1539
4
5 6 I 8
9 10
n
iEmax[u,,
11 12 13 14 15 16 17 18 19 20
0.1442 0.1358 0.1283 0.1217 0.1157 0.1104 0.1055 0.1011 0.0971 0.0934
u2 ,..., u,]
The multinomial logit derivative is
&Pdij = &exp{ X,P,}/ I
I
= /?Pr[i](l
Ii ew{X,PL}
j=l
- Pr[i]),
(3.4)
which, when evaluated at X, = X, for all j, becomes &Pr[i] I
= PLn-‘(l
- K’).
(3.5)
The rule of thumb follows from comparing eq. (3.3) to eq. (3.5): 1 /3’;Emax[u,,
u,,...,u,]=P,~~l(l-n-‘).
(3.6)
This can be written in the form BP = a,,PL where a, is presented in table 2 for n = 2, 3,. . . ,20.
Table 2 Comparison factors: BP = a,PL. n
a”
n
0”
2 3 4 5 6 7 8 9 10
0.8862 0.7877 0.7284 0.6879 0.6576 0.6338 0.6145 0.5986 0.5848
11 12 13 14 15 16 17 18 19 20
0.5731 0.5625 0.5534 0.5450 0.5378 0.5307 0.5248 0.5190 0.5135 0.5086
238
S.Stern/ Rulesof thumb for logit andprobit coefficients
There are two things to notice in table 2. First, the standard rule of thumb for n = 2 is a special case of this rule. To see this, note that probit and logit are written in terms of the distribution of the difference of errors. Thus, the variance of the difference is two times the variance of the errors for the notation of this paper. If the entry in table 2 for n = 2 is divided by a, the proper normalizing factor, it becomes 0.6266, which is the reciprocal of 1.596. The second thing is that the comparison factors decline with n. Thus, using the ‘1.6’ rule of thumb for polychotomous discrete choice models will result in errors.
4. Conclusions The advantage of multinomial probit estimation over multinomial logit estimation is that it allows for a general covariance matrix. The rule described in this paper provides a way to compare multinomial probit and logit estimates and to decompose the difference into a component due to the general covariance matrix and a component due to normality-extreme value differences. Multinomial logit coefficients can be translated using table 2 and then be directly compared to multinomial probit estimates.
References Amemiya, Takeshi, 1981, Qualitative response models: A survey, Journal of Economic Literature, Dec. Butler, J.S. and Robert Moffitt, 1982, A computationally efficient quadratic procedure for the one-factor multinomial probit model, Econometrica, May. Dubin, Jeffrey and Daniel McFadden, 1984, An econometric analysis of residential electric appliance holdings and consumption, Econometrica, March. Hausman, Jerry and David Wise, 1978, A conditional probit model for qualitative choice: Discrete decisions recognizing interdependence and heterogeneous preferences, Econometrica, March. Heckman, J., 1981, Statistical models for discrete panel data, in: D. McFadden and C. Manski, eds., The econometrics of panel data (MIT Press, Cambridge, MA). Owen, D.B., 1962, Handbook of statistical tables (Addison-Wesley, Reading, MA).