It is a common practice to rescale data to assist the model estimation process. This paper describes a case where care is called upon when interpreting the results after rescaling. The case is shown as associated with the lognormally distributed coefficient in a mixed logit model. Implications of rescaling data on a normally distributed coefficient are also given for comparison.Research Methods/ Statistical Methods,

Hu, Wuyang

English

Research Papers in Economics

 
 
 
 
RURAL ECONOMY 
 
 
Dollars or Cents: Impacts of Rescaling Data on a Mixed Logit  
Model with Normally and Lognormally Distributed Coefficients  
 
Wuyang Hu 
 Staff Paper 04-01 
 
 
 
Staff  Paper 
 
 
 
 
 
 
 
 
 
 
 
 
 
  Department of Rural Economy 
Faculty of Agriculture, Forestry  
and Home Economics 
University of Alberta 
Edmonton, Canada 
  
 
 
 
Dollars or Cents: Impacts of Rescaling Data on a Mixed Logit Model with Normally 
and Lognormally Distributed Coefficients 
 
Wuyang Hu 
 
Staff Paper 04-01 
 
 
November 2004 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
The author is Postdoctoral Fellow, Department of Rural Economy, University of Alberta.  
 
The author thanks Wiktor Adamowicz, Kenneth Train, and David Hensher for helpful 
comments.  The author is responsible for any remaining errors. 
 
The purpose of the Rural Economy ‘Staff Papers’ series is to provide a forum to 
accelerate the presentation of issues, concepts, ideas and research results within the 
academic and professional community. Staff Papers are published without peer review.  
 
Copyright 2004 by Wuyang Hu. All rights reserved. Readers may 
make verbatim copies of this document for non-commercial purposes by any means, 
provided that this copyright notice appears on all such copies. 
 
Abstract 
It is a common practice to rescale data to assist the model estimation process.  This paper 
describes a case where care is called upon when interpreting the results after rescaling.  
The case is shown as associated with the lognormally distributed coefficient in a mixed 
logit model.  Implications of rescaling data on a normally distributed coefficient are also 
given for comparison.   
 
 
 
   1
Introduction 
The mixed logit (ML) model, sometimes also referred to as the random parameter logit or 
error component logit model (Train 2003), has proved to be flexible and useful in 
modeling taste heterogeneity in random utility models.  Also due to its simplicity 
compared with other flexible models (e.g., probit or mixed probit model), it has drawn a 
significant amount of attention in the economics, marketing, and transportation literature.  
This paper describes a problem in interpretation of the model results after rescaling the 
data that is specified to be associated with a lognormally distributed random utility 
coefficient.  Rescaling is a common practice used to alleviate numerical problems in 
estimation or simply for ease of interpretation.  However, in this paper we show that the 
t-ratio associated with the estimate of one particular parameter in a lognormally 
distributed coefficient is not independent from the weighting factor used in rescaling 
although the likelihood function and model fit should be identical for the mixed logit 
model before and after rescaling the data.  This finding prevents the attempt to interpret 
that estimated parameter directly.   
 
Mixed Logit Model 
Choice models can be used to describe under what conditions consumers are more likely 
to make a purchasing/participating decision versus not.  In these models, researchers 
often wish to assume, for example, the coefficient associated with the price variable to be 
non-positive.  If the price coefficient is to be specified as random corresponding to the 
belief that consumers may be different in their “taste” or sensitivity towards price, the 
density function of the coefficient cannot have any mass on negative values.  In the   2
literature researchers have applied a series of different types of distributions to satisfy this 
restriction, including lognormal, gamma, Rayleigh, and various truncated symmetric 
distributions.  Among these the lognormal distribution is by far the most commonly used 
distribution (e.g., Train 1998; Hensher and Greene 200).   
 
In a random utility model (RUM), the utility of consumer n choosing alternative i among 
J total available alternatives can be written as: 
  ni ni ni e V U + = ,  J i ,..., 2 , 1 =        ( 1 )  
where ni V and ni e are respectively the deterministic and stochastic portion of the utility 
from the perspective of the researcher.  Suppose three variables X1, X2, and X3 are used to 
directly describe ni V , then in a parametric context, one can write: 
  3 3 2 2 1 1 X X X V n n ni β β β + + =         ( 2 )  
X1 to X3 are respectively assumed to be an alternative specific constant (ASC), a 
continuous quality index, and the price variable multiplied by negative one.  To allow the 
model to be representative and incorporate both fixed and random coefficients, we 
specify 1 β  as a fixed coefficient and n 2 β  and n 3 β as random coefficients with normal and 
lognormal distributions respectively.  We further assume that a variable Z (e.g., a 
demographic characteristic) can be used to explain the heterogeneity in the “average” of 
each of the two random coefficients across the sampled individuals.  The most popular 
parameterization for n 2 β  and n 3 β  in the literature are as follows:  
  a a n Z a a ε σ β + + = 1 0 2 ,  ( ) 1 , 0 ~ Normal a ε   
 a n d            ( 3 )    3
() b b n Z b b ε σ β + + = 1 0 3 exp ,  ( ) 1 , 0 ~ Normal b ε  
 
For simplicity, we assume that a ε and b ε are independent.  Parameters 0 α , 1 α , 0 b , 1 b , a σ  
and b σ are to be estimated.  Substituting (3) into (2) and then (2) into (1), one generates 
the complete indirect utility function.  If ni e is assumed to be iid Gumbel distributed, the 
implied choice probability can be represented by a mixed logit model with choice 
probability: 
  () () ∫ = 3 2 3 2 β β β β d d f f P P n n ni ni        ( 4 )  
 where  ()
() ∑
=
=
J
j
nj
ni
ni
V
V
P
1
exp
exp
 
This probability does not have a closed form but it can be simulated.  Suppose
d β is the 
d-th draw of n 2 β and n 3 β  from their corresponding density functions,  ( )
d
n
d
n
d
3 2 ,β β β =  
(draws of
d
n 2 β and
d
n 3 β are independent).  Given the total D numbers of draws, the 
simulated probability can be written as: 
  () ∑
=
=
D
d
d
ni ni P
D
P
1
|
1 ~
β          ( 5 )  
The simulated log-likelihood function for individual n is: 
  () ∑
=
=
J
i
ni ni P c SLL
1
~
ln          ( 6 )  
where 1 = ni c if alternative i is chosen by individual n.   
 
   4
The Problem of Rescaling  
It is well known that when the magnitudes of the variables in a model differ very 
significantly (say by a factor of 1000), the numerical procedure for maximization may 
have difficulty achieving convergence.  In this case, researchers usually proceed by 
scaling up (down) the excessively small (large) variables to help the maximization 
algorithm.  The requirement for scaling may come from the data directly.  For example, 
for a study on consumers’ choices of food items, the price variable may be measured by a 
small unit in the original data, such as cents rather than dollars, or in a small currency 
unit such as the Japanese Yen.  These indicate that the magnitude of the measures in the 
price data will be relatively large.  However, in the same model, all other variables may 
be relative small (e.g., dummy variables).  In these cases, scaling is often necessary.   
 
The model fit (LL value and
2 ρ statistic) should be identical before and after rescaling 
and if any X1 to X3 is scaled up (down) by a certain factor, the associated 
coefficient 1 β , n 2 β or n 3 β will be scaled down (up) by the same factor.  It is simple to show 
that the impact of rescaling X1 on 1 β  is the same as if in a linear regression model.  To 
examine the impact of scaling on the “deeper” parameters of the random coefficients, one 
can write:  
  = 2 2 X n β 2 2 1 2 0 X ZX a X a a aε σ + +  
 a n d            ( 7 )  
  = 3 3 X n β () [] 3 1 0 exp X Z b b b bε σ + + = ( ) ( ) ( ) ( ) ( ) 3 1 0 exp exp exp X b b
b
b
Z ε σ  
   5
For the normally distributed coefficient, it can be easily seen that holding Z unchanged, if 
X2 is scaled up by a factor f, then the estimated parameters a0, a1, and  a σ  will all be 
scaled down by f.  However, this is not obvious in the case of lognormally distributed 
coefficient.  Suppose X3 is scaled up by a factor f, then the expression 
() () () ( ) ()
b
b
Z b b
ε σ exp exp exp 1 0 will be scaled down by f.  In contrast to the case of the 
normal distribution, these terms are multiplicative rather than additive.  By further 
observing the three terms involved, it can be seen that parameter b1 is directly associated 
with variable Z which is not affected by rescaling, and  b σ is directly associated with the 
random variable b ε , which follows a standard normal distribution.  The only “free” 
parameter that can be scaled down in order for n 3 β to be scaled down by f is b0.  In other 
words, the estimated b0 after scaling must satisfy  
  ( )
()
f
b
b
scaling after
scaling before
=
0
0
exp
exp
         ( 8 )  
 
In simulated maximum likelihood estimation, the asymptotic variance of parameters is 
the corresponding diagonal of the inverse Hessian matrix.  It is known that the exact 
Hessian for a mixed logit model often cannot be inverted due to the complexity of the 
model.  Usually the approximated Hessian is computed instead and by far the most 
commonly used method is the BHHH algorithm.  Under this algorithm, the Hessian 
matrix is approximated by the outer product of the first-order derivative of the SLL with 
respect to the parameters: 
 
′
∂
∂
∂
∂
= ′
θ θ
SLL SLL
H          ( 9 )    6
where H’ is the approximated Hessian andθ stands for all the parameters involved in the 
model.  For parameters defining the normally distributed coefficient, it is not difficult to 
show that 
  ()
=
∂
∂
=
∂
∂
a a a
SLL SLL
n σ θ β , , 1 0 3 () ∑∑
== ∂
∂
⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
−
D
d a
n
J
j
njd nid
ni a a
X P X P
D P 1 1 0
2
1
2 2 , ,
1
~
1
σ
β
 
           = ∑∑
==⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
−
D
d
a
J
j
njd nid
ni
Z X P X P
D P 11
2 2
1
1
~
1
ε
      ( 1 0 )  
Since scaling the data does not affect the overall model result, ni P
~
 and nid P  will remain 
unchanged after scaling.  It can be seen that if X2 is scaled up by f, then  () a a a
SLL
σ , , 1 0 ∂
∂
will 
be scaled up by f as well.  This implies that the inverse of the H’ matrix will be scaled 
down by
2 f , which in turn implies that the estimated standard errors of the parameters 
in n 2 β will be scaled down by f.  Combining the result we have on the impact of scaling 
on the parameter estimates themselves, the conclusion is that the t-ratios of these 
parameters will remain unchanged before and after rescaling the data.   
 
Similarly, for the lognormally distributed coefficient n 3 β , the first-order derivative of the 
SLL with respect to the parameters is: 
  ()
=
∂
∂
b b b
SLL
σ , , 1 0 () ∑∑
== ∂
∂
⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
−
D
d b
n
J
j
njd nid
ni b b
X P X P
D P 1 1 0
3
1
3 3 , ,
1
~
1
σ
β
 
             = () b
n
b b σ
β
, , 1 0
3
∂
∂
Γ         ( 1 1 )  
However, given the expression of n 3 β in (3), it follows that:   7
  ()
n n b n
n
n
b
n Z
b b
3 3 ˆ 3
3
3
1 0
3
, ,
β β σ β
β
β
σ
β
= ⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
∂
∂
        ( 1 2 )  
If X3 is scaled up by f, then Γwill be scaled up by f as well.  However, since 
() b
n
b b σ
β
, , 1 0
3
∂
∂
is a function of n 3 ˆ β ,  n 3 ˆ β is scaled down by f due to the scaling in X3.  
Therefore, the expression () b b b
SLL
σ , , 1 0 ∂
∂
will not be affected by the scaling in X3.  This 
further implies that the estimated standard errors for the parameters in n 3 β will remain 
unchanged as well.  However, it is noted previously that the estimated parameter b0 will 
be changed to maintain the relationship in (8).  The resulting effect is that the t-ratio for 
parameter b0 will not be constant before and after rescaling of the data.  In other words, 
the t-ratio for b0 is dependent on the scaling factor f.  To complicate the situation, there is 
no “correct” scaling factor f one can use.  Any arbitrary rescaling may be used, dollars or 
cents for example, and still generate identical model fit but with different t-ratios 
associated with the parameter b0.   
 
Concluding Remarks 
The findings of this paper can be summarized as follows: 
1.  Scaling data will not affect the overall model fit and estimation.   
2.  Scaling variable with a normally distributed random coefficient changes the 
magnitude of the estimated parameters of the random coefficient but does not 
change their corresponding t-ratios.     8
3.  Scaling variables with a lognormally distributed random coefficient will not 
change the magnitude of the estimated parameters of the random coefficient 
except the constant term in the distribution.   
4.  Scaling variables with a lognormally distributed random coefficient will not 
change the standard errors associated with the parameters associated with the 
random coefficient.  However, the implied t-ratio of the constant term will be 
affected.   
 
It is noticeable that these results hold regardless of whether variables explaining 
heterogeneity in mean random coefficients are specified (the Z variable).  Researchers 
who wish to estimate a lognormally distributed coefficient in a mixed logit model should 
be aware of these impacts of scaling and not interpret the constant term separately but 
rather interpret the overall mean or median of the lognormal term, which in this study are 
given as ( ) ( ) 2 exp
2
1 0 b Z b b σ + +  and  ( ) 0 exp b .  The standard errors of these two measures 
can be obtained by the delta method or simulation.   
 
 
   9
References 
 
Hensher, D. A. and W. H. Greene (2003) “Mixed Logit Models: State of Practice” 
Transportation 30(2): 133-176.  
 
Train, K. (2003) Discrete Choice Models with Simulation Cambridge University Press, 
Cambridge.  
 
Train, K. (1998) “Recreation Demand Models with Taste Variation over People” Land 
Economics 74(2): 230-239.   
 
 

Dollars or Cents: Impacts of Rescaling Data on a Mixed Logit Model with Normally and Lognormally Distributed Coefficients

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics