Download Analyzing Household Choice: McFadden's Logit Model for Dryer Purchases and more Study notes Introduction to Econometrics in PDF only on Docsity!
Econ 513, USC, Fall 2005
Lecture 16. Discrete Response Models: McFaddenâs Conditional Logit Model for Gas/Electric Dryer Purchases
McFadden (1982) is interested in analyzing the choice by households to purchase an electric dryer, a gas dryer or no dryer at all. He uses a conditional logit model. The starting point is a indirect utility function that depends on the operating and capital cost of the device and interactions of the indicators for the choices with some individual characteristics.
The utility for the electric dryer for household i is
Ui,elec = β 0 ,elec + β 1 ,elec ¡ owni + β 2 ,elec ¡ personsi + β 3 ,elec ¡ gasavi
+βoper ¡ elecoperi + βcap ¡ eleccapi + ξi,elec.
The utility for the gas dryer for household i is
Ui,gas = β 0 ,gas + β 1 ,gas ¡ owni + β 2 ,gas ¡ personsi + β 3 ,gas ¡ gasavi
+βoper ¡ gasoperi + βcap ¡ gascapi + ξi,gas.
The utility for no dryer for household i is
Ui,no = β 0 ,no + β 1 ,no ¡ owni + β 2 ,no ¡ personsi + β 3 ,no ¡ gasavi + ξi,no.
(The operating and capital cost of no dryer are assumed to be zero by McFadden. He probably has not done much hand washing.) McFadden assumes that the three disturbances are independent, and identically distributed with extreme value distribution with cdf
F (Îľ) = exp(â exp(âÎľ)).
Household i chooses the electric dryer if
Ui,elec = max(Ui,elec, Ui,gas, Ui,no),
and similarly for the other options.
Define
U (^) i,elecâ = β 0 ,elec + β 1 ,elec ¡ own + β 2 ,elec ¡ persons + β 3 ,elec ¡ gasav
+βoper ¡ elecoper + βcap ¡ eleccap,
and similarly U (^) i,gasâ and U (^) i,noâ. The implication of the model is that the probability of buying an electric dryer is
Pr(elec) =
exp(U (^) i,elecâ ) exp(U (^) i,elecâ ) + exp(U (^) i,gasâ ) + exp(U (^) i,noâ )
and similarly
Pr(gas) =
exp(U (^) i,gasâ ) exp(U (^) i,elecâ ) + exp(U (^) i,gasâ ) + exp(U (^) i,noâ )
Pr(no) =
exp(U (^) i,noâ ) exp(U (^) i,elecâ ) + exp(U (^) i,gasâ ) + exp(U (^) i,noâ )
From data on the operating and capital cost, and the individual characteristics we cannot identify all parameters. Suppose we subtract an individual specific, choice-invariant ci from Ui,elec, Ui,gas, and Ui,no. That would not change the ranking, so we cannot tell that apart from the original model. So, choose
ci = âβ 0 ,gas â β 1 ,gas ¡ own â β 2 ,gas ¡ persons â β 3 ,gas ¡ gasav.
That would amount to fixing in the original model β 0 ,gas = β 1 ,gas = β 2 ,gas = β 3 ,gas = 0.
McFaddenâs estimates are given in Table 1.
Even more than in the binary logit and probit models these coefficients are difficult to interpret. So instead McFadden reports some elasticities. For example consider the elasticity of the probability of buying an electric dryer with respect to the operating cost of an electric dryer:
elec,elecoper =
âPr(elec) âelecoper
elecoper Pr(elec)
This elasticity will depend on the values of the covariates. We will evaluate the elasticities at the means of the variables, given in Table 2.
The derivative of the probability of buying an electric dryer with respect to the operating cost of an electric dryer is
âPr(elec) âelecoper
= βopercost ¡
exp(U (^) i,elecâ ) exp(U (^) i,elecâ ) + exp(U (^) i,gasâ ) + exp(U (^) i,noâ )
Table 2: Means
variable electric gas no
CHOICE 0.447 0.235 0.
CDOPCOST 31.17 7.56 0
CDCPCOST 233.20 258.80 0
GASAV 0.
OWN 0.
PERSONS 3.
from the paper, and it is likely to be very similar to the elasticities evaluated at the average values for the covariates. References
McFadden, D., (1982), âQualitative Response Models,â in Hildenbrand (ed.), Advances in Econometrics, Econometric Society Monographs, Cambridge University Press.
