For our purpose, let’s first define a ”KT Model”. Let s⇤ = (s⇤1, s⇤2, ..., s⇤M) denote the observed
costshares. Let Z denote the price variables plus other exogenous variables, and generally Zm=↵m + PMj mjlnpzj + O
where O represents other exogenous variables. Assume
sm = Zm+ ✏m (3)and✏ = (✏1, ✏2, ..., ✏M)|Z ⇠ N(0, ⌃) (4)
which implies ✏ is independent of Z.
Whether the observed costshares s⇤m equals 0 or the notional costshare sm are determined by KuhnTucker
conditions. More formally, s⇤ = Sc if conditions on s = (s1, s2, ..., sM) for regime c is satisfied. Sc
and corresponding KT regime conditions are characterized in more detail later. Assumptions (3),(4) and
the regime conditions for s⇤ define the ”KT Model”. Let D(s⇤|Z) = KT(Z,⌃) denote the conditional
distribution of observed costshares, and the density is denoted as g(⌘|Z,⌃) where ⌘ is a generic argument.
Step 2
Now, for m=2, we characterize the regime condition and D(s⇤|Z) = KT(Z,⌃) following the setup in
Step 1, and obtain g(⌘|Z,⌃). This part largely follows Wales and Woodland(1983), Lee and Pitt(1986),
and Lee and Pitt(1987). For simplicity, assume Z contains only price variables, but eventually the density
functions can be written in terms of Z which can include prices and other exogenous variables. For m=2,
there are 22
1 = 3 regimes.
(1) s⇤ = S1 ⌘ (s⇤1 > 0, s⇤2 > 0)
Regime conditions:
s⇤1 = s1 = Z1+ ✏1 (5)
So the likelihood function for this observation is
r1 = f(s⇤1Z1)
(6)
where f is a normal density function for ✏1.
(2) s⇤ = S2 ⌘ (s⇤1 = 0, s⇤2 = 1)
Regime conditions:s⇤1 = 0 , virtual price ⇠1 s.t.s1 = 0 and ln1 lnp1, ln1= (Z111lnp1+ ✏1)/11 lnp1, ✏1 Z1
(7)
The last equivalence holds because 11
< 0. Otherwise, (✏1, ✏2, ✏3) that satisies regime (2) would
overlap with that satisfies regime (1).
So we have the likelihood function as
r2 = P r(✏1 Z1)=Z Z11f(✏1)d✏1(8)
(3) s⇤ = S3 ⌘ (s⇤1 = 1, s⇤2 = 0)2
Similarly as regime(2), we have the likelihood function as
r2 = P r(✏2 Z2)=Z Z21f(✏2)d✏2
(9)
Then the density function corresponding to D(s⇤|Z) is given by
g(⌘|Z,⌃) = Y3，c=1[rc]1[s⇤=Sc] (10)
Step 3
Having defined KT Model and derived the corresponding distributions and density functions, the last
step is to adapt it to accommodate endogenous variables to use one-step MLE procedure. The model for
notional costshares with endogenous variables modeled as probit on the right-hand side can be written as:
sm = Z11m + ⇠my + um (11)y = 1[Z02+ v > 0] (12)
where (u, v) ⇠ N(0, ⌃) are independent of Z0, and V ar(v) = 1 as a normalization. Rewrite um as the
following:
um = ⇢mv + em (13)
where ⇢m ⌘ Cov(v, um), and em is independent of v2, v3 with a zero-mean Normal distribution. Since
(u, v) are independent of Z0, e = (e1, e2) are also independent of Z0. Therefore we have
e|Z1, y, v ⇠ N(0, ⌃e) (14)
and elements in ⌃e could be derived from ⌃.
If we rewrite sm as the following:
sm = Z11
m + ⇠my + ⇢mv + em (15)
then (14) and (15) fits the KT Model we defined in Step 1. This can be seen if we write as Z = (Z1, y, v)
and ✏ = e. So we have D(s⇤|Z1, y, v) = KT(Z11
+ ⇠y + ⇢v, ⌃e), and the corresponding density function
can be expressed as g(⌘|Z11
+ ⇠y + ⇢v, ⌃e). Then the log-likelihood for an individual i is given by
li(✓) = logf(s⇤i , yi|Z0i)= logf(s⇤i |yi, Z0i) + logf(yi|Z0i)= X
d2(0,1)1[yi = d]ldi(✓)
(16)
where ldi(✓) ⌘ logf(s⇤i |yi = d, Z0i) + logf(yi = d|Z0i). ldi(✓) can be obtained as the following: ld(✓) =8<:log hR Z02
1g(⌘|Z11+ ⇠y + ⇢v, ⌃e)(v)dvi
if d = 0
log hR 1Z02
g(⌘|Z11+ ⇠y + ⇢v, ⌃e)(v)dvi
if d = 1 (17)
where (v)
is the density function of normal distribution.
Parameter Constraints and Parameter Identification
Now, having derived the log-likelihood function for the Kuhn-Tucker model with endogenous variable,
we are able to apply it and proceed with MLE to estimate the model described by eq.(1) and (2) with
corner solutions.
The complete set of parameters to estimate is:1m = (↵m, m1,m2,m,⌧m, dm1, dm2, dm3, dm4)0, ⇠m, m = 1, 22= (↵y, y1,y2,&,y,⌧y, dy1, dy2, dy3, dy4)
and paramters in the variance-covariance matrix⌃ =0@2
However, there are constraints on the parameters. First, the shares summing to one results in the
following:
Applying also the symmetry in variance-covariance matrix
and homogeneous in prices (which implies m1+ m2= 0 for m = 1, 2), we are left with three sets of free
parameters: share equation parameters (↵1, 11,1,⌧1, d1j , ⇠1), endogenous variable equation parameters2,
and var-cov matrix parameters (2u1 , u1v),
that is, 21 free parameters in total to estimate. The model
is programmed in Gauss.