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3TH0D OF SIMULATED MOMENTS FOE ESTIMATION OF DISCRETE
RESPONSE MODELS WITHOUT NUMERICAL INTEGRATION
Daniel McFadden
No. 464
August 1987
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
A METHOD OF SIMULATED MOMENTS FOE ESTIMATION OF DISCEETE
RESPOHSE MODELS VITHOUT NUMERICAL INTEGRATION
Daniel McFadden
No. 464
August 1987
...T,
MAR
LIBRARIES
i
6
mo
A METHOD OF SIMULATED MOMENTS FOR ESTIMATION OF
DISCRETE RESPONSE MODELS WITHOUT NUMERICAL INTEGRATION
by
Daniel McFadden
Department of Economics
Massachusetts Institute of Technology
Cambridge, MA 02139
March 1986 (Revised July 1986, August 1987)
ABSTRACT
This paper proposes a simple modification of a conventional method of moments
estimator for a discrete response model, replacing response probabilities that
require numerical integration with estimators obtained by Monte Carlo
simulation.
This method of simulated moments (MSM) does not require precise
estimates of these probabilities for consistency and asymptotic normality,
relying instead on the law of large numbers operating across observations to
control simulation error, and hence can use simulations of practical size.
The method is useful for models such as high-dimensional multinomial probit
(MNP), where computation has restricted applications.
ACKNOWLEDGEMENTS
This research grows out of joint work with Kenneth Train on the estimation of
choice models containing variables measured with error.
I have particularly
benefited from discussions with Ariel Pakes and David Pollard, who pointed out
a lacuna in my original analysis of this problem, and whose independent
investigation of the asymptotic behavior of simulation experiments motivated
several critical steps in my proofs.
I have also benefited from suggestions
made by Chungrung Ai, Moshe Ben-Akiva, Chris Cavanagh, Vassilis Hajivassiliou,
Robert Hall, James Heckman, Hidehiko Ichimura, Charles Manski, Dan Nelson,
Peter Phillips, and Paul Ruud.
This research was supported in part by
National Science Foundation Grant No. SES-860S349.
KEYWORDS:
METHOD OF MOMENTS, SIMULATION, MULTINOMIAL PROBIT, DISCRETE RESPONSE
forthcoming, Econometrica
2009750
A Method of Simulated Moments for Discrete Response Models
McFadden
1.
INTRODUCTION
A classical method of moments estimator 8
mm
of an unknown parameter
*
vector 6
minimizes the (generalized) distance from zero of empirical moments
1
E
(1)
observations
Instrument
Vector
Expected
Response at
Observed
Response
6
mm
For some problems, the expected response function may be difficult to express
analytically or compute, but relatively easy to simulate.
When this function
is replaced by an asymptotically unbiased simulator such that the simulation
errors are independent across observations and sufficiently regular in
B,
the
variance introduced by simulation will be controlled by the law of large
numbers operating across observations, making
estimate each expected response.
it
unnecessary to consistently
This is the basis for the estimation method
developed in this paper, the method of simulated moments (MSM).
This paper focuses on application of MSM to the multinomial probit model.
However,
the method is more general and can be applied to most moment
estimation problems.
In a related paper,
Pakes and Pollard (1987) have
independently proposed minimum distance estimators using simulation, and have
established their statistical properties using combinatorial empirical process
methods.
Most of the statistical results in this paper could be obtained by
application of their methods.
Section 2 of this paper gives definitions and notation for discrete
response models.
Section 3 defines the MSM estimator and gives an informal
argument that it is consistent asymptotically normal (CAN).
Section 4
discusses issues of computation and statistical efficiency.
Sections 5-7
)
A Method of Simulated Moments for Discrete Response Models
McFadden
2
discuss applications of the method to discrete panel data with autoregressive
errors,
to discrete response models with measurement errors in explanatory
variables, and to non-normal discrete response problems.
An Appendix contains
formal statements of assumptions and results.
2.
DEFINITIONS AND NOTATION
Define C = {l,...,m> to be a set of mutually exclusive and exhaustive
alternatives.
=
u
(2)
A latent variable model for response from C is defined by
cxx
e C,
i
,
i
where a is a row vector of individual weights distributed randomly in the
population,
response
i
is a column vector of measured attributes of alternative
x.
is observed if u.
and
i,
£ u. for j e C (with zero probability of ties).
equal to one for the observed response,
Let d. denote a response indicator,
zero otherwise.
Assume a = a(6,T)) is a smooth parametric function of a random vector
with unknown parameter vector 6 taking true value 6
density of
and g (a|e) the induced density of a
75,
the mean and covariance matrix of
a.
rT
Define X„ =
for alternative
x
l
i,
(3)
(4)
1
= (u -u
u_
C-i
1
1
.
X_
.
C-i
m
)
and u_ = (u,,...,u
L
P (i|6,X_),
vector u_ with u, s u, for
L
let T(6)
it
g(Tj)
fS[B)
denote the
and
fl(6)
denote
is often convenient
be a upper triangular
= n.
(x,
L
f2:
Let
Let
.
In applications,
to work with a Cholesky factorization of
matrix satisfying
.
7),
j
1
€ C,
given X_.
U
,-u.,u.
1-1
The response probability
).
equals the probability of drawing a latent
J
u.
m
i'
1+1
Define
-u.,...,u
-u.
1'
m
1
= (x -x.,...,x.
-x.,...,x -x.).
-x.,x.
1'
1
i'i+l 1'
m 1
i-1
'
,
McFadden
A Method of Simulated Moments for Discrete Response Models
Then u__, has a multivariate density g (u
covariance matrix X
probability of u
'QX
9
,
X
)
with mean 0X
,
Kuc _ ii o)g u (u c _
i
|e,x )du _
c i
c
Hate.nJx^sOgCTjMTi,
= S
where 1(Q) denotes an indicator function for the event
Q.
When a is multivariate normal, one obtains the MNP model.
model,
For this
a can be written
a = a(6,7)) h £(6) + vT(6)
(6)
with
and
and P (i|e,X ) equals the nonpositive orthant
r
r
= a(8,T))X
P (i|e,x ) = S
c
c
(5)
,
1
3
j)
a row vector of independent standard normal variates.
In economic applications,
the latent variables u
often have the
interpretation of utility or profit, and P_(i|e,X_) is the choice probability
for a population of optimizing agents.
The attributes x
are functions of
observed characteristics of the alternatives and of the decision-makers, with
ax.
interpreted as an approximation to a general economic function of observed
and unobserved characteristics and of the deep parameter
Alternative-specific dummy variables may be included in
components of
8.
x.
;
the associated
can be interpreted as alternative-specific additive
a.
disturbances.
Let n = 1.....N index a random sample from the population,
observations (d_ X_ ) with d_ =
Cn Cn
Cn
(d,
,
In
log likelihood of the sample is
N
(7)
L(9) = E
£
„
n=l leC
.
d.
in
.
The associated score is
2
Ln p r (i|e,x r ).
C
Cn
'
d
mn
)
and X„ = (x,
Cn
In
yielding
x
mn
).
The
A Method of Simulated Moments for Discrete Response Models
McFadden
4
N
5L(e)/se =
(8)
E
Z
_
,
n=l ieC
,
W.
in
[d.
in
- P (i|e,x
)],
r
r
Cn
C
'
where
O)
w.
in
= aLn p_(i|e,x_ )/ae.
L
Ln
'
The primary impediment to practical maximum likelihood estimation of S
for the MNP model is computation of the (m-1) dimensional orthant
probabilities for u„_. to obtain P_(i|0,X_).
Direct numerical integration is
practical for m ^ 4 using a method of Owen (1956), modified by Hausman and
Wise (1978),
or expansions due to Dutt (1976).
Otherwise,
unless a has a
factor-analytic covariance structure with less than four factors,
it
is
usually impractical to carry out the large number of numerical integrations
Lerman and Manski (1981) suggest a
required to iteratively maximize (6).
Monte Carlo procedure for estimating P(i|e,X
models with large
m;
)
that can be applied to MNP
but find that it requires an impractical number of Monte
Carlo draws to estimate small probabilities and their derivatives with
acceptable precision.
Daganzo (1980) has developed approximate maximum
likelihood estimators for MNP using a normal approximation to maxima of normal
This approach has the drawbacks that the
variates suggested by Clark (1961).
accuracy of the approximation cannot be refined with increasing sample size,
and the method can be inaccurate when components have unequal variances; see
Horowitz,
3.
Sparmonn, and Daganzo (1982).
THE METHOD OF SIMULATED MOMENTS
The conventional method of moments estimator of a k x
6
in the discrete response model P
(
i
j
9,
X
)
satisfies
1
parameter vector
)
,
A Method of Simulated Moments for Discrete Response Models
McFadden
(10)
e
mm
= argmin Q (d-P(B)
)
'
a
where d-P(0) denotes the mN x
W W(d-P(9)
)
vector of residuals
1
d.
-
In
P_(i|9,X„ ) stacked
C
Cn
by observation and by alternative within observation, and where W is a K x mN
array of instruments of rank K £
evaluated at some fixed 6
)
6,
but are
in forming first-order conditions for solution of
The instrument array (9), evaluated at 6
(10).
of 6
The instruments may depend on
k.
(or at a consistent estimator
yields a method of moments estimator asymptotically equivalent to the
maximum likelihood estimator for
and hence asymptotically efficient.
6,
If
computation makes exact calculation of the efficient instruments impractical,
nevertheless provides a template for instruments that with relatively
(9)
crude approximations to P and its mN
moderately efficient estimators.
x k
array of derivatives P
will yield
3
Under mild regularity assumptions, sufficient conditions for classical
method of moments estimation to be CAN are (i) and (ii):
(
i
)
The instruments are asymptotical ly correlated with the score;
i
,
e,
,
«
-1
the array R =
1 i
m N
WP (8
9
J
is of maximum rank.
(ii) The conditional expectation of the residuals d-P(9),
instruments,
is zero if and only if 6 =
In the remainder of this section,
I
given the
.
will assume the instruments W are a
computationally practical fixed array, defined independently of
6.
(Approximation of the optimal instruments (S) is considered in Section
4.
The method of simulated moments (MSM) avoids the computation of P(6) required
for (10), replacing
it
with a simulator f(0) that is (asymptotically)
conditionally unbiased, given W and
d,
and independent across observations.
The MSM estimator is given by any argument
satisfying
A Method of Simulated Moments for Discrete Response Models
McFadden
(11)
(d-f(e
sm
))'WW(d-f(e
sm
s inf
))
8
(d-f(e))'WW(d-f(e))
+ 0(1).
Simulators for the Response Probabilities
An unbiased frequency simulator f(8) is readily calculated from the
latent variable model (2):
Draw one or more vectors
independently for each observation
the analysis.
Given trial
frequency f r (i|8,X
n,
from the density
g(r)),
and fix these draws for the remainder of
calculate u_ = a(8,7))X_ and calculate the
Cn
Cn
6,
with which component
)
7)
i
of u_
is largest.
This
simulator has discontinuities at values of 8 where there are ties for the
maximum component of u
.
For the MNP model,
the frequency simulator is
computed economically from (6) by drawing standard normal vectors
calculating u_ = (0(8)
Cn
It
is also
+ 7)r(6))X_
Cn
77
and
.
possible to construct smooth unbiased simulators f(8).
This
simplifies the iterative computation of the estimator, and its statistical
analysis.
Let y(u__,
)
denote a density chosen for the simulation that has the
nonpositive orthant as its support.
(12)
P (i|e,X
)
h
T h(u
,e,X
Then (5) can be rewritten
Mu
)du
where h(u_ — ,6,X_ — .) = g .(u„ _ |e,X_)/r(u_ .).
L 1
L 1
U
l,
1
L
L~l
.
T
Average h(u,6,X_ — .) for an
C 1
.
'
observation, using one or more Monte Carlo draws from y(u__.) that are taken
independently across observations and fixed for different
8.
This gives a
smooth positive unbiased estimator of P_(i|8,X_), provided f has sufficiently
thick tails so the expectation of h exists.
The density z can be chosen to
facilitate Monte Carlo draws and reduce simulation variance.
r is independent exponential
in each component,
For example,
if
then random variates from this
distribution can be calculated from logarithms of uniform random numbers from
A Method of Simulated Moments for Discrete Response Models
McFadden
(0,1).
7
Choices of y that make h flatter can reduce simulation error, as in
Monte Carlo importance sampling.
.,9,Xr _.) =
For MNP, h(u
n(u _ -p(e)X _ ,X^_ r(e)'r(e)X _ )/y(u
c 1
c i
i
c i
),
where n(v,A) denotes a
multivariate normal density centered at zero with covariance matrix
is exponential,
A.
When r
this h is uniformly bounded.
A potential drawback of smooth simulators based on (12)
not constrained to sum up to one for
i
is that they are
An alternative class of
e C.
kernel -smoothed frequency simulators are defined in Section 4 that satisfy
but are only asymptotically unbiased.
summing-up,
Section 4 also defines
special unbiased smooth frequency simulators for MNP.
Statistical Properties
I
shall argue that MSM estimators are CAN under mild regularity
The main result on the asymptotic properties of these estimators
conditions.
Theorem
is given in
1
below.
closed convex subset of [0,1]
I
will assume that the parameter space
,
that the true 8
is a
*
k
is in the interior of 0,
the explanatory variables have a distribution with compact support,
that
that the
response probabilities are uniformly bounded and twice continuously
differentiate with respect to
functions.
e,
Define a simulation bias B(6) =
if unbiased simulators are used,
(13)
sup
and that the instruments are smooth
Q
|B(6)
|
= o
'
N"
/£
Vf'(£f
(6)-?(6)
)
;
this is zero
and more generally is assumed to satisfy
4
(1).
Define a simulation residual process £(e) =
N*
1/£
W(f (9)-£f (6)
)
.
These
simulation residuals are by construction the normalized sum over observations
of independent identically distributed terms,
independent of d and uniformly
A Method of Simulated Moments for Discrete Response Models
McFadden
bounded, with E(C,(6) |W) =
for each
8 that by a standard central
I
8
Then £{B) is an empirical process in
6.
limit theorem is pointwise asymptotically normal.
shall need the following critical stochastic boundedness and stochastic
equicontinuity assumptions:
=
0(1),
(14)
sup
(15)
su P _.
lc(e)-C(e*)l = o (l),
0£A
P
N
ICC©)
Q
1
V2 |8-8
where A^ = {e\ N
I
|stO(-l)>.
prove these properties for smooth simulators in Proposition
this section;
Suppose
— —
1
at the end of
the technically more difficult case of simulators with
discontinuities is handled in Appendix Lemma
Theorem
1
.
the MSM estimator 8
defined by
11
(
satisfies Appendix
)
(These are stated informally as the assumptions
assumptions [Al] to [A10].
in the preceding paragraph.
sm
7.
)
Then 8
i_s
consistent
,
with N
1/2
(8
-8
*
)
converging in distribution to a normal vector with mean zero and covariance
matrix
G
I
sm
=
(R'rVr'G
sm
R(R'R)"
1
1
,
with R = lim N' WP„(6
8
)
and
1
sm
Proof:
= lim N" EW(d-f(e*))(d-f(e*))'W'
.
The argument parallels that of Pakes and Pollard (19S7).
The vector
W(d-f(8)) entering the defining condition (11) for the MSM estimator can be
decomposed into four terms,
(15)
N"
V2w(d-f(e)) h
+
[N"
1/2
[<;(e*)
- c(e)]
- B(e)
w(d-f(e*)) + B(e*)] -
[N'
V2w(P(e)-P(e*))].
The asymptotic properties of the estimator are argued by applying conditions
(13)-(15) to the first two terms in (16), and applying the following arguments
A Method of Simulated Moments for Discrete Response Models
McFadden
9
to the last two terms.
(a)
By construction of the simulator, N
expectation zero, given
-1/2.
%/(d-f(8
*
))
+ B(8
*
)
has
Random sampling plus the independence of the
W.
simulators across observations implies by a central limit theorem that this
term converges in distribution to a multivariate normal vector Z with mean
zero and covariance matrix G
(b)
The expression
5
sm
- N
<<\j(6)
probability to a smooth function
1/2
W(d-f(6
6
sm
is of rank k.
)
=
if and
6
Argument (a) implies
.
(11) satisfies
Hence,
(1).
u>(8)
«
= lim u„ V! (8
6N
)
e
=
))
converges uniformly in
))
with the properties that
Consider first the consistency of
N"
*
*
and that R = u (8
,
W(P(6)-P(6
co(8)
—
»
G = G
only if
-1
P
1
(17)
(d-f(6
N"
£
sm
))'W'W(d-f(6
1
N" inf
£ [N"
Q
sm
))
(d-f(8))'W'W(d-f(e)) + 0(N'
1/2
W(d-f(e*))]'[N"
1/2
W(d-f(e*))]
1
)
+ 0(N'
1
)
=
(1),
P
implying
(13),
N'
1/2
W(d-f (B
)
(14) and argument
=
)
Then,
(1).
one has u,,(8
N sm
(b),
uniformly outside each neighborhood of
zero.
Hence,
multiplying (16) by
the probability that 8
sm
)
= o (1).
p
But
N'
u,,
N
1/2
and using
converges
8
to a function bounded away from
is
contained in any neighborhood of
approaches one.
Next,
I
1/2
argue that N
(8
the condition N" V2W(d-f (8*
(18)
(1)
p
= N'
1/2
W(P(8
) )
sm
A Taylor's expansion yields
sm
=
-8
*
)
(1)
)-P(8*)).
is
stochastically bounded.
plus (13),
(14),
From (15),
and (17) imply
8
A Method of Simulated Moments for Discrete Response Models
McFadden
(19)
N"
1/2
w(P(e
sm
8
(8
= [R+o (1)]N
p
p
V2
(8
=0
k,
-e )]N
1/2
i
imply
+ o (1)
•
1/?
'
(8
sm
-8
*
)
=
(1).
p
/?
~
Let
8=8
— —
.1—
*
)
and then 8
= (R'R)
*
R'Z
o (1) =
+
C,{Q
)
- C,{6)
W(d-f(8
Then N
(1).
R'G
sm
R(R'R)
implies
1/2
(8-8*)
W(d-f(e)) = Z - [R + 0(8-8*). ]N
+
o
(1)
P
1
= [I-R(R'R)' R']Z + o (1).
P
From (13),
(22)
(15),
A = N"
1/2
and argument (b),
W(f(e)-f(6
= N"
V2w(P(e)-?(e
= N'
V2
v(P(e)-p(e
sm
sm
sm
))
))
+ c(e)
))
+ o
p
+ 3(e)
(l)
= RN
- ?(e
1/2
(e-e
sm
sm
)
)
- B(e
sm
+ o (1)
p
1
= R(R'R)" R'Z + o3 (1)
P
(
°
1
(23)
N'
(d-f(e))'WW(d-f(e))
+ o
sm
(l)
p
1
* N" (d-f(e
sm
)).
.
1/?
'
*
~
(8-8
Also,
Substituting 8 in (16) and applying the
= o (1).
Taylor's expansion (19) with 8 in place of 8
1/2
*
.1
R'N
(R'R)
An
is shown to be
sm
.i+
asymptotically normal with covariance matrix (R'R)
N"
).
*
is defined,
8
it.
Argument (a) implies N " (8-8
(21)
-e
-8*).
this implies N
asymptotically equivalent to
implies
sm
sm
asymptotically normal statistic
(15)
(e
consider the asymptotic normality of the MSM estimator.
Finally,
is
sm
R
—
Since R is of rank
+ o(e
)
S
= R + o (1) and 6
sm
p
)
(1)
(20)
= [n" wp (e
))
-
«
.1
Then N WP
1
)-p(e
10
))'WW(d-f(e
sm
))
1
= N' (d-f(e))'W'W(d-f(8))
+ 2N"
V2 (d-f(e))'W'A + A'A.
)
)
:
.
A Method of Simulated Moments for Discrete Response Models
McFadden
From (21) and (22),
But then A s RN
1/2
1/2
N'
(0-0
sm
)
(d-f (8)
+ o
p
)
'
(1)
WA
= o (1),
P
= o (1),
p
11
and (23) implies A'A = o (1).
P
implying
r * ° 6 sm and 6 asymptotically
* r
*
'
equivalent. P
The stochastic boundedness and equicontinuity conditions used in the
proof of Theorem
can be demonstrated for smooth simulators by the .following
1
argument
Proposition
If the simulator f (G)
1
j_s
uniformly bounded and twice
continuously differentiable, then (14) and (15) hold.
Proof:
A second-order Taylor's expansion of
(24)
c(e)-C(e*) = c D (e*HeV)
where
C,
00
is a k x k
+
(i/2)[n'
1/2
<
The array
C,
9
satisfies
E(C,
9
(8
))
= 0,
C
ee
=
=
)
p
8
(1).
sup.
(25)
1/2
(e-e*)]
),
with independence across
so a central limit theorem implies C„(&
contribution of each observation to the array
V2
ivec([ (e-e*)]' [N
•
.
observations,
N'
Do
array of second derivatives evaluated at points between
*
9 and
yields
about 6
C,
.
e€A
Hence,
(24)
implies,
C,
00
(1).
The
uniformly bounded, so
is
V2
|e-6*| i 0(1)},
for A^ = {0| N
1/2
ic(e)-c(e)| = o ti) -o (n'
),
?
?
N
establishing (15).
I
next establish (14),
cover [0,1]
k
with 2
ki
cubes with sides 2
-i
,
and let
point selected from each cube that intersects
0.
to be the nearest point in 0.;
<
From this construction,
Given an integer
using a "chaining" argument.
6.
For
be a set containing one
€ 0,
2
define 9
= 0. (6)
_1
_1
then |9-9.(9)|
i,
and
|0
(9)-0 (6)1
<
2
.
A Method of Simulated Moments for Discrete Response Models
McFadden
(26)
*
lc(e)l
ic(e
Z K(e
+
)|
)-c(e )|.
=l
i
I
12
shall need a version of Bernstein's inequality, giving an exponential bound
for sums of independent random variables:
distributed with
|Y
N
(27)
P{N"
1/2
£ Y
|
i
Let M £
1
EX
£ c and
|
>
I
t}
are independently Identically
If Y,
then for
= 0,
t
> 0,
7
2
2
V2
£ 2exp[-t /(2c +2ct/3N
)].
=l
be a uniform bound for 2, ^W, (f_(i|e,X- )-Ef„(i|e,X„ )) and for its
ieC in C
Cn
C
Cn
00
derivative with respect to
Note that
6.
£
i2
-i-3
for any
Then,
= 1/4.
1=1
C > 48M+8kM Ln 2,
(28)
P{sup lC(G)|
Q
C>
>
CO
1
_1 " 3
,
(e)-<(e.(e))|>i2
£ p<sup lc(e
Q
i+1
* p{|<(e )|>c/2> +
c>
i=l
CD
(29)
£ P{|c(e )|>c/2>
£ 2
+
1
i
(30)
ki
=l
sup p{K(e.
(e)-c(e.(e))|>i2"
+1
1
~3
c}
2
2
£ 2exp[-C /4(2M +MC/3)]
CO
+
£ 2
kl
2exp[-C
2 2
i
X
4
3
2
/(2M
*+2M2
4
x
Ci2
*
3
/3)]
i=l
CO
(31)
< 2exp[-C/4M]
+
£ 2exp[-iC/8M] i 5exp[-C/8M].
1=1
The inequalities (28) and (23) hold since left-hand-side events are contained
in the union of the right-hand-side events,
of the Bernstein inequality,
while (30) follows by application
and (31) by use of the bound on C and
manipulation of the exponential terms.
Given
c
>
0,
C can then be chosen
sufficiently large to make the right-hand-side of (31) less than
proves (14).
c.
This
A Method of Simulated Moments for Discrete Response Models
McFadden
13
The preceding arguments also apply to the classical method of moments
estimator by setting f(e) = P(8) and <(6) =
covariance matrix of the estimator is Z
the asymptotic
Then,
0.
1
mm
= (R'R)' R'G
mm
R(R'R)'
1
,
with
N
G
(32)
w
mm
= lim N"
1
)w; W
Z ( Z_ P C
r (i|e\x r
Cn in in
_1 L
'
Cn=i
E P c (i|e .x Cn )w in
W'
Cn
W
-
Cn_
.
c
Define
N
G
(33)
= lim
E
E
where C = C(9
).
Then G
W, £(<, C, ),
in Jn
in Jn
W;
n=l i.jeC
sm
= G
mm
+ G
ss
,
simulation to the asymptotic variance.
and G
ss
If f(6)
is the
is the
contribution of the
frequency simulator
obtained by r independent Monte Carlo draws for each observation, then G
= r
G
mm
and I
sm
= (1+r
)Z
mm
In this case,
.
one draw per observation gives
fifty percent of the asymptotic efficiency of the corresponding classical
method of moments estimator, and nine draws per observation gives ninety
percent relative efficiency.
Use of Monte Carlo variance reduction techniques
such as antithetic variates, or use of smooth simulators, may improve further
the relative efficiency of MSM.
4.
COMPUTATIONAL ISSUES AND STATISTICAL EFFICIENCY
Practical use of the MSM estimator requires that easily calculated,
moderately efficient instruments W be available, that the Monte Carlo
simulation of the probabilities and their derivatives be economical, that
iterative algorithms to compute the estimators be fairly stable and efficient,
,,
A Method of Simulated Moments for Discrete Response Models
McFadden
14
and that estimators for the asymptotic covariance matrix of the estimators be
computable.
Choice of Instruments
Consider first the question of suitable instruments.
The classical
method of moments estimator is asymptotically efficient if and only
if,
for stochastically negligible terms, W is proportional to SLnP(9 )/dd.
except
The
asymptotic efficiency of MSM relative to the classical moments estimator
approaches one if the expected response in (11) is simulated consistently.
The issue is how to construct W to obtain good asymptotic efficiency for MSM
without excessive computation.
For the MNP model,
the integral
(12) defining the response probability
can be differentiated with respect to £ and
(34)
T,
yielding
SP (i|e,X)/33 = XtX'nXj'V (u-£X)n(u-/3X,X'nX)du
L
u*0
1
X(X'nX)' T (u-£X)h(u,X,e)y(u)du,
u^O
(35)
sp (i|e,x)/sr
c
= rx(X'nx)"
1
s rx(X'nx)
where n =
TT, X
= X
,
S [(U-/3X)' (u-0X)-X nX]-n(u-/3X,X nX)du
u^O
,
-
(x'nxfV
S [(u-0X)'(u-/3X)-X'nX]-h(u,X,e)r(u)du}- CX'QX)
u^O
,
X'
and as before h(u,X,6) is the ratio of the
multivariate normal density to a Monte Carlo sampling distribution y(u) on the
nonpositive orthant.
g
Applying the chain rule to
/3(e)
and T(e) then yields
derivatives of the response probabilities with respect to the deep parameters
e.
For discrete response models other than MNP,
analogues of (34) and (35),
.
A Method of Simulated Moments for Discrete Response Models
McFadden
involving derivatives of the density g
with respect to
0,
15
can be defined, and
smooth Monte Carlo simulators constructed.
Economical Simulators
Simulation of (34) and (35) based on Monte Carlo draws from the density
y(u) yields smooth unbiased estimates of the derivatives.
simulator (12) of P_(i|e) is positive,
(35)
Since the smooth
the ratios of simulators of (34) and
to the simulator of (12) provides an approximation to the ideal
instruments 3Ln P /38.
The number of draws per observation must go to
infinity with sample size if the ideal instruments are to be estimated
consistently, permitting MSM to be asymptotically efficient.
However,
moderately efficient instruments can be obtained with relatively few draws.
It
is essential for the asymptotic statistics of the MSM estimator that the
simulation of (12) and the derivatives used to construct instruments be
independent of the draws used to simulate the expected response in (11);
however,
use of common draws from y(u) to simulate P_,
5P_/Sj3,
and 8P-./8T at
observation n may improve the approximation of the ideal instruments.
The frequency simulator of P (i|8,X_)
the smooth simulators (12),
permits easy draws.
(34),
For MNP,
is economical to compute,
as are
and (35) when the Monte Carlo density y(u)
a practical choice is y independent exponential,
allowing u to be drawn as a vector of logarithms of uniform random deviates.
However,
more accurate smooth simulators may be obtained with suitable
transf ormat i ons
First consider kernel -smoothed frequency simulators that satisfy
summing-up.
The construction of these simulators starts from a perturbation
of the latent variable model
(2),
A Method of Simulated Moments for Discrete Response Models
McFadden
u
(36)
= u
c
is a
where v
^b N
+
c
,
vector whose components are independently distributed with a
distribution function * and b
is a simulation parameter.
a finite moment generating function
/i(t)
for
Choose
N
t\,
so that for some c > 0,
b..
N
in a neighborhood of zero.
t
Associated with (36) is a
0.
->
N
Assume * has
response probability, obtained by first conditioning on u„_,
P (i|e,X
c
c
(37)
IB
)
=
K(u
J"
c_ i
/b )G(u _ |e,X )du _ =
c 1
N
c i
c
J"
,
K(a(8,
T^X^/b^g^di)
m-1
y m-1 ) = J
with K(y
1
(
*(v-y,)) *(dv).
II
,_.
As
b.,
N
J
approaches zero, K(u_ ./b M )
C-i
N
approaches the indicator function l(u__,s0), and P_(i|s,X r ) approaches
defined by
The kernel-smoothed frequency simulator is an
P_(i|9,X
)
average,
over Monte Carlo draws from
(5).
g(i))
of K(a(6,7j)X_ ,/b
,
If the simulators for all
simulator is nonnegative.
from common Monte Carlo draws,
i
€ C
are constructed
then they satisfy summing-up.
strictly positive if the support of * is the real line.
This
).
They are
Choosing * to be type
v
extreme value distributed yields a multinomial logit form K(v
m
m-1
g
1/(1 + "Z exp(-vj), and (37) is a multivariate normal mixture of logits.
I
)
=
A
J
j=l
polynomial kernel such as *(v) = [6+5v+(2-
computationally economical, and for small
|
v|
b.,
N
)
v
3
]/12 for
|v|
£
1
yields a smoothed simulator that
for most draws coincides with the unsmoothed frequency simulator.
variant of the kernel -smoothed frequency simulator is unbiased:
the form u_ = u_ + vb.„
C
C
N
10
For MNP, a
Write (2) in
with v a standard normal vector and u_ ~ W(|3X_,A'A),
C
with A upper triangular and
small enough so
is
X'rTX_-b
I
A' A =
X'rTX.-b
I;
C
this can be done provided b
is positive definite.
As in (37),
conditioning
is
.
A Method of Simulated Moments for Discrete Response Models
McFadden
on u
c
17
,
(38)
P (i|e,x ) = S
c
c
K(OXc _ i +TjA c _ 1 )/b N )g(u)d7i,
K(y
= S
m-1
y
.)
n *(v+y
(
»'(v)dv,
)
with g the multivariate standard normal density, * the standard normal
distribution, and A
column of
the array with columns A -A
Li
j
j
*
i
,
where A
is a
J
An average of K in (38) over Monte Carlo draws from g yields an
A.
unbiased positive smooth frequency simulator.
are used for
for
i
i
Adding-up holds if common draws
e C.
For MNP, construction of economical simulators is aided by the use of
spherical transformations.
Each of the expressions (12),
(34),
and (35)
involves simulation of integrals of the generic form
m
k
Q=;(Du.J
(39)
J
)n(u+u,A)du,
u£0 j=l
m
where
£ k.
is 0,
or
1,
2,
r
j
r
and A = X-.OX'
,
and s
0=1
nt
v.i
=
C(n,a,b)
j3X
Make the
..
o/2
m
transformation r =
c/ini
(40)
u =
n
J
e
= u ./v.
.
J
'
Define
J
-(r-b/a) a/2,
dr;
this is proportional to a parabolic cylinder function (Spanier and Oldham,
1987),
(41)
and satisfies the recursion
C(O.a.b) = (2:i/a)
1/2
*(b/a
V2
),
C(l,a,b) = C(0,a,b)b/a + e
C(n,a,b) = C(n-l,a,b)b/a
Then,
+
b /2a
/a,
C(n-2,
a, b) (n-1
)/a
(n
2:
2).
,
A Method of Simulated Moments for Discrete Response Models
McFadden
(42)
m
m
k
J
=
k,+m-l,a,b)(
n
s,
c
£
C(
c
Q
Z
S
1
J
J
°
j-1
J-l
18
),
where s is distributed uniformly on the intersection of the unit sphere and
the nonnegative orthant,
and
a = s'(X^_ nX _ )G i
i
1
s,
- " <a . (xj_ Qx . )-V
i
c i
c l
*»
1/2„-3m/2, _.. -1/2_,
,~%-1
= ,„
2
r ( m/2 )
( 2n )
n
*
c
i
o
i
,
1
= exp(-[px(x , nx)" x , p , -(£X(x ,
c
with X = X
and c
,
the distribution of
independent of X and
s,
m
Sj =
(43)
|u
(x nx)" s]/2),
To generate uniform draws from
s.
draw a standard normal random vector
j=l J
J
recursion (41) to calculate
variates for
and take
u,
„ ,1/2
is simulated by drawing one or more s,
(43)
1
/
/
|/
L
Then,
1
nxf s)/s
C.
and for each s using the
A further refinement is to use control
12
C.
The spherical transformation can also be used to calculate an economical
unbiased smooth frequency simulator for MNP.
unit sphere in R
K
,
and let A
2
Let s be a uniform draw from the
be a random variable with a Chi-squared
distribution with K degrees of freedom, denoted
variable model for MNP can be written u r =
computation yields a partition of
intervals may be degenerate.)
P„(i|e,X
O+AsDX
,
vector srX
s)
.
O
also smooth.
2
)
Given
.
where
j
is the
an easy
],
=
j
(Some of the
i,
given
s,
ascending rank of sTx.
The A. are smooth in 6 for almost all X
J
s,
is maximum.
The probability of response
= EL(A, .) - Hj,(X,),
the latent
Then,
into intervals [A., A.
[0,+co]
on which each of the components of u
0,...,m,
IL,(A
,
L*
so P (i|e,X
L*
,
w
s)
is
in the
is
The simulator is an average of the P (i|6,X ,s) for r random
A Method of Simulated Moments for Discrete Response Models.
McFadden
draws of
19
s.
The accuracy of simulators for MNP that are based on spherical
transformations can be improved substantially by use of antithetic variates.
IK
For uniform draws from the sphere
Deak (1980) gives an effective procedure:
in
K
IR
,
first draw a random basis s
,
.
.
.
,s
.
This can be done for example by
drawing K standard normal vectors and applying a Gram-Schmidt
Then use the 2K vectors ±s
orthonormalization.
(±s
± s
)
for
i
<
j,
,
or the 2K(K-1) vectors
as directions for the simulation.
Iterative Estimation methods
A practical estimation procedure is first to use relatively crude
to iterate to an initial consistent
instruments,
defined independently of
estimator
second to simulate the ideal instruments using (12),
at 6,
(35)
9,
6,
(34),
and
and third to carry out one or more iterations using the
approximately ideal instruments.
Good candidates for crude instruments are
low-order polynomials in the explanatory variables; e.g., X__. for
SLnP
c
(i
|e,X )/33 and X _.X^._. for SLnP
C
(
i
|6,X )/3r.
14
Consider iterative algorithms for calculation of MSM estimators.
When
smooth simulators are used for f(6) in (12), and the instruments W are defined
independently of
6,
then estimates can be computed by Newton-Raphson iteration
or a similar second-order method applied to minimize the criterion
(44)
(d-f(e))'W'W(d-f(e)).
This criterion may be irregular;
simulators may have local flats.
in particular,
Then,
kernel-smoothed frequency
optimization methods that use
McFadden
A Method of Simulated Moments for Discrete Response Models
20
non-local information, such as simulated annealing, may be more reliable; see
Press et al (1986).
When a frequency simulator is used,
(44)
constant in
is piecewise
and non-local methods must be used in iteration.
I
6,
have tried random search
algorithms and pseudo-gradient methods that adapt ively approximate slopes
using long baselines; the former have performed better.
For discrete
response models that can be parameterized in terms of mean and variance,
such as MNP, convergence can be accelerated using a method due to Manski:
Suppose r simulations per observation, and that starting from a trial 6
with A a step
* size to be determined.
a jump in (44) from draw
j,
observation
a
Consider (44) as a function of
search direction A8 has been determined.
6 +AA6,
o
,
n,
is
The value A
,
nj
at which there is
easily calculated. Then,
practical to enumerate the values of (44) at all the jump points X
.
it
is
and
choose a global minimum along this search direction.
Generally,
iteration using smooth simulators is faster that that using
frequency simulators.
However,
in applications where the number of
alternatives is very large, the burden of computing f(6) or approximations to
the optimal instruments for all alternatives may be excessive.
Then,
a
frequency simulator f(8) with r repetitions will be non-zero for at most r
alternatives, and the instruments need be computed only for these alternatives
plus the observed one.
For example, a single Monte Carlo draw for each
observation requires calculation of the instruments only for the observed and
drawn alternatives, and yields fifty percent of the efficiency of the
classical method of moments estimator, no matter how large the set of possible
alternatives.
Comparable reductions in computation can be achieved using a
kernel-smoothed frequency simulator with a kernel of bounded support.
A Method of Simulated Moments for Discrete Response Models
McFadden
21
Asymptotic Covariance Matrix
Consider estimation of the asymptotic covariance matrix of the MSM
estimator, £
(R'rYVg sm R(R'R)'
=
sm
1
A consistent estimator of G
.
sm
is
N
G
(45)
1
= N"
sm
E
Z
.
.
W.
.
_
n=l i.jeC
-
in
in
-f(i|e
'
sm
,X_ ))(d. -f(i|e
Cn
jn
'
sm
,X_ ))W'
Cn
jn
.
*
-1
The matrix R = lim N
Cd.
WP (9
)
is
consistently estimated by
1
R = N' WP Q
(46)
where P
is
,
an unbiased simulator of the array P
,
evaluated at
sm
or any
»
initially consistent estimator
of
,
obtained using one or more draws per
observation in (34) and (35) or their analogues in models other than MNP,
independent of any simulation used to compute
To show (45),
1.
note first that this expression with
in place of
sm
by a law of large numbers; see part (a) of the proof of
converges to G
Theorem
W.
Second,
by (13) and (15),
terms involving the difference of
*
f(0
and f(0
)
sm
)
are o (1).
p
The argument for (46)
necessary to use versions of (13) and (15) for P
simulators using the argument of Proposition
5.
.
is the same,
but it is
These hold for smooth
1.
DISCRETE PANEL DATA WITH AUTOREGRESSIVI ERRORS
Consider longitudinal discrete response data
(d,
,
v.n
= 1,...,N observed over t = 1.....T r
periods,
binary response and x
has 2
T
is a
where
d,
tn
x,
tn
)
for subjects n
= ±1 indicates a
vector of explanatory variables.
alternative response patterns,
large for long panels.
variable model that may be appropriate for such data is
This proble m
A latent
•
A Method of Simulated Moments for Discrete Response Models
McFadden
u
(47)
d
with c,
tn
tn
tn
= £
=
/3x
tn
+ c
= sign(u
n
+ v.
tn
,
tn
,
),
tn
£
22
a normal subject-specific disturbance,
n
first-order autoregressive disturbance, and
and independent across subjects.
£,
tn
a normal
v independent of each other
with variance
is stationary,
If c
v,
normalized to one, then
(48)
c.
tn
2
with X
= (l-X
2
1/2
)
7,
nn
On
t-2
2,V
m
+ X
t-1
J
the proportion of the variance in the autoregressive error,
serial correlation, and
probability P(d
|x
ri
.
jn
independent standard normal variates.
of d
,j3,X,p)
n
=
(d
ln
>
•
•
d
Tn
^
given x
equals the probability of a (T+l) dimensional draw
u
tn ttn
d,
>
for
t
=
(tj
,
.
(x
=
n
.
.
,
p the
The
x
Jn
t)_
)
Jn
)
such that
T.
1
Full maximum likelihood estimation of this model requires
to evaluate P(d
T-dimensional numerical integration
&
computationally impractical for T
>
4.
n
|x
1
n
,B,\,p),
which is
Ruud (1981) has developed practical
consistent estimators using partial likelihoods for small numbers of
adjacent periods;
see also Chamberlain (1984).
from initially consistent estimators,
The MSM method,
starting
offers a computationally practical way
A frequency simulator or a kernel-smoothed
to increase efficiency.
frequency simulator with a finite-support kernel, can be computed using
(41),
and with a moderate number of repetitions requires simulation of the
instruments for a practical number of alternatives per subject, even for
large
T.
Alternately,
it
may be possible to compute directly an unbiased
simulator of the score SLn P(d le.x )/8Q for the observed response pattern
n
d
.
1
n
From the analogues of (34) and (35) for the discrete panel data
1
A Method of Simulated Moments for Discrete Response Models
McFadden
23
this requires that one obtain asymptotically unbiased or consistent
problem,
simulators for the conditional expectation of first and second order
polynomials given draws from the nonpositive orthant.
Unbiased simulators
can be obtained by use of acceptance/rejection methods, or asymptotically
unbiased simulators by allowing the number of repetitions in the simulation
of the probability in the denominator of P dP/89 to go to infinity with
<
These approaches have been investigated by Ruud and McFadden
sample size.
(1987) and Hajivassiliou and McFadden (1987).
MSM estimation of discrete panel data models extends readily to more
general time-series covariance structures,
so long as it is practical to
Cholesky-f actor and invert the covariance matrix to obtain a representation
analogous to (48) for the c
in terms of independent normal variates,
and
so long as it is practical to construct instrument arrays for the deep
parameters of the problem.
The estimator can also be applied to models
with general state dependence, provided the initial value problem (Heckman,
1981) can be handled.
(49)
with £
u,
= £x.
d
= Sign{u
tn
n
+ \p±
r
.
t-l,n
tn
tn
For example,
tn
^n
+
v.
tn
,
]
a subject-soecif ic disturbance and v
°
disturbances are normal, and
(50)
+ £
consider the model
P(d
n
|x
1
n
,d n
On
,p,£
n
)
v,
tn
of the x process.
;
independent across
T
= n *(d. Ox,
+
tn
tn
L—
n
this is justified if x
ifjd,
If the
t.
-
has unit variance,
Suppose the conditional distribution of £
depend only on d
tn
+ £
.
t-l,n
given x
then
n
n
)).
and d n can be assumed to
On
is independent of the past
Suppose the inverse distribution of £
given d
flexible parametric form that spans the true inverse distribution.
history
is given a
Then the
A Method of Simulated Moments for Discrete Response Models
McFadden
response probabilities are given by the expectation of (50) with respect to
which can be simulated economically from the inverse distribution.
serial correlation to the disturbances
tn
£,
Adding
(50) a T-dimensional
whose simulation by MSM can be handled jointly with simulation of
integral,
the expectation with respect to
6.
in (49) makes
v.
24
£,.
DISCRETE RESPONSE MODELS WITH MEASUREMENT ERROR
Suppose discrete response for a random sample n = 1,...,N satisfies a
latent variable model
u
(51)
d
n
n
=
|3z
+ c
n
= H(u
n
n
,
),
where H maps
e the row vector u
for the observed category,
n
into m discrete categories with d
and H
To simplify notation,
category.
(d
)
the set of u
assume
(3
n
an indicator
yielding the observed
is a scalar;
generalization merely
requires that the construction below be carried out component by component.
Suppose z
is not
observations x
x
(52)
where £
z
n
observed directly, but is related to a vector of
by
= 2 A + C
n
n
~ N^O,^),
.
independent of
We interpret the x as observations on
c.
measured with error, or as indicators for
z.
In form,
this is a multiple
indicator or factor-analytic latent variable model, with A giving the
factor loadings.
Suppose in the population
z
~ N{f±,r)
conditional distribution of z given
(53)
z ~
Nin
+
,
x,
,
independent of £
.
suppressing subscripts,
(x-uA)(A rA+i<)"Vr,r-rA(A'rA+>i')"Vr).
Then the
is
McFadden
A Method of Simulated Moments for Discrete Response Models
If the e ~ W(0,fi)
(54)
in (51),
25
then
2
,
u ~ w(p/i+p(x-MA)(ArA +*)" A'r,p [n+rA(A'rA+*)" A r]).
1
,
1
and the response probabilities given x are of MNP form.
The MSM estimator for
the general MNP model can be adapted directly to this problem,
the main
practical difficulty being calculation of the derivatives of the Cholesky
factor of the covariance matrix with respect to the deep parameters in order
to calculate a relatively efficient instrument matrix.
A number of variants of the measurement error model (51) may be
encountered in applications,
including variables measured with error that are
common to several alternatives or interact with alternative-specific dummys,
multiple variables measured with possibly correlated errors, and simultaneity
between the latent variables and observed indicators.
These may alter the
details of (52) and (53), but give the same basic structure for the response
probabilities and MSM estimator.
It
is also
possible to treat measurement
error in discrete response models such as multinomial logit by allowing the
to have an appropriate distribution.
For the logit example,
e
MSM estimation
can be used by simulating the expectations of the logit formulas with respect
to the conditional distribution of the true explanatory variables.
topics are studied in greater detail in McFadden
(
These
1985a, 1985b) and Train,
McFadden, and Goett (1987).
7.
NON-NORMAL DISCRETE RESPONSE MODELS
This paper has focused on estimation of the MNP model.
However,
the MSM
estimator can be applied to any latent variable model in which unbiased
estimates of the response probabilities can be obtained economically by Monte
McFadden
A Method of Simulated Moments for Discrete Response Models
Carlo methods.
For example,
in the latent variable model
it
(1),
26
may be
reasonable to assume that some components of a are always non-negative, giving
monotonicity.
This could be modeled by taking the density of
a.
to be
multivariate truncated normal, or by giving some components non-negative
densities such as gamma.
This complicates the analytic representation of
response probabilities, but is .readily accommodated in Monte Carlo draws from
the latent variable model to obtain frequency estimators.
The MSM estimator also permits analysis of discrete response data
generated by more complex partial observation functions than the maximum
indicator appearing in (1).
alternatives.
For example, consider data on ranks of
With the exception of the multinomial logit model,
is
it
impossible to obtain analytically tractable expressions for probabilities of
more than the first few ranks in terms of response probabilities;
(1978),
Barbara and Pattanaik (1985), and McFadden (1986a).
see Falmange
However,
Monte
Carlo drawings from the latent variable model provides unbiased frequency
estimators of the ranking probabilities that can be used in MSM estimation.
8.
SUMMARY
This paper has proposed a simple modification of a classical method of
moments estimator for discrete response models that avoids the necessity for
accurate numerical integration to calculate response probabilities, using
instead asymptotically unbiased simulators of these probabilities.
This
method of simulated moments is practical for problems where direct numerical
integration is computationally intractable.
A Method of Simulated Moments for Discrete Response Models
McFadden
9.
THEOREMS AND PROOFS
APPENDIX:
I
27
use the following notation, mostly collected from Sections 2 and
3 of the paper:
C = {1
= ax.
u.
the set of possible responses.
m}
or u
= aX
a latent variable model,
x)aKxm
m
(x.
1
a = a(G,7)) a
x
1
= l(u.iu_)
l
C
l
P (ile.X
T)
given X
i,
g,
,
independent of X„.
u.
t;
observed response),
(=
such that a(8,T))X
maximized
is
.
a simulator for P (i|e,X
)
).
instrument vector, determined as a function
1
= w.(e.X-).
l
C
a random sample.
N
mN x
f(8)
1
vectors formed by stacking
or f (i|e,X r
)
by alternative,
d.
k x mN array formed by stacking W.
P (8)
mN x k array of derivatives of P(8).
6
anv vector satisfying
o
sm
IIW(8
o
P (i|8,X
,
),
then by observation.
W = W(6)
)(d-f(8
.
)
sm
some 8
)
II
£ infJIVtG )(d-f(9))li +
(1),
8
o
o
p
density for X_, with associated measure p.
)
P^te),
8
parameter vector with true value 8
probability of
l
p(X
),
a random vector with density g(ii) and
W.
P(8),
m
and
k x
d,
u
l
l
)
1
L
1
indicator for maximum
W.
n =
u„ = (u.
array,
a k x
at
f (i|8,X
with X_ =
K vector function, with
associated measure
d.
e C,
i
R(8),
R
k x k array of sample covariances,
R(e) = lim F^.O). R = R(e*).
.
^(6)
1
= N' W? (8)
15
The response probabilities are invariant under monotone transformations of the
latent variable model.
= 0,
so X
is
Hence,
without loss of generality, we may normalize x
contained in a K(m-l) dimensional space.
Further,
a may be
McFadden
A Method of Simulated Moments for Discrete Response Models
1
defined without loss of generality to have a compact domain.
28
R
and 9 to have regular domains, and
The first assumptions made require X
guarantee a zero probability that the latent variables for different
alternatives are equal, so the response probabilities are well-defined without
additional tie-breaking rules:
t
Al
is
[
A2
]
The parameter space S is a compact convex subset of R
in the interior of
]
The domain t of the attributes X
The random vector
independent of X
is
,
continuous on
i_s
tj
IN,
For an open set X
such that a (
each 8 €
,
7) )
X
and 6
is.
a compact subset of a
.
finite-dimensional with domain
is
IN,
The function a = a (6,
.
7))
and is twice differentiable in 6 with these
derivatives continuous on
[A4]
j_s_
and has a finite mean
x
'*
,
0.
K(m-l) dimensional space
[A3]
k
£
x
5?
IN.
with p(X
)
=
1,
the subset IN(6,X
)
of
IN
distinct in every component has probabi 1 ity one for
0.
The last assumption is usually imposed in the definition of discrete response
models,
and can be derived from more basic structural conditions.
following lemma covers common applications,
including MNP.
The
When the model
contains alternative-specific random effects, the array A„„ in this result is
a (m-1) dimensional identity matrix.
A Method of Simulated Moments for Discrete Response Models
McFadden
Lemma
X
A
f0
=
C
Suppose there is a partition
Suppose [Al] and [A2].
1.
such that A „
12
is.
29
(m-1) x (m-1) and almost surely nonsingular.
A
22
Suppose a = a(8,T)) =
S uppose a
6.
[a
(55)
1
a
is.
2
=
]
+ Tjr(6)
/3(B)
twice continuously differentiable in
is.
partitioned commensurate ly with X„, so
2
1^(6)
(G)]
r
2
1
+
[tj
i)
il
]
f
°
Suppose T
2
7)
1
conditioned on
r\
and suppose
tj
R
,
Proof:
1
aX
c
= pX
the term
one,
7)
nonsingular for 8 e
i_s
T)
T
k
7)
22
Suppose the density of
8.
has a finite mean
[0,
i2
uniformly bounded and continuous
is
1
Q +
r
(
r
11
A
12
+r
A
i
Then [A3] and [A4] hold.
.
2 22
with support
,
)
+ 7,2r
A
]22 22
With P robabilit y
has a continuous density with support R
,
given
implying that the probability of a hyperplane where components of aX f
,
are tied is zero.
The next assumption guarantees that the response probabilities are
well-behaved:
[A5]
6,
The probabi litv P (i|e,X_)
is.
positive,
and twice differentiable in
and the probabi litv and its derivatives are cent ir.uous
,
or.
6 x X.
The following result gives a sufficient condition for [A5] which holds in
particular for the MNP model with alternative-specific dummys:
Lemma
2.
Suppose the hypotheses of Lemma
1,
with
A„
always nonsingular
.
Then [A5] holds.
Proof:
By the symmetry of the problem in alternatives,
it
is sufficient to
)
.
A Method of Simulated Moments for Discrete Response Models
McFadden
consider P (lle.X
so
= [0,s]
aX
2
(56)
1
given
Then,
tj
= Pr(ctX
Usin g the decomposition of Lemma
°| x r ^-
=£
1,
implies
p\ 2
= [s -
r,
)
30
the set
,
2
- e A
B(tj
^(r n A l2+ r l2 A 22 )](r 22 A 22 )-
-
12
22
of
)
satisfying aX
tj
1
.
intersection
is the
£
of m-1 linearly independent half-spaces, with each bounding hyperplane
twice continuously differentiable in (9,X
J"
2
g»
J"
.(t|
|k)
12
)dij
g
(t)
11
P (l|B,X
)
=
continuously differentiable in
is twice
)dT)
Hence,
).
1
1
m—
(S,X
1
has support R
Since g
).
the probability is positive,
,
n
The next assumptions concern the instruments and identification of 9
The function
[A6]
continuous
(
Let M
[A7]
,
bounded
,
= w. (B,X_)
determining the instruments is
and twice continuously differentiable on 6 x
denote a bound on
w
and its B derivative
w.
The instruments identify B
Sy,
Z
w (e,x
ieC
c
,
uniform in
i,
6,
X_.
C
with
HP c (i|e,x
)-P (i|e ,x )]dp(X
*
eoual to zero if and only if 9 = 6
[A3]
,
1.
l
u(e,e) =
(57)
W.
:
-
,
-
for any 9 €
c
)
17
6.
R is of max i mum rank
To satisfy [AS],
instruments constructed by simulation require the use of
smooth simulators such as (12),
(34),
and (35) in the case of MNP.
If
[A5]
holds and the instruments equal the score of the likelihood evaluated at each
trial
9,
w.
(9,X
)
= 5Ln P
(i|9,X )/59,
then
6=9
and u reduces to
X
A Method of Simulated Moments for Discrete Response Models
McFadden
«(e,e) =
P (i|e*,x )[5LnP (i|e,x )/ae]dp(x )
c
c
c
c
c
S
Sy,
31
1
the expected score of an observation under maximum likelihood estimation.
this case,
[A7]
requires that
For
be the only critical point of the expected
6
log likelihood, a standard identification condition.
Also,
R
in this case,
*
equals the information matrix evaluated at 6
which is symmetric and
,
nonnegative definite, and by [A7] is definite at some point in every
neighborhood of G
Then,
.
[A8]
adds only a regularity requirement.
case of more general instruments,
In the
and [A8] are standard assumptions for
[A7]
the identification and regularity of classical method of moments estimators.
Hence,
the identification conditions for MSM are the same as for the
corresponding classical method of moments estimator.
The next assumption concerns the simulator f
[A9]
Vectors
otherwise
(17,
In
,
.
.
.
,
t)
rn
are drawn
)
independently of W and
,
so that each
tj
(
|
i
9 X
):
,
by simple random sampl ing or
,
and independently for different
d,
has marginal density g
(
tj )
Define
.
ip
(
i
1
6
,
X
)
frequency in the r draws for observation n of the event that a
is maximized at component
bounded function of
(58)
£{f (i|e,x
c
for some c
>
0;
Cn
e
i
.
Define f
X_ and
Cn
,
(
In
,
....
i
77
|
6 X_
rn
,
and for some M
,
M„,
X > 0,
1
,
17
.
)
be anv uniformly
to.
)
8
satisfying
)
+ 0(N*
)|w,d} = ? (i|e,x
)
c
Cn
cp
(59)
73 ,
(
be the
to.
(
n,
'
C
"
V2
)
and all 6,6 € 8 and X_ €
Cn
X,
|f_(i|e,Xp )-f r (i|e,xr )| s M Vr (i|e,x_ )-p_(i|e,xr )l + M,je-e|\
C
f
Cn
C
Cn
Cn
C
Cn
C
<p
|
'
'
'
'
Condition (58) requires that the simulator be asymptotically unbiased,
while (59) requires that it be at least as smooth in 6 as the frequency
simulator.
Condition (59) is satisfied trivially by either the frequency
.
A Method of Simulated Moments for Discrete Response Models
McFadden
simulator, or by a smooth simulator such as that based on (12).
simulator is different iable, then X -
If the
the assumption also allows
1;
32
<
X <
1,
A simulator satisfying
corresponding to "polynomial" non-differentiability.
(59) will be termed X-Lipschitz in neighborhoods where
is constant.
<p
The
following lemma establishes sufficient conditions for a kernel-smoothed
frequency simulator to satisfy (58).
Lemma
3.
Suppose the assumptions of Lemma
nonsingular
hold
1_
with A
,
always
Suppose a kernel -smoothed frequency simulator (37) with a
.
distribution function * having a finite moment generating function in a
neighborhood of the origin
and with N
,
b
Then
0.
->
Let u(t) be the moment generating function for
Proof:
such that *(v) £ e
TV
u(-t) for all v
and l-*(v) ^ e
<
(58) holds
.
.
There exists t
*.
"TV
for all v
/i(t)
>
>
0.
m-1
Then,
for c = max. y./2
J
>
1
m-1
+ J
(
II
y m-1, )= f
K(y.
0,
J
*(v-y.)) ¥(dv) ^ e
+ e
/i(-t)
with the first term the result
mCt),
J
of bounding the product at negative arguments,
bounding the measure at positive arguments.
yields K(y ,...,y _ )
1
B 1
J\|K(u_ ./b.,)
A
C-i
N
-
- e~
(1
2=
T|cl
u(T))
m
and the second the result of
A similar argument for c
^
1
l(u_ .sO)|G(u_
|e,X_)du_ ..
C-i
C-i
C
C-i
.
-
me"
T|c|
Define
in at least one component,
ProM
.
.
1(A)
|u
i e
.-"u. |=s
-tM
u(T)m and I(A_) ^ e
Mb,,).
But (55)
single component u.-u. of u_
conditional density of
Then,
..
C-i
i
2
-n
= R
given
-tM
m-1
-A -A
(u(t)+u(-t)
.
)
in the proof of Lemma 2
holds when the second partition is of dimension one and
J
greater than Mo
.
with M a positive constant, and A
the bounds on K imply
I(A„) £ £.
to be the set of u_
C-i
1
to be the set of u_
A.
<
Define 1(A) =
u(x).
A,
'
less than -Mb., in every component,
Further,
j
°
_
v>c J-l
Then,
¥(v-y.)) ¥(dv)
II
(
v£c
s
is the value of a
be a uniform bound on the
letting M
r
1
tj
,
Prob( |u .-u. s
j
l
Mb.,)
N
£
2Mb„M
N z
•.
Therefore,
A Method of Simulated Moments for Discrete Response Models
McFadden
I(A
£ 2mMb M
N
)
3
I(A„)) s N
(e"
TM
N
1/2
u(T)m + e~
|P
(
C
TM
|G,
i
X
c
)
- P (i|e,X )|
(u(T)+u(-r)
)
c
c
+
2mMb M
* N
V2
(I(A
KAg)
+
)
The condition N
0.
Choose M =
).
1
x"
Ln
N.
(Ln N)b
implies the required limit.
-»
b..
+
1
the right-hand-side of the last inequality goes to zero if N
Then,
->
1/2
Then,
.
33
N
The next result characterizes the regularity in 8 of the simulated
and guarantees that with probability one, the condition defining e
moments,
has a solution with |W(d-f(G
Lemma
sm
Suppose [A1]-[A9].
4.
£ mM M /r:
w (p
))|
Then
almost surely
,
—
Define I(S,X
Proof:
and I(e,X
,
=
(60)
=
7})
J"
1
,
tj)
=
if the components of a(e,T))X
otherwise.
uniformly
are bounded by
jl mM M /r.
w y
and the *jjumps
in this function on
*
,
»
ijs
of 9 with Lebesgue measure
X- Lipschitz in 6 except for a closed subset
zero
W(d-f(6))
.
For each 9 e
0,
[A4]
are all distinct,
r
implies
I(e,X ,7))dg(T)) dp(X ),
N S%
c
c
and hence
o = S
(61)
Q
s^ S
t
Ke,x
c
,7))dg(T))
d P (x
c
)
de.
Applying Fubini's theorem to (54), there exists a set X
probability measure one, for X_ e X
measure one, and for (X
,tj)
measure on which 1(6, X _,
t?)
if I(6,X
,
7))
= 0,
e X
= 0.
x
a set N(X
£
)
N(X„) a set
with probability
IN
(X
£ X with
,7))
of full Lebesgue
£
The continuity of a(6,T)]
in 6 implies that
then this is also true in a neighborhood of
6,
so
(X_,7))
is upcji.
The function W(d-f(6)) is defined by N independent draws X„
density p(X_), and for each
l
density
g(7j).
Hence,
n,
r draws
(tj,
ln
with probability one, X_
Cn
tj
rn
),
6 X,
1
with
each with marginal
and
tj
.
jn
e (N(X_
Cn
)
for
A Method of Simulated Moments for Discrete Response Models
McFadden
j =
r and n =
1
of full measure.
constant M_ on
f
Suppose
.
.
.
implying
,N,
6
r
N
p|
("|
(X
l
n=l j=l
,
tj
Jn
)
is
an open set
uniformly X-Lipschitz with
is
)
Ln
0...
N
6*0,
so 9 i
i(
xr .V-
(X„ ,,T).,
,)
)
With probability
for some (n,j).
for
the discontinuity in |W(d-f(8))|
(52),
N
=
by [A9], f (i|6,X
But,
is contained in
one,
one.
1,
34
(n, 'j')
Hence,
* (n,j).
is at most mM M /r
w
using
with probability
tp
n
Assumption [A4] implied that the set of
for which there are ties in the
7)
components of a(6,7))X_ has probability zero for all G and almost all X
.
The next assumption requires that the geometry of a(6,i)) be such that the
exceptional set IN(6,X
of
)
-q
where ties occur varies smoothly in (6,X_).
[A10] There exists M g and X >
the set B_(e,X_)
o
L.
{tj
such that for X_ e X
C
heN(e, X,J
C
L
'
1
o
1
L
o
most a polynomial singularity.
conditions,
L
g
8
0,
18
X
.
ABOUT HERE)
illustrates the construction of B_(6,X_).
if the set-valued function IN(G,X_)
and almost all 6 e
for some |e-9|s5} has g(B_(G,X_)) * M
(INSERT FIGURE
Figure
o
The assumption holds
is transversal at G or if there
is at
The next result shows that with regularity
the case a(6,X„) = 3(G) + T)T{e) for the MNP model satisfies
[A10].
Lemma
5.
Suppose the hypotheses of Lemma
and [AB]-[A9].
Proof:
i,
with A
always non-singular
Then [A10] holds.
Suppose a tie between alternatives
1
and
2,
so ax
= 0.
Using the
,
A Method of Simulated Moments for Discrete Response Models
McFadden
notation of (55) and (56), partition
= (a, ,a„ ...,a
a.
the second component.
2
(62)
2
This function
p\ 2
-
and let a„ denote
)
<L
,
tj
^(r n A l2+ r l2 A 22 )](r 22 A22 )-
-
1
= 0(0, X
tj
m
Then,
p\ 2
= [-
-n
J,
l
1
35
1
.
is continuously different iable in (0,X
)
and
),
hence has a Taylor's expansion
e-e
0(6, ^.tj
(63)
where A
1
-
)
0(e,X
1
,T)
c
t^A
= [A
)
and (0,X
)
X^n
1
Then the set [^(0,
2
7)
)
g^T)
J"
1
c
{r>
(0,
X )-(0, X
M.d+ElT)
g2
1
!
)6M
r
bounds g
7)
-0(0, X
)
)
(7)
I
tj
7)
X_)-(0, X_)
|
7)
6,
)5.
)
I
£ M (1+|t
1
)
|)6>
contains all
)dT)
1
)
= 2M 5/m(m-l),
g
There are m(m-l) possible combinations of tied
.
alternatives, each of which can with permutations of components of X„,
and
^
2
1
1
1
,
(0,
1
1
c
|
and satisfies
s 6,
|
2
J
N (0,X ,7)
2
C
i
2
I
\
such that for
^ M (1+It)
)|
~ ~
[
)dT)
T)
where M
,T)
2
=
)
11
1
1
0(0,X
-
X-,7}
solving (55) for
(65)
c
Then uniform continuity on compact
).
x X implies there exists a constant M
|0(0,
x
are vectors of continuous derivatives of 0(0,X„,7)
and A
evaluated between (6,X
(64)
V
]
and relocation of X
be put in the form above.
c.,
The sum of the bounds
for each combination gives [A10].
Given
c >
0,
a finite family of random functions F
is
said to bracket
a family of random functions F if for each Y € F there exist Y,Y
such
e F
c
that Y ^ Y ^ Y and E(Y-Y)
in the smallest set F
c
<
c.
The logarithm of of the number of elements
that brackets
F,
denoted H(e),
is
called metric
A Method of Simulated Moments for Discrete Response Models
McFadden
entropy with bracketing
36
The following result establishes stochastic
.
equicontinuity conditions for families whose metric entropy does not rise
too rapidly as c falls.
Lemma
Assume F is a uniformly bounded family of measurable random
6.
functions satisfying
with bracketing
.
H(e
J"
Suppose y
copies of Y-£Y for Y e
lim limsup Pr{sup
8-»0
N-»0
F
(66)
Proof:
,
y_,
.
.
Define
F.
sup
..
de finite
)
N"
.
,
where H is the metric entropy
are Independent identical ly distributed
IIYII
Then for every X
= £|Y-£Y|.
>
0,
N
1/2
|
~ IUe _
£ (y -y
n=l
n
n
)
|
X}
>
= 0.
Dudley (1984), Theorem 6.2.1, establishes (59).
I
use a
restatement from Alexander (1987), Theorem 2.1.
The next result establishes that the simulation residuals satisfy
stochastic equicontinuity and boundedness conditions sufficient for the MSM
estimator to be CAN.
The critical step is to show that these residuals
satisfy the assumption on metric entropy required by Lemma
Lemma
Suppose [A1]-[A10].
7.
B(B) = N'
V2
w"(£:f
(67)
limsup
(68)
sup
Then given X,5
(e)-P(e)).
sup |B(G)|
PHsup
K(e)|
Define £(e) =
>
=0,
>
M}
<
X,
and for A, = {G| N V2 |e-6 |£6>,
(69)
limsup
Pr{sup \C,{B)-^{6
6eA
N
)|
>
A}
= 0.
N"
0,
19
6.
V2
W(f (6)-£f (6)
)
and
there exists M such that
1
A Method of Simulated Moments for Discrete Response Models
McFadden
Proof:
Condition (67) is immediate from [A8].
integer
j,
cover [0,1]
k
with 2
k
i
Assume 8 € [0,1]
cubes with sides 2
-
For any
.
and for each X
,
37
€-X
v.*
let 0.(X
J
L
be a set containing one point selected from each cube that
)
intersects
|e-e.(e)|
Define 9.(9,X
).
C
2~
J
^
= 5..
Let Y(9,X
draws
the selection can be made so that #(13.(9, X„)
C
°
By [A10],
0.
for 6 € 8 (X
J
L
J
s
L
to be point in 8 (X
)
J
= E W.f (ile.X ).
L
L
1
i€C
)
for s = l,...,r with
tj
,
O
Define q (9,X
J
e B.
o
t?
s
)
nearest to
)
< M 5
g
then
6;
to be the number of
Using the notation of (59),
(9.X.J.
C
.
L
L
)
J
and X satisfying [A9] and [A10],
Y°(9,X
(70)
J
Then,
define
= mM (M
WI |e-e.(e)|
)
^
A
+M q.(e,X )/r).
y5JL
J
by [A10], £q.(e,X„) s rg(B s (6.X.J),
C
j
8.
EY°(e,X_) £ mM M Je-e.(G)|
(71)
wf
C
J
implying
C
X
mM M g(B. (e,X r ))
w <p° 5
C
+
j
.
i mM (M„+M M )5.
w f <p g J
X
= M S
o
A
j
From (59),
|Y(e,x_)-Y(e.(e),x rj| * mM (Mje-e
(72)
+
£ mM
Y.(e,X
i
A
.(e)
|
)-(p
(i|e.(6),X
)|
c(e,x„)/r)
-rM
^
J
)
J
(i|e,X
max|<p
x
\
C
"J
'CjC
JC
).
Given
c
>
0,
choose
j
to be the smallest integer such that 2
>-
J
c.
(MJe-e
.(e)
jCj'C
C
J
<
w
mM M
I
Y.(9,X_) = Y(e.(6),X_) - Y°(6,X_) * Y(6,X_) £ Y.(e,X_) = Y(9.(9),X_)
Hence,
+
W
L
J
L-
Then the 2
k ^ +1
functions Y. and Y. bracket Y(9,X_),
6 e 8.
This
>-
J
J
implies that the metric entropy with bracketing for F = <Y(9, X_) |9e8}
satisfies H(e)
T
12
==
(kj+l)Ln2 * (-Lnc)k/A
1
H(e )dc £ (k+l)Ln2 - 2(k/X)T Lne
of Lemma
5,
so (66) holds.
de
+
<
(k+l)Ln2,
co.
and hence si H(e
2
1/2
)
dc
±
This establishes the assumptions
A Method of Simulated Moments for Discrete Response Models
McFadden
For any 5
< 6
>
forming the expectation of (59) and using [A10],
0,
implies E| Y(e,X_)-Y(e X_)
>
L
mM (M„+M M )5 = M 5.
O
W I <p g
<
I
L
Hence,
38
|6-0|
can be
(66)
written in the form
IC(6)-C(0)| > X} =
sup
lim limsup Pr{sup
(73)
0.
i
Taking 5 =
2~\
= 1,2
j
one has
1
0-0
5 for N £ 4^
<
1
and N
V2
|0-0|
<
1.
Then (73) implies (69).
Next prove (68).
limsup Pr{sup
(73)
N ^°
N
E
(6
.
.|" e -)
X}
IC(6)-<(6)|
>
)|
M
o
<
}
>
<
with 6. = (j/J)6
+
Pr{|C(6)|
>
can be written
(l-j/J)G
°
6 = 6
o
and J the smallest integer
M +XJ}
Q
*'
Then (67) holds with M = M
In this lemma,
o
>
& lc(e
M
**
)-c(e
)|
<
x,
j=o
< x.
J}
'
vJ
\J
+ XJ.
the construction in (70) and the following arguments
hold even if the number of repetitions r is a random function of
Then,
such
Then,
* Pr{|<;'(e J|
N.
such that
>
A.
But any 6 e
X.
J
exceeding 1/6.
(74)
o
J
J
sup
there exists 5
0,
A central limit theorem implies there exists M
e 0.
that sup., Pr{|C(6
j=0
>
16-6 1*5
Choose any 6
+
From (73), given X
in particular,
6,
X_,
and
the lemma holds for simulators formed by
acceptance/rejection methods with random stopping rules, and for consistent
simulators where r increases with sample size.
Let 8
be a sequence in
and assume that the instruments are
A Method of Simulated Moments for Discrete Response Models
McFadden
evaluated at 6
for each
N.
The 8
of initially consistent estimators,
In the last case,
(1).
P
9
sm
solves HW(G
sm
might be non-stochastic,
or a sequence
or might equal the MSM estimator
)(d-f(6
Lemma 7 holds in all these cases.
sm
))ll
£
infJIWO
6
sm
39
)(d-f(6))ll
sm
+
A Method of Simulated Moments for Discrete Response Models
McFadden
40
FOOTNOTES
1.
The idea of simulating response probabilities from an underlying
latent variable model, generating the response probabilities by stochastic
integration,
is standard in the area of computer simulation;
see Hammersley
and Handscomb (1964), Fishman (1973), and Lerman and Manski (1981).
This
literature has concentrated on simulating the response probabilities to a
level of accuracy that enables their use in standard maximum likelihood
procedures.
2.
o =
Use the identity
£ sp (i|e,x )/ae
ieC
3.
c
Z [3LnP c (i |e,x )/ae]P c (i |e,x
c
).
ieC
Starting from any K
x mN
array of instruments Z
,
and taking account
of the structure of the covariance matrix of the residuals,
one can show by a
standard argument from non-linear least squares that the asymptotic minimum
variance estimator in the class using linear combinations of instruments in Z
*
is attained by W =
P(6
8
*
—i
"
)'Z'(ZPZ'
)
Z,
where P Q (6
6
)
is the mN x k
»
derivatives 5P (i|e,X
Z.
in
If Z
o
)/50,
evaluated at 9
*
,
P = diag P(6
),
and
2? - S Z° P_(j|e*,X r ).
in
_
jn C
Cn
jeC
.
= 5LnP(e
*
*
*
)/S6,
functions P and P
then W = 5LnP(6 )/8Q.
Approximations to 6
and to the
yield approximations to the minimum variance instruments
that can be constructed from Z
4.
array of
Appendix Lemma
6
.
and assumption [A3] give sufficient conditions for
simulators to satisfy (13).
5.
The independence of the simulators across observations can be relaxed
to any process that is sufficient to give the term in (a) asymptotically
,
A Method of Simulated Moments for Discrete Response Models
McFadden
41
normal and to give stochastic equicontinuity of the simulation residuals.
Continuous differentiability and compactness imply
6.
M|
e-e
1
Given c
where M £ max |R(6)|.
,
>
for N
each
N
>
P{max
,
e
'
|u..(e)-u(e)
P{max
Hence,
e/3> <
>
|
|
u>
C
(e)-u(G)
By construction of 8
c.
|
N
> c}
<
i
sufficiently large so that
such that |u.,(e)-u(8)
|
|
let 8 denote the finite set of
Then choose N
N
c
there is a 6 e 8
6 e 8,
2e/3.
8
u (G)-co (6)
extract a finite covering of
0,
with neighborhoods of radius less than e/3M;
centers of these neighborhoods.
|
c
for
,
^ |u.,(e)-<4e)
N
|
+
Regularity condition [A8]
c.
*
implies R(6
nonsingular.
)
The inequality, from Gine and Zinn (1986; Lemma 3.2),
7.
N
I
P{
Y.
>
is
2
2
s exp[-t /(2Ncc + 2ct/3)],
tt
1
i=l
where
2
2
= EY.
cr
;
see also Pollard (1984) or Shorack and Wellner (1986).
l
Replace
8.
t
by N
and use
t
Consider the normal density
XTTX
l/2
=
nCu-n.A)
with A =
to obtain (24).
£ c
cr
and
(2Tt)-
p.
|Ar
= £X.
1/2
(u
e
^ ),A
"
1(U '^ /2
,
The following matrix differentiation formulas
are derived by writing out terms from the familiar expressions 5Ln|A|/6A A
and dk /ok =
-
©A
A
cwhich hold when A is symmetric,
but identity
of cross-terms is not imposed in the differentiation):
aLn|XT'rx|/ar
=
2rx(XT'rx)*
1
1
x',
1
1
3(z' (xT'rx)" z)/6r = -2rx(x / r rx)" zz' (x'r'rx)" x'.
,
The derivatives of Ln n(u-|3X,
SLn n/sp =
X(XTTX)"
sLn n/ar
-
1
X'
TTX)
(u-pX)
rxu'rrxrV
are then
'
+
1
1
rx(x'r'rx)" (u-3X)' (u-px)(x'r'rx)" x'.
A Method of Simulated Moments for Discrete Response Models
McFadden
42
This construction was suggested by Paul Ruud.
9.
A mixture of multinomial
with the mixture interpreted
logit models,
as the result of taste variations in the population, has been of independent
interest as a discrete choice model; see Westin (1974) and McFadden (1984).
The polynomial kernel limits the number of alternatives for which
10.
calculations must be done.
greater than 2b
If an observation has
in magnitude,
probability of the converse
is
then K(u
0(b).
)
every component of u
coincides with l(u„ .^0); the
If for a
draw of u
IN
for an
L.•
in magnitude,
observation, all component differences exceed 2b
the
kernel-smoothed frequency simulator coincides with the simple frequency
simulator.
in a sample of size N with r Monte Carlo draws per
Then,
observation,
the expected number of alternatives for which further
calculation is required to obtain the simulator and corresponding
instruments is bounded by (r+l)N
+
mrN0(b
)
^
(r+l)N + 0(mrN
+
This
).
makes the calculation practical even if the number of alternatives is large.
Discussions with Jim Heckman contributed to the formulation of
11.
kernel-smoothed frequency simulators.
The unbiased kernel-smoothed frequency
simulator for the MNP model was suggested by Steve Stern.
Moran (1984) suggests several control variates.
12.
Peter Phillips ana
Vasillis Kajivassiliou suggested the use of spherical transformations for this
and Dan Nelson developed many of the details.
problem,
To generate a denser set of antithetic points, for any integer T
13.
i
and each pair
s
for
T-l.
t
=
1
and
s
J
with
i
<
j,
construct the directions (±ts
Combined with the points ±s
,
When £ =
(T-t)s
this gives 4(T+1) evenly
spaced points on each great circle, for a total of 2K
14.
±
>
+
2TK(K-1) directions.
and T consists of an identity submatrix corresponding to
1,
)
A Method of Simulated Moments for Discrete Response Models
McFadden
43
alternative-specific dummy variables and a zero submatrix corresponding to the
and X__,X'_. are, except for proportional
remaining variables, then X
If the model
constants, a superset of the ideal instruments.
is identified,
then it will always be possible to find low-order polynomials in X_
,
that
have an asymptotic correlation matrix with 5P„(i|e)/59 that is of full rank.
the crude instruments proposed may not be grossly inefficient.
Thus,
In the
third step when smooth simulators are being used, one iteration from 6 using
Newton's method achieves the maximum asymptotic efficiency attainable from
better instruments simulated at
15.
for each
16.
With random sampling,
6.
R»,(8)
by application of a strong law of large numbers.
8,
For any continuous function a = a(6,7j),
variable model u
=
the transformed latent
yields the same response probabilities,
aX
(1+llall)
and is uniformly Lipschitz in X
17.
converges almost surely to a limit R(8),
.
If crude instruments independent of 6 are used to obtain an initially
consistent estimator, then w in (50) is independent of
18.
then it is sufficient that [A7] hold for
Loo
a closed set,
in a neighborhood of G
X
X
:
If
V
e B r and
for some (6 ,X^) in a closed
<5
.
L
limit point
I
(0
L
o
77^
-»
tj
,
then
tj^
and hence
e M(e
J
,X^)
c
L
L
i)
€
N(6 ,X„)
u
C
for each
,X°) of (8 J ,X^).
am indebted to Ariel Pakes and David Pollard for discussions that
led to the formulation of this lemma.
.
neighborhood of (e,X_), by [A3].
using the continuity of a(8,7))X r in (8,X_),
19.
6
The following argument establishes that B_(8,X_) is closed,
measurable, for (8,X_) €
Hence,
If approximations
instruments are calculated, starting from an initially consistent
to the ideal
estimator,
9.
,
A Method of Simulated Moments for Discrete Response Models
McFadden
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McFadden
A Method of Simulated Moments for Discrete Response Models
e-s
e
FIGURE
1
e+s
47
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