Post on 18-Nov-2015
description
Motivation: The Hansen-Singleton problemThe non-linear GMM estimator
Asymptotic properties
GMM for non-linear models
Walter Sosa-Escudero
Econ 507. Econometric Analysis. Spring 2009
April 30, 2009
Walter Sosa-Escudero GMM for non-linear models
Motivation: The Hansen-Singleton problemThe non-linear GMM estimator
Asymptotic properties
Prelude: Hansen-Singletons consumer problem
Consider the following optimization problem for a representativeconsumer:
maxct+i,At+i Et
i=0
U(ct+i)(1 + )i
subject to:
At+i = (1 + r)At+i1 + yt+i ct+ilimi
EtAt+i(1 + r)i = 0
y = labor income, c = consumption of non-durables, A = wealth(a financial asset) with return r. U is a utility function. is therate of intertemporal preferences.
Walter Sosa-Escudero GMM for non-linear models
Motivation: The Hansen-Singleton problemThe non-linear GMM estimator
Asymptotic properties
Assume that:
U(ct+i) =c1t+i1
(CRRA specification). represents consumers risk aversion.
First order (Euler) conditions are:
Et
(1 + r1 +
ct+1 ct
)= 0
We will use this knowledge to obtain consistent estimates for and, the deep parameters of the problem.
Walter Sosa-Escudero GMM for non-linear models
Motivation: The Hansen-Singleton problemThe non-linear GMM estimator
Asymptotic properties
We can write the FOC as follows:
Et [g(xt+i, 0)zt] = 0
where g(xt+i, 0) (
1+r1+0
c0t+1 c0t
), = (, ) and zt is a
vector of n variables that are orthogonal to g(xt+i, 0).
This is a set of non-linear momment conditions.
If there is available a sample of size T , and n > p, we will use anon-linear GMM strategy, that is, solve:
min
(Tt=1
g(xt+i, )zt
)A
(Tt=1
g(xt+i, )zt
)
Walter Sosa-Escudero GMM for non-linear models
Motivation: The Hansen-Singleton problemThe non-linear GMM estimator
Asymptotic properties
Structure and assumptions
1 Random Sample: vi is an i.i.d. sequence of random variables.
2 Regularity: f(vi, ) : V 7
Motivation: The Hansen-Singleton problemThe non-linear GMM estimator
Asymptotic properties
4 Identification:
1 Global identification: E[f(vi, )] 6= 0 for all 6= in .2 Regularity on derivatives: f(vi; )/ is p q matrix that
exists and is continuous on for each vi V ; ii) 0 is aninterior point of ; iii) E[f(vi; )/]=0 exists and is finite.
3 Local identification: (E[f(vi; )/]=0
)= p.
5 Weighting: Wn is psd which converges in probability to a pdmatrix W .
6 Compactness: is compact.7 Domination: E[sup ||f(vi, )||]
Motivation: The Hansen-Singleton problemThe non-linear GMM estimator
Asymptotic properties
Identification
The global identification condition E[f(vi, )] 6= 0 for all 6= 0 isdifficult to characterize. Remember that in the IV case we tiedidentification to a rank condition.
Local identification refers to conditions that hold in a smallneighborhood of 0.
Under the assumed regularity conditions, we can expand f(.)around 0 in a small neighborhood of 0:
f(vi, ) ' f(vi, 0) +f(vi, 0)
( 0)
Walter Sosa-Escudero GMM for non-linear models
Motivation: The Hansen-Singleton problemThe non-linear GMM estimator
Asymptotic properties
f(vi, ) ' f(vi, 0) +f(vi, 0)
( 0)
Taking expectations and using the moment conditions
E[f(vi, )] ' E[f(vi, 0)
]( 0)
which under the local identification condition is zero whenever 6= 0.
Note the close similarity of the local identification condition andthe rank condition in the IV case.
Walter Sosa-Escudero GMM for non-linear models
Motivation: The Hansen-Singleton problemThe non-linear GMM estimator
Asymptotic properties
The GMM estimator
Recall that the GMM objective function is:
Qn() ={n
t=1 f(vi, )n
}Wn
{nt=1 f(vi, )
n
}and the GMM estimator is defined as:
n = argmin Qn()
The FOCs for this problem are:{1n
nt=1
f(vi, n)
}Wn
{1n
nt=1
f(vi, n)
}= 0
a possibly non-linear system of q equations with p unknowns.
Walter Sosa-Escudero GMM for non-linear models
Motivation: The Hansen-Singleton problemThe non-linear GMM estimator
Asymptotic properties
Asymptotics 1: Consistency
The GMM estimator is based on minimizing:
Qn() ={n
t=1 f(vi, )n
}Wn
{nt=1 f(vi, )
n
}Define the population version of this function as:
Q0() = E[f(vi, )]WE[f(vi, )]
Walter Sosa-Escudero GMM for non-linear models
Motivation: The Hansen-Singleton problemThe non-linear GMM estimator
Asymptotic properties
Result 1: Q0() achieves a unique minimum at 0By the moment condition E[f(vi, 0)] = 0 and by theidentification condition and the pd of W , E[f(vi, )] 6= 0 forany 6= 0.
Result 2: Q0()p Qn() uniformly on .
Intuition: if we fix at some point, by the LLNnt=1 f(vi, )
p E[f(vi, )] and by assumption Wn W ,then by continuity the result follows.
It is more difficult to make the uniform statement.
Walter Sosa-Escudero GMM for non-linear models
Motivation: The Hansen-Singleton problemThe non-linear GMM estimator
Asymptotic properties
Intuition of consistency:
n minimizes Qn().
Qn()p Q0() uniformly.
0 minimizes Q0()
Then np 0
We will work out through a detailed proof in the homework.
Walter Sosa-Escudero GMM for non-linear models
Motivation: The Hansen-Singleton problemThe non-linear GMM estimator
Asymptotic properties
Asymptotics 2: Normality
Now we are in much more familiar territory...
Let gn() 1nn
t=1 f(vi, ) and Gn() 1n
ni=1 f(vi, )/
.Then, the FOCs of the GMM problem are:
Gn(n) Wn gn(n) = 0
Now take a mean value expansion of gn(n) at 0:
gn(n) = gn(0) +Gn()(n 0
)where lies between and 0. Now replace above:
Gn(n) Wngn(0) +Gn(n) WnGn()(n 0
)= 0
Walter Sosa-Escudero GMM for non-linear models
Motivation: The Hansen-Singleton problemThe non-linear GMM estimator
Asymptotic properties
Gn(n) Wngn(0) +Gn(n) WnGn()(n 0
)= 0
Multiply byn and solve for
n(n 0
)n(n 0
)=
(Gn(n) WnGn()
)Gn(n) Wn
n gn(0)
= Mnn gn(0)
with Mn (Gn(n) WnGn()
)Gn(n) Wn. Sounds familiar?.
Walter Sosa-Escudero GMM for non-linear models
Motivation: The Hansen-Singleton problemThe non-linear GMM estimator
Asymptotic properties
Now we are definitely at home.
n(n 0
)= Mn
n gn(0)
We will show that Mn does not explode and thatn gn(0) is
asymptotically normal
We start the Mn. Under the continuity assumptions we get byan appropriate LLN:
Gn(n)p G0 and Gn()
p G0
where G0 E[f(vi; 0)/]. Then:
(Gn(n) WnGn()
)Gn(n) Wn
p(G0 WG0
)G0 W M0
which is a finite matrix.Walter Sosa-Escudero GMM for non-linear models
Motivation: The Hansen-Singleton problemThe non-linear GMM estimator
Asymptotic properties
Regardingngn(0), by the iid and regularity assumptions, we can
apply the CLT to show:
ngn(0) =
n
nt=1 f(vi; 0)
n
d N(0, S)
Then by Slutzkys theorem
n(n 0
)= Mn
n gn(0)
d N(0,M0SM 0)
Walter Sosa-Escudero GMM for non-linear models
Motivation: The Hansen-Singleton problemThe non-linear GMM estimator
Asymptotic properties
We can write the FOC as follows:
Et [g(xt+i, 0)zt] = 0
where g(xt+i, 0) (
1+r1+0
c0t+1 c0t
), = (, ) and zt is a
vector of n variables that are orthogonal to g(xt+i, 0).
This is a set of non-linear momment conditions.
If there is available a sample of size T , and n > p, we will use anon-linear GMM strategy, that is, solve:
min
(Tt=1
g(xt+i, )zt
)A
(Tt=1
g(xt+i, )zt
)
Walter Sosa-Escudero GMM for non-linear models
Motivation: The Hansen-Singleton problemThe non-linear GMM estimator
Asymptotic properties
Empirical example: Hansen-Singleton reloaded
Data:
x1t = ct1/ctx2t = 1 + rt
In these terms, the moment conditions can be written as:
Et
( x1,t+1 x2,t+1 1
)zt = 0
with (1 + )1. Following Hansen and Singleton, for zt we willtake a constant, rt and ct1/ct.
Walter Sosa-Escudero GMM for non-linear models
Motivation: The Hansen-Singleton problemThe non-linear GMM estimator
Asymptotic properties
The estimated discount factor is: 0.998 (exponential)
Risk aversion parameter: 0.89 (risk adverse), careful, notsignificant.
Walter Sosa-Escudero GMM for non-linear models
Motivation: The Hansen-Singleton problemThe non-linear GMM estimatorAsymptotic properties