Nonlinear g Mm

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



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.




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.




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

)




Structure and assumptions

1 Random Sample: vi is an i.i.d. sequence of random variables.

2 Regularity: f(vi, ) : V 7



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, )||]



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)




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.




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.




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, )]




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.




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.




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




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?.




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



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)




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

)




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.




The estimated discount factor is: 0.998 (exponential)

Risk aversion parameter: 0.89 (risk adverse), careful, notsignificant.


Motivation: The Hansen-Singleton problemThe non-linear GMM estimatorAsymptotic properties

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Nonlinear g Mm