hw6.htm

Homework 6

Empirical likelihood

(Due 4/2)

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Let
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be a $k\times 1$ vector of population moment conditions implied by economic theory, where $\QTR{bf}{Z}_{i}$ is a set of iid data observations with $Z_{i}\in R^{P}$ and unknown continuos distribution function , $\beta$ is a unique $m\times 1$ vector of unknown parameters, and g $_{i}(.)$ is a known function
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with $k\geq m.$ Let
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Consider the following instrumental variable model:
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where $x_{i}$ is a $m\times 1$ endogenous variables. Suppose the endogenous variables are linearly related to a set of instrument variables,
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where $z_{i}$ is a $k\times 1$ with
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and
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A fundamental assumption of the instrumental variable model is that the structure errors are mean independent of the instrumental variables:

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where denotes the matrix of instrumental variables since we could use any power of z $_{i}$ as an instrument. For now, assume $A_{i}=z_{i}$
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The 2SLS estimator is based on the orthogonality condition
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with . The 2SLS estimators of $\beta _{0}$ takes the form
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where is $n\times m,$ is $n\times 1,$ and is $n\times k.$ 2SLS essentially uses the exogenous variation in X (i.e., the variation in X explained by Z) to explain y. The convention generalized method of moment (GMM) estimator utilizes an analogy principle theory, thereby minimizing the distance between the sample moment conditions and their hypothesized values. The objective function achieves this via the quadratic form:
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where
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$F_{n}$ denotes the empirical distribution placing probability mass $n^{-1}$ on each data point, and $W_{n}$ is a positive definite weight matrix. The first-order necessary conditions for an optimum are
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Since
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and
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the first-order conditions become
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so that
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The optimal GMM estimator sets
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where $\widehat{\beta }$ is an initial consistent estimator of $\beta _{0}.$ Hence the optimal GMM estimator takes the form
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Let
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be the Jacobian matrix of partial derivative and
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be the optima weight matrix. Then
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and
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Thus the optimal GMM estimator may be rewritten as
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Empirical likelihood (EL). The EL estimator, in contrast to the GMM estimator, utilizes an alternative form of the analogy principle, minimizing a distance between probability measures rather than the distance of the population moment conditions from their sample counterparts. That is, the EL estimator assigns multinomial weights to each of the observations, so that

This allows the EL estimator to choose probability weights so that the sample moment conditions exactly satisfy the moment restrictions implied by economic theory. The optimal $p_{i}^{\prime }s$ then maximize the empirical log-likelihood

subject to the movement restrictions

and constraints on the empirical likelihood probabilities
1. Show that the optimal empirical likelihood probabilities, , take the form
  
  where $\lambda$ is the vector of Lagrange multipliers for the constraints
2. Show the EL estimator of $\beta _{0}$ is
3. Show the first-order necessary conditions for an optimum of the empirical log-likelihood function are (w.r.t $\lambda )$
  
  and (w.r.t $\beta )$
  
  where and are the implicit solutions to the first-order conditions.
4. How would you find and
Semiparametric Efficient Estimation. Consider the efficient estimation of an arbitrary expectation

say, under the semiparametric assumption

Brown and Newey (1998) show that the semiparametric efficient estimate of takes the form

where

and $\widehat{\beta }$ is an initial consistent estimator of $\beta _{0}.$ The semiparametric efficient estimator of the expectation function may be written in a simpler form. Let

and

Then

The weights $\frac{1}{n}w_{i}$ are the optimal probability weights of the discrete points of the empirical distribution.
1. Show that the weights in () are a first-order Taylor expansion of the optimal empirical likelihood probabilities, $p_{i}^{\ast }$ in ().
2. The weights $w_{i}$ have an intuitive interpretation. Suppose, for a moment, that
  
  and (one instrumental variable). Consider observations contributing to be negative mean of the sample moment condition, Since
  
  $w_{i}<1.$ On the other hand, observations with will have $w_{i}>1.$ Therefore, the weights proposed in () downweight observations that contribute to the sample moment restriction not be satisfied, while up-weighting observations that cause the sample moment condition to hold more closely. An analogous result holds for the case in which . This is the mechanism by which the weighted sample moment conditions exactly hold.
  
  Show that if
  
  then
3. Quasi Empirical Likelihood Estimation. Following equation (), the semiparametric efficient estimates of
  
  and
  
  under
  
  are therefore
  
  and
  
  where $\widehat{\beta }$ is an initial consistent estimator of $\beta _{0}.$ Brown and Newey show that is semiparametric efficient relative to if
  
  Likewise, is semiparametric efficient relative to if
  
  Recall that a close-form solution does not exist with respect to and in the first-order conditions for the empirical log-likelihood function, even with $g_{i}(\beta )$ is linear in $\beta .$ So consider using the weights of Brown and Newey, $\frac{1}{n}w_{i},$ in place of the optimal empirical likelihood probabilities, $p_{i}^{\ast }.$ These weights are a first-order equivalent. Making this substitution in the first-order conditions with respect to $\lambda$ in () yields
  
  and with respect to $\beta$
  
  Show that
4. Substitute () into the modified first-order condition with respect to $\beta$ in (), yielding
  
  or
  
  where is the semiparametric efficient estimator of and is the optimal GMM weight matrix, both of which are based on an initial consistent estimator of $\beta _{0},$ $\widehat{\beta }.$
5. Show that
  
  if
  
  where $\widehat{\beta }$ is an initial consistent estimator of $\beta _{0}.$
  
  Note: Let
  
  so that $w_{i}=1-q_{i}.$ Note that
  
  That is, the Brown and Newey weights make an linear adjustment to In addition, Brown and Newey show that is typically semiparametric efficient relative to So it is reasonable to expect some good statistical properties, such as consistency, to carry out to relative to or

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