Appendix

A.2 Generalized Method of Moments

The Generalized Method ol Moments (Hansen 110821) can he imdersiood as an extension ol the linear IV regression we have discussed. Suppose now that we have a model which defines a vector f, = f (X). 0), where x, now includes all (he data relevant for the model (thai is, we have dropped the distinction between v, and x,). and 0 is a vector of Np coefficients, litis lorinttlalioii generalizes the lineal IV regression in three wavs. First, e(\,. 0) can be a column vet tor with ,V, elements rather than a scalar. Second, fix,. 0) can be a nonlineai rather than a linear function of the data and the parameters. Third. c(x,.0) can he heteroskedastic and serially correlated rather than homoskedastic white noise. Our model tells us only thai there is some true set of parameters 0 for which e(x 0 ) is orthogonal to a set ol insirumenis; as before these are written in an (rV,/x 1) column vector h,. Ну analogy with (Л.1.2) we define

1,(0) = h,®f(x 0). (Л.2.1)

fhe notation W denotes the Kroueeliei product of the two vectors. Thai is. f is a vector containing die cross-product of each instrument in h with each clement ofc. Г is therefore a column vector with Nf = N, N elements, and thi model implies by analog)with (A. 1.3) that

l-IMfli.U = 0. (Д.9.2)

Just as in (Л. I. I), we define a vector g,(0) containing the sample averages corresponding lo the elements off in (A. 1.10):

g,(0) = 7 £]f,<0). (Д.2..Ч)

Ну analogy with (A.l.fi), GMM minimizes the quadratic form

Q,(0) = g/(0)W,g,(0). (Д.2.-1)

Since the problem is now nonlinear, this ininimizalion must be performed numerically, fhe fust-order condition is

D,(0,)W, g,(0,) = (). (Д.2.Г,)

where D, (0) is a matrix of partial derivatives defined bv

/(0) = .)g,(0)/;i0. file (i. /) eleineni ol 1), ( ) is i)g, ,(0l/V,.

(Л.2.1.)

/1.2. (lenniditsd Method of Moments 5SS, ];%if Asymptotic Distribution Theory

fhe asympioiic distribution of (he coefficient estimate 07 is

ч/7(0, -0 ) 4- AV((),V). (A.i.7)

where

V = (DWDttJ-DSWDofDVVD )-1. (A.2.8)

lhesc expressions are directly analogous to (A. 1.12) and (A. 1.13) for thelin-ear instrumental variables case. D() is a generalization of My/.y in those equations and is defined by Do = E[9f(x 0(i)/<10t,]. Dr(0) converges asymptotically lo Do. S is defined as in (A. 1.11) by

S = lim Var

Y-TO

7 * J)f,(e )

lim Var[7~l/2gr(0 )]. (A.2.9)

/-DC

Optimal Weighting Matrix

Just as in the linear IV case, the optimal weighting matrix that minimizes V is any positive scalar times S~. With an optimal weighting matrix the asympioiic variance of 7l/2 times the coefficient estimate 0T is

(DS-D )-1. (A.2.10)

Also, when the weighting matrix S 1 is used, T times the minimized objective function is distributed x2 with {Nj - Np) degrees of freedom, where Nf is the number of orthogonality conditions and Np is the number of parameters to be estimated.

In practice, of course, S and the other quantities in (A.2.8) must be estimated. To do this, one starts with an arbitrary weighting matrix Wr, this could be the identity matrix or could be chosen using some prior information about the relative variances of the different orthogonality conditions. Using W one minimizes (A.2.4) lo get an initial consistent estimate Qy. To estimate V in (A.2.8), one replaces its elements by consistent estimates. Dti can be replaced by D; (0,), W can be replaced by Wr, and S can be replaced by a consistent estimate S; (0; ). Given these estimates, one can construct a new weighting matrix W*, = S/(0,) l and minimize (A.2.4) again to get a second-stage estimate в r. The asympioiic variance of V1 times ihe second-stage estimate can be estimated as

V*. = (D,(07.)S,-(0vrD/(0v.)r

(A.2.11)

and the second-stage minimized objective function is distributed x with (.V/ - Np) degrees of freedom. Although a two-stage procedure is asymptotically efficient, it is also possible lo iterate the procedure further until

the parameter estimates ami minimized objective function converge. This eliminates any dependence of the estimator on the initial weighting matrix, and it appears to improve the finite-sample performance of GMM when the number of parameters is large (Ferson and Foerster [1994]).

A.3 Serially Correlated and Heteroskedastic Errors

One of the most important steps in implementing GMM estimation is estimating the matrix S. From (A.2.9),

lim E

Г-.00

= ГО(0О) + Х](Г;(19О)+Г;(0 )),

(A.3.1)

where

Гу(0о) = E[f,(0o)f(+1->(0o)] (A.3.2)

is thj: jth autocovariance matrix of f<(0o)- The matrix S is the variance-cova iance matrix of the time-average of f,(t90); equivalently, it is the spectral density matrix of f;(0o) at frequency zero. It can be written as an infinite sum af autocovariance matrices of f,(0H).

If the autocovariances of f((0o) are zero beyond some lag, then one can simp ify the formula (A.3.1) forS bydropping the zero autocovariances. The autocovariances of f/(0o) are zero if the corresponding autocovariances of e(x 90) are zero. In. the linear IV case with serially uncorrelated errors discussed earlier, for example, (x 0()) is white noise and so f;(0n) is white noisq; in this case one can drop all the autocovariances for j > 0 and S is just Го, the variance of f,(t90). The same result holds in the consumption CAPM with one-period returns studied in Chapter 8. However in regressions with fC-period returns, like those studied in Chapter 7, К- 1 autocovariances of f,(0n) are nonzero and the expression for S is correspondingly more complicated.

щр- The Newey-West Estimator

~W$ ° estmalc woud seem natural to replace the true autocovariances of f/(0o), Г;((9о) with sample autocovariances

Г,.Т(0Т) = Г1 f,v0r)f,-,(f},), r=;+t

(Л.З.З)

and substitute into (A.3.1). However there are two difficulties that must be faced. First, in a finite sample one can estimate only a finite number

of autocovariances; and to get a consistent estimator of S one cannot allow the number of estimated autocovariances lo increase too rapidly with the sample size. Second, there is no guarantee lhat an estimator of S formed by substituting (Л.З.З) into (A.3.1) will be positive delinilc. To handle these two problems Newey and West (1987) suggested the following estimator;

ST(q. 0T) = Га.тфт) + () {rii(Or) + r;.,.(flv-)) (Л.3.4)

where ц increases with the sample size but not loo rapidly. q- 1 is tlie maximum lag length thai receives a nonzero weight in the Newey and West (1987) estimator. The estimator guarantees positive deliniteness by down-weighting higher-order autocovariances, and it is consistent because the downweighting disappears asymptotically.

In models where autocovariances arc known to be zero beyond lag K- I, il is tempting to use the Newey and West (1987) estimator with q = K. Tin.- is legitimate when K-\, so that only the variance Г() y(0y) appears in the estimator; but when K> 1 this approach can severely downweight some nonzero autocovariances; depending on the sample size, it may be better to use q>K in this situation.

Although die Newey and West (1987) weighting scheme is the most commonly used, there arc several alternative estimators in the literature including those of Andrews (1991), Andrews and Monahan (1992), and Gallant (1987). Hamilton (1994) provides a useful overview.

The Linear Instrumental Variables Case

The general formulas given here apply in both nonlinear and linear models, but they can be understood more simply in linear IV regression models. Return to the linear model of Section A.l, but allow the error term e,(0n) to be serially correlated and hclcroskcdastic. Equation (A.1.1(1) becomes

S = lim Var[7 1/1!He(0o)] =-- lim 7 Н{Л(0 )Н, (A.3.5)

/ -. oo /-+ oo

where f2(0<>) is the variance-covariance matrix of £(0 ). This can be estimated by

S,(0V) = У-нТыёЫН. (A.3.6)

where S~iy(0y) is an estimator of П(0ц). Equation (A.2.1!) now becomes

irr = (7~1XH(H y(0y)H) lHX)-1. (A.3.7)

In the homoskedastic white noise case considered earlier, ii - a2!/ so we used an estimate Пу(0у) з 621, where cr2 = / £, t (f(0/ ). Substituting this into (A.3.7) gives (A.1.18).

A/t/irntlix

When 11 it- error lenn is serially iineorrelaiecl bill heteroskedastic, then il is a diagonal matrix wiih different variances on each element of the main diagonal. One can constrttci a sample equivalent of eacli element as follows, for each element on the main diagonal of the matrix, SI7. (0-;)=(,{() 1 )-, while each off-diagonal element il,. (0/) = 0 for л/Л Substituting the resulting matrix ii / (0/) into (Л.З.Ь) one gets a consistent estimatoi olS, (0), and substituting it into (A.3.7) one gels a consistent estimator V*(. This is true even though die matrix i1(0r) is nol itself a consistent estimator of П because the- number of elements of JI that must be estimated equals the sample si/e.

When 11 if error lei 111 is serially correlated and homoskedastic, then one can construct each clement of the matrix fly(0y) as follows:

T E,Ui () d-dOr) if / = /-.v < 4. 0 otherwise

(Л.3.8)

where the Newev anil West (1.I87) weighting scheme with maximum lag length ц is used. Alternatively, one can replace the triangular weights ( - /)/</will) unit weights to gel die estimator of I lansen and Hodrick (1980), bill this is not guaranteed to be positive definite.

When the error term is serially correlated and heteroskedastic, then one can construct Ji/ (0,) as:

[0 otherwise

where the Nfwey and West (1987) weighting scheme is used. Again one can replace the triangular weights with unit weights 10 get the estimator of I lansen and 1 lodi i< к (1980). hi each case substituting fi(0,) into (A.3.6) gives a consistent estimate olS, and substituting it into equation (A.3.7) gives a consistent estimator V*,-, even though the matrix fl(0/) is not itself a consistent estimator ol il because the number of nonzero elements increases too rapidly with the sample size. While (1981) gives a comprehensive treatment ol die linear model with serially correlated and heteroskedastic errors.

A.4 GMM and Maximum Likelihood

following 11.million (199 I), we now show how some well-known properties nf Maximum Likelihood estimators (MI.K) can be understood in relation to GMM. We lirsl lay out some notation. We use /., lo denote the density of

A.4. GMM and Maximum Likelihood

x,+ i conditional on the history of x, and a parameter vector 0:

L,(x,+1,0) = Цхж I x x, ,.....в). (A.4.1)

We use the notation (, for the log of L the conditional log likelihood:

c,(xl+1>0) = logl,(x(+1,0). (A.4.2)

The log likelihood С of the whole data set X,..., xy is just the sum of the conditional log likelihoods:

с = (M-3)

Since is a conditional density, it must integrate to 1:

Mx,+ 0)dx/+, = 1. (A.4L4)

Given certain regularity conditions, it follows that the partial derivative of I., with respect to 0 must integrate to zero. A series of simple manipulations

then shows thai

f 3L,(x,+1,0) Г 3L,(x,+I,0) 1 . .

0 = / -£0-dxi+i - I ---- L,dx,+\

30 L,

3c,(x,+ 1,0)

Li dx,+ i

3t (x,+1,0) 3c,(x(+1,0)

= E,- = E---. (A.4.5)

30 30

The partial derivative of the conditional log likelihood with respect to the parameter vector, dl,{x,+\, 0)/30, is a vector with Np elements. It is known as the score vector. From (A.4.5), it has conditional expectation zero when the dala are generated by (A.4.1). It also has unconditional expectation zero and thus plays a role analogous to the vector of orthogonality conditions f, in GMM analysis. The sample average (1/T) £1=1 Э£/(хц-1, в)/дв plays a role analogous to gr(0) in GMM analysis.

The maximum likelihood estimate of the parameters is just the solution 0 to Max£(0) = 1=1 ti- The first-order condition for the maximization can be written as

gr(0) = 7-1 £эЛ(х,+ 1,0)/Э0 = 0, (A.4.6)

which also characterizes the GMM parameter estimate for a just-identified model. Thus the MLE is the same as GMM based on the orthogonality conditions in (A.4.5).