12.3 Use kernel regression lo estimate ihe relation between the monthly returns оГ IBM and the S&P 500 from 1005:1 to 1994:12. How would a conventional beta be calculated from the results of the kernel estimator. Construct at least two measures lhat capture the incremental value of kernel estimation over ordinary least squares.

me

Appendix

Tins APPENDIX PROVIDES a brief introduction to ihe most commonly used estimation techniques in financial econometrics. Many other good reference texts cover ibis malerial in more detail, hence we focus only on those aspects lhat are most relevant for our immediate purposes. Readers looking for a more systematic and comprehensive treatment should consult Hall (1992), Hamilton (1994), Ogaki (1992), and White (1984).

We begin by following Halls (1992) exposition of linear instrumental variables (IV) estimation in Section A.l as an intuitive introduction to Hansens (1982) Generalized Method of Moments (GMM) estimator. Wc develop the GMM method itself in Section A.2, and discuss methods for handling serially correlated and heteroskedastic errors in Section A.3. In Section A.4 we relate GMM to maximum likelihood (MI.) estimation.

A.l Linear Instrumental Variables

Consider a linear relationship between a scalar yt and a vector x,: y, = x,6\> + c,(f?i)), t = 1 ... T. Stacking ihe 7 observations, this can be written as

у = X6Y,+ е ? ). (АЛ.I)

where у is a (7x I) vector containing 7 observations of y X is a (7x Лтд-) matrix containing 7observations of the N\ independent variables in x (9 is an {N\x 1) parameter vector, and e[0()) is a (7x 1) vector containing 7 observations of the error term e,. The error term is written as a function of the true parameter vector so lhat the notation с can be used for both the true equation error and the residual of an estimated equation. For simplicity, assume that the error lerm is serially uncorrelated and homoskedastic, with variance аг\ thus the variance of e(t9 ) is Var[e(t9 )] = о*\т, where Iy is a (7x 7) identity matrix.

/\pp<lt<ltX

Thoi-с arc also available Nn instruments in an (/V x 1) column vector h,. The 7observations of this vcctoi form a [TxN ) matrix H. Tbe instruments have the properly dial K(h,c, (0 )) is an (/V x I) vector of zeroes; that is, the instruments are contemporaneously uncorrelated with ilie error The statement that a particular insiriimenl is uncorrelated with the equation error is known as an orthogonality condition, and IV regression uses the .V available orthogonality conditions lo estimate the model.

(liven an arbitral у coefficient vector 0, we can form the corresponding residual c,(0) ~ y, - x,0. Slacking this residual into a vector, we get e(0) = у - X0. We can also deline an (N x I) column vector containing the cross-product oldie insiriimenl vector with the residual,

MO) - h,t,{0). (A. 1.2)

The expectation ol this cross-product is an (Л /Х I) vector of zeroes at the true parameter vector:

l.f,(0o)l = 0. (A. 1.3)

The basic idea ol IV estimation is to choose coefficients to satisfy this condition as closely as possible. Of course, we do not observe the true expectation ol f and so we must work instead with the sample average off. We write this as g/ (0), using the subscript / lo indicate dependence on the sample:

g,(0) = 7-f,(0) = / £V,(0) = 7-lHe(0). (Л.1.4)

/=i i-i

Minimum Distance Criterion

In general there may be more elements of g, (0) than (here are coefficients, and so in general it is not possible to set all the elements of g,(0) to /ею. Instead, we minimize a quadratic form, a weighted sum of squares and cross-products of die elements of g, (0). We define the quadratic form Q,(0) as

Q/(0) = g,(0)W, g,(0) = 7, lc:<0)HW-;-7,-,He(0)]. (A.l.o)

where W, is an [N/, x ,V;/) s> inmcii ic, positive definite weighting matrix.

IV regression 1 booses 0-, as the value of 0 that minimizes Q,(0). Substituting ihedelintliou olс(0) into (Л.1.Г)), the lirsl-order condition for the

In iii.tnt applii .ilions.. , is.1 lutei .imcihh 1I1.11 iMiniiirrrliittil with anv variables known in advance. In litis case ilie iusti iiint-oi ve, 101 li, will iik litilc only lagged vatialilcs dial ate known 11..... 1 1,1 e.uliei. Niini-ilielrss we wtite ii as Ii, lor nolalional siniplit iiv and generality.

A. I. Linear Instrumental Variables

minimization problem is

XHWTHy = XHWrHX07-. (A. 1.6)

When the number of instruments, N/ equals the number of parameters to be estimated, Nx, and the matrix HX is nonsingular, then XHWr cancels out of the left- and right-hand sides of (A. 1.6) and the minimization gives

07- = (HX)-Hy. (A. 1.7)

This estimate is independent of the weighting matrix W7-, since with Nn = N\ all tbe orthogonality conditions can be satisfied exactly and there is no need to trade off one against another. It is easy to see that (A. 1.7) gives the usual formula for an OI.S regression coefficient when the instruments H are the same as the explanatory variables X.

More generally, Nh may exceed Nx so that there are more orthogonality conditions to be satisfied than parameters to be estimated. In this case the model is overidentified and the solution for Or is

0r = (XHWrHXrXHWj-Hy. (AJ.8)

Asymptotic Distribution Theory

The next step is to calculate the asymptotic distribution of the parameter estimate 0-j. Substituting in for у from (A. 1.1) and rearranging, we find that

у/Т(вг-во) = (TXHWr ГНХ)- rXHWT rl/2He(0o).

(A. lb)

Now suppose that as 7 increases, T~HH converges to Мцн> a ndn-singular moment matrix, and T~XH converges to Мхи, a moment matrix of rank Nx- Suppose also that as 7 increases the weighting matrix W7-converges to some symmetric, positive definite limit W. Because we have assumed that the error £(0o) is serially uncorrelated and homoskedastic, when the orthogonality conditions hold Т~1*Не(в0) converges in distribution io a normal vector with mean zero and variance-covariance matrix а2Мцц. We use the notation S for this asymptotic variance-covariance matrix:

S = lim Var[jr1/2He(0Q)] = ст2Мнн. (A.1.10)

7- 00

Using (A.1.4), S can be interpreted more generally as the asymptotic variance of T]/2 times the sample average off, that is, T12 times gr:

S = lim Var

7-00

7 *£f,<0 )

= lim Var[rl/2gr<0o)]- (A.l.U)

7-00

Appendix

With these convergence assumptions, (A. 1.9) implies that

/Тф-г-ви) N(0,V), (AJ.12)

where

V = (MJtf,WM xr1MA7,WSWM x(MwWM,/.vr

= ct2(ma7/WM/,a) 1MA WM vm A(M.wWM .v) 1, (A.l.13)

and M x = Мш.

Optimal Weighting Matrix

We have now shown that the estimator вт is consistent and asymptotically normal. The final step оГ the analysis is lo pick a weighting matrix W that minimizes the asymptotic variance matrix V and hence delivers an asymptotically efficient estimator. It turns out that V is minimized by picking W equal lo any positive scalar times S-1. Recall that S is the asymptotic variance-covariance matrix of the sample average orthogonality conditions \kr(0). Intuitively, one wants to downweight noisy orthogonality conditions jand place more weight on orthogonality conditions that arc precisely measured. Since here S l = <т~2М;, it is convenient to set W equal lo

i W* = М- ; (A. 1.14)

he formula for V then simplifies to \

V* = fftMmMMjw)-1. (A.l.15)

n practice one can choose a weighting matrix

W*;. = (7 HH) I. (A.l.lb)

\s 7 increases, W*r will converge to W*.

With this weighting matrix the formula for в/ becomes

I вт = [ХЩННГНХрХЩННГНу = (ХХГХу, (Л. 1.17)

Where X = H(HH)~HX is the predicied value of X in a regression olX on H. This is the well-known two-stage least squares (2SI.S) estimator. It can be thought of as a two-stage procedure in which one first regresses X on H, then regresses у on the fitted value from the first stage lo estimate the parameter vector 6V

Alternatively, one can think of 2S1.S as regressing both X ami у on H in the first stage, and then regressing the fitted value of у on the fitted value of X;

A.l. Linear Instrumental Variables

exactly the same coefficient estimate (A. 1.17) is implied. Nole that under this alternative interpretation, the second-stage regression asymptotically has an Rl statistic of unity because the error term in (A.I.I) is orthogonal to the instruments and therefore has a fitted value of/его when projected on the instruments. This implies thai asymptotically, if (Л. 1.1) and the orthogonality conditions hold, then the coefficient estimates should not depend on which variable is chosen to be the dependent variable in (A. 1.1) and wliich are chosen lo be regressors. Asymptotically, the same coefficient estimates will be obtained (up to a normalization) whichever way the regression is written.

The variance-covariance matrix ol 2SI.S coefficient estimates, V*, can be estimated by substituting consistent estimates of the various moment matrices into (A. 1.15) to obtain

irr = ffV-XHtHHrHXr1, (A.l.18)

where cri is a consistent estimate of the variance of the equation error. This formula is valid for just-identified IV and OLS coefficient estimates as well.

In place of the weighting matrix W* defined above, it is always possible to use /;W* where к is any positive scalar. Similarly, in place of the weighting matrix W*r one can use AyWy-, where /</ is any positive scalar that converges lo k. This rescalingdocs not affect the formula for the instrumental variables estimator (A. 1.17). One possible choice for die scalar/{is a-2, the reciprocal of the variance of the equation error e,; this makes the weighting matrix equal to S-1. The corresponding choice for the scalar k-f is some consistent estimate a-2 of a-2. Hansen (H)82) has shown that with this scaling, 7times the minimized value of the objective function is asymptotically distributed X with (N/i - Nx) degrees of freedom under the null hypothesis that (A.I.I) holds and the instruments arc orthogonal to the equation error.

Hansens test of die null hypothesis is related to die intuition dial under die null, the residual from the IV regression equation should be uncorrelated with the instruments and a regression of the residual on the instruments should have a small /< 2 statistic. To understand this, note that when W; = ((T27~HH)~, the minimized objective function is

[7-e(0*;.)H](<727 lHH)-l7 lHe(0*.)l. (A. 1.19)

Now consider regressing the residual ((<),) on die instruments H. The filled value is H(H H) H е(вт), and the R statistic of the regression converges to the same limit as (A.l.19). Thus Hansens result implies that 7 limes die R2 in a regression of the residual on the instruments is asymptotically x2 wiih (Nii - Nx) degrees of freedom. This is a standard lest of overideiitifying restrictions in two-stage least squares (Logic [1984]).