Category Springer Texts in Business and Economics

The General Linear Model: The Basics

7.1 Invariance of the fitted values and residuals to non-singular transformations of the independent variables.

The regression model in (7.1) can be written as y = XCC-1" + u where Cisa non-singular matrix. LetX* = XC, theny = X*"* + u where "* = C-1".

a. PX* = X* (X*0X*)-1 X*0 = XC [C0X0XC]-1 C0X0 = XCC-1 (X0X)-1 c0-1 C0X0 = PX.

Hence, the regression of y on X* yields

y = X*" *ls = PX* y = PXy = X" ols which is the same fitted values as those from the regression of y on X. Since the dependent variable y is the same, the residuals from both regressions will be the same.

b. Multiplying each X by a constant is equivalent to post-multiplying the matrix X by a diagonal matrix C with a typical k-th element ck. Each Xk will be multiplied by the constant ck for k = 1,2,.., K...

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Regression Diagnostics and Specification Tests

8.1 Since H = PX is idempotent, it is positive semi-definite with b0H b > 0 for any arbitrary vector b. Specifically, for b0 = (1,0,.., 0/ we get hn > 0. Also, H2 = H. Hence,

n

hii =J2 hb2 > h2i > 0.

j=i

From this inequality, we deduce that hjy — h11 < 0 or that h11(h11 — 1/ < 0. But h11 > 0, hence 0 < h11 < 1. There is nothing particular about our choice of h11. The same proof holds for h22 or h33 or in general hii. Hence, 0 < hii < 1 for i = 1,2,.., n.

8.2 A Simple Regression With No Intercept. Consider yi = xi" + ui for i = 1,2,.., n

a. H = Px = x(x0x)_1x0 = xx0/x0x since x0x is a scalar. Therefore, hii =

n

x2/ x2 for i = 1,2,.., n. Note that the xi’s are not in deviation form as

i=1

in the case of a simple regression with an intercept. In this case tr(H/ =

n

tr(Px/ = tr(xx0//x0x ...

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Generalized Least Squares

9.1 GLS Is More Efficient than OLS.

a. Equation (7.5) of Chap. 7 gives "ois = " + (X’X)-1X’u so that E("ois) = " as long as X and u are uncorrelated and u has zero mean. Also,

var("ols) = E("ols – ")("ols – ")’ = E[(X, X)_1X, uu, X(X, X)_1]

= (X’X)-1X’ E(uu’)X(X’X)-1 = CT2(X, X)-1X’fiX(X’X)-1.

b. var("ols) – var("gls) = o2[(X’X)-1X’fiX(X’X)-1 – (X’fi-1X)-1]

= CT2[(X, X)-1X, fiX(X, X)-1 – (X’^-1X)-1X’^-1fifi-1 X(X’fi-1X)-1]

= ct2[(X’X)-1X’ – (X’fi-1X)-1X’fi-1]fi[X(X’X)-1 – fi-1X(X’fi-1X)-1]

= o2 AfiA’

where A = [(X’X)-1X’ – (X’fi-1X)-1X’fi-1]. The second equality post multiplies (X’fi-1X)-1 by (X’fi-1X)(X’fi-1X)-1 which is an identity of dimension K. The third equality follows since the cross-product terms give -2(X’fi-1X)-1...

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Seemingly Unrelated Regressions

10.1 When Is OLS as Efficient as Zellner’s SUR?

a. From (10.2), OLS on this system gives

p /Р 1,ols I

p ols = "

P2,ols I

"(X1X1) 1 0

11

"(x! xO 1 х1уГ

о (x2X2)-1_

х2у2 )

_(x2x^ 1 X2y2_

This is OLS on each equation taken separately. For (10.2), the estimated var("ols) is given by

where s2 = RSS/(2T — (K1 + K2)) and RSS denotes the residual sum of squares of this system. In fact, the RSS = e,1e1 + e2e2 = RSS1 + RSS2 where

ei = yi — Xi P i, ols fori = 1,2.

If OLS was applied on each equation separately, then

var (p 1,ol^ = s2 (X1X1)-1 with s2 = RSS1/(T — K1)

and

var (p2,ol^ = s2 (X2X2)-1 with s2 = RSS2/(T — K2).

Therefore, the estimates of the variance-covariance matrix of OLS from the system of two equations differs from OLS on each equation s...

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Simultaneous Equations Model

11.1 The Inconsistency of OLS. The OLS estimator from Eq. (11.14) yields 8ols =

T T

E Ptqt/ E Pt2 where pt = Pt — l3 and qt = Qt — Q. Substituting qt = 8pt C

t=i t=i

TT

(u2t — U2) from (11.14) we get 8ois = 8 C E Pt(u2t — N/Y, P2. Using (11.18),

t=i t=i

T

we get plim £ Pt(U2t — U2)/T = (012 — 022)/(8 — ") where Oij = cov. Uit, Ujt)

t=1

for i, j = 1,2 and t = 1,2,.., T. Using (11.20) we get

Plim 8ois = 8 C [(CT12 — 022)/(8 — ")]/[(011 C CT22 — 2ст12)/(8 — ")2] = 8 C (o12 — 022)(8 — ")/(o11 C 022 — 2°12).

11.2 When Is the IVEstimator Consistent?

a. ForEq.(11.30)y1 = a^y2 C ‘1зУз C "11X1 C "12X2 C U1.Whenweregress

T

У2 on X1, X2 and X3 to get У2 = у2 C V2, the residuals satisfy E y2tV2t = 0

t=1

TTT

and 22 V2tXu = 22 V2tX2t = 22 V2tX3t = 0.

t=1 t=1 t=1

Simi...

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Pooling Time-Series of Cross-Section Data

12.1 Fixed Effects and the Within Transformation.

a. Premultiplying (12.11) by Q one gets Qy = «Qint + QX" + QZpp + Qv

But PZp = Zp and QZp = 0. Also, PiNT = iNT and Qint = 0. Hence, this

transformed equation reduces to (12.12)

Qy = QX" + Qv

Now E(Qv) = QE(v) = 0 and var(Qv) = Q var(v)Q0 = o2Q, since var(v) = ov2Int

and Q is symmetric and idempotent.

b. For the general linear model y = X" + u with E(uu0) = Й, a necessary and sufficient condition for OLS to be equivalent to GLS is given by X0 fi_1PX where PX = I – PX and PX = X(X0X)_1 X0, see Eq.(9.7) of Chap.9. For Eq. (12.12), this condition can be written as

(X0Q)(Q/o2)P qx = 0

using the fact that Q is idempotent, the left hand side can be written as (X0Q)P qx/ov2

which is clearly 0, since PqX is the orthogonal projection of QX.

One ca...

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