Category Springer Texts in Business and Economics

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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Variance-Covariance Matrix of Random Effects

a. From (12.17) we get

Й = ct^In <8> Jt) + c^.In <8> It)

Replacing JT by TJT, and IT by (Et + JT) where ET is by definition (It — JT), one gets

Й = Tc^(In <8> Jt) + c^(In <8> Et) + c^.In <8> Jt)

collecting terms with the same matrices, we get

Й = (Tc^ C c2)(In <S> Jt) C cv2(In <S> Et) = стуР + cv2Q where Cj2 = Tc2 C c2.

b. p = z2(z;z2)“ z; = IN <S> JT is a projection matrix of Z2. Hence,

it is by definition symmetric and idempotent. Similarly, Q = INT — P is the orthogonal projection matrix of Z2. Hence, Q is also symmetric and idempotent. By definition, P + Q = INT. Also, PQ = P(Int—P) = P—P2 = P — P = 0.

c. From (12.18) and (12.19) one gets

П ^-1 = (ci2P C cv2Q) (%P C q) = P C Q = Int

Vc12 cv2 J

since P2 = P, Q2 = Q and PQ = 0 as verified in part (b)...

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Moment Generating Function (MGF)

a. For the Binomial Distribution,

Подпись: n-XMx(t) = E(eXt) = XX) eX‘0X (1 – 0)

X=0

Подпись: = E X (1 - ™)n X=0 X/ tn -X

= [(1 – 0) + 0e‘]

where the last equality uses the binomial expansion (a + b)n =

P ( v I aXbn-X with a = 0e‘ and b = (1 – 0). This is the fundamental x=o X /

image095

relationship underlying the binomial probability function and what makes it a proper probability function. b. For the Normal Distribution,

completing the square

Mx(t) = 1 f °° e-222{[x-(^+ta2)]2-(^+ta2)2+^1 dx

os/2rr J-i

_ e"2O2 [^2-^2-2^ta2-tV]

The remaining integral integrates to 1 using the fact that the Normal den­sity is proper and integrates to one. Hence Mx(t) = e^*+ 2°2*2 after some cancellations.

image096

c. For the Poisson Distribution,

= e-X X ^ _ X = eA(e‘-!)

^ X!

X=0

X

where the fifth equality follows from the fact that X = ea and...

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For parts (b) and (c), SAS will automatically compute confidence intervals for the mean (CLM option) and for a specific observation (CLI option), see the

SAS program in 3.17.

95% CONFIDENCE PREDICTION INTERVAL

Dep Var

Predict

Std Err

Lower95%

Upper95%

Lower95%

Upper95%

COUNTRY

LNEN1

Value

Predict

Mean

Mean

Predict

Predict

AUSTRIA

14.4242

14.4426

0.075

14.2851

14.6001

13.7225

15.1627

BELGIUM

15.0778

14.7656

0.077

14.6032

14.9279

14.0444

15.4868

CYPRUS

11.1935

11.2035

0.154

10.8803

11.5268

10.4301

11.9770

DENMARK

14.2997

14.1578

0.075

14.0000

14.3156

13.4376

14.8780

FINLAND

14.2757

13.9543

0.076

13.7938

14.1148

13.2335

14.6751

FRANCE

16.4570

16.5859

0.123

16.3268

16.8449

15.8369

17.3348

GREECE

14.0038

14.2579

0.075

14.1007

14.4151

13.5379

14.9780

ICELAND

11.1190

10.7795

0.170

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