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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Dependent Variable: LNRGDP

Analysis of Variance

Sumof

Mean

Source

DF

Squares

Square

F Value Prob > F

Model

1

53.88294

53.88294

535.903

0.0001

Error

18

1.80983

0.10055

C Total

19

55.69277

RootMSE 0.31709 R-square 0.9675 DepMean 10.60225 Adj R-sq 0.9657

C. V.

2.99078

Parameter Estimates

Parameter

Standard

T for HO:

Variable

DF

Estimate

Error

Parameter=0

Prob > |T|

INTERCEP

1

1.070317

0.41781436

2.562

0.0196

LNEN

1

0.932534

0.04028297

23.150

0.0001

e. Log-log specification

Dependent Variable: LNEN1

Analysis of Variance

Sum of

Mean

Source

DF

Squares

Square

F Value

Prob > F

Model

1

59.94798

59.94798

535.903

0.0001

Error

18

2.01354

0...

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The backup regressions are given below

a. OLS regression of consumption on a constant and Income using EViews Dependent Variable: Consumption Method: Least Squares

Sample: 1959 2007 Included observations: 49

b. The Breusch-Godfrey Serial Correlation LM Test for serial correlation of the first order is obtained below using EViews. An F-statistic as well as the LM statistic which is computed as T* R-squared are reported, both of which are significant. The back up regression is also shown below these statistics. This regression runs the OLS residuals on their lagged values and the regressors in the original model. We cannot reject first order serial correlation.

Breusch-Godfrey Serial Correlation LM Test:

F-statistic 168.9023 Prob. F(1,46) 0.0000

Obs*R-squared 38.51151 Prob. Chi-Square(1) 0.0000

Test Equation:

Dependent Variable: ...

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