Category Springer Texts in Business and Economics

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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Limited Dependent Variables

13.1 The Linear Probability Model

Уі

u;

Prob.

1

1 — x0"

к;

0

CD.

УС

1

1 — к;

a. Let к і = Pr[y; = 1], then y; = 1 when u; = 1 — x0" with probability к; as shown in the table above. Similarly, y; = 0 when u; = —x0" with probability 1 — к ;. Hence, E(u;) = к; (1 — x[") + (1 — к 😉 (—x0").

For this to equal zero, we get, к; — к ;xi" + к ;xi" — x0" = 0 which gives к ; = xi" as required.

b. var(u;) = E(u2) = (1 — xi")2 к ; + (—x0")2 (1 — к 😉

1 — 2×0" + (x0")2 к; + (xi")2 (1 — к i)

= к ; — 2×0" к ; + (x0")2 = к ; — к 2 = к ;(1 — к 😉 = x0" (1 — xi") using the fact that к ; = xi".

13.2 a. Since there are no slopes and only a constant, x0" = a and (13.16) becomes

n

log ‘ = J]{y; logF(a) +...

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What is Econometrics?

This chapter emphasizes that an econometrician has to be a competent mathematician and statistician who is an economist by training. It is the unification of statistics, economic theory and mathematics that constitutes econometrics. Each view point, by itself is necessary but not sufficient for a real understanding of quantitative relations in modern economic life, see Frisch (1933).

Econometrics aims at giving empirical content to economic relationships. The three key ingredients are economic theory, economic data, and statistical methods. Neither ‘theory without measurement’, nor ‘measurement without theory’ are sufficient for explaining economic phenomena. It is as Frisch emphasized their union that is the key for success in the future development of econometrics.

Econometrics p...

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Using EViews, Qt+i is simply Q(1) and one can set the sample range from 1954-1976

a. The OLS regression over the period 1954-1976 yields RSt = -6.14 + 6.33 Qt+1 – 1.67 Pt

(8.53) (1.44) (1.37)

with R2 = 0.62 and D. W. = 1.07. The t-statistic for у = 0 yields

t = -1.67/1.37 = -1.21 which is insignificant with a p-value of 0.24.

Therefore, the inflation rate is insignificant in explaining real stock returns.

LS // Dependent Variable is RS Sample: 1954 1976 Included observations: 23

Variable

Coefficient

Std. Error t-Statistic

Prob.

C

-6.137282

8.528957 -0.719582

0.4801

Q(1)

6.329580

1.439842 4.396024

0.0003

P

-1.665309

1.370766 -1.214875

0.2386

R-squared

0.616110

Mean dependent var

8.900000

Adjusted R-squared

0.577721

S. D. dependent var

21.37086

S. E. of regression

13.88743

Akaike info criterion

5...

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