Category Springer Texts in Business and Economics

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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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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Moment Generating Function Method

a. If Xi,.., Xn are independent Poisson distributed with parameters (Xi) respec­tively, then from problem 2.14c, we have

MXi (t) = eAi(e-1) for i = 1,2,… ,n

n n

Y = Xi has My(t) = П MXi (t) since the X/s are independent. Hence

i=i i=i

n

Ai (e‘-l)

MY(t) = ei=1

n

which we recognize as a Poisson with parameter ^ Xi.

i=i

b. IfXi, ..,Xn are IIN (^i, a2), then from problem 2.14b, we have MXi(t) = ew‘+ 1ai2‘2 for i = 1,2,.., n

nn

Y = ^ Xi has MY(t) = ]""[ MXi (t) since the X/s are independent. Hence

i=i i=i

image101

MY(t) = e 1=i

c. If Xi,.., Xn are IIN(|a,, a2),thenY = J2 Xi is N(np,,na2) from part b using

i=i

the equality of means and variances. Therefore, X = Y/nis N(p,, a2/n).

d. If Xi,.., Xn are independent x2 distributed with parameters (ri) respectively, then from problem 2...

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Multiple Regression Analysis

4.1 The regressions for parts (a), (b), (c), (d) and (e) are given below.

a. Regression of LNC on LNP and LNY Dependent Variable: LNC

Analysis of Variance

Sum of

Mean

Source

DF

Squares

Square

F Value

Prob>F

Model

2

0.50098

0.25049

9.378

0.0004

Error

43

1.14854

0.02671

C Total

45

1.64953

Root MSE

0.16343

R-square

0.3037

Dep Mean

4.84784

Adj R-sq

0.2713

C. V.

3.37125

Parameter Estimates

Parameter

Standard

T for HO:

Variable

DF

Estimate

Error Parameter=0

Prob>|T|

INTERCEP 1

4.299662

0.90892571

4.730

0.0001

LNP

1

-1.338335

0.32460147

-4.123

0.0002

LNY

1

0.172386

0.19675440

0.876

0.3858

Regression of LNC on LNP Dependent Variab...

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