Springer Texts in Business and Economics

Distributed Lags and Dynamic Models

6.1 a. Using the Linear Arithmetic lag given in Eq. (6.2), a 6 year lag

on income gives a regression of consumption on a constant and

6

Zt = J2 (7 — i) Xt_i where Xt denotes income. In this case,

i=0

Zt = 7Xt C 6Xt_i + .. + Xt_6,

The Stata regression output is given below:

. gen ^6=7*ly+6*l. ly+5*l2.ly+4*l3.ly+3*l4.ly+2*l5.ly+l6.ly (6 missing values generated)

. reg lc z_6

Source

SS

df

MS

Number of obs F(1,41)

Prob > F R-squared Adj R-squared Root MSE

= 43 = 3543.62 = 0.0000 = 0.9886 = 0.9883 = .03037

Model

Residual

3.26755259

.037805823

1

41

3.26755259

.000922093

Total

3.30535842

42

.07869901

lc

Coef.

Std. Err.

t

P>|t|

[95% Conf. Interval]

z_6

.cons

.0373029 .0006266 -.4950913 .1721567

59.53

-2.88

0.000

0.006

.0360374

-.8427688

.0385684

-.1474138

From Eq. (6.2)

Pi = [(s C 1) — iP for i = 0,.., 6 with P estimated as the coefficient of Zt (which is z_6 in the regression). This estimate is 0.037 and is statistically significant.

Now we generate the regressors for an Almon lag first-degree polynomial with a far end point constraint using Stata:

. gen Z0= ly+l. ly+l2.ly+l3.ly+l4.ly+l5.ly+l6.ly (6 missing values generated)

B. H. Baltagi, Solutions Manual for Econometrics, Springer Texts in Business and Economics, DOI 10.1007/978-3-642-54548-1—6, © Springer-Verlag Berlin Heidelberg 2015

. gen Z1=0*ly+l. ly+2*l2.ly+3*l3.ly+4*l4.ly+5*l5.ly+6*l6.ly (6 missing values generated)

.gen Z=Z1-7*Z0 (6 missing values generated)

. reg lc Z

Source

SS

df

MS

Number of obs

= 43

Model

Residual

3.26755293

.037805483

1

41

3.26755293

.000922085

F(1,41) Prob > F R-squared

= 3543.66 = 0.0000 = 0.9886

Total

3.30535842

42

.07869901

Adj R-squared Root MSE

= 0.9883 = .03037

lc

Coef.

Std. Err.

t

P>|t|

[95% Conf. Interval]

Z

-.0373029

.0006266

-59.53

0.000

-.0385684

-.0360374

_cons

-.4950919

.1721559

-2.88

0.006

-.8427679

-.147416

The EViews output for PDL(Y,6,1,2) which is a sixth order lag, first degree polynomial, with a far end point constraint is given by:

Dependent Variable: LNC Method: Least Squares

Sample (adjusted): 1965 2007 Included observations: 43 after adjustments

Coefficient

Std. Error

t-Statistic

Prob.

C -0.495091

0.172156

-2.875817

0.0064

PDL01 0.149212

0.002507

59.52843

0.0000

R-squared

0.988562

Mean dependent var

9.749406

Adjusted R-squared

0.988283

S. D. dependent var

0.280533

S. E. of regression

0.030366

Akaike info criterion

-4.105595

Sum squared resid

0.037806

Schwarz criterion

-4.023678

Log likelihood

90.27028

Hannan-Quinn criter.

-4.075386

F-statistic

3543.634

Durbin-Watson stat

0.221468

Prob(F-statistic)

0.000000

b. Using an Almon-lag second degree polynomial described in Eq. (6.4), a

6 year lag on income gives a regression of consumption on a constant,

6 6 6

Z0 = 2^ Xt_i, Zi = 2^ iXt_i and Z2 = i2Xt_i. This yields the Almon-lag

i=0 i=0 i=0

without near or far end-point constraints. A near end-point constraint imposes

0 in Eq. (6.1) which yields a0 — a1 + a2 = 0 in Eq. (6.4). Substituting

for a0 in Eq. (6.4) yields the regression in (6.5).

The following Stata code, generates the variables needed to estimate an Almon lag second-degree polynomial with a near end point constraint:

.gen Z2=0*ly+l. ly+2‘2*l2.ly+3‘2*l3.ly+4‘2*l4.ly+5‘2*l5.ly+6‘2*l6.ly (6 missing values generated)

. gen Z01= Z1+Z0 (6 missing values generated)

. gen Z02= Z2-Z0

(6 missing values generated) . reg lc Z01 Z02

43

1701.72

0.0000

0.9884

0.9878

.03098

lc

Coef.

Std. Err.

t

P>|t|

[95% Conf. Interval]

Z01

.1708636

.0260634

6.56

0.000

.1181875

.2235397

Z02

-.0441775

.0085125

-5.19

0.000

-.0613819

-.026973

_cons

-.8139917

.2306514

-3.53

0.001

-1.280156

-.3478277

The EViews output for PDL(Y,6,2,1) which is a sixth order lag, second degree polynomial, with a near end point constraint is given by:

Dependent Variable: LNC Method: Least Squares

Sample (adjusted): 1965 2007 Included observations: 43 after adjustments

Coefficient Std. Error t-Statistic Prob.

C -0.813961 0.230649

-3.529009

0.0011

PDL01 0.259216 0.043085

6.016333

0.0000

PDL02 -0.044177 0.008512

-5.189671

0.0000

R-squared

0.988384

Mean dependent var

9.749406

Adjusted R-squared

0.987803

S. D. dependent var

0.280533

S. E. of regression

0.030982

Akaike info criterion

-4.043589

Sum squared resid

0.038396

Schwarz criterion

-3.920714

Log likelihood

89.93716

Hannan-Quinn criter.

-3.998276

F-statistic

1701.718

Durbin-Watson stat

0.377061

Prob(F-statistic)

0.000000

Lag Distribution of LNY

i

Coefficient

Std. Error

t-Statistic

0

0.21504

0.03457

6.21968

1

0.34172

0.05213

6.55566

2

0.38006

0.05266

7.21676

3

0.33003

0.03621

9.11501

4

0.19166

0.00402

47.7225

*. |

5

-0.03508

0.04810

0.72915

* . I

6

-0.35016

0.11562

3.02843

Sum of

1.07327

0.02249

47.7225

Lags

c. The far end-point constraint imposes "7 = 0. This translates into the fol­lowing restriction a0 + 7ai + 49a2 = 0. Substituting for a0 in (6.4) yields the regression in (6.6) with s = 6, i. e., the regression of consumption on a constant, (Z1 — 7Z0) and (Z2 — 49Z0).

The following Stata code, generates the variables needed to estimate an Almon lag second-degree polynomial with a far end point constraint:

. gen Z10_far=Z1-7*Z0 (6 missing values generated) . gen Z20_far=Z2-7‘2*Z0 (6 missing values generated) . reg lc Z10_farZ20_far

Source

Model

Residual

Total

------- l-c---

Coef.

Std. Err.

t

P>|t|

[95% Conf. Interval]

Z10_far

-.3833962

.0538147

-7.12

0.000

-.4921598

-.2746326

Z20_far

.0381843

.0059371

6.43

0.000

.0261849

.0501837

_cons

-1.237493

.1681062

-7.36

0.000

-1.577249

-.8977381

The EViews output for PDL(Y,6,2,2) which is a sixth order lag, second degree polynomial, with a far end point constraint is given by:

Dependent Variable: LNC Method: Least Squares

Sample (adjusted): 1965 2007 Included observations: 43 after adjustments

Coefficient

Std. Error

t-Statistic

Prob.

C

-1.237555

0.168112

-7.361483

0.0000

PDL01

0.006206

0.022306

0.278216

0.7823

PDL02

-0.154297

0.018196

-8.479577

0.0000

. gen Z_NF=-47* Z0-6* Z1 + Z2 (6 missing values generated)

. reg lc Z_NF

43

2663.90

0.0000

0.9848

0.9845

.03496

----- lc---

Coef.

Std. Err.

t

P>|t|

[95% Conf. Interval]

Z_NF

-.0028049

.0000543

-51.61

0.000

-.0029147

-.0026952

_cons

-.2411805

.1936405

-1.25

0.220

-.6322455

.1498845

The EViews output for PDL(Y,6,2,3) which is a sixth order lag, sec­ond degree polynomial, with both near and far end point constraints is given by:

Dependent Variable: LNC Method: Least Squares

Sample (adjusted): 1965 2007 Included observations: 43 after adjustments

Coefficient

Std. Error

t-Statistic

Prob.

C -0.233622

0.197176

-1.184841

0.2429

PDL01 0.097163

0.001918

50.64935

0.0000

R-squared

0.984269

Mean dependent var

9.749406

Adjusted R-squared

0.983886

S. D. dependent var

0.280533

S. E. of regression

0.035612

Akaike info criterion

-3.786892

Sum squared resid

0.051996

Schwarz criterion

-3.704975

Log likelihood

83.41817

Hannan-Quinn criter.

-3.756683

F-statistic

2565.357

Durbin-Watson stat

0.214478

Prob(F-statistic)

0.000000

Lag Distribution of LNY

i

Coefficient

Std. Error

t-Statistic

0

0.08502

0.00168

50.6494

1

0.14574

0.00288

50.6494

2

0.18218

0.00360

50.6494

3

0.19433

0.00384

50.6494

4

0.18218

0.00360

50.6494

5

0.14574

0.00288

50.6494

6

0.08502

0.00168

50.6494

Sum of Lags

1.02021

0.02014

50.6494

e. The RRSS for the Chow test for the arithmetic lag restrictions is given by the residual sum of squares of the regression in part (a), i. e., .037805823. The URSS is obtained from running consumption on a constant and six lags on income. The corresponding Stata regression is given below:

. reg lc ly l. ly l2.ly l3.ly l4.ly l5.ly l6.ly

Source

SS

df

MS Number of obs =

................... F(7 35)

43

1093.59

0.0000

0.9954

0.9945

.02073

Model

Residual

3.29031477

.015043647

7

35

=

.470044967 Prob > F = .000429818 R-squared =

Total

3.30535842

42

.07869901 Root MSE =

|c

Coef.

Std. Err.

t

P>|t|

[95% Conf. Interval]

ly

1.237818

.2192865

5.64

0.000

.7926427

1.682993

L1.

.2504519

.310222

0.81

0.425

-.3793323

.8802361

L2.

-.203472

.3005438

-0.68

0.503

-.8136084

.4066644

L3.

-.0279364

.3041055

-0.09

0.927

-.6453034

.5894306

L4.

.0312238

.3049614

0.10

0.919

-.5878808

.6503284

L5.

-.0460776

.3048432

-0.15

0.881

-.6649422

.572787

L6.

-.1270834

.2028801

-0.63

0.535

-.5389519

.2847851

_cons

-1.262225

.1667564

-7.57

0.000

-1.600758

-.9236913

URSS = .015043647. The number of restrictions given in (6.2) is 6. Hence, the Chow F-statistic can be computed as follows:

. display (.037805483-.015043647)*35/(6*.015043647)

8.8261428

and this is distributed as F(6,35) under the null hypothesis. This rejects the arithmetic lag restrictions.

f. Similarly, the Chow test for the Almon lag second-degree polynomial with a near end point constraint can be computed using RRSS = .038395978 from part (b). URSS = .015043647 from the last regression in part (e), and the number of restrictions is 5:

. display (.038395978-.015043647)*35/(5*.015043647)

10.866136

and this is distributed as F(5,35) under the null hypothesis. This rejects the Almon lag second-degree polynomial with a near end point constraint.

g. The Chow test for the Almon lag second-degree polynomial with a far end point constraint can be computed using RRSS = .018586036 from part (c). URSS = .015043647 from the last regression in part (e), and the number of restrictions is 5:

. display (.018586036-.015043647)*35/(5*.015043647)

1.6483186

and this is distributed as F(5,35) under the null hypothesis. This does not reject the Almon lag second-degree polynomial with a far end point constraint.

Finally, The Chow test for the Almon lag second-degree polynomial with both near and far end point constraints can be computed using RRSS = .050101463 from part (d). URSS = .015043647 from the last regression in part (e), and the number of restrictions is 6:

. display (.050101463-.015043647)*35/(6*.015043647)

13.594039

and this is distributed as F(6,35) under the null hypothesis. This rejects the Almon lag second-degree polynomial with both near and far end point constraints.

6.2 a. For the Almon-lag third degree polynomial "i = a0 C a1i C a2i2 + a3i3 for i = 0,1,.., 5.

In this case, (6.1) reduces to

5

Yi = a C У (a0 + a1i + a2i2 + a3i3) Xt_i + ut

i=1

5 5 5 5

= a C a0 C У Xt_i C a1 У iXt_i C a2 У i2Xt_i C a3 У i3Xt_i C ut,

i= 0 i= 0 i= 0 i= 0

Now a, a0,a1,a2 and a3 can be estimated from the regression of Yt on a

5 5 5 5

constant, Z0 = Yl Xt_i, Z1 = Yl iXt_i, Z2 = Yl i2Xt_i and Z3 = Yl i3Xt_i.

i=0 i=0 i=0 i=0

The following Stata code generates the variables to run the OLS regression:

. gen Z5_0= ly+l. ly+l2.ly+l3.ly +l4.ly +l5.ly (5 missing values generated)

. gen Z5_1=l. ly+2*l2.ly +3*l3.ly +4*l4.ly +5*l5.ly (5 missing values generated)

. gen Z5_2=l. ly+2''2*l2.ly +3“2*l3.ly +4“2*l4.ly +5“2*l5.ly (5 missing values generated)

. gen Z5_3=l. ly+2‘3*l2.ly +3‘3*!3.ly +4‘3*!4.ly +5‘3*!5.ly

(5 missing values generated) . reg lc Z5_0 Z5_1 Z5_2 Z5.3

44

2063.28

0.0000

0.9953

0.9948

.02086

----- lc---

Coef.

Std. Err.

t

P>|t|

[95% Conf. Interval]

Z5_0

1.27317

.1911863

6.66

0.000

.8864591

1.659881

Z5_1

-1.623359

.5714614

-2.84

0.007

-2.779249

-.4674696

Z5_2

.5693275

.2955923

1.93

0.061

-.0285644

1.167219

Z5_3

-.0599977

.0390623

-1.54

0.133

-.1390087

.0190133

_cons

-1.128979

.1533499

-7.36

0.000

-1.439158

-.8187994

b. The estimate of "3 is "3 = a0 + 3a3 + 9a2 + 27a3 with var 3^ = var (a0) + 9 var(ai) + 81 var (a2) + 272var (a 3)

+ 2.3. cov (a0, a1) + 2.9. cov (a0, a2) + 2.27.cov (a0, a3) + 2.3.9. cov (ai, a2) + 2.3.27 cov (ai, a3) + 2.9.27 cov (a2, a3) .

The estimate of ^3 can be computed from this regression results as follows: . *beta3

. display 1.27317+3*(-1.623359)+9*.5693275+27*(-.0599977)

-.0928974

The var-cov matrix of the regression estimates are given by:

. matrix list e(V) symmetric e(V)[5,5]

Z5_0

Z5_1

Z5_2

Z5_3

_cons

Z5_0

.0365522

Z5_1

-.09755986

.32656818

Z5_2

.04577699

-.16633866

.08737482

Z5_3

-.00565526

.0214819

-.0114859

.00152587

_cons

-.01320609

.02236579

-.00962469

.00120366

.02351619

The EViews output for PDL(Y,5,3) which is a fifth order lag, third degree polynomial, with no end point constraint is given by:

Dependent Variable: LNC Method: Least Squares

Sample (adjusted): 1964 2007 Included observations: 44 after adjustments

Coefficient Std. Error

t-Statistic

Prob.

C -1.129013 0.153388

-7.360517

0.0000

PDL01 -0.176078 0.122030

-1.442904

0.1570

PDL02 -0.066325 0.162878

-0.407206

0.6861

PDL03 0.209243 0.066943

3.125704

0.0033

PDL04 -0.059926 0.039088

-1.533091

0.1333

R-squared

0.995295

Mean dependent var

9.736786

Adjusted R-squared

0.994813

S. D. dependent var

0.289615

S. E. of regression

0.020859

Akaike info criterion

-4.795464

Sum squared resid

0.016968

Schwarz criterion

-4.592715

Log likelihood

110.5002

Hannan-Quinn criter.

-4.720275

F-statistic

2062.698

Durbin-Watson stat

0.393581

Prob(F-statistic)

0.000000

Lag Distribution of LNY

i

Coefficient

Std. Error

t-Statistic

■ *|

0

1.27295

0.19130

6.65411

■ * I

1

0.15942

0.15649

1.01868

* ■ I

2

-0.17608

0.12203

-1.44290

*■ I

3

-0.09309

0.11943

-0.77943

■* |

4

0.04883

0.15599

0.31306

*■ I

5

-0.10987

0.17870

-0.61485

Sum of

1.10217

0.01492

73.8706

Lags

c. Imposing the near end-point constraint = 0 yields the following restriction on the third degree polynomial in a’s:

ao _ ai C a2 — a3 — 0.

solving for a0 and substituting above yields the following constrained regression:

Yt — a C ai. Zi C Zo) C a2(Z2 — Zo) C a3.Zi C Z3) C ut
The corresponding Stata regression is reported below.

. gen ZZ01= Z5_0+Z5_1 (5 missing values generated) . gen ZZ03= Z5_0+Z5_3 (5 missing values generated) . gen ZZ02= Z5_2-Z5_0 (5 missing values generated) . reg lc ZZ01 ZZ03 ZZ02

44

2205.97

0.0000

0.9940

0.9935

.02327

lc

Coef.

Std. Err.

t

P>|t|

[95% Conf. Interval]

ZZ01

.2551128

.0259506

9.83

0.000

.2026646

.307561

ZZ03

.0632448

.0123507

5.12

0.000

.0382831

.0882065

ZZ02

-.3852599

.0629406

-6.12

0.000

-.5124676

-.2580523

_cons

-.9865419

.1641747

-6.01

0.000

-1.318351

-.6547324

d. Test the near end point constraint with a Chow test. TheURSS = .016963297 from part (a) and RRSS = .021668704 from part (c) and there is one restriction.

. display (.021668704-.016963297)*39/(1*.016963297)

10.818114

and this is distributed as F(1,39) under the null hypothesis. This rejects the near end point constraint.

e. The Chow test for the Almon 5 year lag third-degree polynomial with no end point constraints can be computed using RRSS = .016963297 from part (a). URSS = .016924337 from the unrestricted 5 year lag regression given below:

44

1308.00

0.0000

0.9953

0.9945

.02139

lc

Coef.

Std. Err.

t

P>|t|

[95% Conf. Interval]

ly

1.303457

.2224286

5.86

0.000

.852774

1.75414

L1.

.090694

.3099447

0.29

0.771

-.5373136

.7187016

L2.

-.1401868

.3081789

-0.45

0.652

-.7646165

.484243

L3.

-.0460399

.3104643

-0.15

0.883

-.6751003

.5830204

L4.

-.0255622

.3129629

-0.08

0.935

-.6596853

.6085608

L5.

-.0800021

.2087812

-0.38

0.704

-.5030331

.3430288

_cons

-1.13104

.1574744

-7.18

0.000

-1.450114

-.8119665

and the number of restrictions is 3:

. display (.016963297- .016924337)*37/(3*.016924337)

.02839146

and this is distributed as F(3,37) under the null hypothesis. This does not reject the Almon 5 year lag third-degree polynomial restrictions.

6.3 a. From (6.18), Yt = "Yt-i C vt. Therefore, Yt_i

pYt-i = p"Yt_2 C pvt-1.

Subtracting this last equation from (6.18) and re-arranging terms, one gets

Yt = (" C p/ Yt-1 — p"Yt-2 C ©t.

Multiply both sides by Yt-1 and sum, we get

T T T T

I>tYt_i = (" C p)£ Y2-i — p"J^ Yt-iYt-2 C £ Yt-iet.

t=2 t=2 t=2 t=2

T

Divide by Yt2 i and take probability limits, we get

t=2

Hence,

T

plim X Х /T = "0 - 2"2/уо - "і/"о = "0 - ("2/"0) = ("0 - Yi) /"о.

t=2

Also, Vt_i = Yt_i — PolsYt_2. Multiply this equationby V and sum, we get

T T T

EVtVt_i = X Yt_i vt - Pol^~^Yt-2vt_i.

t=2 t=2 t=2

T

But, by the property of least squares Yt_iV = 0, hence

t= 2

X VtVt_i /T = - "ol^Yt_2Vt /T = - olsJ2 Yt_2Yt/T

t=2 t=2 t=2

T

C P 2l^Yt_2Yt_i/T

t2

and

From part (a), we know that

Yi = (P + p) Yt_i - p"Yt_2 + ©t multiply both sides by Yt_2 and sum and take plim after dividing by T, we get

"2 = (P C p) "i - pP"0 so that

"0"2 = (P C p)"i"0 C pP"2 and "2 - "0У2 = "2 - (P C p) "i"0 C pP"2.

But from part (a), plim"ols = yi/yo = (" + p)/(1 + p"). Substituting

(" + P)"o = (1 + Р")Уі above we get

- У0У2 = У? - (1 + Pp) У? + p""0 = P" ("2 - У?) ■

Hence,

plimp = — • p" = p" (" + p) / (1 + p")

У0

and

plim (p - p) = (p"2 + p2" - p - p2") / (1 + p") = p ("2 - 1) / (1 + p")

= - plim ("ols - .

The asymptotic bias of p is negative that of "ols.

T T T

d. Since d = ^2 (Vt - O-1) / ^2 and as T! 1, vt2 is almost identical

t=2 t=2 t=2

T

to ^2 v2_1, then plim d ^ 2(1 - plimp) where p was defined in (c). But

t=2

plimp = p - p(1-p2)/(1 + p") = ( p2" + pP2)/(1 + pP) = p"( p+")/(1 + p"). Hence, plim d = 2 1 - p"1(_C'"pp) -

e. Knowing the true disturbances, the Durbin-Watson statistic would be

T

d* = К (v‘- v‘-1)2 ^Zvt2

t=2 t=2

and its plim d* = 2(1 - p). This means that from part (d)

plim (d - d*) = 2 (1 - plim p) - 2 (1 - p) = 2 [ p - " p ( p + ") / (1 + p")]

= 2 [ p + p2" - " p2 - "2p ] / (1 + p") = 2 p (1 - "2) / (1 + p") = 2plim("ols - ")

from part (a). The asymptotic bias of the D. W. statistic is twice that of "ols. The plim d and plim d* and the asymptotic bias in d can be tabulated for various values of p and ".

6.4 a. From (6.18) with MA(1) disturbances, we get Yt = "Yt-1 + ©t + 0©t_1 with |"| < 1.

In this case,

T T T T

P ols =J2 YtYt-1^; Y_1 = P +T, Yt-i©t /J2 Y?-1

t=2 t=2

TT

C 0J2 Yt_1©t_1/£ Y2_1

t=2

so that plim ols - ") = plim ^ E Yt_1©t /Tj /plim ^E Y2_1 /Tj

+ 0plim IE Yt_1©t_1 /^ /plim ^ X Y2_1 /^ . Now the above model can be written as (1 - "L) Yt = (1 C 0L)©

or

Yt = (1 C 0L)^]piLi©t

i=0

Yt = (1 C 0L) (©t C P©t_1 + "2©t_2 + ..)

Yt = ©t C (0 C ") [©t_1 c "©t_2 c "2©t_3 c ..]

From the last expression, it is clear that E(Yt) = 0 and

T

var(Yt) = oe2[1 C (0 C ")2/(1 - P2)] = plim£ Y2_1/T.

t=2

Also,

TT

Yt_1©t/T = ©t [©t_1 C (0 C P) (©t_2 C "©t_3 C ..)] /T

t= 2 t= 2

Since the ©t’s are not serially correlated, each term on the right hand side has

T

zero plim. Hence, plim Yt_1©t/T = 0 and the first term on the right hand

t= 2

side of plim (pols — is zero. Similarly,

T T

£Yt_i8t_i/T = E ©2-1/T c (0 c P)

=2

TT

©t-1 ©t-2/T C P ©t-1 ©t-3/T C..

t=2

T

which yields plim£ Yt-i©t-i/T = a©2 since the second term on the right

t= 2

hand side has zero plim. Therefore,

plim (pols — P) = 0a©2 /a©2 [l + (0 + P)2 / (1 — p2)"

= 0 (1 — P2) / (1 — P2 C 02 C P2 C 20p)

= 0 (1 — P2) / (1 C 02 C 20P) = 8 (1 — P2) / (1 C 2P8)

where 8 = 0/(1 C 02).

The asymptotic bias of Pols derived in part (a) can be tabulated for various values of P and 0 < 0 < 1.

Let vt = Yt — PolsYt-1 = PYt-1 — PolsYt-1 + vt = vt — (pols — P) Yt-1. But

TT

P ols — P = E Yt-1V^]Y2-1.

t=2 t=2

Therefore,

T T / T 2 T T

E v 2 = E v2 + E Yt-1 v^EY2-1 — 2 (E Yt-1 vt)2 / E Y2-1

t=2 t=2 t=2 t=2 t=2

T / T 2 T

= Ev2 — EYt-1 vt I /E! Y2-1.

t=2 t=2 t=2

But, vt = ©t C 0©t-1 withvar(vt) = a©2(1 C 02). Hence,

T

pHmEv2/T = a2 +02 aE2 = ae2(1 +02).

t= 2

Also,

T T T

plim^^Yt-iVt/T = plim£Yt_iEt/T + 9plim EY‘-1©‘/T

t=2 t=2 t=2

= 0 C 0a©2 = 0a©2

from part (a). Therefore,

T

1 C (0 C ")2/(1 - "2)

t=2

= a©2 [1 C 02 - 02 (1 - "2) / (1 - 02 C 20")]

= a©2 [1 C 02 - 08 (1 - "2) / (1 C 28")] = a©2 [1 C 02 - 00*]

where 8 = (1_|0Q2) and 0* = 8(1 - "2)/(1 C 2"8).

6.5 a. The stata commands to generate the Durbin h statistic from scratch are the

following:

. gen lc_lag=l. lc (1 missing value generated) . reg lc ly lcJag

Source

Model

Residual

Total

----- lc---

Coef.

Std. Err.

t

P>|t|

[95% Conf. Interval]

ly

.3091105

.0816785

3.78

0.000

.1446015

.4736194

lcJag

.7045042

.0765288

9.21

0.000

.5503673

.8586411

_cons

-.1475238

.0870474

-1.69

0.097

-.3228462

.0277986

. display.0765288*2 .00585666 . predict miu, resid (1 missing value generated) . reg miu l. miu, noconstant

. * Durbin’s h

. display.5065964*(48*(1-.00585666))“0.5 3.4995099

This is asymptotically distributed as N(0,1) under the null hypothesis of p = 0. This rejects Ho indicating the presence of serial correlation. b. The EViews output for the Breusch and Godfrey test for first-order serial correlation is given below. The back up regression is given below the test statistics. This regresses the OLS residuals on their lagged value and the regressors in the model including the lagged dependent variable. This yields an R2 = 0.278576. The number of observations is 48. Therefore, the LM statistic = TR2 = 48(0.278576) = 13.372. This is asymptotically dis­tributed as Chi-Square(1) under the null hypothesis of p = 0. The p-value is 0.0003, and we reject the null hypothesis.

Breusch-Godfrey Serial Correlation LM Test:

F-statistic 16.99048 Prob. F(1,44) 0.0002

Obs*R-squared 13.37165 Prob. Chi-Square(1) 0.0003

Test Equation:

Dependent Variable: RESID Method: Least Squares

Sample: 1960 2007 Included observations: 48

Presample missing value lagged residuals set to zero.

Coefficient

Std. Error

t-Statistic

Prob.

C

-0.061130

0.076227

-0.801941

0.4269

LNC(-1)

-0.081685

0.068658

-1.189747

0.2405

LNY

0.086867

0.073256

1.185810

0.2421

RESID(-1)

0.552693

0.134085

4.121951

0.0002

R-squared

0.278576

Mean dependent var

-5.00E-16

Adjusted R-squared

0.229388

S. D. dependent var

0.013797

S. E. of regression

0.012112

Akaike info criterion

-5.909614

Sum squared resid

0.006455

Schwarz criterion

-5.753680

Log likelihood

145.8307

Hannan-Quinn criter.

-5.850686

F-statistic

5.663494

Durbin-Watson stat

1.773229

Prob(F-statistic)

0.002272

c. The EViews output for the Breusch and Godfrey test for second-order serial correlation is given below. The back up regression is given below the test statistics. This regresses the OLS residuals et on et_i and et_2 and the regres­sors in the model including the lagged dependent variable. This yields an R2 = 0.284238. The number of observations is 48. Therefore, the LM statistic = TR2 = 48(0.284238) = 13.643.

This is asymptotically distributed as Chi-Square(2) under the null hypothesis of no second-order serial correlation. The p-value is 0.0011, and we reject the null hypothesis.

Breusch-Godfrey Serial Correlation LM Test:

F-statistic 8.537930 Prob. F(2,43) 0.0008

Obs*R-squared 13.64344 Prob. Chi-Square(2) 0.0011

Test Equation:

Dependent Variable: RESID Method: Least Squares

Sample: 1960 2007 Included observations: 48

Presample missing value lagged residuals set to zero.

Coefficient

Std. Error

t-Statistic

Prob.

C

-0.049646

0.079289

-0.626136

0.5345

LNC(-1)

-0.067455

0.073355

-0.919582

0.3629

LNY

0.071648

0.078288

0.915192

0.3652

RESID(-1)

0.591123

0.150313

3.932602

0.0003

RESID(-2)

-0.092963

0.159390

-0.583241

0.5628

R-squared

0.284238

Mean dependent var

-5.00E-16

Adjusted R-squared

0.217656

S. D. dependent var

0.013797

S. E. of regression

0.012204

Akaike info criterion

-5.875827

Sum squared resid

0.006404

Schwarz criterion

-5.680910

Log likelihood

146.0198

Hannan-Quinn criter.

-5.802167

F-statistic

4.268965

Durbin-Watson stat

1.867925

Prob(F-statistic)

0.005364

Ordinary Least Squares Estimates

SSE

0.014269

DFE

32

MSE

0.000446

Root MSE

0.021117

SBC

-167.784

AIC

-175.839

Reg Rsq

0.9701

Total Rsq

0.9701

Durbin h

2.147687

PROB>h

0.0159

Godfrey's Serial Correlation Test Alternative LM Prob>LM

AR(+ 1)

4.6989

0.0302

AR(+ 2)

5.1570

0.0759

Variable

DF

B Value

Std Error

t Ratio

Approx Prob

Intercept

1

0.523006

1.1594

0.451

0.6550

RGNP. POP

1

0.050519

0.1127

0.448

0.6571

CAR. POP

1

-0.106323

0.1005

-1.058

0.2981

PMG. PGNP

1

-0.072884

0.0267

-2.733

0.0101

LAG. DEP

1

0.907674

0.0593

15.315

0.0001

c. LM Test for AR(1) by BREUSCH & GODFREY

Dependent Variable: RESID

Analysis of

Variance

Sum of

Mean

Source

DF

Squares

Square

F Value

Prob>F

Model

5

0.00178

0.00036

0.863

0.5172

Error

30

0.01237

0.00041

C Total

35

0.01415

Root MSE

0.02031

R-square

0.1257

Dep Mean

-0.00030

Adj R-sq

-0.0200

C. V.

-6774.59933

Parameter Estimates

Variable

DF

Parameter

Estimate

Standard

Error

T for HO: Parameter=0

Prob > |T|

INTERCEP

1

-0.377403

1.15207945

-0.328

0.7455

RESID.1

1

0.380254

0.18431559

2.063

0.0479

RGNP. POP

1

-0.045964

0.11158093

-0.412

0.6833

CAR. POP

1

0.031417

0.10061545

0.312

0.7570

PMG. PGNP

1

0.007723

0.02612020

0.296

0.7695

LAG. DEP

1

-0.034217

0.06069319

-0.564

0.5771

SAS PROGRAM Data RAWDATA;

Input Year CAR QMG PMG POP RGNP PGNP; Cards;

Data USGAS; set RAWDATA; LNQMG=LOG(QMG);

LNCAR=LOG(CAR);

LNPOP=LOG(POP);

LNRGNP=LOG(RGNP);

LNPGNP=LOG(PGNP);

LNPMG=LOG(PMG);

QMG_CAR=LOG(QMG/CAR);

RGNP_POP=LOG(RGNP/POP);

CAR_POP=LOG(CAR/POP);

PMG_PGNP=LOG(PMG/PGNP);

LAG_DEP=LAG(QMG_CAR);

Proc reg data=USGAS;

Model QMG_CAR=RGNP_POP CAR. POP PMGPGNP; TITLE ‘ STATIC MODEL’;

Proc autoreg data=USGAS;

Model QMG_CAR=RGNP_POP CAR_POP PMG_PGNP

LAG_DEP/LAGDEP=LAG_DEP godfrey=2; OUTPUT OUT=MODEL2 R=RESID;

TITLE ‘ DYNAMIC MODEL’;

RUN;

DATA DW_DATA; SET MODEL2;

RESID_1=LAG(RESID);

PROC REG DATA=DW_DATA;

MODEL RESID=RESID_1 RGNP. POP CAR. POP PMG. PGNP LAG. DEP; TITLE ‘LM Test for AR(1) by BREUSCH & GODFREY’;

RUN;

6.7 a. Unrestricted Model Autoreg Procedure Dependent Variable = QMG. CAR

Ordinary Least Squares Estimates

SSE

0.031403

DFE

22

MSE

0.001427

Root MSE

0.037781

SBC

-96.1812

AIC

-110.839

Reg Rsq

0.9216

Total Rsq

0.9216

Durbin-Watson

0.5683

Std

t

Approx

Variable DF

B Value

Error

Ratio

Prob

Intercept

1

-7.46541713

3.0889

-2.417

0.0244

RGNP. POP

1

-0.58684334

0.2831

-2.073

0.0501

CAR. POP

1

0.24215182

0.2850

0.850

0.4046

PMG. PGNP

1

-0.02611161

0.0896

-0.291

0.7734

PMPG.1

1

-0.15248735

0.1429

-1.067

0.2975

PMPG.2

1

-0.13752842

0.1882

-0.731

0.4726

PMPG.3

1

0.05906629

0.2164

0.273

0.7875

PMPG.4

1

-0.21264747

0.2184

-0.974

0.3408

PMPG.5

1

0.22649780

0.1963

1.154

0.2609

PMPG.6

1

-0.41142284

0.1181

-3.483

0.0021

Almon Lag (S = 6,

P =

: 2)

PDLREG Procedure

Dependent Variable =

QMG. CAR

Ordinary Least Squares Estimates

SSE

0.04017

DFE

26

MSE

0.001545

Root MSE

0.039306

SBC

-102.165

AIC

-110.96

Reg Rsq Durbin-Watson

0.8998

0.5094

Total Rsq

0.8998

Variable

DF

B Value

Std

Error

t

Ratio

Approx

Prob

Intercept

1

-5.06184299

2.9928

-1.691

0.1027

RGNP. POP

1

-0.35769028

0.2724

-1.313

0.2006

CAR. POP

1

0.02394559

0.2756

0.087

0.9314

PMG_PGNP**0

1

-0.24718333

0.0340

-7.278

0.0001

PMG_PGNP**1

1

-0.05979404

0.0439

-1.363

0.1847

PMG_PGNP**2

1

-0.10450923

0.0674

-1.551

0.1331

Estimate of Lag Distribution

Variable -0.184 0

c. Almon Lag (S=4,P=2) PDLREG Procedure Dependent Variable = QMG_CAR

Ordinary Least Squares Estimates

SSE

0.065767

DFE

28

MSE

0.002349

Root MSE

0.048464

SBC

-94.7861

AIC

-103.944

Reg Rsq

0.8490

Total Rsq

0.8490

Durbin-Watson

0.5046

Std

t

Approx

Variable

DF

B Value

Error

Ratio

Prob

Intercept

1

-6.19647990

3.6920

-1.678

0.1044

RGNP_POP

1

-0.57281368

0.3422

-1.674

0.1053

CAR_POP

1

0.21338192

0.3397

0.628

0.5351

PMG_PGNP**0

1

-0.19423745

0.0414

-4.687

0.0001

PMG_PGNP**1

1

-0.06534647

0.0637

-1.026

0.3138

PMG_PGNP**2

1

0.03085234

0.1188

0.260

0.7970

Estimate of Lag Distribution

Variable -0.116 0

ALMON LAG(S=8,P=2)

PDLREG Procedure Dependent Variable = QMG_CAR

Ordinary Least Squares Estimates

SSE

0.020741

DFE

24

MSE

0.000864

Root MSE

0.029398

SBC

-112.761

AIC

-121.168

Reg Rsq

0.9438

Total Rsq

0.9438

Durbin-Watson

0.9531

Variable

DF

B Value

Std

Error

t

Ratio

Approx

Prob

Intercept

1

-7.71363805

2.3053

-3.346

0.0027

RGNP. POP

1

-0.53016065

0.2041

-2.597

0.0158

CAR_POP

1

0.17117375

0.2099

0.815

0.4229

PMG_PGNP**0

1

-0.28572518

0.0267

-10.698

0.0001

PMG_PGNP**1

1

-0.09282151

0.0417

-2.225

0.0358

PMG_PGNP**2

1

-0.12948786

0.0512

-2.527

0.0185

Parameter

Std

t

Approx

Variable

Value

Error

Ratio

Prob

PMG_PGNP(0)

-0.11617

0.028

-4.09

0.0004

PMG_PGNP(1)

-0.07651

0.016

-4.73

0.0001

PMG_PGNP(2)

-0.05160

0.015

-3.34

0.0027

PMG_PGNP(3)

-0.04145

0.018

-2.30

0.0301

PMG_PGNP(4)

-0.04605

0.017

-2.63

0.0146

PMG_PGNP(5)

-0.06541

0.013

-4.85

0.0001

PMG_PGNP(6)

-0.09953

0.012

-8.10

0.0001

PMG_PGNP(7)

-0.14841

0.025

-5.97

0.0001

PMG_PGNP(8)

-0.21204

0.047

-4.53

0.0001

Estimate of Lag Distribution

Variable

-0.212

0

PMG_PGNP(0)

1

**********************1

PMG_PGNP(1)

1

***************j

PMG_PGNP(2)

1

**********j

PMG_PGNP(3)

1

PMG_PGNP(4)

1

*********j

PMG_PGNP(5)

1

*************j

PMG_PGNP(6)

1

*******************j

PMG_PGNP(7)

1

*****************************j

PMG_PGNP(8)

*****************************************

d. Third Degree Polynomial Almon Lag(S = 6,P = 3/ PDLREG Procedure

Dependent Variable = QMG_CAR

Ordinary Least Squares Estimates

SSE

0.034308

DFE

25

MSE

0.001372

Root MSE

0.037045

SBC

-103.747

AIC

-114.007

Reg Rsq

0.9144

Total Rsq

0.9144

Durbin-Watson

0.6763

Std

t

Approx

Variable

DF

B Value

Error

Ratio

Prob

Intercept

1

-7.31542415

3.0240

-2.419

0.0232

RGNP_POP

1

-0.57343614

0.2771

-2.069

0.0490

CAR_POP

1

0.23462358

0.2790

0.841

0.4084

PMG_PGNP**0

1

-0.24397597

0.0320

-7.613

0.0001

PMG_PGNP**1

1

-0.07041380

0.0417

-1.690

0.1035

PMG_PGNP**2

1

-0.11318734

0.0637

-1.778

0.0876

PMG_PGNP**3

1

-0.19730731

0.0955

-2.067

0.0493

Parameter

Std

t

Approx

Variable

Value

Error

Ratio

Prob

PMG_PGNP(0)

-0.03349

0.059

-0.57

0.5725

PMG_PGNP(1)

-0.14615

0.041

-3.54

0.0016

PMG_PGNP(2)

-0.12241

0.043

-2.87

0.0082

PMG_PGNP(3)

-0.04282

0.026

-1.64

0.1130

PMG_PGNP(4)

-0.01208

0.045

0.27

0.7890

PMG_PGNP(5)

-0.03828

0.043

-0.90

0.3788

PMG_PGNP(6)

-0.27443

0.067

-4.10

0.0004

Estimate of Lag Distribution

Variable -0.274 0.0121

PMG_PGNP(0)

PMG_PGNP(1)

PMG_PGNP(2)

PMG_PGNP(3)

PMG_PGNP(4)

PMG_PGNP(5)

PMG_PGNP(6)

e. Almon Lag(S = 6,P = 2/ with Near End-Point Restriction

PDLREG Procedure

Dependent Variable = QMGCAR

Ordinary Least Squares Estimates

SSE

0.046362

DFE

27

MSE

0.001717

Root MSE

0.041438

SBC

-101.043

AIC

-108.372

Reg Rsq

0.8843

Total Rsq

0.8843

Durbin-Watson

0.5360

Variable

DF

B Value

Std Error

t Ratio

Approx Prob

Intercept

1

-3.81408793

3.0859

-1.236

0.2271

RGNP_POP

1

-0.28069982

0.2843

-0.988

0.3322

CAR_POP

1

-0.05768233

0.2873

-0.201

0.8424

PMG_PGNP**0

1

-0.21562744

0.0317

-6.799

0.0001

PMG_PGNP**1

1

-0.07238330

0.0458

-1.581

0.1255

PMG_PGNP**2

1

0.02045576

0.0268

0.763

0.4519

Restriction

DF

L Value

Std Error

t Ratio

Approx Prob

PMG. PGNP(-l)

-1

0.03346081

0.0176

1.899

0.0683

Variable

Parameter Value

Std Error

t Ratio

Approx Prob

PMG_PGNP(0)

-0.02930

0.013

-2.31

0.0286

PMG_PGNP(1)

-0.05414

0.020

-2.75

0.0105

PMG_PGNP(2)

-0.07452

0.021

-3.49

0.0017

PMG_PGNP(3)

-0.09043

0.018

-4.91

0.0001

PMG_PGNP(4)

-0.10187

0.015

-6.80

0.0001

PMG_PGNP(5)

-0.10886

0.022

-4.88

0.0001

PMG_PGNP(6)

-0.11138

0.042

-2.66

0.0130

Estimate of Lag Distribution

Variable -0.111 0

PMG_PGNP(0)

PMG_PGNP(1)

PMG_PGNP(2)

PMG_PGNP(3)

PMG_PGNP(4)

PMG_PGNP(5)

PMG_PGNP(6)

ALMON LAG(S=6,P=2) with FAR END-POINT RESTRICTION

PDLREG Procedure Dependent Variable = QMG_CAR

Ordinary Least Squares Estimates

SSE

0.050648

DFE

27

MSE

0.001876

Root MSE

0.043311

SBC

-98.2144

AIC

-105.543

Reg Rsq

0.8736

Total Rsq

0.8736

Durbin-Watson

0.5690

Std

t

Approx

Variable

DF

B Value

Error

Ratio

Prob

Intercept

1

-6.02848892

3.2722

-1.842

0.0764

RGNP_POP

1

-0.49021381

0.2948

-1.663

0.1079

CAR_POP

1

0.15878127

0.2982

0.532

0.5988

PMG_PGNP**0

1

-0.20851840

0.0337

-6.195

0.0001

PMG_PGNP**1

1

-0.00242944

0.0418

-0.058

0.9541

PMG_PGNP**2

1

0.06159671

0.0240

2.568

0.0161

Restriction

DF

L Value

Std Error t Ratio

Approx Prob

PMG_PGNP(7)

-1

0.03803694

0.0161 2.363

0.0256

Estimate of Lag Distribution Variable -0.106

PMG_PGNP(0)

PMG_PGNP(1)

PMG_PGNP(2)

PMG_PGNP(3)

PMG_PGNP(4)

PMG_PGNP(5)

PMG_PGNP(6)

SAS PROGRAM Data RAWDATA;

Input Year CAR QMG PMG POP RGNP PGNP; Cards;

Data USGAS; set RAWDATA; LNQMG=LOG(QMG);

LNCAR=LOG(CAR);

LNPOP=LOG(POP);

LNRGNP=LOG(RGNP);

LNPGNP=LOG(PGNP);

LNPMG=LOG(PMG);

QMG_CAR=LOG(QMG/CAR);

RGNP_POP=LOG(RGNP/POP);

CAR_POP=LOG(CAR/POP);

PMG_PGNP=LOG(PMG/PGNP);

PMPG_1=LAG1(PMG_PGNP);

PMPG_2=LAG2(PMG_PGNP);

PMPG_3=LAG3(PMG_PGNP);

PMPG_4=LAG4(PMG_PGNP);

PMPG_5=LAG5(PMG_PGNP);

PMPG_6=LAG6(PMG_PGNP);

Proc autoreg data=USGAS;

Model QMG_CAR=RGNP_POP CAR_POP PMG_PGNP PMPG_1 PMPG_2 PMPG_3 PMPG_4 PMPG.5 PMPG_6;

TITLE‘UNRESTRICTED MODEL’;

PROC PDLREG DATA=USGAS;

MODEL QMG_CAR=RGNP_POP CAR_POP PMG_PGNP(6,2);

TITLE ‘ALMON LAG(S=6,P=2)’;

PROC PDLREG DATA=USGAS;

MODEL QMG_CAR=RGNP_POP CAR_POP PMG_PGNP(4,2);

TITLE ‘ALMON LAG(S=4,P=2)’;

PROC PDLREG DATA=USGAS;

MODEL QMG_CAR=RGNP_POP CAR_POP PMG_PGNP(8,2);

TITLE ‘ALMON LAG(S=8,P=2)’;

PROC PDLREG DATA=USGAS;

MODEL QMG_CAR=RGNP_POP CAR_POP PMG_PGNP(6,3);

TITLE ‘Third Degree Polynomial ALMON LAG(S=6,P=3)’;

PROC PDLREG DATA=USGAS;

MODEL QMG_CAR=RGNP_POP CAR_POP PMG_PGNP(6,2,,FIRST); TITLE ‘ALMON LAG(S=6,P=2) with NEAR END-POINT RESTRICTION’;

PROC PDLREG DATA=USGAS;

MODEL QMG_CAR=RGNP_POP CAR_POP PMG_PGNP(6,2,,LAST); TITLE ‘ALMON LAG(S=6,P=2) with FAR END-POINT RESTRICTION’;

RUN;

CHAPTER 7

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Springer Texts in Business and Economics

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