Time-Series Analysis

14.1 The AR(1) Model. yt = pyt_i + ©t with |p| <1 and ©t ~ IIN (0, a©2). Also, yo – N (0, o©2/1 – p2).

a. By successive substitution

yt = pyt_i + ©t = p(pyt_2 + ©t_1) + ©t = P2yt_2 + P©t_1 + ©t

= p2(pyt_3 + ©t_2) + p©t_1 + ©t = p3yt_3 + p2 ©t_2 + p©t_1 + ©t

= ••• = pVo c pt 1©i c pt 2©2 C ••• C ©t Then, E(yt) = ptE(yo) = 0 for every t, since E(yo) = E(©t) = 0.

var(yt) = p2tvar(yo) C p2(t 1)var(©i) C p2(t 2)var(©2) C———————————— C var(©t)

If p = 1, then var(yt) = „©2/0 ! 1. Also, if |p| > 1, then 1 — p2 < 0 and var(yt) < 0.

b. The AR(1) series yt has zero mean and constant variance „2 = var(yt), for t = 0, 1 , 2, … In part (a) we could have stopped the successive substitution at yt_s, this yields yt = psyt_s C pS 1©t_s+1 C • • C©t

Therefore, cov(yt, yt_s) = cov(psyt_s C Ps 1©t_s+1 C • • C©t, yt_s) = psvar(yt_s) = ps„2 which only depends on s the distance between t and t-s. Therefore, the AR(1) series yt is said to be covariance-stationary or weakly stationary.

B. H. Baltagi, Solutions Manual for Econometrics, Springer Texts in Business and Economics, DOI 10.1007/978-3-642-54548-1_14, © Springer-Verlag Berlin Heidelberg 2015

c. First one generates yo = 0.5 N(0,1)/(1 — p2)1/2 for various values of p. Then yt = pyt_i + et where et ~ IIN(0,0.25) for t = 1,2,.., T.

14.2 The MA(1) Model. yt = et + 0et_1 with et ~ IIN (0, о2)

a. E(yt) = E(et) + 0E(et_1) = 0 for all t. Also,

var(yt) = var(et) + 02var(et_1) = (1 + 02)oe2 for all t.

Therefore, the mean and variance are independent of t.

b. cov(yt, yt—1) = cov(et + 0St_1, St_1 + 0St_2) = 0var(8t_1) = 0oe2

0a©2 when s = 1 0 when s >1

0a©2 when s = 1 0 when s >1

for all t and s, then the MA(1) process is covariance stationary.

c. First one generates et ~ IIN(0,0.25). Then, for various values of 0, one generates yt = et + 0et_1.

14.3

a. The sample autocorrelation function for Income using EViews is as follows:

Correlogram of Income Sample: 1959 2007 Included observations: 49

Partial

Autocorrelation

Correlation

AC

PAC

Q-Stat

Prob.

1 *******

|******* 1

0.935

0.935

45.538

0

1 *******

■ 1 ■ 2

0.869

-0.043

85.714

0

1 ******

■ 1 ■ 3

0.804

-0.031

120.81

0

1 ******

■ 1 ■ 4

0.737

-0.05

150.95

0

1 *****

■ 1 ■ 5

0.671

-0.026

176.52

0

1 *****

■ 1 ■ 6

0.607

-0.026

197.93

0

1****

■ 1 ■ 7

0.545

-0.022

215.62

0

1****

■ 1 ■ 8

0.484

-0.034

229.91

0

1***

■ 1 ■ 9

0.428

-0.006

241.33

0

1***

■ 1 ■ 10

0.373

-0.023

250.25

0

I**

■ 1 ■ 11

0.324

0.005

257.17

0

I**

■ 1 ■ 12

0.278

-0.022

262.37

0

I**

■ 1 ■ 13

0.232

-0.025

266.12

0

■ |*.

■ 1 ■ 14

0.189

-0.022

268.68

0

■ |*.

■ 1 ■ 15

0.15

-0.009

270.33

0

|*.

.*| .

16

0.105

-0.077

271.17

0

| .

. | .

17

0.061

-0.033

271.46

0

| .

. | .

18

0.016

-0.056

271.48

0

| .

. | .

19

-0.029

-0.038

271.56

0

*| .

. | .

20

-0.072

-0.024

272

0

The sample autocorrelation function for differenced Income is as follows:

Correlogram of Differenced Income Sample: 1959 2007 Included observations: 48

Partial

Autocorrelation

Correlation

AC

PAC

Q-Stat

Prob.

1

0.021

0.021

0.0234

0.879

. |*

. |*.

2

0.145

0.145

1.1253

0.57

*| .

.*| .

3

-0.141

-0.15

2.1907

0.534

4

-0.008

-0.022

2.1944

0.7

*| .

.*| .

5

-0.145

-0.106

3.3743

0.642

1*

6

0.112

0.11

4.0884

0.665

**i

**| .

7

-0.309

-0.306

9.6689

0.208

. | .

. | .

8

0.05

0.026

9.8171

0.278

**i

*1

9

-0.232

-0.166

13.135

0.157

*| .

**i

10

-0.129

-0.228

14.18

0.165

. |*

1*

11

0.077

0.182

14.561

0.203

. |*

. | .

12

0.137

0.024

15.804

0.2

. | .

13

-0.019

-0.056

15.83

0.258

14

0.209

0.11

18.905

0.169

. | .

15

-0.057

-0.022

19.145

0.207

. |*

. | .

16

0.113

0.043

20.371

0.256

*| .

.*| .

17

-0.06

-0.149

20.094

0.216

*| .

. | .

18

-0.074

-0.054

20.813

0.289

. |*

19

0.083

0.185

21.386

0.316

. |*

. | .

20

0.089

0

22.069

0.337

b. The Augmented Dickey-Fuller test statistic for Income using a constant and

linear trend is as follows:

Null Hypothesis: Y has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 0 (Automatic based on SIC, MAXLAG=10)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -1.843738 0.6677

Test critical values: 1% level -4.161144

5% level -3.506374

10% level -3.183002

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation Dependent Variable: D(Y)

Method: Least Squares

Sample (adjusted): 1960 2007 Included observations: 48 after adjustments

This does not reject the null hypothesis of unit root for Income. c. The Augmented Dickey-Fuller test statistic for differenced Income using a

constant and linear trend is as follows:

Null Hypothesis: D(Y) has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 0 (Automatic based on SIC, MAXLAG=10)

F-statistic 22.58242 Durbin-Watson stat 2.018360 Prob(F-statistic) 0.000000

This rejects the null hypothesis of unit root for differenced Income. We conclude that Income is I(1).

d. Let R1 denote the ols residuals from the regression of Consumption on Income and a constant. This tests R1 for unit roots: This ADF includes a constant

Null Hypothesis: R1 has a unit root Exogenous: Constant

Lag Length: 0 (Automatic based on SIC, MAXLAG=10)

Augmented Dickey-Fuller test statistic

t-Statistic Prob.‘ -1.502798 0.5237

Test critical values: 1 % level

-3.574446

5% level

-2.923780

10% level

-2.599925

‘MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation Dependent Variable: D(R1)

Method: Least Squares

Sample (adjusted): 1960 2007 Included observations: 48 after adjustments

Coefficient

Std. Error t-Statistic

Prob.

R1(-1)

-0.094140

0.062643 -1.502798

0.1397

C

-1.110596

26.49763 -0.041913

0.9667

R-squared

0.046798

Mean dependent var

0.149870

Adjusted R-squared

0.026076

S. D. dependent var

185.9291

S. E. of regression

183.4889

Akaike info criterion

13.30296

Sum squared resid

1548737.

Schwarz criterion

13.38093

Log likelihood

-317.2710

Hannan-Quinn criter.

13.33242

F-statistic

2.258401

Durbin-Watson stat

2.408726

Prob(F-statistic)

0.139726

This ADF includes a constant and a linear trend.

Null Hypothesis: R1 has a unit root Exogenous: Constant, Linear Trend Lag Length: 1 (Automatic based on SIC, MAXLAG=10)

t-Statistic Prob.‘

Augmented Dickey-Fuller test statistic

-1.000195 0.9342

Test critical values: 1% level

-4.165756

5% level

-3.508508

10% level

-3.184230

‘MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation Dependent Variable: D(R1)

Method: Least Squares

Sample (adjusted): 1961 2007 Included observations: 47 after adjustments

Coefficient

Std. Error t-Statistic

Prob.

R1(-1)

-0.062470

0.062458 -1.000195

0.3228

D(R1(-1))

-0.324625

0.145177 -2.236060

0.0306

C

-121.4508

54.46854 -2.229742

0.0310

@TREND(1959)

4.793820

1.937792 2.473856

0.0174

R-squared

0.209855

Mean dependent var

-0.103081

Adjusted R-squared

0.154728

S. D. dependent var

187.9309

S. E. of regression

172.7812

Akaike info criterion

13.22319

Sum squared resid

1283693.

Schwarz criterion

13.38065

Log likelihood

-306.7451

Hannan-Quinn criter.

13.28245

F-statistic

Prob(F-statistic)

3.806787

0.016614

Durbin-Watson stat

2.032797

Both ADF tests do not reject the null hypothesis of unit roots in the ols residuals.

f. Correlogram of log(consumption)

Correlogram of log(consumption)

Sample: 1959 2007 Included Observations: 49

Partial

Autocorrelation

Correlation

AC

PAC

Q-Stat

Prob

і*******

I*******

1

0.937

0.937

45.664

0

і*******

■*I ■

2

0.87

-0.059

85.895

0

і******

■ I ■

3

0.801

-0.053

120.74

0

і******

■ I ■

4

0.733

-0.027

150.61

0

і*****

■ I ■

5

0.667

-0.03

175.86

0

і*****

■ I ■

6

0.603

-0.021

196.97

0

і****

■ I ■

7

0.541

-0.017

214.41

0

і****

■ I ■

8

0.482

-0.024

228.6

0

I***

■ I ■

9

0.425

-0.027

239.89

0

I***

■ I ■

10

0.373

0.003

248.81

0

I**

■ I ■

11

0.324

-0.016

255.73

0

I**

■ I ■

12

0.275

-0.041

260.85

0

I**

■ I ■

13

0.228

-0.027

264.44

0

1*

■ I ■

14

0.183

-0.011

266.84

0

1*

■ I ■

15

0.142

-0.01

268.33

0

1*

■*I ■

16

0.099

-0.063

269.07

0

■ |.

■ I ■

17

0.055

-0.038

269.3

0

■ I ■

■ I ■

18

0.012

-0.036

269.32

0

■ I ■

■ I ■

19

-0.029

-0.028

269.39

0

■*I ■

■ I ■

20

-0.068

-0.022

269.78

0

For log(consumption), the ADF with a Constant and Linear Trend yields:

Null Hypothesis: LOGC has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 1 (Automatic based on SIC, MAXLAG=10)

t-Statistic

Prob.*

Augmented Dickey-Fuller test statistic

-3.201729

0.0965

Test critical values: 1% level

-4.165756

5% level

-3.508508

10% level

-3.184230

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation Dependent Variable: D(LOGC)

Method: Least Squares

Sample (adjusted): 1961 2007

Included observations: 47 after adjustments

Coefficient

Std. Error t-Statistic

Prob.

LOGC(-1)

-0.216769

0.067704 -3.201729

0.0026

D(LOGC(-1))

0.447493

0.129784 3.447996

0.0013

C

1.987148

0.614623 3.233118

0.0024

@TREND(1959)

0.004925

0.001590 3.097180

0.0034

R-squared

0.311464

Mean dependent var

0.024014

Adjusted R-squared

0.263427

S. D. dependent var

0.015980

S. E. of regression

0.013715

Akaike info criterion

-5.659384

Sum squared resid

0.008088

Schwarz criterion

-5.501924

Log likelihood

136.9955

Hannan-Quinn criter.

-5.600131

F-statistic

6.483785

Durbin-Watson stat

1.978461

Prob(F-statistic)

0.001019

We do not reject the null hypothesis that log(consumption) has unit root at the 5% level, but we do so at the 10% level.

For differenced log(consumption), the ADF with a Constant and Linear Trend yields:

Null Hypothesis: D(LOGC) has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 1 (Automatic based on SIC, MAXLAG=10)

t-Statistic

Prob.*

Augmented Dickey-Fuller test statistic

-5.143889

0.0006

Test critical values: 1 % level

-4.170583

5% level

-3.510740

10% level

-3.185512

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation Dependent Variable: D(LOGC,2)

Method: Least Squares

Sample (adjusted): 1962 2007

Included observations: 46 after adjustments

Coefficient

Std. Error t-Statistic

Prob.

D(LOGC(-1))

-0.848482

0.164950 -5.143889

0.0000

D(LOGC(-1),2)

0.253834

0.143686 1.766584

0.0846

C

0.026204

0.006566 3.990611

0.0003

@TREND(1959)

-0.000215

0.000165 -1.307036

0.1983

R-squared

0.403764

Mean dependent var

0.000300

Adjusted R-squared

0.361175

S. D. dependent var

0.018320

S. E. of regression

0.014642

Akaike info criterion

-5.526861

Sum squared resid

0.009005

Schwarz criterion

-5.367849

Log likelihood

131.1178

Hannan-Quinn criter.

-5.467294

F-statistic

Prob(F-statistic)

9.480625

0.000066

Durbin-Watson stat

2.028804

We do reject the null hypothesis that differenced log(consumption) has unit root at the 5% level.

Correlogram of log Income Sample: 1959 2007 Included Observations: 49

Autocorrelation

Partial

Correlation

AC

PAC

Q-Stat

Prob

|*******

і*******

1

0.935

0.935

45.507

0

|*******

. | .

2

0.867

-0.057

85.463

0

|******

. | .

3

0.798

-0.042

120.07

0

|******

. | .

4

0.729

-0.041

149.58

0

|*****

. | .

5

0.661

-0.035

174.36

0

|*****

. | .

6

0.596

-0.012

194.98

0

|****

. | .

7

0.534

-0.016

211.96

0

|****

. | .

8

0.475

-0.026

225.69

0

1***

. | .

9

0.418

-0.015

236.64

0

. | .

10

0.365

-0.021

245.17

0

1**

. | .

11

0.315

-0.009

251.7

0

1**

. | .

12

0.268

-0.023

256.54

0

1**

. | .

13

0.222

-0.023

259.97

0

1*

. | .

14

0.179

-0.017

262.26

0

1*

. | .

15

0.141

0.002

263.72

0

1*

■*|.

16

0.098

-0.079

264.44

0

. | .

17

0.055

-0.035

264.68

0

. | .

18

0.011

-0.043

264.69

0

. | .

19

-0.031

-0.034

264.77

0

| .

. | .

20

-0.071

-0.018

265.2

0

For log(Income), the ADF with a Constant and Linear Trend yields:

Null Hypothesis: LOGY has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 0 (Automatic based on SIC, MAXLAG=10)

t-Statistic

Prob.*

Augmented Dickey-Fuller test statistic

-1.567210

0.7912

Test critical values: 1% level

-4.161144

5% level

-3.506374

10% level

-3.183002

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation Dependent Variable: D(LOGY)

Method: Least Squares

Sample (adjusted): 1960 2007

Included observations: 48

after adjustments

Coefficient

Std. Error

t-Statistic

Prob.

LOGY(-1)

-0.079684

0.050844

-1.567210

0.1241

C

0.765061

0.469055

1.631070

0.1099

@TREND(1959)

0.001460

0.001136

1.285021

0.2054

R-squared

0.118240

Mean dependent var

0.022569

Adjusted R-squared

0.079051

S. D. dependent var

0.016010

S. E. of regression

0.015364

Akaike info criterion

-5.453080

Sum squared resid

0.010623

Schwarz criterion

-5.336130

Log likelihood

133.8739

Hannan-Quinn criter.

-5.408884

F-statistic

3.017149

Durbin-Watson stat

1.746698

Prob(F-statistic)

0.058936

We do not reject the null hypothesis that log(income) has unit root at the 5% level.

For differenced log(income), the ADF with a Constant and Linear Trend yields:

Null Hypothesis: D(LOGY) has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 0 (Automatic based on SIC, MAXLAG=10)

t-Statistic

Prob.*

Augmented Dickey-Fuller test statistic

-6.336919

0.0000

Test critical values : 1% level

-4.165756

5% level

-3.508508

10% level

-3.184230

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation Dependent Variable: D(LOGY,2) Method: Least Squares

Sample (adjusted): 1961 2007 Included observations: 47 after adjustments

Coefficient

Std. Error

t-Statistic

Prob.

D(LOGY(-1))

-0.924513

0.145893

-6.336919

0.0000

C

0.029916

0.006480

4.616664

0.0000

@TREND(1959)

-0.000347

0.000172

-2.019494

0.0496

R-squared

0.477988

Mean dependent var

0.000278

Adjusted R-squared

0.454260

S. D. dependent var

0.020895

S. E. of regression

0.015436

Akaike info criterion

-5.442528

Sum squared resid

0.010484

Schwarz criterion

-5.324434

Log likelihood

130.8994

Hannan-Quinn criter.

-5.398088

F-statistic

20.14464

Durbin-Watson stat

2.057278

Prob(F-statistic)

0.000001

We do reject the null hypothesis that differenced log(income) has unit root at the 5% level.

The OLS regression of log(consumption) on log(income) and a constant yields:

Dependent Variable: LOGC Method: Least Squares

Sample: 1959 2007 Included observations: 49

Coefficient

Std. Error

t-Statistic

Prob.

C

-0.625988

0.107332

-5.832253

0.0000

LOGY

1.053340

0.010972

95.99861

0.0000

R-squared

0.994926

Mean dependent var

9.672434

Adjusted R-squared

0.994818

S. D. dependent var

0.335283

S. E. of regression

0.024136

Akaike info criterion

-4.570278

Sum squared resid

0.027379

Schwarz criterion

-4.493061

Log likelihood

113.9718

Hannan-Quinn criter.

-4.540982

F-statistic

9215.733

Durbin-Watson stat

0.205803

Prob(F-statistic)

0.000000

Let R2 denote the ols residuals from the regression of Log(Consumption) on log(Income) and a constant.

This tests R2 for unit roots:

Null Hypothesis: R2 has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic based on SIC, MAXLAG=10)

t-Statistic

Prob.*

Augmented Dickey-Fuller test statistic

-1.898473

0.6399

Test critical values: 1 % level

-4.161144

5% level

-3.506374

10% level

-3.183002

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation Dependent Variable: D(R2)

Method: Least Squares

Sample (adjusted): 1960 2007 Included observations: 48 after adjustments

Coefficient

Std. Error

t-Statistic

Prob.

R2(-1)

-0.122648

0.064603

-1.898473

0.0641

C

-0.005115

0.003074

-1.663739

0.1031

@TREND(1959)

0.000201

0.000109

1.838000

0.0727

We do not reject the null hypothesis of unit roots in these ols residuals.

Johansen’s Cointegration Tests for Log(consumption) and log(Y)

Sample (adjusted): 1961 2007

Included observations: 47 after adjustments

Trend assumption: Linear deterministic trend

Series: LOGC LOGY

Lags interval (in first differences): 1 to 1

Unrestricted Cointegration Rank Test (Trace)

Hypothesized Trace

No. of CE(s) Eigenvalue Statistic

None 0.253794 14.26745

At most 1 0.010751 0.508018

Trace test indicates no cointegration at the 0.05 level * denotes rejection of the hypothesis at the 0.05 level **MacKinnon-Haug-Michelis (1999) p-values

Unrestricted Cointegration Rank Test (Maximum Eigenvalue)

Max-eigenvalue test indicates no cointegration at the 0.05 level * denotes rejection of the hypothesis at the 0.05 level **MacKinnon-Haug-Michelis (1999) p-values

Both tests indicate no cointegration between Log(consumption) and log(Y) at the 5% level

GARCH(1,1) for Log(consumption) and log(Y)

Dependent Variable: LOGC

Method: ML – ARCH (Marquardt) – Normal distribution

Sample: 1959 2007 Included observations: 49 Convergence achieved after 20 iterations Presample variance: backcast (parameter = 0.7) GARCH = C(3) + C(4)*RESID(-1)"2 + C(5)*GARCH(-1)

Coefficient

Std. Error

z-Statistic

Prob.

C

-0.712055

0.077795

-9.152987

0.0000

LOGY

1.063091

0.007986

133.1172

0.0000

Variance Equation

C

0.000127

0.000114

1.112565

0.2659

RESID(-1)"2

1.332738

0.542199

2.458025

0.0140

GARCH(-1)

-0.195201

0.120479

-1.620211

0.1052

R-squared

0.994060

Mean dependent var

9.672434

Adjusted R-squared

0.993521

S. D. dependent var

0.335283

S. E. of regression

0.026989

Akaike info criterion

-4.800472

Sum squared resid

0.032049

Schwarz criterion

-4.607429

Log likelihood

122.6116

Hannan-Quinn criter.

-4.727232

F-statistic

1841.004

Durbin-Watson stat

0.178014

Prob(F-statistic)

0.000000

14.5 Data Description: This data is obtained from the Citibank data base.

Ml: is the seasonally adjusted monetary base. This a monthly average series. We get the quarterly average of M1 by using (M1t + M1t+i + M1t+2)/3. TBILL3: is the T-bill-3 month-rate. This is a monthly series. We

calculate a quarterly average of TBILL3 by using (TBILL3t + TBILL3t+1 + TBILL3t+2)/3. Note that TBILL3 is an annualized rate (per annum).

GNP: This is Quarterly GNP. All series are transformed by taking their natural logarithm.

a. VAR with two lags on each variable

Sample(adjusted): 1959:3 1995:2

Included observations: 144 after adjusting endpoints

Standard errors & t-statistics in parentheses

LNGNP

LNM1

LNTBILL3

LNGNP(-1)

1.135719

-0.005500

1.437376

(0.08677)

(0.07370)

(1.10780)

(13.0886)

(-0.07463)

(1.29751)

LNGNP(-2)

-0.130393

0.037241

-1.131462

(0.08750)

(0.07431)

(1.11705)

(-1.49028)

(0.50115)

(-1.01290)

LNM1(-1)

0.160798

(0.07628)

(2.10804)

1.508925

(0.06478)

(23.2913)

1.767750

(0.97383)

(1.81525)

LNM1(-2)

-0.163492

(0.07516)

(-2.17535)

-0.520561

(0.06383)

(-8.15515)

-1.892962

(0.95951)

(-1.97284)

LNTBILL3(-1)

0.001615

(0.00645)

(0.25047)

-0.036446

(0.00547)

(-6.65703)

1.250074

(0.08230)

(15.1901)

LNTBILL3(-2)

-0.008933

(0.00646)

(-1.38286)

0.034629

(0.00549)

(6.31145)

-0.328626

(0.08248)

(-3.98453)

C

-0.011276

(0.07574)

(-0.14888)

-0.179754

(0.06433)

(-2.79436)

-1.656048

(0.96696)

(-1.71264)

R-squared

0.999256

0.999899

0.946550

Adj. R-squared

0.999223

0.999895

0.944209

Sum sq. resids

0.009049

0.006527

1.474870

S. E. equation

0.008127

0.006902

0.103757

Log likelihood

492.2698

515.7871

125.5229

Akaike AI C

-9.577728

-9.904358

-4.484021

Schwarz SC

-9.433362

-9.759992

-4.339656

Mean dependent

8.144045

5.860579

1.715690

S. D. dependent

0.291582

0.672211

0.439273

Determinant Residual Covariance 2.67E-11 Log Likelihood 1355.989

Akaike Information Criteria -24.24959

Schwarz Criteria -24.10523

b. VAR with three lags on each variable

Sample(adjusted): 1959:4 1995:2

Included observations: 143 after adjusting endpoints

Standard errors & t-statistics in parentheses

LNGNP

LNM1

LNTBILL3

LNGNP(-1)

1.133814

-0.028308

1.660761

(0.08830)

(0.07328)

(1.11241)

(12.8398)

(-0.38629)

(1.49295)

LNGNP(-2)

-0.031988

0.103428

0.252378

(0.13102)

(0.10873)

(1.65053)

(-0.24414)

(0.95122)

(0.15291)

LNGNP(-3)

-0.105146

(0.08774)

(-1.19842)

-0.045414

(0.07281)

(-0.62372)

-1.527252

(1.10526)

(-1.38180)

LNM1(-1)

0.098732

(0.10276)

(0.96081)

1.375936

(0.08528)

(16.1349)

1.635398

(1.29449)

(1.26335)

LNM1(-2)

-0.012617

(0.17109)

(-0.07375)

-0.134075

(0.14198)

(-0.94432)

-3.555324

(2.15524)

(-1.64962)

LNM1(-3)

-0.085778

(0.09254)

(-0.92693)

-0.253402

(0.07680)

(-3.29962)

1.770995

(1.16577)

(1.51917)

LNTBILL3(-1)

0.001412

(0.00679)

(0.20788)

-0.041461

(0.00564)

(-7.35638)

1.306043

(0.08555)

(15.2657)

LNTBILL3(-2)

-0.013695

(0.01094)

(-1.25180)

0.039858

(0.00908)

(4.38997)

-0.579077

(0.13782)

(-4.20158)

LNTBILL3(-3)

0.006468

(0.00761)

(0.84990)

0.000144

(0.00632)

(0.02281)

0.207577

(0.09588)

(2.16504)

C

0.037812

(0.07842)

(0.48217)

-0.166320

(0.06508)

(-2.55566)

-2.175434

(0.98789)

(-2.20210)

R-squared

0.999271

0.999907

0.950041

Adj. R-squared

0.999222

0.999901

0.946661

Sum sq. resids

0.008622

0.005938

1.368186

S. E. equation

0.008051

0.006682

0.101425

Log likelihood

491.8106

518.4767

129.5215

Akaike AIC

-9.576480

-9.949432

-4.509499

Schwarz SC

-9.369288

-9.742240

-4.302307

Mean dependent

8.148050

5.866929

1.718861

S. D. dependent

0.288606

0.670225

0.439160

Determinant Residual Covariance 2.18E-11 Log Likelihood 1360.953

Akaike Information Criteria -24.40808

Schwarz Criteria -24.20088

d. Pairwise Granger Causality Tests Sample: 1959:1 1995:2 Lags: 3

Obs F-Statistic Probability

LNTBILL3 does not Granger Cause LNM1 143 20.0752 7.8E-11

LNM1 does not Granger Cause LNTBILL3 1.54595 0.20551

e. Pairwise Granger Causality Tests Pairwise Granger Causality Tests Sample: 1959:1 1995:2 Lags: 2

Obs F-Statistic Probability

LNTBILL3 does not Granger Cause LNM1 144 23.0844 2.2E-09

LNM1 does not Granger Cause LNTBILL3 3.99777 0.02051

14.6 The Simple Deterministic Time Trend Model. This is based on Hamilton (1994). yt = a + "t + ut t = 1,… ,T where ut ~ IIN(0, a2).

In vector form, this can be written as y = X® + u, X = [1,t], ® =

a. ®ols = (X’X)-1X’y and ®ols – ® = (X’X)-1X’u Therefore,

1 (T + 1)/2

(T + 1)/2 (T + 1)(2T + 1)/6

Therefore, plim(XjX) diverges as T! 1 and is not a positive definite matrix.

c. Note that

TT

TEi Т^Е t

t=i t=i

TT

T2Et ТзЕt2

L t=1 t=1 .

d. Show that z1 = – E ut ~ N(0, ct2). ut ~ N(0, ct2), so that Eut

VT t=i t=i

N(0,Tct2).

Therefore, —T P ut ~ N (0, – T • Tct2 • —t) = N(0, ct2).

T2

Also, show that z2 = t-t E tut ~ N 0, 6T. • (T + 1)(2T + 1) .

has an asymptotic distribution N(0, ct2Q). Hence, [A 1(X, X)A J] 1 [A-1(X0u)] has an asymptotic distribution N(0,Q-1ct2QQ-1) or N(0, o2Q-1). Thus,

T ((a ols – a)

TPT " ols – " ,

has an asymptotic distribution N(0, o2Q-1). Since "ols has the factor T/T rather than the usual VT, it is said to be superconsistent. This means that not only does ("ols — ") converge to zero in probability limits, but so does T("ols — "). Note that the normality assumption is not needed for this result. Using the central limit theorem, all that is needed is that ut is White noise with finite fourth moments, see Sims et al. (1990) or Hamilton (1994).

Test of Hypothesis with a Deterministic Time Trend Model. This is based on Hamilton (1994).

t

a. Show thatplim s2 = Tzy S(yt — ‘ok — "olst)2 = o2. By the law of large

T

Hence, plim s2 = plimTij u2 = var(ut) = ct2

t=1

-y has the same asymptotic N(0,1)

2

s2(1,0)(X’X)-

distribution as t* = VT(a ois_a°). Multiplying both the numerator and

o qll

denominator of ta by VT gives

Q=

Now, [P’T, 0] in p can be rewritten as [1,0]A, because [1,0]A = [1,0] f TPT = [PT, 0].

Therefore, ta = —————– VT(Sols~a°)————- r-

s2[1,0]A(X’X)-1A

(aols ao) , *

oVq^ " ‘

Л/Т (a ols ao)

_WT(Pols – "o)

totic distribution N(0, o2Q-1), so that pT (aols — ao) is asymptotically distributedN(0, o2q11). Thus,

, * Л/Т (‘ols ao)

a-~^/qr~

is asymptotically distributed as N(0,1) under the null hypothesis of a = ao. Therefore, both ta and t* have the same asymptotic N(0,1) distribution. c. Similarly, fortesting Ho; " = "o, show that

t" = (" ols — "o)/[s2(0,1)(X0X)-1(0,1)0]1/2

has the same asymptotic N(0,1) distribution as t* = T/T(pols—p^/o^q22. Multiplying both numerator and denominator of tp by T/T we get,

tp = tVT(P ois — po)/[s2(0,TVT)(X’X)—1(0,tVT)’]1/2

= tVT(|° ois — Po)/[s2(0,1)A(X, X)_1A(0,1)0]1/2

Now [0,TVT] in tp can be rewritten as [0,1]A because

Therefore, plim tp = TVT (pols — "o) /[o2(0,1)Q“1(0,1),]1/2 = TVT (pols — Po^ /oy^q22- Using plim s2 = o2 and plim A(X0X)_1A = Q_1, we get that plim [s2(0,1)A(X0X)_1A(0,1)0]1/2 = [o2(0,1)Q_1(0,1)0]1/2 = o^q22 where q22 is the (2,2) element of Q_1. Therefore, tp has the same asymptotic distribution as

TVT (pols — p^ /^q22 = tp*

From problem 14.6, part (e), T/T ols — po^ has an asymptotic distribu­tion N(0, o2q22). Therefore, both tp and t* have the same asymptotic N(0,1) distribution – Also, the usual OLS t-tests for a = ao and p = po will give asymptotically valid inference-

14.8 A Random Walk Model. This is based on Fuller (1976) and Hamilton (1994). yt = yt_1 + ut, t = 0,1,.., T where ut ~ IIN(0, o2) and yo = 0.

a. Show that yt = u1 h————– hut with E(yt) = 0 and var(yt) = to2. By successive

substitution, we get

yt = yt—1 + ut = yt—2 + ut-1 + ut = •• = yo + u1 + u2 h hut

substituting yo = 0 we get yt = u1 + •• +ut.

Hence, E(yt) = E(u1) + • • +E(ut) = 0

var(yt) = var(u1) + • • +var(ut) = to2

and yt – N(0, to2).

2 1

prj – 222 • T u2 is asymptotically distributed as 2 (x2 – 1).

c. Show that E ^P у2-^ = T(T2-1)o2. Using the results in part (a), we get

yt-1 = yo + u1 + u2 + •• +ut-1

Substituting yo = 0, squaring both sides and taking expected values, we get E (y2-i) = E (u2) C CE (u2-J = (t – 1)ct2 since the ut’s are independent.

Therefore,

E (E y?-t! = X E (yf-x) = E(t – V = T^T2L^-2

U=1

T

where we used the fact that t = T(T C 1)/2 from problem 14.6.

t=i

d. For the AR(1) model, yt = pyt-i C ut, show that OLS estimate of p satisfies

T

From part (b), T;? P yt-1ut has an asymptotic distribution 1 (x2 — 1).

° t=1

This implies that J2 yt-1ut/o2 converges to an asymptotic distribution of

2 (x? — 1) at the rate of T. Also, from part (c), E ^P y2-1^ = g T(T ^

T

t=1

to 2 at the rate of T2. One can see that the asymptotic distribution of p when p = 1 is a ratio of a 2 (xf — 1) random variable to a non-standard distribution in the denominator which is beyond the scope of this book, see Hamilton (1994) or Fuller (1976) for further details. The object of this exercise is to show that if p = 1, VT(p — p) is no longer Normal as in the standard stationary least squares regression with |p| < 1. Also, to show that for the non-stationary (random walk) model, p converges at a faster rate (T) than for the stationary case (VT). From part (c) it is clear that one has to divide the denominator of p by T2 rather than T to get a convergent distribution.

14.9 Cointegration Example

and

, ————– Ut — ———— V

(‘ – P) t (‘ – P)t (‘ – P)

In this case, ut is I(0) and vt is I(1). Therefore both Yt and Ct are I(1). Note

that there are no excluded exogenous variables in (14.13) and (14.14) and only one right hand side endogenous variable in each equation. Hence both equations are unidentified by the order condition of identification. However, a linear combination of the two structural equations will have a mongrel dis­turbance term that is neither AR(1) nor random walk. Hence, both equations are identified. If p = 1, then both u, and v, are random walks and the mon­grel disturbance is also a random walk. Therefore, the system is unidentified. In such a case, there is no cointegrating relationship between Ct and Yt. Let (с’ – yY,) be another cointegrating relationship, then subtracting it from the first cointegrating relationship, one gets (y – P)Yt which should be I(0). Since Y’ is I(1), this can only happen if y = P. Differencing both equations in (14.13) and (14.14) we get

Ac, – pAY’ = Au, = (p – 1)u,-i + ©, = ©, + (p – 1)(C’_i – PY’_i)

= s, + (p – 1)C’-1 – P(p – 1)Y’-1

and AC, – aAY, = Av, = q,. Writing them as a VAR, we get (14.17)

"1 – p"

ac,

"s, + (p – 1)C’-1 – "(p – 1)Y’-1_

1 -“_

ay,

q,

Post-multiplying by the inverse of the first matrix, we get

ac,

=

1 Ї

1—-

CD.

3

1

1___

"s, + (p – 1)C’-1 – "(p – 1)Y’-1_

ay,

P -‘)

-1 1

-aet – a(p – 1)Ct_і + a"(p – 1)Yt_i +

—©t — (p — 1)Ct-i + "(p — 1)Yt-i + ht

where ht and gt are linear combinations of et and rp. This is Eq. (14.18). This

8 = (p — 1)/(" — a) and Zt = Ct — "Yt.

These are Eqs. (14.19) and (14.20). This is the Error-Correction Model (ECM) representation of the original model. Zt-1 is the error correction term. It represents a disequilibrium term showing the departure from long – run equilibrium. Note that if p = 1, then 8 = 0 and Zt-1 drops from both ECM equations.

T T

CtYt Ytut

b. "ols = " C tD1

T

Since ut is I(0) if p ф 1 and Yt is I(1), we have plim p Yt2/T2 is O(1),

t=1

T

while plimp; Ytut/T is O(1). Hence T("ols — ") is O(1) or ("ols — ") is O(T).

t=1

References

Fuller, W. A. (1976), Introduction to Statistical Time Series (John Wiley and Sons: New York).

Hamilton, J. D. (1994), Time Series Analysis (Princeton University Press: Princeton, New Jersey).

Sims, C. A., J. H. Stock and M. W. Watson (1990), “Inference in Linear Time Series Models with Some Unit Roots,” Econometrica, 58: 113-144.

P X’/o?] [POA)] – [p( Vo? r

b. From the regression equation Y, = a C "X, C u, one can multiply by w,*

and sum to get P w*Y, = a P w* C " P w*X, C P w*u,. Now divide

i= 1 i i= 1 i i= 1 i i= 1 i n

by w* and use the definitions of Y* and X* to get Y* = a C "X* C u*

i=1 i

nn

where u * = w*u, w*.

i=1 i i=1 i

[1]

P wi* (Xi — X*)2

i=1

_ 1 P wi* (Xi — X*)2

i=1

where the third equality uses the fact that the ui’s are not serially correlated and heteroskedastic and the fourth equality uses the fact that w* = (t/o2).

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