Recursive Residuals

In Section 8.1, we showed that the least squares residuals are heteroskedastic with non-zero co­variances, even when the true disturbances have a scalar covariance matrix. This section studies recursive residuals which are a set of linear unbiased residuals with a scalar covariance matrix. They are independent and identically distributed when the true disturbances themselves are independent and identically distributed.2 These residuals are natural in time-series regressions and can be constructed as follows:

1. Choose the first t > k observations and compute f3t = (X’tXt)-1X’tYt where Xt denotes the t x k matrix of t observations on k variables and Yt’ = (y1,…,yt). The recursive residuals are basically standardized one-step ahead forecast residuals:

Table 8.2 Diagnostic Statistics for the Cigarettes Example

OBS

STATE

LNC

LNP

LNY

PREDICTED

e

e

e*

Cook’s D

Leverage

DFFITS

COVRATIO

1

AL

4.96213

0.20487

4.64039

4.8254

0.1367

0.857

0.8546

0.012

0.0480

0.1919

1.0704

2

AZ

4.66312

0.16640

4.68389

4.8844

-0.2213

-1.376

-1.3906

0.021

0.0315

-0.2508

0.9681

3

AR

5.10709

0.23406

4.59435

4.7784

0.3287

2.102

2.1932

0.136

0.0847

0.6670

0.8469

4

CA

4.50449

0.36399

4.88147

4.6540

-0.1495

-0.963

-0.9623

0.033

0.0975

-0.3164

1.1138

5

CT

4.66983

0.32149

5.09472

4.7477

-0.0778

-0.512

-0.5077

0.014

0.1354

-0.2009

1.2186

6

DE

5.04705

0.21929

4.87087

4.8458

0.2012

1.252

1.2602

0.018

0.0326

0.2313

0.9924

7

DC

4.65637

0.28946

5.05960

4.7845

-0.1281

-0.831

-0.8280

0.029

0.1104

-0.2917

1.1491

8

FL

4.80081

0.28733

4.81155

4.7446

0.0562

0.352

0.3482

0.002

0.0431

0.0739

1.1118

9

GA

4.97974

0.12826

4.73299

4.9439

0.0358

0.224

0.2213

0.001

0.0402

0.0453

1.1142

10

ID

4.74902

0.17541

4.64307

4.8653

-0.1163

-0.727

-0.7226

0.008

0.0413

-0.1500

1.0787

11

IL

4.81445

0.24806

4.90387

4.8130

0.0014

0.009

0.0087

0.000

0.0399

0.0018

1.1178

12

IN

5.11129

0.08992

4.72916

4.9946

0.1167

0.739

0.7347

0.013

0.0650

0.1936

1.1046

13

IA

4.80857

0.24081

4.74211

4.7949

0.0137

0.085

0.0843

0.000

0.0310

0.0151

1.1070

14

KS

4.79263

0.21642

4.79613

4.8368

-0.0442

-0.273

-0.2704

0.001

0.0223

-0.0408

1.0919

15

KY

5.37906

-0.03260

4.64937

5.1448

0.2343

1.600

1.6311

0.210

0.1977

0.8098

1.1126

16

LA

4.98602

0.23856

4.61461

4.7759

0.2101

1.338

1.3504

0.049

0.0761

0.3875

1.0224

17

ME

4.98722

0.29106

4.75501

4.7298

0.2574

1.620

1.6527

0.051

0.0553

0.4000

0.9403

18

MD

4.77751

0.12575

4.94692

4.9841

-0.2066

-1.349

-1.3624

0.084

0.1216

-0.5070

1.0731

19

MA

4.73877

0.22613

4.99998

4.8590

-0.1202

-0.769

-0.7653

0.018

0.0856

-0.2341

1.1258

20

MI

4.94744

0.23067

4.80620

4.8195

0.1280

0.792

0.7890

0.005

0.0238

0.1232

1.0518

21

MN

4.69589

0.34297

4.81207

4.6702

0.0257

0.165

0.1627

0.001

0.0864

0.0500

1.1724

22

MS

4.93990

0.13638

4.52938

4.8979

0.0420

0.269

0.2660

0.002

0.0883

0.0828

1.1712

23

MO

5.06430

0.08731

4.78189

5.0071

0.0572

0.364

0.3607

0.004

0.0787

0.1054

1.1541

24

MT

4.73313

0.15303

4.70417

4.9058

-0.1727

-1.073

-1.0753

0.012

0.0312

-0.1928

1.0210

25

NE

4.77558

0.18907

4.79671

4.8735

-0.0979

-0.607

-0.6021

0.003

0.0243

-0.0950

1.0719

26

NV

4.96642

0.32304

4.83816

4.7014

0.2651

1.677

1.7143

0.065

0.0646

0.4504

0.9366

27

NH

5.10990

0.15852

5.00319

4.9500

0.1599

1.050

1.0508

0.055

0.1308

0.4076

1.1422

28

NJ

4.70633

0.30901

5.10268

4.7657

-0.0594

-0.392

-0.3879

0.008

0.1394

-0.1562

1.2337

29

NM

4.58107

0.16458

4.58202

4.8693

-0.2882

-1.823

-1.8752

0.076

0.0639

-0.4901

0.9007

30

NY

4.66496

0.34701

4.96075

4.6904

-0.0254

-0.163

-0.1613

0.001

0.0888

-0.0503

1.1755

31

ND

4.58237

0.18197

4.69163

4.8649

-0.2825

-1.755

-1.7999

0.031

0.0295

-0.3136

0.8848

32

OH

4.97952

0.12889

4.75875

4.9475

0.0320

0.200

0.1979

0.001

0.0423

0.0416

1.1174

33

OK

4.72720

0.19554

4.62730

4.8356

-0.1084

-0.681

-0.6766

0.008

0.0505

-0.1560

1.0940

34

PA

4.80363

0.22784

4.83516

4.8282

-0.0246

-0.153

-0.1509

0.000

0.0257

-0.0245

1.0997

35

RI

4.84693

0.30324

4.84670

4.7293

0.1176

0.738

0.7344

0.010

0.0504

0.1692

1.0876

36

SC

5.07801

0.07944

4.62549

4.9907

0.0873

0.555

0.5501

0.008

0.0725

0.1538

1.1324

37

SD

4.81545

0.13139

4.67747

4.9301

-0.1147

-0.716

-0.7122

0.007

0.0402

-0.1458

1.0786

38

TN

5.04939

0.15547

4.72525

4.9062

0.1432

0.890

0.8874

0.008

0.0294

0.1543

1.0457

39

TX

4.65398

0.28196

4.73437

4.7384

-0.0845

-0.532

-0.5271

0.005

0.0546

-0.1267

1.1129

40

UT

4.40859

0.19260

4.55586

4.8273

-0.4187

-2.679

-2.9008

0.224

0.0856

-0.8876

0.6786

41

VT

5.08799

0.18018

4.77578

4.8818

0.2062

1.277

1.2869

0.014

0.0243

0.2031

0.9794

42

VA

4.93065

0.11818

4.85490

4.9784

-0.0478

-0.304

-0.3010

0.003

0.0773

-0.0871

1.1556

43

WA

4.66134

0.35053

4.85645

4.6677

-0.0064

-0.041

-0.0404

0.000

0.0866

-0.0124

1.1747

44

WV

4.82454

0.12008

4.56859

4.9265

-0.1020

-0.647

-0.6429

0.011

0.0709

-0.1777

1.1216

45

WI

4.83026

0.22954

4.75826

4.8127

0.0175

0.109

0.1075

0.000

0.0254

0.0174

1.1002

46

WY

5.00087

0.10029

4.71169

4.9777

0.0232

0.146

0.1444

0.000

0.0555

0.0350

1.1345

Table 8.3 Regression of Real Per-Capita Consumption of Cigarettes

Dep

Obs

Var

LNC

Predict

Value

Std Err Predict

Lower95%

Mean

Upper95%

Mean

Lower95%

Predict

Upper95%

Predict

Std Err Residual

Student

Residual

Residual –

to

1

о

2 Cook’s D

і

4.9621

4.8254

0.0.36

4.75.32

4.8976

4.4880

5.1628

0.1.367

0.159

0.857

к

0.012

2

4.6631

4.8844

0.029

4.8259

4.9429

4.5497

5.2191

-0.221.3

0.161

-1.376

к к

0.021

3

5.1071

4.7784

0.048

4.6825

4.874.3

4.4.351

5.1217

0.3287

0.156

2.102

ккк к

0.1.36

4

4.5045

4.6540

0.051

4.5511

4.7570

4.3087

4.999.3

-0.1495

0.155

-0.96.3

к

0.0.33

5

4.6698

4.7477

0.060

4.6264

4.8689

4.3965

5.0989

-0.0778

0.152

-0.512

к

0.014

6

5.0471

4.8458

0.0.30

4.786.3

4.905.3

4.5109

5.1808

0.2012

0.161

1.252

к к

0.018

7

4.6564

4.7845

0.054

4.6750

4.8940

4.4.372

5.1.318

-0.1281

0.154

-0.8.31

к

0.029

8

4.8008

4.7446

0.0.34

4.6761

4.81.30

4.4079

5.0812

0.0562

0.160

0.352

0.002

9

4.9797

4.94.39

0.0.33

4.8778

5.0100

4.6078

5.2801

0.0.358

0.160

0.224

0.001

10

4.7490

4.865.3

0.0.33

4.798.3

4.9.32.3

4.5290

5.2016

-0.116.3

0.160

-0.727

к

0.008

11

4.8145

4.81.30

0.0.33

4.7472

4.8789

4.4769

5.1491

0.00142

0.160

0.009

0.000

12

5.111.3

4.9946

0.042

4.9106

5.0786

4.6544

5.3.347

0.1167

0.158

0.7.39

к

0.01.3

13

4.8086

4.7949

0.029

4.7.368

4.8529

4.4602

5.1295

0.01.37

0.161

0.085

0.000

14

4.7926

4.8.368

0.024

4.7876

4.8860

4.50.36

5.1701

-0.0442

0.162

-0.27.3

0.001

15

5.3791

5.1448

0.07.3

4.9982

5.291.3

4.7841

5.5055

0.2.34.3

0.146

1.600

ккк

0.210

16

4.9860

4.7759

0.045

4.6850

4.8668

4.4.340

5.1178

0.2101

0.157

1.3.38

к к

0.049

17

4.9872

4.7298

0.0.38

4.652.3

4.8074

4.3912

5.0684

0.2574

0.159

1.620

ккк

0.051

18

4.7775

4.9841

0.057

4.8692

5.0991

4.6.351

5.3.332

-0.2066

0.15.3

-1.349

к к

0.084

19

4.7.388

4.8590

0.048

4.7625

4.9554

4.5155

5.2024

-0.1202

0.156

-0.769

к

0.018

20

4.9474

4.8195

0.025

4.7686

4.870.3

4.4860

5.15.30

0.1280

0.161

0.792

к

0.005

21

4.6959

4.6702

0.048

4.57.3.3

4.7671

4.3267

5.01.37

0.0257

0.156

0.165

0.001

22

4.9.399

4.8979

0.049

4.8000

4.9959

4.5541

5.2418

0.0420

0.156

0.269

0.002

23

5.064.3

5.0071

0.046

4.9147

5.0996

4.6648

5.3495

0.0572

0.157

0.364

0.004

24

4.7.3.31

4.9058

0.029

4.8476

4.9640

4.5711

5.2405

-0.1727

0.161

-1.07.3

к к

0.012

25

4.7756

4.87.35

0.025

4.8221

4.9249

4.5.399

5.2071

-0.0979

0.161

-0.607

к

0.00.3

26

4.9664

4.7014

0.042

4.6176

4.7851

4.361.3

5.0414

0.2651

0.158

1.677

ккк

0.065

27

5.1099

4.9500

0.059

4.8.308

5.0692

4.5995

5.3005

0.1599

0.152

1.050

к к

0.055

28

4.706.3

4.7657

0.061

4.6427

4.8888

4.41.39

5.1176

-0.0594

0.152

-0.392

0.008

29

4.5811

4.869.3

0.041

4.7859

4.9526

4.529.3

5.2092

-0.2882

0.158

-1.82.3

к к к

0.076

30

4.6650

4.6904

0.049

4.5922

4.7886

4.3465

5.0.34.3

-0.0254

0.156

-0.16.3

0.001

31

4.5824

4.8649

0.028

4.808.3

4.9215

4.5.305

5.199.3

-0.2825

0.161

-1.755

к к к

0.0.31

32

4.9795

4.9475

0.0.34

4.8797

5.015.3

4.6110

5.2840

0.0.320

0.160

0.200

0.001

33

4.7272

4.8.356

0.0.37

4.7616

4.9097

4.4978

5.17.35

-0.1084

0.159

-0.681

к

0.008

34

4.80.36

4.8282

0.026

4.7754

4.8811

4.4944

5.1621

-0.0246

0.161

-0.15.3

0.000

35

4.8469

4.729.3

0.0.37

4.655.3

4.80.3.3

4.3915

5.0671

0.1176

0.159

0.7.38

к

0.010

36

5.0780

4.9907

0.044

4.9020

5.0795

4.6494

5.3.320

0.087.3

0.157

0.555

к

0.008

37

4.8155

4.9.301

0.0.33

4.8640

4.996.3

4.5940

5.266.3

-0.1147

0.160

-0.716

к

0.007

38

5.0494

4.9062

0.028

4.8497

4.9626

4.5718

5.2406

0.14.32

0.161

0.890

к

0.008

39

4.6540

4.7.384

0.0.38

4.6614

4.8155

4.4000

5.0769

-0.0845

0.159

-0.5.32

к

0.005

40

4.4086

4.827.3

0.048

4.7.308

4.92.37

4.48.39

5.1707

-0.4187

0.156

-2.679

к к ккк

0.224

41

5.0880

4.8818

0.025

4.8.304

4.9.3.32

4.5482

5.2154

0.2062

0.161

1.277

к к

0.014

42

4.9.307

4.9784

0.045

4.8868

5.0701

4.6.36.3

5.3205

-0.0478

0.157

-0.304

0.00.3

43

4.661.3

4.6677

0.048

4.5708

4.7647

4.3242

5.011.3

-0.006.38

0.156

-0.041

0.000

44

4.8245

4.9265

0.044

4.8.387

5.014.3

4.5854

5.2676

-0.1020

0.158

-0.647

к

0.011

45

4.8.30.3

4.8127

0.026

4.7602

4.865.3

4.4790

5.1465

0.0175

0.161

0.109

0.000

46

5.0009

4.9777

0.0.39

4.9000

5.055.3

4.6.391

5.316.3

0.02.32

0.159

0.146

0.000

Sum of Residuals 0

Sum of Squared Residuals 1.1485

Predicted Resid SS (Press) 1.3406

 

2. Add the (t + 1)-th observation to the data and obtain i3t+l Compute wt+2.

3. Подпись:

Подпись: (X+iXt+i)-1 Подпись: (X[Xt)-1 Подпись: (X[Xt)-iXt+ix't+i(X[Xt)-i/[1+xt+i(X[Xt)-iXt+i] (8.31)

Repeat step 2, adding one observation at a time. In time-series regressions, one usually starts with the first ^-observations and obtain (T — k) forward recursive residuals. These recursive residuals can be computed using the updating formula given in (8.11) with A = (X’tXt) and a = —b = x’t+i. Therefore,

and only (XtX^ i have to be computed. Also,

Подпись: (8.32)&+1 = & + (XtXt) ixt+i(yt+i — x’t+iPt)//t+i where /t+i = 1 + xt+i(Xt/Xt)-ixt+i, see problem 13.

Alternatively, one can compute these residuals by regressing Yt+i on Xt+i and dt+i where dt+i = 1 for the (t + 1)-th observation, and zero otherwise, see equation (8.5). The estimated coefficient of dt+i is the numerator of wt+i. The standard error of this estimate is s^ times the denominator of wt+i, where s^ is the standard error of this regression. Hence, wt+i can be retrieved as s^ multiplied by the t-statistic corresponding to dt+i. This computation has to be performed sequentially, in each case generating the corresponding recursive residual. This may be computationally inefficient, but it is simple to generate using regression packages.

It is obvious from (8.30) that if ut ~ IIN(0, a2), then wt+i has zero mean and var(wt+i) = a2. Furthermore, wt+i is linear in the y’s. Therefore, it is normally distributed. It remains to show that the recursive residuals are independent. Given normality, it is sufficient to show that

Подпись: (8.33)cov(wt+i, ws+i) = 0 for t = s; t, s = k,…,T — 1

This is left as an exercise for the reader, see problem 13.

Подпись: C image315 Подпись: i л/ fk+1 Подпись: i Vft image318 Подпись: (8.34)

Alternatively, one can express the T — k vector of recursive residuals as w = Cy where C is of dimension (T — k) x T as follows:

xtT (XT— iXT — i) iX’T-i i

V fT V fT

Problem 14 asks the reader to verify that w = Cy, using (8.30). Also, that the matrix C satisfies the following properties:

Подпись: (8.35)(i) CX = 0 (ii) CCC = It-k (iii) C’C = Px

This means that the recursive residuals w are (LUS) linear in y, unbiased with mean zero and have a scalar variance-covariance matrix: var(w) = CE(uu’)C’ = а2Іт-к. Property (iii) also
means that w’w = y’C’Cy = y’PxУ = e’e. This means that the sum of squares of (T — k) recursive residuals is equal to the sum of squares of T least squares residuals. One can also show from (8.32) that

RSSt+i = RSSt + w2t+l for t = k,…,T — 1 (8.36)

where RSSt = (Yt — Xtf3t)'(Yt — Xt/3t), see problem 14. Note that for t = k; RSS = 0, since with k observations one gets a perfect fit and zero residuals. Therefore

RSSt = £Tt=k+l wt = £T=1 e2 (8.37)

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