Applications of Recursive Residuals

Recursive residuals have been used in several important applications:

Подпись: F image321 Подпись: (8.38)

(1) Harvey (1976) used these recursive residuals to give an alternative proof of the fact that Chow’s post-sample predictive test has an F-distribution. Recall, from Chapter 7, that when the second sample nt had fewer than k observations, Chow’s test becomes

wt[11]

Подпись: e'e Подпись: ЕП1+П2 t=k+1 Подпись: and e1e1 = £ n= k+1 wt Подпись: (8.39)

where e’e = RSS from the total sample (n1 + nt = T observations), and e’1e1 = RSS from the first n1 observations. Recursive residuals can be computed for t = k + 1, …,n1, and continued on for the extra nt observations. From (8.36) we have

Therefore,

Подпись: (8.40)£ 1 wt/nt

TZ k+1 wt/(n 1 — k)

But the wt’s are ~IIN(0, a2) under the null, therefore the F-statistic in (8.38) is a ratio of two independent chi-squared variables, each divided by the appropriate degrees of freedom. Hence, F ~ F(nt, n1 — k) under the null, see Chapter 2.

(2) Harvey and Phillips (1974) used recursive residuals to test the null hypothesis of homoskedasticity. If the alternative hypothesis is that a2 varies with Xj, the proposed test is as follows:

1) Order the data according to Xj and choose a base of at least k observations from among the central observations.

3) Under the null hypothesis, it follows that

F = w2W2/wlwi ~ Fm, m (8.41)

Harvey and Phillips suggest setting m at approximately (n/3) provided n > 3k. This test has the advantage over the Goldfeld-Quandt test in that if one wanted to test whether varies with some other variable Xs, one could simply regroup the existing recursive residuals according to low and high values of Xs and compute (8.41) afresh, whereas the Goldfeld-Quandt test would require the computation of two new regressions.

Подпись: MVNR Подпись: ELH-2(wt — wt-i)2/(T — k — 1) T=k+i w?/(T — k) Подпись: (8.42)

(3) Phillips and Harvey (1974) suggest using the recursive residuals to test the null hy­pothesis of no serial correlation using a modified von Neuman ratio:

This is the ratio of the mean-square successive difference to the variance. It is arithmetically closely related to the DW statistic, but given that w ~ N(0,a2IT-k) one has an exact test available and no inconclusive regions. Phillips and Harvey (1974) provide tabulations of the significance points. If the sample size is large, a satisfactory approximation is obtained from a normal distribution with mean 2 and variance 4/(T — k).

(4) Harvey and Collier (1977) suggest a test for functional misspecification based on recursive residuals. This is based on the fact that w N(0,a2IT-k). Therefore,

w/(sw/^T — k) ~ tT-k-l (8.43)

where w = ^2T=k+l wt/(T — k) and s2w = Y)’T=k+l(wt — w)2/(T — k — 1). Suppose that the true functional form relating y to a single explanatory variable X is concave (convex) and the data are ordered by X. A simple linear regression is estimated by regressing y on X. The recursive residuals would be expected to be mainly negative (positive) and the computed t-statistic will be large in absolute value. When there are multiple X’s, one could carry out this test based on any single explanatory variable. Since several specification errors might have a self-cancelling effect on the recursive residuals, this test is not likely to be very effective in multivariate situations. Wu (1993) suggested performing this test using the following augmented regression:

y = X@ + zy + v (8.44)

where z = CiT-k is one additional regressor with C defined in (8.34) and iT-k denoting a vector of ones of dimension T — k. In fact, the F-statistic for testing Ho; y = 0 turns out to be the square of the Harvey and Collier (1977) t-statistic given in (8.43), see problem 15.

Alternatively, a Sign test may be used to test the null hypothesis of no functional misspecifi – cation. Under the null hypothesis, the expected number of positive recursive residuals is equal to (T — k)/2. A critical region may therefore be constructed from the binomial distribution. However, Harvey and Collier (1977) suggest that the Sign test tends to lack power compared with the t-test described in (8.43). Nevertheless, it is very simple and it may be more robust to non-normality.

image331 Подпись: (8.45)

(5) Brown, Durbin and Evans (1975) used recursive residuals to test for structural change over time. The null hypothesis is

where et is the vector of coefficients in period t and a2 is the disturbance variance for that period. The authors suggest a pair of tests. The first is the CUSUM test which computes

Wr — Yn=k+1 Wt/sw for r — k + 1,…,T (8.46)

where s2w is an estimate of the variance of the wt’s, given below (8.43). Wr is a cumulative sum and should be plotted against r. Under the null, E(Wr) — 0. But, if there is a structural break, Wr will tend to diverge from the horizontal line. The authors suggest checking whether Wr cross a pair of straight lines (see Figure 8.1) which pass through the points {k, ±a/T — k}and {T, ±3a/T — k} where a depends upon the chosen significance level a. For example, a — 0.850, 0.948, and 1.143 for a — 10%, 5%, and 1% levels, respectively.

If the coefficients are not constant, there may be a tendency for a disproportionate number of recursive residuals to have the same sign and to push Wr across the boundary. The second test is the cumulative sum of squares (CUSUMSQ) which is based on plotting

Wr* — E r=k+i w2/ E T=k+i w2 for t — k + 1,…,T (8%7)

against r. Under the null, E(Wr*) — (r — k)/(T — k) which varies from 0 for r — k to 1 for r — T. The significance of the departure of Wr) from its expected value is assessed by whether W. Г crosses a pair of lines parallel to E(Wr*) at a distance co above and below this line. Brown, Durbin and Evans (1975) provide values of c0 for various sample sizes T and levels of significance a.

The CUSUM and CUSUMSQ should be regarded as data analytic techniques; i. e., the value of the plots lie in the information to be gained simply by inspecting them. The plots contain more information than can be summarized in a single test statistic. The significance lines constructed are, to paraphrase the authors, best regarded as ‘yardsticks’ against which to assess the observed plots rather than as formal tests of significance. See Brown et al. (1975) for various examples. Note that the CUSUM and CUSUMSQ are quite general tests for structural change in that they do not require a prior determination of where the structural break takes place. If this is known, the Chow-test will be more powerful. But, if this break is not known, the CUSUM and CUSUMSQ are more appropriate.

Example 2: Table 8.5 reproduces the consumption-income data, over the period 1959-2007, taken from the Economic Report of the President. In addition, the recursive residuals are computed as in (8.30) and exhibited in column 5, starting with 1961 and ending in 2007. The CUSUM given by Wr in (8.46) is plotted against r in Figure 8.2. The CUSUM crosses the upper 5% line in 1998, showing structural instability in the latter years. This was done using EViews 6.

The post-sample predictive test for 1998, can be obtained from (8.38) by computing the RSS from 1950-1997 and comparing it with the RSS from 1950-2007. The observed F-statistic is 5.748 which is distributed as F(10,37). Using EViews, one clicks on stability diagnostics and then selects Chow forecast test. You will be prompted to enter the break point period which

image333

Figure 8.1 CUSUM Critical Values Table 8.4 Chow Forecast Test

Specification: CONSUM C Y

Test predictions for observations from 1998 to 2007

Value

df

Probability

F-statistic

5.747855

(10,37)

0.0000

Likelihood ratio

45.93529

10

0.0000

F-test summary:

Sum of Sq.

df

Mean Squares

Test SSR

5476210.

10

547621.0

Restricted SSR

9001348.

47

191518.0

Unrestricted SSR

3525138.

37

95273.99

LR test summary:

Value

df

Restricted LogL

-366.4941

47

Unrestricted LogL

-343.5264

37

in this case is 1998. EViews gives the back up regression which is not shown here, and also performs a likelihood ratio test, see Table 8.4.

The reader can verify that the same F-statistic can be obtained from (8.40) using the recursive residuals in Table 8.5. In fact,

F = (£wt2/10)/(£ 1997)6! wf/37) = 5.748

Table 8.5 Recursive Residuals for the Consumption Regression

Year

CONSUM

Income

RESID

Recursive RES

1959

8776

9685

635.4909

NA

1960

8837

9735

647.5295

NA

1961

8873

9901

520.9776

-30.06109

1962

9170

10227

498.7493

53.63333

1963

9412

10455

517.4853

57.07454

1964

9839

11061

351.0732

-14.42043

1965

10331

11594

321.1447

40.23840

1966

10793

12065

321.9283

72.59054

1967

10994

12457

139.0709

-58.72718

1968

11510

12892

229.1068

88.63871

1969

11820

13163

273.7360

125.0883

1970

11955

13563

17.04481

-88.54736

1971

12256

14001

-110.8570

-123.0740

1972

12868

14512

0.757470

68.23355

1973

13371

15345

-311.9394

-118.2972

1974

13148

15094

-289.1532

-100.8288

1975

13320

15291

-310.0611

-72.86693

1976

13919

15738

-148.7760

148.9270

1977

14364

16128

-85.67493

231.2810

1978

14837

16704

-176.7102

178.9840

1979

15030

16931

-205.9950

147.8067

1980

14816

16940

-428.8080

-80.37207

1981

14879

17217

-637.0542

-229.1660

1982

14944

17418

-768.8790

-296.0910

1983

15656

17828

-458.3625

86.49899

1984

16343

19011

-929.7892

-205.6594

1985

17040

19476

-688.1302

111.3357

1986

17570

19906

-579.1982

251.5306

1987

17994

20072

-317.7500

479.8759

1988

18554

20740

-411.8743

405.8181

1989

18898

21120

-439.9809

366.8060

1990

19067

21281

-428.6367

347.8156

1991

18848

21109

-479.2094

243.0261

1992

19208

21548

-549.0905

195.0177

1993

19593

21493

-110.2330

588.3097

1994

20082

21812

66.39330

731.2551

1995

20382

22153

32.47656

660.7508

1996

20835

22546

100.6400

696.2055

1997

21365

23065

122.4207

689.6197

1998

22183

24131

-103.4364

474.3981

1999

23050

24564

339.5579

870.8977

2000

23862

25472

262.4189

751.2861

2001

24215

25697

395.0926

808.6041

2002

24632

26238

282.3303

639.0555

2003

25073

26566

402.1435

700.0686

2004

25750

27274

385.8501

633.1310

2005

26290

27403

799.5297

970.8717

2006

26835

28098

663.9663

760.6385

2007

27319

28614

642.6847

673.7335

image334

——- CUSUM ———– 5% Significance

Figure 8.2 CUSUM Plot of the Consumption Regression

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