Durbin-Watson Test

In this subsection we shall consider the test of the hypothesis p = 0 in Model 6 with {u,} following AR(1). The Durbin-Watson test statistic, proposed by Durbin and Watson (1950, 1951), is defined by

itflr-a,-,)2

d = s=2-^————— , (6.3.13)

Efi?

/-1

where (fi() are the least squares residuals. By comparing (6.3.13) with (6.3.3), we can show that

d = 2-2p + oiT-1). (6.3.14)

From this we know that plim d = 2 — 2p, and the asymptotic distribution of d follows easily from the asymptotic distribution of p derived in Section 6.3.2.

Tо derive the exact distribution of d under the null hypothesis, it is useful to rewrite (6.3.13) as

Подпись: (6.3.15)u’MAMu u’Mu ’

where M = I — X(X’X)_,X’ and A is а Г X Г symmetric matrix defined by

1

-1

0

• 0

-1

2

-1

0

0

0

0

-1

2

-1

, о

0

-1

1

Because MAM commutes with M, there exists an orthogonal matrix H that diagonalizes both (see Theorem 8 of Appendix 1). Let v,,v2,. . . , vr_A-bethe nonzero characteristic roots of MAM. Then we can rewrite (6.3.15) as

Подпись: T-K (6.3.17)

where (Q are i. i.d. N(0, 1). In the form of (6.3.17), the density of d can be represented as a definite integral (Plackett, 1960, p. 26). The significance points of d can be computed either by the direct evaluation of the integral (Imhof, 1961) or by approximating the density on the basis of the first few moments. Durbin and Watson (1950, 1951) chose the latter method and suggested two variants: the Beta approximation based on the first two mo­ments and the Jacobi approximation based on the first four moments. For a good discussion of many other methods for evaluating the significance points of d, see Durbin and Watson (1971).

The problem of finding the moments of d is simplified because of the independence of d and the denominator of d (see Theorem 7 of Appendix 2), which implies

image444(6.3.18)

for any positive integer s. Thus, for example,

1 T-K

Подпись: (6.3.19)Ed= T-K,?{ v<

and

image446

(6.3.20)

 

where the term involving v, can be evaluated using

T-K i—1


The Beta approximation assumes that jc = d/4 has a density given by

Подпись:B{p, Qr v ^

so that Ed = 4p/(p + q) and Vd = 16pq/[(p + q)2(p + q+ 1)]. The Jacobi approximation assumes that x = d/4 has a density given by

image448

(6.3.23)

 

where G3 and G4 are the third – and fourth-order Jacobi polynomials as defined by Abramowitz and Segun (1965, Definition 22.2.2). The parameters p and q are determined so that the first two moments of (6.3.23) are equated to those of d, and a3 and aA are determined so that the third and fourth moments match. In either approximation method the significance points can be com­puted by a straightforward computer program for integration.

The distribution of d depends upon X. That means investigators must calculate the significance points for each of their problems by any of the approximation methods discussed in the previous section. Such computa­tions are often expensive. Tо make such a computation unnecessary in certain cases, Durbin and Watson (1950,1951) obtained upper and lower bounds of d that do not depend on X and tabulated the significance points for these. The bounds for d are given by

Подпись: T-KПодпись: T-Kimage451(6.3.24)

where A, = 2( 1 — cos inT~l) are the positive characteristic roots of A. (The remaining characteristic root of A is Xq = 0.) These bounds were obtained under the assumption that X contains a vector of ones and that no other column of X is a characteristic vector of A.2 Durbin and Watson (1951) calculated the significance points (at the 1,2.5, and 5% levels) of dL and dv for

K= 1,2……….. 6 and T= 15, 16,. . . , 100 by the Jacobi approximation.

Let d^ and dUa be the critical values of these bounds at the a% significance level. Then test H0: p = 0 against Hx: p > 0 by the procedure:

Reject H0 if d^d^,,

Accept H0 if d^dVa.

If іс/ц* < d < dVa, the test is inconclusive. To test Ho:p = 0 against Ht:p< 0, Durbin and Watson suggested the following procedure:

Reject tf0 if — (6.3.26)

Accept #0 if ^-4 dVa.

If these tests are inconclusive, the significance points of (6.3.17) must be evaluated directly.

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