Distribution of the Durbin–Watson Statistic in Near Integrated Processes

Abstract

This paper analyzes the Durbin–Watson (DW) statistic for near-integrated processes. Using the Fredholm approach the limiting characteristic function of DW is derived, in particular focusing on the effect of a “large initial condition” growing with the sample size. Random and deterministic initial conditions are distinguished. We document the asymptotic local power of DW when testing for integration.

Appendix

First we present a preliminary result. Lemma 2 contains the required limiting distributions in terms of Riemann integrals.

Lemma 2

Let $\left \{y_{t}\right \}$  be generated according to ( 1 ) and satisfy Assumptions  1 and  2 . It then holds for the test statistics from ( 2 ) asymptotically

$$\displaystyle{ \mathit{DW }_{j,T}\mathop{ \rightarrow }\limits^{ D}\mathit{DW }_{j} = \left (\int \left \{w_{c}^{j}\left (r\right )\right \}^{2}dr\right )^{-1}\text{, }j =\mu,\ \tau \text{,} }$$

where under Assumption  3 b)

$$\displaystyle\begin{array}{rcl} w_{c}^{\mu }\left (r\right )& =& w_{ c}\left (r\right ) -\int w_{c}\left (s\right )\mathit{ds}, {}\\ w_{c}^{\tau }\left (r\right )& =& w_{ c}^{\mu }\left (r\right ) - 12\left (r -\frac{1} {2}\right )\int \left (s -\frac{1} {2}\right )w_{c}\left (s\right )\mathit{ds}, {}\\ \end{array}$$

with $w_{c}\left (r\right ) = w\left (r\right )$  for c = 0 and

$$\displaystyle{ w_{c}\left (r\right ) =\delta \, \left (e^{-cr} - 1\right )\left (2c\right )^{-1/2} + J_{ c}\left (r\right )\ \mbox{ for }c > 0 }$$

 and the standard Ornstein–Uhlenbeck process $J_{c}\left (r\right ) =\int _{ 0}^{r}e^{-c\left (r-s\right )}\mathit{dw}\left (s\right )$ .

Proof

The proof is standard by using similar arguments as in Phillips (1987) and Müller and Elliott (2003).

1.1 Proof of Proposition 1

We set $\upsilon = \sqrt{\lambda -c^{2}}$. For DW _μ we have $w_{c}^{\mu }\left (s\right ) = w_{c}\left (r\right ) -\int w_{c}\left (s\right )\mathit{ds}$, thus

$$\displaystyle\begin{array}{rcl} \mathit{Cov}\left [w_{c}^{\mu }\left (s\right ),w_{ c}^{\mu }\left (t\right )\right ]& =& \mathit{Cov}\left [J_{ c}\left (s\right ) -\int J_{c}\left (s\right )\mathit{ds},J_{c}\left (t\right ) -\int J_{c}\left (s\right )\mathit{ds}\right ] {}\\ & =& K_{1}\left (s,t\right ) -\int \mathit{Cov}\left [J_{c}\left (s\right ),J_{c}\left (t\right )\right ]\mathit{ds} -\int \mathit{Cov}\left [J_{c}\left (s\right ),J_{c}\left (t\right )\right ]\mathit{dt} {}\\ & & +\int \int \mathit{Cov}\left [J_{c}\left (s\right ),J_{c}\left (t\right )\right ]\mathit{ds} {}\\ & =& K_{1}\left (s,t\right ) - g\left (t\right ) - g\left (s\right ) +\omega _{0} {}\\ \end{array}$$

where

$$\displaystyle\begin{array}{rcl} g\left (t\right )& =& \frac{e^{-c\left (1+s\right )}\left (1 - e^{\mathit{cs}}\right )\left (1 - 2e^{c} + e^{\mathit{cs}}\right )} {2c^{2}}, {}\\ \omega _{0}& =& -\frac{3 - 2c + e^{-2c} - 4e^{-c}} {2c^{3}}. {}\\ \end{array}$$

For DW _τ we have $w_{c}^{\tau }\left (s\right ) = w_{c}^{\mu }\left (s\right ) - 12\left (s -\frac{1} {2}\right )\int \left (u -\frac{1} {2}\right )w_{c}\left (u\right )\mathit{du}$, thus

$$\displaystyle\begin{array}{rcl} \mathit{Cov}\left [w_{c}^{\tau }\left (s\right ),w_{ c}^{\tau }\left (t\right )\right ]& =& \mathit{Cov}\left [w_{ c}^{\mu }\left (s\right ),w_{ c}^{\mu }\left (t\right )\right ] - 12\left (t -\frac{1} {2}\right )\int \left (u -\frac{1} {2}\right ) {}\\ & & \mathit{Cov}\left [J_{c}\left (s\right ) -\int J_{c}\left (v\right )dv,J_{c}\left (u\right )\right ]\mathit{du} {}\\ & & -12\left (s -\frac{1} {2}\right )\int \left (u -\frac{1} {2}\right )\mathit{Cov}\left [J_{c}\left (u\right ),J_{c}\left (t\right )\right. {}\\ & & \left.-\int J_{c}\left (v\right )dv\right ]\mathit{du} {}\\ & & +144\left (s -\frac{1} {2}\right )\left (t -\frac{1} {2}\right )\int \int \left (u -\frac{1} {2}\right )\left (v -\frac{1} {2}\right ) {}\\ & & \times \mathit{Cov}\left [J_{c}\left (u\right ),J_{c}\left (v\right )\right ]\mathit{dudv} {}\\ & =& \mathit{Cov}\left [w_{c}^{\mu }\left (s\right ),w_{ c}^{\mu }\left (t\right )\right ] {}\\ & & -12\left (t -\frac{1} {2}\right )\int \left (u -\frac{1} {2}\right )\mathit{Cov}\left [J_{c}\left (s\right ),J_{c}\left (u\right )\right ]\mathit{du} {}\\ & & +12\left (t -\frac{1} {2}\right )\int \int \left (u -\frac{1} {2}\right )\mathit{Cov}\left [J_{c}\left (v\right ),J_{c}\left (u\right )\right ]\mathit{dvdu} {}\\ & & -12\left (s -\frac{1} {2}\right )\int \left (u -\frac{1} {2}\right )\mathit{Cov}\left [J_{c}\left (u\right ),J_{c}\left (t\right )\right ]\mathit{du} {}\\ & & +12\left (s -\frac{1} {2}\right )\int \int \left (u -\frac{1} {2}\right )\mathit{Cov}\left [J_{c}\left (u\right ),J_{c}\left (v\right )\right ]\mathit{dvdu} {}\\ & & +144\left (s -\frac{1} {2}\right )\left (t -\frac{1} {2}\right )\int \int \left (u -\frac{1} {2}\right )\left (v -\frac{1} {2}\right ) {}\\ & & \times \mathit{Cov}\left [J_{c}\left (u\right ),J_{c}\left (v\right )\right ]\mathit{dudv} {}\\ \end{array}$$

With some calculus the desired result is obtained. In particular we have with $\phi _{1}\left (s\right ) = -1$, $\phi _{2}\left (s\right ) = -g\left (s\right )$, $\phi _{3}\left (s\right ) = -3f_{1}\left (s\right )$, $\phi _{4}\left (s\right ) = -3\left (s - 1/2\right )$, $\phi _{5}\left (s\right ) = 3\omega _{1}$, $\phi _{6}\left (s\right ) = 3\omega _{1}\left (s - 1/2\right )$, $\phi _{7}\left (s\right ) = 6\omega _{2}\left (s - 1/2\right )$, $\phi _{8}\left (s\right ) =\omega _{0}$, $\psi _{1}\left (t\right ) = g\left (t\right )$, $\psi _{2}\left (t\right ) =\psi _{6}\left (t\right ) =\psi _{8}\left (t\right ) = 1$, $\psi _{3}\left (t\right ) =\psi _{5}\left (t\right ) =\psi _{7}\left (t\right ) = t - 1/2$ and $\psi _{4}\left (t\right ) = f_{1}\left (t\right )$ while

$$\displaystyle{ f_{1}\left (s\right ) = \frac{e^{-c\left (1+s\right )}} {c^{3}} \times \left [2 + c + 2ce^{c} -\left (2 + c\right )e^{2cs} + 2ce^{c+cs}\left (2s - 1\right )\right ], }$$

$$\displaystyle\begin{array}{rcl} \omega _{1}& =& \frac{e^{-2c}\left (e^{c} - 1\right )} {c^{4}} \times \left [2 + c + \left (c - 2\right )e^{c}\right ]\text{,} {}\\ \omega _{2}& =& \frac{e^{-2c}} {c^{5}} \times \left [-3\left (c + 2\right )^{2} - 12c\left (2 + c\right )e^{c} + \left (12c - 9c^{2} + 2c^{3} + 12\right )e^{2c}\right ]. {}\\ \end{array}$$

This completes the proof.

1.2 Proof of Proposition 2

Let $\mathcal{L}(X) = \mathcal{L}(Y )$ stand for equality in distribution of X and Y and set $A =\delta \left (2c\right )^{-1/2}$. To begin with, we do the proofs conditioning on $\delta$. Consider first DW _μ . To shorten the proofs for DW _μ we work with the following representation for a demeaned Ornstein–Uhlenbeck process given under Theorem 3 of Nabeya and Tanaka (1990b), for their $R_{1}^{\left (2\right )}$ test statistic, i.e. we write

$$\displaystyle{ \mathcal{L}\left (\int \left \{J_{c}\left (r\right ) -\int J_{c}\left (s\right )\mathit{ds}\right \}^{2}dr\right ) = \mathcal{L}\left (\int \int K_{ 0}\left (s,t\right )\mathit{dw}\left (t\right )\mathit{dw}\left (s\right )\right ), }$$

where $K_{0}\left (s,t\right ) = \frac{1} {2c}\left [e^{-c\left \vert s-t\right \vert } - e^{-c\left (2-s-t\right )}\right ] - \frac{1} {c^{2}} p\left (s\right )p\left (t\right )$ with $p\left (t\right ) = 1 - e^{-c\left (1-t\right )}$. Using Lemma 2, we find that

$$\displaystyle{ n_{\mu }\left (t\right ) = A\left (e^{-ct} - 1\right ) - A\int \left (e^{-ct} - 1\right )\mathit{dt}. }$$

For DW _μ we will be looking for $h_{\mu }\left (t\right )$ in

$$\displaystyle{ h_{\mu }\left (t\right ) = m_{\mu }\left (t\right ) +\lambda \int K_{0}\left (s,t\right )h_{\mu }\left (t\right )\left (s\right )\mathit{ds}, }$$

(15)

where $m_{\mu }\left (t\right ) =\int K_{0}\left (s,t\right )n_{\mu }\left (s\right )\mathit{ds}$. Equation (15) is equivalent to the following boundary condition differential equation

$$\displaystyle{ h_{\mu }^{{\prime\prime}}\left (t\right ) +\upsilon ^{2}h_{\mu }\left (t\right ) = m_{\mu }^{{\prime\prime}}\left (t\right ) - c^{2}m_{\mu }\left (t\right ) +\lambda b_{ 1}\text{,} }$$

(16)

with

$$\displaystyle\begin{array}{rcl} h_{\mu }\left (1\right )& =& m_{\mu }\left (1\right ) - \frac{1} {c^{2}}\lambda b_{1}p\left (1\right ),{}\end{array}$$

(17)

$$\displaystyle\begin{array}{rcl} h_{\mu }^{{\prime}}\left (1\right )& =& m_{\mu }^{{\prime}}\left (1\right ) -\lambda e^{-c}b_{ 2} - \frac{1} {c^{2}}\lambda p^{{\prime}}\left (1\right )b_{ 1},{}\end{array}$$

(18)

where $b_{1} =\int p\left (s\right )h_{\mu }\left (s\right )\mathit{ds}$ and $b_{2} =\int e^{\mathit{cs}}h_{\mu }\left (s\right )\mathit{ds}$. Thus have

$$\displaystyle{ h_{\mu }\left (t\right ) = c_{1}^{\mu }\cos \upsilon t + c_{ 2}^{\mu }\sin \upsilon t + g_{\mu }\left (t\right ) + b_{ 1}g_{1}\left (t\right ), }$$

where $g_{\mu }\left (t\right )$ is a special solution to $g_{\mu }^{{\prime\prime}}\left (t\right ) +\upsilon ^{2}g_{\mu }\left (t\right ) = m_{\mu }^{{\prime\prime}}\left (t\right ) - c^{2}m_{\mu }\left (t\right )$ and $g_{1}\left (t\right )$ is a special solution to $g_{1}^{{\prime\prime}}\left (t\right ) +\upsilon ^{2}g_{1}\left (t\right ) =\lambda$. Boundary conditions (17) and (18) together with $h_{\mu }\left (t\right )$ imply

$$\displaystyle\begin{array}{rcl} c_{1}^{\mu }\cos \upsilon + c_{ 1}^{\mu }\sin \upsilon + \left [g_{ 1}\left (1\right ) + \frac{1} {c^{2}}\lambda p\left (1\right )\right ]b_{1}& =& m_{\mu }\left (1\right ) - g_{\mu }\left (1\right ), {}\\ -c_{1}^{\mu }\upsilon \sin \upsilon + c_{ 1}^{\mu }\upsilon \cos \upsilon + \left [ \frac{1} {c^{2}}\lambda p^{{\prime}}\left (1\right ) + g_{ 1}^{{\prime}}\left (1\right )\right ]b_{ 1} +\lambda e^{-c}b_{ 2}& =& m_{\mu }^{{\prime}}\left (1\right ) - g_{\mu }^{{\prime}}\left (1\right ), {}\\ \end{array}$$

while expressions for b ₁ and b ₂ imply that

$$\displaystyle\begin{array}{rcl} & & c_{1}^{\mu }\int p\left (s\right )\cos \upsilon \mathit{sds} + c_{ 1}^{\mu }\int p\left (s\right )\sin \upsilon \mathit{sds} + b_{ 1}\left (\int p\left (s\right )g_{1}\left (s\right )\mathit{ds} - 1\right ) {}\\ & & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = -\int p\left (s\right )g_{\mu }\left (s\right )\mathit{ds} {}\\ & & c_{1}^{\mu }\int e^{\mathit{cs}}\cos \upsilon \mathit{sds} + c_{ 1}^{\mu }\int e^{\mathit{cs}}\sin \upsilon \mathit{sds} + b_{ 1}\int e^{\mathit{cs}}g_{ 1}\left (s\right )\mathit{ds} - b_{2} {}\\ & & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = -\int e^{\mathit{cs}}g_{\mu }\left (s\right )\mathit{ds} {}\\ \end{array}$$

These form a system of linear equations in c ₁ ^μ, c ₂ ^μ, b ₁, and b ₂, which in turn identifies them. With some calculus we write

$$\displaystyle\begin{array}{rcl} \int n_{\mu }\left (t\right )h_{\mu }\left (t\right )\mathit{dt}& =& \frac{Ae^{-2c}} {2c^{2}\lambda \upsilon } \times {}\\ & &\left [-A\left (-1 + e^{c}\right )\left (2 + c + (-2 + c)e^{c}\right )\right. {}\\ & & +2e^{c}\left (cc_{ 2}^{\mu }\lambda - ce^{c}\left (c^{2}c_{ 2}^{\mu } - c^{2}c_{ 1}^{\mu } - (-1 + c)c_{ 2}^{\mu }\upsilon ^{2}\right )\right ) {}\\ & & +2ce^{c}\left (c^{3}c_{ 2}^{\mu } - cc_{ 2}^{\mu }\lambda + c_{ 2}^{\mu }\left (-1 + e^{c}\right )\lambda - c^{2}c_{ 1}^{\mu }\upsilon \right )\cos \upsilon {}\\ & & \left.-2ce^{c}\left (c^{3}c_{ 1}^{\mu } - cc_{ 1}^{\mu }\lambda + c_{ 1}^{\mu }\left (-1 + e^{c}\right )\lambda + c^{2}c_{ 2}^{\mu }\upsilon \right )\sin \upsilon \right ]. {}\\ \end{array}$$

Solving for c ₁ ^μ and c ₂ ^μ we find that they are both a multiple of A, hence

$$\displaystyle{ \varPsi _{\mu }\left (\theta;c\right ) = \frac{1} {A^{2}}\int n_{\mu }\left (t\right )h_{\mu }\left (t\right )\mathit{dt}\text{,} }$$

is free of A. Now with $\varTheta _{\mu } =\int n_{\mu }^{2}\left (t\right )\mathit{dt}$, an application of Lemma 1 results in

$$\displaystyle{ E\left [e^{i\theta \int \left \{w_{c}^{\mu }\left (r\right )\right \}^{2}dr }\vert \delta \right ] = \left [D_{\mu }\left (2i\theta \right )\right ]^{-1/2}\exp \left [i\theta A^{2}\varTheta _{ \mu } - 2\theta ^{2}A^{2}\varPsi _{ \mu }\left (\theta;c\right )\right ]. }$$

As $\sqrt{2c}A =\delta \sim N\left (\mu _{\delta },\sigma _{\delta }^{2}\right )$, standard manipulations complete the proof for j = μ.

Next we turn to DW _τ . Using Lemma 2 we find that

$$\displaystyle{ n_{\tau }\left (t\right ) = A\left [\left (e^{-ct} - 1\right ) -\int \left (e^{-\mathit{ct}} - 1\right )\mathit{dt} - 12\left (t - 1/2\right )\int \left (t - 1/2\right )\left (e^{-\mathit{ct}} - 1\right )\mathit{dt}\right ]. }$$

Here we will be looking for $h_{\tau }\left (t\right )$ in the

$$\displaystyle{ h_{\tau }\left (t\right ) = m_{\tau }\left (t\right ) +\lambda \int K_{\tau }\left (s,t\right )h_{\tau }\left (t\right )\left (s\right )\mathit{ds}, }$$

(19)

where $m_{\tau }\left (t\right ) =\int K_{\tau }\left (s,t\right )n_{\tau }\left (s\right )\mathit{ds}$ and $K_{\tau }\left (s,t\right )$ is from Proposition 1. Equation (19) can be written as

$$\displaystyle{ h_{\tau }^{{\prime\prime}}\left (t\right ) +\upsilon ^{2}h_{\tau }\left (t\right ) = m_{\tau }^{{\prime\prime}}\left (t\right ) - c^{2}m_{\tau }\left (t\right ) +\lambda \sum \nolimits _{ k=1}^{8}b_{ k}\left [\psi _{k}^{{\prime\prime}}\left (t\right ) - c^{2}\psi _{ k}\left (t\right )\right ], }$$

(20)

with the following boundary conditions

$$\displaystyle\begin{array}{rcl} h_{\tau }\left (0\right )& =& m_{\tau }\left (0\right ) +\lambda \sum \nolimits _{ k=1}^{8}b_{ k}\psi _{k}\left (0\right ),{}\end{array}$$

(21)

$$\displaystyle\begin{array}{rcl} h_{\tau }^{{\prime}}\left (0\right )& =& m_{\tau }^{{\prime}}\left (0\right ) +\lambda \sum \nolimits _{ k=1}^{8}b_{ k}\psi _{k}^{{\prime}}\left (0\right ) +\lambda b_{ 9},{}\end{array}$$

(22)

where

$$\displaystyle{ b_{k} =\int \phi _{k}\left (s\right )h_{\tau }\left (s\right )\mathit{ds}\text{, }k = 1,\ldots,8\text{ and }b_{9} =\int e^{-cs}h\left (s\right )\mathit{ds}. }$$

(23)

The solution to (20) is

$$\displaystyle{ h_{\tau }\left (t\right ) = c_{1}^{\tau }\cos \upsilon t + c_{ 2}^{\tau }\sin \upsilon t + g_{\tau }\left (t\right ) +\sum \nolimits _{ k=1}^{8}b_{ k}g_{k}\left (t\right ) }$$

(24)

where $g_{k}\left (t\right )$, $k = 1,2,\ldots,8$, are special solutions to the following differential equations

$$\displaystyle{ g_{k}^{{\prime\prime}}\left (t\right ) +\upsilon ^{2}g_{ k}\left (t\right ) =\lambda \left [\psi _{k}^{{\prime\prime}}\left (t\right ) - c^{2}\psi _{ k}\left (t\right )\right ]\text{, }k = 1,2,\ldots,8, }$$

and $g_{\tau }\left (t\right )$ is a special solution of $g_{\tau }^{{\prime\prime}}\left (t\right ) +\upsilon ^{2}g_{\tau }\left (t\right ) = m_{\tau }^{{\prime\prime}}\left (t\right ) - c^{2}m_{\tau }\left (t\right )$. The solution given in (24) can be written as

$$\displaystyle\begin{array}{rcl} h_{\tau }\left (t\right )& =& c_{1}^{\tau }\cos \upsilon t + c_{ 2}^{\tau }\sin \upsilon t + g_{\tau }\left (t\right ) - b_{ 1} \frac{\lambda } {\upsilon ^{2}} -\frac{\lambda c^{2}} {\upsilon ^{2}} (b_{2} + b_{6} + b_{8}) \\ & +& \left (b_{3} + b_{5} + b_{7}\right )\lambda c^{2}\frac{1 - 2t} {2\upsilon ^{2}} + b_{4}\lambda \frac{1 - 2t} {2\upsilon ^{2}}. {}\end{array}$$

(25)

The boundary conditions in (21) and (22) imply

$$\displaystyle\begin{array}{rcl} m_{\tau }\left (0\right ) - g_{\tau }\left (0\right )& =& c_{1}^{\tau } + \frac{\lambda } {2\upsilon ^{2}}\left (-2b_{1} + b_{4}\right ) {}\\ & & + \frac{\lambda } {2}\left (\frac{c^{2}} {\upsilon ^{2}} + 1\right )\left (-2b_{2} + b_{3} + b_{5} - 2b_{6} + b_{7} - 2b_{8}\right ) {}\\ \end{array}$$

$$\displaystyle\begin{array}{rcl} m_{\tau }^{{\prime}}\left (0\right ) - g_{\tau }^{{\prime}}\left (0\right )& =& c_{ 2}^{\tau }\mu -\lambda b_{ 1}g^{{\prime}}\left (0\right ) -\lambda \left (\frac{c^{2}} {\upsilon ^{2}} + 1\right )\left (b_{3} + b_{5} + b_{7}\right ) {}\\ & & -\left (\frac{1} {\upsilon ^{2}} + f^{{\prime}}\left (0\right )\right )\lambda b_{ 4} -\lambda b_{5}, {}\\ \end{array}$$

while expressions given under (23) characterize nine more equations. These equations form a system of linear equations in unknowns $c_{1}^{\tau },\ c_{2}^{\tau },\ b_{1},\ \ldots,\ b_{9}$, which can be simply solved to fully identify (25). Let $\varTheta _{\tau } =\int n_{\tau }\left (t\right )^{2}\mathit{dt}$. Also as for the constant case we set

$$\displaystyle{ \varPsi _{\tau }\left (\theta;c\right ) = \frac{1} {A^{2}}\int n_{\tau }\left (t\right )h_{\tau }\left (t\right )\mathit{dt}\text{,} }$$

whose expression is long and we do not report here. When solving this integral we see that $\varPsi _{\tau }\left (\theta;c\right )$ is free of A. As before we apply Lemma 1 to establish the following

$$\displaystyle{ E\left [e^{i\theta \int \left \{w_{c}^{\tau }\left (r\right )\right \}^{2}dr }\vert \delta \right ] = \left [D_{\tau }\left (2i\theta \right )\right ]^{-1/2}\exp \left [i\theta A^{2}\varTheta _{ \tau } - 2\theta ^{2}A^{2}\varPsi _{ \tau }\left (\theta;c\right )\right ]. }$$

Now, using $E\left [e^{i\theta \int \left \{w_{c}^{\tau }\left (r\right )\right \}^{2}dr }\right ] = EE\left [e^{i\theta \int \left \{w_{c}^{\tau }\left (r\right )\right \}^{2}dr }\vert \delta \right ]$, standard manipulations complete the proof.