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.
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Appendix
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
where under Assumption 3 b)
with \(w_{c}\left (r\right ) = w\left (r\right )\) for c = 0 and
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
where
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
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
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
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
For DW μ we will be looking for \(h_{\mu }\left (t\right )\) in
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
with
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
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
while expressions for b 1 and b 2 imply that
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
Solving for c 1 μ and c 2 μ we find that they are both a multiple of A, hence
is free of A. Now with \(\varTheta _{\mu } =\int n_{\mu }^{2}\left (t\right )\mathit{dt}\), an application of Lemma 1 results in
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
Here we will be looking for \(h_{\tau }\left (t\right )\) in the
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
with the following boundary conditions
where
The solution to (20) is
where \(g_{k}\left (t\right )\), \(k = 1,2,\ldots,8\), are special solutions to the following differential equations
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
The boundary conditions in (21) and (22) imply
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
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
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.
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Hassler, U., Hosseinkouchack, M. (2015). Distribution of the Durbin–Watson Statistic in Near Integrated Processes. In: Beran, J., Feng, Y., Hebbel, H. (eds) Empirical Economic and Financial Research. Advanced Studies in Theoretical and Applied Econometrics, vol 48. Springer, Cham. https://doi.org/10.1007/978-3-319-03122-4_26
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