A model specification test for nonlinear stochastic diffusions with delay

Abstract

This paper investigates model specification problems for nonlinear stochastic differential equations with delay (SDDE). Compared to the model specification for conventional stochastic diffusions without delay, the observed sequence does not admit a Markovian structure so that the classical testing procedures may not be applicable. To overcome this difficulty, a moment estimator is newly proposed based on the ergodicity of SDDEs and its asymptotic properties are established. Based on the proposed moment estimator, a testing procedure is proposed for our model specification testing problems. Particularly, the limiting distributions of the proposed test statistic are derived under null hypotheses and the test power is examined under some specific alternative hypotheses. Finally, a Monte Carlo simulation is conducted to illustrate the finite sample performance of the proposed test.

Appendix: General results on SDDEs

In this appendix, recall the ergodicity theory for SDDEs for our problem from Bao et al. (2020). Note that the following theorem is from Bao et al. (2020) concerning about the exponential ergodicity of SDDEs.

Theorem 5

Suppose Assumption 1 holds. Then, the followings are true.

(i) The Markov process $\{X_t\}$ admits a unique invariant measure $\mu$ on $\mathscr {C}$ with for any $p\ge 1$

$$\begin{aligned} \sup _{t\ge 0}\mathbb {E}\Vert X_t\Vert _{\mathscr {C}}^{2p}<L_p, \end{aligned}$$

(15)

and

$$\begin{aligned} \sup _{t\ge 0}\delta ^{-p}\mathbb {E}w_{\delta }^{2p}(X_t)<L_p, \end{aligned}$$

(16)

where $L_p$ is a constant independent of $\delta$.

(ii) If $|g(\eta )|\le L \Vert \eta \Vert _{\mathscr {C}}^2$ for some $L>0$, the following law of large numbers holds

$$\begin{aligned} \frac{1}{T}\int _\tau ^Tg(X_t)dt\rightarrow m(g)=\int _{\mathscr {C}}g(\eta )\mu (d\eta ;\theta ) \end{aligned}$$

(17)

almost surely.

(iii) For any $h:\mathscr {C}\mapsto \mathbb {R}$ satisfying

$$\begin{aligned} |h(\eta )-h(\xi )|\le L\Vert \eta -\xi \Vert _\mathscr {C}, \end{aligned}$$

one has

$$\begin{aligned} A_{T}(h;\theta ^*)=\frac{1}{\sqrt{T}}\int _\tau ^T [h(X_t)-m(h)]dt\rightarrow N\left( 0, v^2(h;\theta )\right) \end{aligned}$$

(18)

in distribution, where $X_t^\eta$ is the solution to (1) with initial $X_0=\eta$,

$$\begin{aligned} R_f(\eta )=\int _0 ^\infty \mathbb {E}f(X_t^\eta )-m(f)dt, \end{aligned}$$

and

$$\begin{aligned} v^2(h;\theta )=\int _{\mathscr {C}}\mu (d\eta ;\theta )\left[ \mathbb {E}\Big | \int _0 ^1f(X^\eta _t)dt+R_f(X_1^\eta )-R_f(\eta )\Big |\right] ^2. \end{aligned}$$

(19)

In particular, if $h(\eta )=\mathcal{A}f(\eta (0),\eta ;\theta ^*)$ some twice continuously differentiable f with bounded second order derivatives, one has

$$\begin{aligned} v^2(\mathcal{A}_0 f;\theta ^*)=\int _\mathscr {C}|\sigma ^\top (\eta ;\theta ^*)\nabla f(\eta (0))|^2\mu (d\eta ;\theta ^*). \end{aligned}$$

(20)

Remark 2

Our statements (i) and (ii) are from (1.2) and Statement (2) in Theorem 1.1 from Bao et al. (2020), respectively. The statement (iii) is from Statement (1) in Theorem 1.2 from Bao et al. (2020).

For our testing problem, we finish the appendix with the following lemma concerning with the continuity of the invariant measure $\mu (\cdot ;\theta )$ with respect to $\theta$.

Lemma 2

Assume Assumption 1 holds and $\sup _{\theta \in \Theta }\sup _{t\ge 0}\mathbb {E}\Vert X_t\Vert _\mathscr {C}^2<\infty$. Further, assume that $|\sigma (\xi ,\theta )-\sigma (\eta ,\theta )|\le \lambda _3|\xi -\eta |_\mathscr {C}$ with $\lambda _1>(\lambda _2+\lambda ^2_3)e^{-\lambda _1\tau }$. Then, as $\theta \rightarrow \theta ^*$, $\mu (\cdot ;\theta )\rightarrow \mu (\cdot ;\theta ^*)$ in distribution with

$$\begin{aligned} \int _\mathscr {C}\Vert \eta \Vert ^2_\mathscr {C}\mu (d\eta ;\theta )\rightarrow \int _\mathscr {C}\Vert \eta \Vert ^2_\mathscr {C}\mu (d\eta ;\theta ^*). \end{aligned}$$

(21)

Proof

Suppose X(t) and Y(t) be the solution of SDDE (1) with same initial and $\theta =\theta ^*$ and $\theta _1$ respectively. Note that

$$\begin{aligned}{} & {} d(X(t)-Y(t))=[b(X_t;\theta ^*)-b(Y_t;\theta _1)]dt+[\sigma (X_t;\theta ^*)-\sigma (Y_t;\theta _1)]dW(t)\\{} & {} \quad =[b(X_t;\theta ^*)-b(Y_t;\theta ^*)]dt+[\sigma (X_t;\theta ^*)-\sigma (Y_t;\theta ^*)]dW(t)\\{} & {} \qquad +[b(Y_t;\theta ^*)-b(Y_t;\theta _1)]dt+[\sigma (Y_t;\theta ^*)-\sigma (Y_t;\theta _1)]dW(t). \end{aligned}$$

Therefore, taking $\delta >0$ such that $\lambda _1>(\lambda _2+\lambda _3^2)e^{-\lambda _1\tau }$ yields that

$$\begin{aligned}{} & {} d|X(t)-Y(t)|^2= \Big [2\langle X(t)-Y(t),b(X_t;\theta ^*)-b(Y_t;\theta ^*)\rangle +|\sigma (X_t;\theta ^*)-\sigma (Y_t;\theta ^*)|^2\\{} & {} \qquad +\Big [2\langle X(t)-Y(t),b(Y_t;\theta ^*)-b(Y_t;\theta _1)\rangle +|\sigma (Y_t;\theta ^*)-\sigma (Y_t;\theta _1)|^2\\{} & {} \qquad +2(\sigma (Y_t;\theta ^*)-\sigma (Y_t;\theta _1))(\sigma (X_t;\theta ^*)-\sigma (Y_t;\theta ^*))dt+dM\\{} & {} \quad \le -\lambda _1|X(t)-Y(t)|^2+\lambda _2\Vert X_t-Y_t\Vert ^2_\mathscr {C}+\lambda _3^2\Vert X_t-Y_t\Vert ^2_\mathscr {C}\\{} & {} \qquad +2L\Vert X_t-Y_t\Vert _\mathscr {C}|\theta ^*-\theta _1|\Vert Y_t\Vert _\mathscr {C}+L|\theta ^*-\theta _1|^2+L|\theta ^*-\theta _1|\Vert X_t-Y_t\Vert _\mathscr {C}+dM\\{} & {} \quad \le -\lambda _1|X(t)-Y(t)|^2{+}(\lambda _2+\lambda ^2_3)\Vert X_t-Y_t\Vert ^2_\mathscr {C}{+}L|\theta ^*-\theta _1|(1+\Vert X_t\Vert _\mathscr {C}^2{+}\Vert Y_t\Vert _\mathscr {C}^2){+}dM, \end{aligned}$$

where M is a martingale. Similar to the proof of Lemma 3.1 in Bao et al. (2020), one has

$$\begin{aligned} \lim _{t\rightarrow \infty }\mathbb {E}\Vert X_t-Y_t\Vert ^2_\mathscr {C}\le L_\delta |\theta _1-\theta ^*|. \end{aligned}$$

As $(X_t,Y_t)$ is an asymptotic coupling of $\mu (\cdot ;\theta ^*)$ and $\mu (\cdot ;\theta _1)$, this proves that $\mu (\cdot ;\theta _1)\rightarrow \mu (\cdot ;\theta ^*)$ in distribution and (21) holds. The proof is complete. $\square$

Finally, a remark should be mentioned here that the condition in the above lemma is sufficient but not necessary. How to get a better condition is beyond the scope of our paper, thus omitted here, and it deserves a further investigation.