Fourier-type estimation of the power GARCH model with stable-Paretian innovations

Abstract

We consider estimation for general power GARCH models under stable-Paretian innovations. Exploiting the simple structure of the conditional characteristic function of the observations driven by these models we propose minimum distance estimation based on the empirical characteristic function of corresponding residuals. Consistency of the estimators is proved, and the asymptotic distribution of the estimator is studied. Efficiency issues are explored and finite-sample results are presented as well as applications of the proposed procedures to real data from the financial markets. A multivariate extension is also considered.

Appendices

Appendix 1: Technical lemmas

Let K and $\varrho$ be generic constants, whose values will be modified along the proofs, such that $K>0$ and $0<\varrho <1$. Let also

$$\begin{aligned} {\Delta }_T(\varvec{\vartheta })=\int _{-\infty }^{\infty }\left| \varphi _T(u,\varvec{\theta }) -\varphi (u,\varvec{\lambda })\right| ^2W(u)du, \end{aligned}$$

where $\varphi _T(u,\varvec{\theta })=\varphi _T\left\{ u; \varepsilon _1(\varvec{\theta }),\ldots ,\varepsilon _T(\varvec{\theta })\right\}$ is the ECF in (2.3) computed from the standardized innovations $\varepsilon _t(\varvec{\theta })=y_t/c_t(\varvec{\theta })$. Note that ${\Delta }_T(\varvec{\vartheta })$ is well defined under A3(0) because $\sup _u\left| \varphi _T(u,\varvec{\theta })-\varphi (u,\varvec{\lambda })\right| ^2<\infty$.

The following lemma shows that the choice of the unknown initial values is asymptotically unimportant for the objective function.

Lemma 8.1

Under the assumptions of Theorem 3.1, we have

$$\begin{aligned} \lim _{T\rightarrow \infty } \sup _{\varvec{\vartheta }\in \Xi }|\Delta _T(\varvec{\vartheta })-\widetilde{\Delta }_T(\varvec{\vartheta })|=0, \quad a.s. \end{aligned}$$

Proof

Using the elementary relation $|\cos x- \cos y|\le |x-y|$, we have

$$\begin{aligned} \left| \text{ Re }\left\{ \widetilde{\varphi }_T(u,\varvec{\theta })- \varphi _T(u,\varvec{\theta })\right\} \right| \le \frac{1}{T}\sum _{t=1}^T\left| uy_t\left( \frac{1}{\widetilde{c}_t(\varvec{\theta })}-\frac{1}{c_t(\varvec{\theta })}\right) \right| . \end{aligned}$$

(8.1)

By the arguments used to show (5.3) in HZ, it is easily shown that

$$\begin{aligned} \sup _{\varvec{\theta }\in \Theta }\left| c^\rho _t(\varvec{\theta })-\widetilde{c}^\rho _t(\varvec{\theta })\right|&\le K\varrho ^t,\quad \forall t. \end{aligned}$$

The mean-value theorem and the fact that $\inf _{\varvec{\theta }\in \Theta }\min \left\{ c_t(\varvec{\theta }),\widetilde{c}_t(\varvec{\theta })\right\} \ge \underline{\mu }^{1/\rho }>0$, then imply that for $c_t^{**}$ between $c_t^\rho (\varvec{\theta })$ and $\widetilde{c}_t^\rho (\varvec{\theta })$,

$$\begin{aligned} \sup _{\varvec{\theta }\in \Theta }\left| c_t(\varvec{\theta })-\widetilde{c}_t(\varvec{\theta })\right|= & {} \sup _{\varvec{\theta }\in \Theta }\left| \left\{ c_t^\rho (\varvec{\theta }) -\widetilde{c}_t^\rho (\varvec{\theta })\right\} \frac{1}{\rho } \left( c_t^{**}\right) ^{1/\rho -1}\right| \\\le & {} K\varrho ^t\left( \max \left\{ c_t(\varvec{\theta }), \widetilde{c}_t(\varvec{\theta })\right\} \right) ^{1/\rho }. \end{aligned}$$

Noting that $\text{ Re }$ can be replaced by $\text{ Im }$ in (8.1), we thus have

$$\begin{aligned} \left| \widetilde{\varphi }_T(u,\varvec{\theta })- \varphi _T(u,\varvec{\theta })\right| \le \frac{K}{T}|u|\sum _{t=1}^{\infty }\varrho ^t|y_t|\left( \max \{c_t(\varvec{\theta }),\widetilde{c}_t(\varvec{\theta })\}\right) ^{1/\rho }. \end{aligned}$$

(8.2)

The strict stationarity in A2 entails that $E|y_t|^{2s}<\infty$ and $E\sup _{\varvec{\theta }\in \Theta }|c_t(\varvec{\theta })|^{2s/\rho }<\infty$ for some small $s>0$ (see Proposition A.1 in HZ). By the same arguments, we also have $E\sup _{\varvec{\theta }\in \Theta }|\tilde{c}_t(\varvec{\theta })|^{2s/\rho }<\infty$. By the Cauchy-Schwarz inequality, the supremum over $\Theta$ of the sum of the right-hand side of the inequality (8.2) admits a moment of order s. Therefore this sum is almost surely finite, uniformly in $\Theta$. It follows that

$$\begin{aligned}&\left| \left| \widetilde{\varphi }_T(u,\varvec{\theta }) -\varphi (u,\varvec{\lambda })\right| ^2-\left| \varphi _T(u,\varvec{\theta }) -\varphi (u,\varvec{\lambda })\right| ^2\right| \\&\quad =\left| \left( \widetilde{\varphi }_T(u,\varvec{\theta }) -\varphi _T(u,\varvec{\theta })\right) \left( \overline{\widetilde{\varphi }}_T(u,\varvec{\theta }) -\overline{\varphi }(u,\varvec{\lambda })\right) \right. \\&\qquad \left. +\left( \overline{\widetilde{\varphi }}_T(u,\varvec{\theta }) -\overline{\varphi _T}(u,\varvec{\theta })\right) \left( \varphi _T(u,\varvec{\theta }) -\varphi (u,\varvec{\lambda })\right) \right| \\&\quad \le \frac{K}{T}|u|. \end{aligned}$$

We then conclude by using A3(1). $\square$

Let $\varepsilon$ and $\sigma$ be two independent random variables, $\sigma >0$ and $\varepsilon$ SP distributed with parameter $\varvec{\lambda }_0$. The following lemma shows that if $\sigma$ admits moments of order 10, then $\sigma \varepsilon$ cannot follow a SP distribution with parameter $\varvec{\lambda }\ne \varvec{\lambda }_0$. Note that the result is not true without the moment assumption (see Remark 8.3 below).

Lemma 8.2

Let $\varphi _{\varvec{\lambda }}(u)= \exp \left\{ - |u|^\alpha \{1-{\mathrm {i}}\beta {{\mathrm{sgn}}}(u) \tan (\pi \alpha /2)\}\right\}$ where $\varvec{\lambda }=(\alpha ,\beta )\in (1,2)\times [-1,1]$. If for some probability measure $\nu$ on $[0,+\infty )$ admitting a moment of order 10

$$\begin{aligned} \int _0^{\infty }\varphi _{\varvec{\lambda }}(xu)\nu (dx)=\varphi _{\varvec{\lambda }_0}(u) \quad \forall u\in {\mathbb {R}}, \end{aligned}$$

then $\varvec{\lambda }=\varvec{\lambda }_0$ and $\nu$ is the Dirac measure at the point 1.

Proof

First order Taylor expansions lead to the following inequalities (see Equation (26.4) in Billingsley 1995)

$$\begin{aligned} \left| e^{-|x|}-1+|x|\right| \le \frac{x^2}{2},\qquad \left| e^{\mathrm {i}x}-1- \mathrm {i}x\right| \le \frac{x^2}{2}. \end{aligned}$$

It follows that

$$\begin{aligned} \varphi _{\varvec{\lambda }_0}(u) =1-|u|^{\alpha _0}\{1-{\mathrm {i}}\beta _0 {{\mathrm{sgn}}}(u) \tan (\pi \alpha _0/2)\}+R_{\varvec{\lambda }_0}(u), \end{aligned}$$

(8.3)

where $|R_{\varvec{\lambda }}(u)|\le K (|u|^{2\alpha }+|u|^{3\alpha })$ for some constant K. Moreover, as $u\rightarrow 0$ we have

$$\begin{aligned} \int _0^{\infty }\varphi _{\varvec{\lambda }}(xu)\nu (dx)=1 -|u|^\alpha \left\{ 1-{\mathrm {i}}\beta {{\mathrm{sgn}}}(u) \tan (\pi \alpha /2)\right\} \int _0^{\infty }|x|^\alpha \nu (dx)+O(|u|^{2\alpha }). \end{aligned}$$

(8.4)

Identifying the right-hand sides of (8.3) and (8.4) as $u\rightarrow 0$, we obtain $\alpha =\alpha _0$ and

$$\begin{aligned} \left\{ 1-{\mathrm {i}}\beta {{\mathrm{sgn}}}(u) \tan (\pi \alpha _0/2)\right\} \int _0^{\infty }|x|^{\alpha _0} \nu (dx)=1-{\mathrm {i}}\beta _0 {{\mathrm{sgn}}}(u) \tan (\pi \alpha _0/2). \end{aligned}$$

The real and imaginary parts of both sides being equal, it follows that

$$\begin{aligned} \int _{0}^{+\infty }|x|^{\alpha _0}\nu (dx)=1\quad \text{ and } \quad \beta \int _0^{+\infty }|x|^{\alpha _0}\nu (dx)=\beta _0, \end{aligned}$$

from which we deduce that $\beta =\beta _0$. This is not sufficient to conclude concerning the measure $\nu$. Doing Taylor expansions of higher orders, we have

$$\begin{aligned} \varphi _{\varvec{\lambda }_0}(u) =&\,1-|u|^{\alpha _0}\left\{ 1-{\mathrm {i}}\beta _0 {{\mathrm{sgn}}}(u) \tan \left( \frac{\pi \alpha _0}{2}\right) \right\} \\&+\frac{|u|^{2\alpha _0}}{2}\left\{ 1-\beta _0^2\tan ^2 \left( \frac{\pi \alpha _0}{2}\right) -2{\mathrm {i}}\beta _0 {{\mathrm{sgn}}}(u) \left( \frac{\pi \alpha _0}{2}\right) \right\} +R^*_{\varvec{\lambda }_0}(u) \end{aligned}$$

where $|R^*_{\varvec{\lambda }}(u)|\le K (|u|^{3\alpha }+|u|^{4\alpha }+|u|^{5\alpha })$ for some constant K. Note that $\int R^*_{\varvec{\lambda }}(xu)\nu (dx)=o(|u|^{2\alpha })$ as $u\rightarrow 0$ because $\int |x|^{5\alpha }\nu (dx)<\infty .$ Identifying the approximations of $\varphi _{\varvec{\lambda }_0}(u)$ and $\int _0^{\infty }\varphi _{\varvec{\lambda }_0}(xu)\nu (dx)$ as $u\rightarrow 0$, we obtain $\int _{0}^{+\infty }|x|^{2\alpha _0}\nu (dx)=\int _{0}^{+\infty }|x|^{\alpha _0}\nu (dx)=1$. It follows that $\nu$ is the Dirac measure at 1, which completes the proof. $\square$

Remark 8.3

The need of moment assumptions on the probability measure $\nu$ is already evident from representation (5.2) which suggests that a symmetric SP random variable can be obtained as a scale mixture of normal distributions, with mixing distribution a SP distribution concentrated on $[0,\infty )$. A more general result involving non-normal mixtures of SP distributions is proved by Samorodnitsky and Taqqu (1994, §1.3). By way of example we consider the random variable $W=X^{1/2} Z$, where Z is standard normal with CF $\varphi _{\varvec{\lambda }}(u)=\exp \{-(1/2) u^2\}$ and X follows the Lévy distribution, which is a totally skewed to the right SP distribution with tail index $\alpha =1/2$ and density $\nu (dx)=1/(\sqrt{2\pi })x^{-3/2}\exp \{-1/(2x)\}dx, \ x>0$. Denote by $\varphi _W(u)$ the CF of W. Then it readily follows that

$$\begin{aligned} \varphi _W(u)=\int _0^\infty \varphi _{\varvec{\lambda }}(x^{1/2}u)\nu (dx)=\frac{1}{\sqrt{2\pi }} \int _0^\infty \frac{1}{x^{3/2}} \exp \left\{ -\frac{1}{2} \left( xu^2+\frac{1}{x}\right) \right\} dx=e^{-|u|}, \end{aligned}$$

which shows that W follows the Cauchy distribution, and consequently that this distribution has a stochastic representation as a mixture of normal distributions with variance following the Lévy distribution.

The following lemma is similar to Lemma 1 in Tauchen (1985) and Lemma 2.4 in Newey and McFadden (1994), except that the assumption of iid observations is relaxed.

Lemma 8.4

Let $(z_t)$ be a stationary and ergodic process. Assume that $\Theta$ is compact, that $\theta \mapsto a(z,\theta )$ is continuous on $\Theta$ for all $z\in \Omega _1$ such that $P(z_1\in \Omega _1)=1$, and that there exists d(z) such that $\Vert a(z,\theta )\Vert \le d(z)$ for all $\theta \in \Theta$ and $Ed(z_1)<\infty .$ Then $\theta \mapsto E a(z_1,\theta )$ is continuous and

$$\begin{aligned} \sup _{\theta \in \Theta }\left\| \frac{1}{T}\sum _{t=1}^Ta(z_t,\theta )-Ea(z_1,\theta )\right\| \rightarrow 0\quad a.s.\quad \text{ as } \quad T\rightarrow \infty . \end{aligned}$$

Proof

Let $V_m(\theta )$ be the open ball of center $\theta$ and radius 1 / m. The dominated convergence theorem entails that for all $\theta _k\in \Theta$ and all $\epsilon >0$ there exists m such that the neighborhood $V(\theta _k)=V_m(\theta _k)$ satisfies

$$\begin{aligned} E\sup _{\theta \in V(\theta _k)\cap \Theta }\left\| a(z_1,\theta )-a(z_1,\theta _k)\right\| \le \epsilon . \end{aligned}$$

(8.5)

By a compactness argument, there exist $\theta _1,\ldots , \theta _K$ such that $\cup _{k=1}^KV(\theta _k) \subseteq \Theta$ where $V(\theta _k)$ satisfies (8.5). Now note that

$$\begin{aligned}&\sup _{\theta \in V(\theta _k)\cap \Theta }\left\| \frac{1}{T}\sum _{t=1}^Ta(z_t,\theta )-Ea(z_1,\theta )\right\| \le \frac{1}{T}\sum _{t=1}^T\sup _{\theta \in V(\theta _k)\cap \Theta }\left\| a(z_t,\theta )-a(z_t,\theta _k)\right\| \\&\qquad +\left\| \frac{1}{T}\sum _{t=1}^Ta(z_t,\theta _k)-Ea(z_1,\theta _k)\right\| +\sup _{\theta \in V(\theta _k)\cap \Theta }\left\| Ea(z_1,\theta _k)-Ea(z_1,\theta )\right\| . \end{aligned}$$

The ergodic theorem and (8.5) entail that, as $T\rightarrow \infty$, the almost sure limit of the first term of the right-hand side of the inequality is bounded by $\epsilon$. The ergodic theorem also shows that the limit of the second term is zero. By (8.5), the last term is bounded by $\epsilon$. Since $\epsilon$ is arbitrarily small, the conclusion follows. $\square$

We now show that the choice of the unknown initial values is asymptotically unimportant for the derivative of the objective function.

Lemma 8.5

Under the assumptions of Theorem 3.3, we have

$$\begin{aligned} \lim _{T\rightarrow \infty } \sqrt{T}\sup _{\varvec{\vartheta }\in V(\varvec{\vartheta }_0)}\left\| \frac{\partial \Delta _T(\varvec{\vartheta })}{\partial \varvec{\vartheta }} -\frac{\partial \widetilde{\Delta }_T(\varvec{\vartheta })}{\partial \varvec{\vartheta }}\right\| =0, \quad a.s. \end{aligned}$$

Proof

In view of (9.4) and (9.5), we have

$$\begin{aligned} \left\| \frac{1}{T}\sum _{s=1}^T\frac{\partial g_s(u,\varvec{\vartheta })}{\partial \varvec{\vartheta }}\right\| \le K+\frac{|u|}{T}\sum _{s=1}^T|\varepsilon _t|\frac{c_t}{c_t(\varvec{\theta })}\left\| \frac{1}{c_t(\varvec{\theta })}\frac{\partial c_t(\varvec{\theta })}{\partial \varvec{\vartheta }}\right\| . \end{aligned}$$

By (3.2), (9.6) and $E\left| \varepsilon _t\right| <\infty$, the random variable of the right hand side of the last inequality is uniformly integrable in a neighborhood of $\varvec{\vartheta }_0$. By the ergodic theorem, it follows that, when $\varvec{\vartheta }$ is sufficiently close to $\varvec{\vartheta }_0$, the right hand side is a.s. bounded by a constant or by u times a constant. Note also that $\left| T^{-1}\sum _{t=1}^T\overline{g_t(u,\varvec{\vartheta })}\right| \le 2$. The dominated convergence theorem and A3(0)–A3(1) thus show that one can take the derivative under the integral symbol to obtain

$$\begin{aligned} \frac{\partial \Delta _T(\varvec{\vartheta })}{\partial \varvec{\vartheta }} =2\text{ Re }\int _{-\infty }^{\infty } \frac{1}{T}\sum _{t=1}^T\overline{g_t(u,\varvec{\vartheta })}\frac{1}{T}\sum _{s=1}^T\frac{\partial g_s(u,\varvec{\vartheta })}{\partial \varvec{\vartheta }} W(u)du, \end{aligned}$$

(8.6)

at least when $\varvec{\vartheta }$ is sufficiently close to $\varvec{\vartheta }_0$. A similar expression holds for ${\partial \widetilde{\Delta }_T(\varvec{\vartheta })}/{\partial \varvec{\vartheta }}$. It follows that

$$\begin{aligned} \sqrt{T}\sup _{\varvec{\vartheta }\in V(\varvec{\vartheta }_0)}\left\| \frac{\partial \Delta _T(\varvec{\vartheta })}{\partial \varvec{\vartheta }} -\frac{\partial \widetilde{\Delta }_T(\varvec{\vartheta })}{\partial \varvec{\vartheta }}\right\| \le a_T+b_T \end{aligned}$$

where, with the obvious notation $\widetilde{g}_t(u,\varvec{\vartheta })=e^{\mathrm {i}u\widetilde{\varepsilon }_t}-\varphi (u,\varvec{\lambda })$,

$$\begin{aligned} a_T= & {} \int _{-\infty }^{\infty }\frac{1}{\sqrt{T}}\sum _{t=1}^{\infty }\sup _{\varvec{\vartheta }\in V(\varvec{\vartheta }_0)}\left| g_t(u,\varvec{\vartheta })-\widetilde{g}_t(u,\varvec{\vartheta })\right| \frac{1}{T}\sum _{s=1}^T\sup _{\varvec{\vartheta }\in V(\varvec{\vartheta }_0)}\left\| \frac{\partial g_s(u,\varvec{\vartheta })}{\partial \varvec{\vartheta }}\right\| W(u)du,\\ b_T= & {} \int _{-\infty }^{\infty }\frac{1}{T}\sum _{t=1}^{T}\sup _{\varvec{\vartheta }\in V(\varvec{\vartheta }_0)}\left| g_t(u,\varvec{\vartheta })\right| \frac{1}{\sqrt{T}}\sum _{s=1}^{\infty }\sup _{\varvec{\vartheta }\in V(\varvec{\vartheta }_0)}\left\| \frac{\partial g_s(u,\varvec{\vartheta })}{\partial \varvec{\vartheta }}-\frac{\partial \widetilde{g}_s(u,\varvec{\vartheta })}{\partial \varvec{\vartheta }}\right\| W(u)du. \end{aligned}$$

By the argument used to show that the series in (8.2) is bounded, we obtain

$$\begin{aligned} \sum _{t=1}^{\infty }\sup _{\varvec{\vartheta }\in V(\varvec{\vartheta }_0)}\left| g_t(u,\varvec{\vartheta })-\widetilde{g}_t(u,\varvec{\vartheta })\right| \le K|u|\quad a.s. \end{aligned}$$

for some neighborhood $V(\varvec{\vartheta }_0)$. By already used arguments, we also have

$$\begin{aligned} \frac{1}{T}\sum _{s=1}^T\sup _{\varvec{\vartheta }\in V(\varvec{\vartheta }_0)}\left\| \frac{\partial g_s(u,\varvec{\vartheta })}{\partial \varvec{\vartheta }}\right\| \le K+K|u|\quad a.s. \end{aligned}$$

It follows that the integrand in $a_T$ is almost surely bounded by $KT^{-1/2}(|u|+u^2)W(u)$, when $V(\varvec{\vartheta }_0)$ is sufficiently small. By the dominated convergence theorem and $\mathbf{A3}(1)$, $\mathbf{A3}(2)$, almost surely $a_T\rightarrow 0$ as $T\rightarrow \infty .$ Similar arguments show that $b_T\rightarrow 0$, and the conclusion follows. $\square$

Lemma 8.6

Let $\varepsilon$ be a random variable with the SP distribution of parameter $\varvec{\lambda }=(\alpha ,\beta )\in (1,2)\times [-1,1]$. For all $\nu \in (-3,\alpha -1)$, there exists a constant K such that for all $c>0$

$$\begin{aligned} \left| E|\varepsilon |^{2+\nu } e^{{\mathrm {i}}c\varepsilon }\right| \le K+\frac{K}{c}. \end{aligned}$$

Proof

The density $f_{\varvec{\lambda }}(x)$ of $\varepsilon$ is bounded and satisfies

$$\begin{aligned} f_{\varvec{\lambda }}(x)\sim \frac{K}{x^{\alpha +1}}\quad \text{ as } \quad |x|\rightarrow \infty , \end{aligned}$$

for some constant $K=K_{\varvec{\lambda }}$ (see e.g. Theorem 1.12 in Nolan 2012). To show the existence of $E|\varepsilon |^{2+\nu } e^{{\mathrm {i}}c\varepsilon }$, it is thus sufficient to show the existence of

$$\begin{aligned} \int _1^{\infty }\cos (cx)\frac{x^{2+\nu }}{x^{\alpha +1}}dx\quad \text{ and } \quad \int _1^{\infty }\sin (cx)\frac{x^{2+\nu }}{x^{\alpha +1}}dx. \end{aligned}$$

An integration by parts shows that the first integral is equal to

$$\begin{aligned} -\frac{\sin c}{c}+\frac{\alpha -\nu -1}{c}\int _1^{\infty }\frac{\sin (cx)}{x^{\alpha -\nu }}dx. \end{aligned}$$

Similarly, is can be seen that the second integral is also bounded by K / c. The conclusion follows. $\square$

Lemma 8.7

Under the assumptions of Theorem 3.3, for any sequence $\varvec{\vartheta }_T$ tending almost surely to $\varvec{\vartheta }_0$ as $T\rightarrow \infty$, we have

$$\begin{aligned} \frac{1}{T}\sum _{s=1}^T\frac{\partial g_s(u,\varvec{\vartheta }_T)}{\partial \varvec{\vartheta }}=\varvec{g}(u)+\varvec{R}_T(u), \end{aligned}$$

where $\Vert \varvec{R}_T(u)\Vert \le (1+|u|+u^2)r_T$ and $r_T$ tends almost surely to 0 as $T\rightarrow \infty$.

Proof

First note that, similarly to (9.7), we have

$$\begin{aligned} E\sup _{\varvec{\theta }\in \Theta }\left\| \frac{1}{c_t(\varvec{\theta })}\frac{\partial ^2 c_t(\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }'}\right\| ^r<\infty \quad \text{ for } \text{ all } r>0\text{. } \end{aligned}$$

(8.7)

Note that $\partial ^2 g_s(u,\varvec{\vartheta })/\partial \varvec{\theta }\partial \varvec{\lambda }'=0$, that $\partial ^2 g_s(u,\varvec{\vartheta })/\partial \varvec{\lambda }\partial \varvec{\lambda }'$ is a non random bounded function of u uniformly in $\varvec{\vartheta }$, and

$$\begin{aligned} \frac{\partial ^2 g_s(u,\varvec{\vartheta })}{\partial \varvec{\theta }\partial \varvec{\theta }'}=ue^{\mathrm {i}u\varepsilon _t}\varepsilon _t\left\{ -u\varepsilon _t\frac{1}{c_t^2}\frac{\partial c_t}{\partial \varvec{\theta }}\frac{\partial c_t}{\partial \varvec{\theta }'}+2\mathrm {i} \frac{1}{c_t^2}\frac{\partial c_t}{\partial \varvec{\theta }}\frac{\partial c_t}{\partial \varvec{\theta }'}-\mathrm {i}\frac{1}{c_t}\frac{\partial ^2 c_t}{\partial \varvec{\theta }\partial \varvec{\theta }'}\right\} (\varvec{\theta }). \end{aligned}$$

(8.8)

Conditioning by the past, using Lemma 8.6 with $c=|u| c_t(\varvec{\theta }_0)/c_t(\varvec{\theta })$ and $\nu =0,-1,-2$, and using (3.2), (9.6) and (8.7), it can be shown that there exist a neighborhood $V(\varvec{\vartheta }_0)$ of $\varvec{\vartheta }_0$ and a constant K independent of u such that

$$\begin{aligned} E\sup _{\varvec{\vartheta }\in V(\varvec{\vartheta }_0)}\left\| \frac{\partial ^2 g_s(u,\varvec{\vartheta })}{\partial \varvec{\vartheta }\partial \varvec{\vartheta }'}\right\| \le K(1+|u|+u^2). \end{aligned}$$

Using this result, Taylor expansions for the real and imaginary parts, and the ergodic theorem, we obtain

$$\begin{aligned} \left\| \frac{1}{T}\sum _{s=1}^T\frac{\partial g_s(u,\varvec{\vartheta }_T)}{\partial \varvec{\vartheta }} -\frac{1}{T}\sum _{s=1}^T\frac{\partial g_s(u,\varvec{\vartheta }_0)}{\partial \varvec{\vartheta }} \right\| \le K(1+|u|+u^2)\left\| \varvec{\vartheta }_T-\varvec{\vartheta }_0\right\| . \end{aligned}$$

Now, in view of (9.4), we have

$$\begin{aligned}&\frac{1}{T}\sum _{s=1}^T \frac{\partial g_s(u,\varvec{\vartheta }_0)}{\partial \varvec{\theta }}-\varvec{g}(u)\\&\quad =\frac{-\mathrm {i}u}{T}\sum _{t=1}^T(e^{\mathrm {i}u\varepsilon _t}\varepsilon _t-Ee^{\mathrm {i}u\varepsilon _t}\varepsilon _t)\frac{1}{c_t}\frac{\partial c_t(\varvec{\theta }_0)}{\partial \varvec{\theta }}\\&\qquad +Ee^{\mathrm {i}u\varepsilon _t}\varepsilon _t\left( \frac{1}{c_t}\frac{\partial c_t(\varvec{\theta }_0)}{\partial \varvec{\theta }}-E\frac{1}{c_t}\frac{\partial c_t(\varvec{\theta }_0)}{\partial \varvec{\theta }}\right) . \end{aligned}$$

Using Lemma 8.6 and (9.6), we then obtain

$$\begin{aligned} \left\| \frac{1}{T}\sum _{s=1}^T \frac{\partial g_s(u,\varvec{\vartheta }_0)}{\partial \varvec{\theta }}-\varvec{g}(u)\right\| \le (1+|u|)o(1),\quad a.s. \end{aligned}$$

Because the derivative of $g_s(u,\varvec{\vartheta })$ with respect to $\varvec{\lambda }$ is non-random, we also have

$$\begin{aligned} \frac{1}{T}\sum _{s=1}^T \frac{\partial g_s(u,\varvec{\vartheta }_0)}{\partial \varvec{\lambda }}-E\frac{\partial g_1(u,\varvec{\vartheta }_0)}{\partial \varvec{\lambda }}=0. \end{aligned}$$

We thus have shown that, a.s.

$$\begin{aligned} \left\| \frac{1}{T}\sum _{s=1}^T\frac{\partial g_s(u,\varvec{\vartheta }_T)}{\partial \varvec{\vartheta }}-\varvec{g}(u)\right\| \le K(1+|u|+u^2)\left\{ \left\| \varvec{\vartheta }_T-\varvec{\vartheta }_0\right\| +o(1)\right\} . \end{aligned}$$

The conclusion follows. $\square$

Lemma 8.8

Under the assumptions of Theorem 3.3, for any sequence $\varvec{\vartheta }_T$ tending almost surely to $\varvec{\vartheta }_0$ as $T\rightarrow \infty$, we have

$$\begin{aligned} \frac{1}{\sqrt{T}}\sum _{t=1}^T \overline{g_t(u,\varvec{\vartheta }_T)}= & {} \frac{1}{\sqrt{T}} \sum _{t=1}^T \overline{g_t(u,\varvec{\vartheta }_0)}+\left\{ \overline{\varvec{g}(u)} +\overline{\varvec{R}^{(1)}_T(u)}\right\} '\sqrt{T}(\varvec{\vartheta }_T-\varvec{\vartheta }_0)\nonumber \\&+ T^{1/4}(\varvec{\vartheta }_T-\varvec{\vartheta }_0)'\left\{ \varvec{H}(u) +\varvec{R}^{(2)}_T(u)\right\} T^{1/4}(\varvec{\vartheta }_T-\varvec{\vartheta }_0),\qquad \quad \end{aligned}$$

(8.9)

where

$$\begin{aligned} \varvec{H}(u)=\frac{1}{2}E\frac{\partial ^2 \overline{g_1(u,\varvec{\vartheta }_0)}}{\partial \varvec{\vartheta }\partial \varvec{\vartheta }'},\quad \Vert \varvec{R}^{(i)}_T(u)\Vert \le (1+|u|+u^2)r^{(i)}_T, \end{aligned}$$

and $r^{(i)}_T$ tends almost surely to 0 as $T\rightarrow \infty$ for $i=1,2$.

Proof

A second-order Taylor expansion yields

$$\begin{aligned} \frac{1}{\sqrt{T}}\sum _{t=1}^T\text{ Re } g_t(u,\varvec{\vartheta }_T)&= \frac{1}{\sqrt{T}}\sum _{t=1}^T\text{ Re }\, g_t(u,\varvec{\vartheta }_0)+\frac{1}{T}\sum _{t=1}^T\frac{\partial \text{ Re }\, g_t(u,\varvec{\vartheta }_0)}{\partial \varvec{\vartheta }'}\sqrt{T}(\varvec{\vartheta }_T-\varvec{\vartheta }_0)\\&\quad +T^{1/4}(\varvec{\vartheta }_T-\varvec{\vartheta }_0)'\frac{1}{2T}\sum _{t=1}^T\frac{\partial ^2\text{ Re }\, g_t(u,\varvec{\vartheta }^*_T)}{\partial \varvec{\vartheta }\partial \varvec{\vartheta }'}T^{1/4}(\varvec{\vartheta }_T-\varvec{\vartheta }_0) \end{aligned}$$

for some $\varvec{\vartheta }^*_T$ between $\varvec{\vartheta }_T$ and $\varvec{\vartheta }_0$. A similar expansion holds when $\text{ Re }(\cdot )$ is replaced by $\text{ Im }(\cdot )$. The result will thus follow from Lemma 8.7, if we show that

$$\begin{aligned} \varvec{R}^{(2)}_T(u):=\varvec{H}(u)-\frac{1}{2T}\sum _{t=1}^T\frac{\partial ^2\overline{g_t(u,\varvec{\vartheta }^*_T)}}{\partial \varvec{\vartheta }\partial \varvec{\vartheta }'} \end{aligned}$$

satisfies the condition of the lemma. Given the arguments of the proof of Lemma 8.7, it will be sufficient to show that

$$\begin{aligned} \left\| \frac{1}{T}\sum _{s=1}^T\frac{\partial ^2 g_s(u,\varvec{\vartheta }^*_T)}{\partial \varvec{\vartheta }\partial \varvec{\vartheta }'} -\frac{1}{T}\sum _{s=1}^T\frac{\partial ^2 g_s(u,\varvec{\vartheta }_0)}{\partial \varvec{\vartheta }\partial \varvec{\vartheta }'} \right\| \le (1+|u|+u^2)o_P(1). \end{aligned}$$

In view of (8.8), the left-hand side of the inequality can be bounded by

$$\begin{aligned} \frac{Ku^2}{T}\sum _{s=1}^T\left| \varepsilon _t(\varvec{\theta }^*_T)-\varepsilon _t(\varvec{\theta }_0)\right| \left\| \frac{1}{c_t^2(\varvec{\theta }^*_T)}\frac{\partial c_t(\varvec{\theta }^*_T)}{\partial \varvec{\theta }}\frac{\partial c_t(\varvec{\theta }^*_T)}{\partial \varvec{\theta }'}\right\| \end{aligned}$$

(8.10)

and by other terms which can be handled similarly. Using the Hölder inequality, the ergodic theorem, (3.2), (9.6), and the consistency of $\varvec{\theta }^*_T$, it can be shown that for any $r>1$ and any sufficiently small neighborhood $V(\varvec{\theta }_0)$ of $\varvec{\theta }_0$, the average (8.10) is almost surely bounded by $Ku^{2}\left\{ E\sup _{\varvec{\theta }\in V(\varvec{\theta }_0)}\left| \frac{c_1(\varvec{\theta }_0)-c_1(\varvec{\theta })}{c_1(\varvec{\theta })}\right| ^{r}\right\} ^{1/r}$. The conclusion follows. $\square$

Appendix 2: Proof of the main results

Proof of Theorem 3.1

We begin to show that the limiting criterion is minimal only at the true value, that is

$$\begin{aligned} \forall \varvec{\vartheta }\ne \varvec{\vartheta }_0,\quad \lim _{T\rightarrow \infty } \widetilde{\Delta }_T(\varvec{\vartheta })> 0\qquad \text{ and } \qquad \lim _{T\rightarrow \infty } \widetilde{\Delta }_T(\varvec{\vartheta }_0)= 0\quad a.s. \end{aligned}$$

(9.1)

First note that Lemma 8.1 shows that $\widetilde{\Delta }_T$ can be replaced by $\Delta _T$ in (9.1). The ergodic theorem shows that, almost surely,

$$\begin{aligned} \varphi _T(u,\varvec{\theta }_0)\rightarrow E\exp \{\mathrm {i}u\varepsilon _t\}=\varphi (u,\varvec{\lambda }_0)\quad \text{ as } \quad T\rightarrow \infty . \end{aligned}$$

The second convergence of (9.1) is thus a direct consequence of the dominated convergence theorem and A3(0). Using $W(\cdot )>0$ and the continuity of the characteristic functions, the same arguments show that $\lim _{T\rightarrow \infty } \Delta _T(\varvec{\vartheta })=0$ iff $\varphi (u,\varvec{\lambda })$ is the characteristic function of

$$\begin{aligned} \varepsilon _1(\varvec{\theta })=\frac{c_1(\varvec{\theta }_0)\varepsilon _1}{c_1(\varvec{\theta })}, \end{aligned}$$

that is iff

$$\begin{aligned} \forall u,\qquad \varphi (u,\varvec{\lambda })=E\varphi \left( u\frac{c_1(\varvec{\theta }_0)}{c_1(\varvec{\theta })},{\varvec{\lambda }_0}\right) . \end{aligned}$$

Lemma 8.2 of “Appendix 1” shows that, under A5, this is only possible if

$$\begin{aligned} \varvec{\lambda }=\varvec{\lambda }_0\qquad \text{ and } \qquad c_1(\varvec{\theta }_0)=c_1(\varvec{\theta })\quad a.s., \end{aligned}$$

which is equivalent to $\varvec{\vartheta }=\varvec{\vartheta }_0$ by A4 (see the proof of Theorem 3.1 in HZ). The proof of (9.1) is complete.

We now show that the inequality in (9.1) holds locally uniformly, i.e. that there exists a neighborhood $V(\varvec{\vartheta }^*)$ of $\varvec{\vartheta }^*=(\varvec{\theta }^{* '},\varvec{\lambda }^{* '})'$ such that

$$\begin{aligned} \liminf _{T\rightarrow \infty }\inf _{\varvec{\vartheta }\in V(\varvec{\vartheta }^*)} \widetilde{\Delta }_T(\varvec{\vartheta })> 0\qquad \text{ if } \quad \varvec{\vartheta }^*\ne \varvec{\vartheta }_0. \end{aligned}$$

(9.2)

In view of Lemma 8.1, we can replace $\widetilde{\Delta }_T$ by $\Delta _T$, which makes possible to apply the ergodic theorem. Let $E\exp \{iu\varepsilon _1(\varvec{\theta })\}$ be the almost sure limit of $\varphi _T(u,\varvec{\theta })$. Lemma 8.4 shows that the convergence is actually uniform:

$$\begin{aligned} \forall u,\quad \sup _{\varvec{\theta }\in \Theta }\left| \varphi _T(u,\varvec{\theta })-E\exp \{\mathrm {i}u\varepsilon _1(\varvec{\theta })\}\right| \rightarrow 0\quad a.s. \end{aligned}$$

(9.3)

Note that

$$\begin{aligned} \Delta _T(\varvec{\vartheta })=a_T(\varvec{\theta })+b(\varvec{\vartheta })+d_T(\varvec{\vartheta }) \end{aligned}$$

with

$$\begin{aligned} a_T(\varvec{\theta })= & {} \int _{-\infty }^{+\infty }\left| \varphi _T(u,\varvec{\theta }) -E\exp \{\mathrm {i}u\varepsilon _1(\varvec{\theta })\}\right| ^2W(u)du,\\ b(\varvec{\vartheta })= & {} \int _{-\infty }^{+\infty }\left| E\exp \{ \mathrm {i}u\varepsilon _1(\varvec{\theta })\}-\varphi (u,\varvec{\lambda })\right| ^2W(u)du,\\ d_T(\varvec{\vartheta })= & {} \int _{-\infty }^{+\infty }2\text{ Re } \left\{ \varphi _T(u,\varvec{\theta })\right. \\&\left. -\exp \{\mathrm {i}u\varepsilon _1(\varvec{\theta })\}\right\} \left\{ E\exp \{-\mathrm {i}u\varepsilon _1(\varvec{\theta })\} -\overline{\varphi (u,\varvec{\lambda })}\right\} W(u)du. \end{aligned}$$

Using (9.3), A3(0) and the bound $\left| \varphi _T(u,\varvec{\theta })-E\exp \{\mathrm {i}u\varepsilon _1(\varvec{\theta })\}\right| \le 2$, we show that $\sup _{\varvec{\theta }\in \Theta }a_T(\varvec{\theta })\rightarrow 0\; a.s.$ By the Cauchy-Schwarz inequality, we similarly obtain $\sup _{\varvec{\vartheta }\in \Xi }d_T(\varvec{\vartheta })\rightarrow 0\; a.s.$ For any positive integer k, let $V_k(\varvec{\vartheta }^*)$ be the open ball of center $\varvec{\vartheta }^*$ and radius 1 / k. By Beppo Levi, as $k\rightarrow \infty$

where the last inequality comes from (9.1). It follows that there exists a neighborhood $V(\varvec{\vartheta }^*)$ such that $\inf _{\varvec{\vartheta }\in V(\varvec{\vartheta }^*)}b(\varvec{\vartheta })>0$. We then obtain (9.2) by noting that

$$\begin{aligned} \inf _{\varvec{\vartheta }\in V(\varvec{\vartheta }^*)} \Delta _T(\varvec{\vartheta })\ge \inf _{\varvec{\vartheta }\in V(\varvec{\vartheta }^*)}b(\varvec{\vartheta }) - \sup _{\varvec{\theta }\in \Theta }a_T(\varvec{\theta }) -\sup _{\varvec{\vartheta }\in \Theta }d_T(\varvec{\vartheta }). \end{aligned}$$

The conclusion follows from (9.1), (9.2) and a standard compactness argument. $\square$

Proof of Lemma 3.2

We have

$$\begin{aligned} \frac{\partial g_t(u,\varvec{\vartheta })}{\partial \varvec{\theta }}=\frac{\partial \exp \{\mathrm {i}u\varepsilon _t(\varvec{\theta }) \}}{\partial \varvec{\theta }}=-\mathrm {i}u\exp \{\mathrm {i}u\varepsilon _t(\varvec{\theta })\}\varepsilon _t(\varvec{\theta })\frac{1}{c_t(\varvec{\theta })}\frac{\partial c_t(\varvec{\theta })}{\partial \varvec{\theta }}, \end{aligned}$$

(9.4)

and

$$\begin{aligned} \frac{\partial g_t(u,\varvec{\vartheta })}{\partial \varvec{\lambda }}=-\frac{\partial \varphi (u,\varvec{\lambda })}{\partial \varvec{\lambda }},\qquad \frac{\partial g_t(u,\varvec{\vartheta })}{\partial \varvec{\lambda }_0}=\varphi (u,\varvec{\lambda }_0)\left( \begin{array}{c}\tau _1(u)\\ \tau _2(u)\end{array}\right) , \end{aligned}$$

(9.5)

with

$$\begin{aligned} \tau _1(u)= & {} |u|^{\alpha _0}\left\{ \log |u|-\mathrm {i}\beta _0 {\mathrm{sgn}} \left( u\right) \left( \log |u|\tan \left( \frac{\pi \alpha _0}{2}\right) +\frac{\pi }{2}\frac{1}{\cos ^2\left( \frac{\pi \alpha _0}{2}\right) }\right) \right\} ,\\ \tau _2(u)= & {} -\mathrm {i}|u|^{\alpha _0}{\mathrm{sgn}} (u)\tan \left( \frac{\pi \alpha _0}{2}\right) . \end{aligned}$$

Since $E\left| \varepsilon _t\right| <\infty$, and $\varepsilon _t$ and $c_t(\varvec{\theta })$ are independent, we have

$$\begin{aligned} E\left\| \frac{\partial g_t(u,\varvec{\vartheta })}{\partial \varvec{\theta }}\right\| \le E\left\| u\frac{c_t}{c_t(\varvec{\theta })}\varepsilon _t\frac{1}{c_t(\varvec{\theta })}\frac{\partial c_t(\varvec{\theta })}{\partial \varvec{\theta }}\right\| \le K|u|E\frac{c_t}{c_t(\varvec{\theta })}\left\| \frac{1}{c_t(\varvec{\theta })}\frac{\partial c_t(\varvec{\theta })}{\partial \varvec{\theta }}\right\| . \end{aligned}$$

In view of (5.20) in HZ and its extension Page 506, we have

$$\begin{aligned} E\sup _{\varvec{\theta }\in \Theta }\left\| \frac{1}{c_t(\varvec{\theta })}\frac{\partial c_t(\varvec{\theta })}{\partial \varvec{\theta }}\right\| ^r<\infty \quad \text{ for } \text{ all } r>0. \end{aligned}$$

(9.6)

Using also (3.2), the Hölder inequality then entails that for some neighborhood $V(\varvec{\vartheta }_0)$ of $\varvec{\vartheta }_0$, we have

$$\begin{aligned} E\sup _{\varvec{\vartheta }\in V(\varvec{\vartheta }_0)}\left\| \frac{\partial g_1(u,\varvec{\vartheta })}{\partial \varvec{\vartheta }}\right\| <\infty , \end{aligned}$$

(9.7)

where the norm of a complex vector denotes the sum of the norms of its real and imaginary parts. By Lebesgue’s dominated convergence theorem, we thus have

$$\begin{aligned} E\frac{\partial g_1(u,\varvec{\vartheta })}{\partial \varvec{\theta }}= & {} \frac{\partial Ee^{\mathrm {i}u\varepsilon _1(\varvec{\theta })}}{\partial \varvec{\theta }} = \frac{\partial }{\partial \varvec{\theta }}E\varphi \left( \frac{c_t(\varvec{\theta }_0)}{c_t(\varvec{\theta })}u,\varvec{\lambda }_0\right) \\= & {} -u E\varphi '\left( \frac{c_t(\varvec{\theta }_0)}{c_t(\varvec{\theta })}u\right) \frac{c_t(\varvec{\theta }_0)}{c_t(\varvec{\theta })}\frac{1}{c_t(\varvec{\theta })}\frac{\partial c_t}{\partial \varvec{\theta }}, \end{aligned}$$

where

$$\begin{aligned} \varphi '(u)=-\varphi (u,\varvec{\lambda }_0)\alpha _0|u|^{\alpha _0-1}{\mathrm{sgn}} \left( u\right) \left\{ 1-\mathrm {i}\beta _0 {\mathrm{sgn}} \left( u\right) \tan \left( \frac{\pi \alpha _0}{2}\right) \right\} . \end{aligned}$$

We then have

$$\begin{aligned} E\frac{\partial g_1(u,\varvec{\vartheta }_0)}{\partial \varvec{\theta }}=\varphi (u,\varvec{\lambda }_0)\varvec{\tau }_3(u), \end{aligned}$$

(9.8)

with

$$\begin{aligned} \varvec{\tau }_3(u)=\tau _3(u)E\frac{1}{c_t}\frac{\partial c_t}{\partial \varvec{\theta }}(\varvec{\theta }_0),\quad \tau _3(u)=\alpha _0|u|^{\alpha _0}\left\{ 1-\mathrm {i}\beta _0 {\mathrm{sgn}} \left( u\right) \tan \left( \frac{\pi \alpha _0}{2}\right) \right\} . \end{aligned}$$

In view of (9.5) and (9.8), each component of $\varvec{g}(u)$ is a bounded function of u (since $\varphi (u,\varvec{\lambda }_0)$ tends to zero at an exponential rate as $|u|\rightarrow \infty$). The existence of $\varvec{G}$ thus follows from A3(0). Since $|g_t(u,\varvec{\vartheta }_0)|\le 2$, the existence of $\varvec{V}$ also follows.

Let us show that $\varvec{G}$ is singular. This is equivalent to proving that there exists $\varvec{a}\ne \varvec{0}$ such that $\varvec{a}'\varvec{g}(u)=0$ for all u (see Theorem 2 in Bryant and Paulson 1979). Letting $\varvec{a}=(\varvec{a}_1',a_2,a_3)'$ with $\varvec{a}_1\in {\mathbb {R}}^{p+q+2}$, and using (9.5) and (9.8), we have

$$\begin{aligned} \varvec{a}'\varvec{g}(u)=\varphi (u,\varvec{\lambda }_0)\left\{ \varvec{a}_1'E\frac{1}{c_t}\frac{\partial c_t}{\partial \varvec{\theta }}(\varvec{\theta }_0)\tau _3(u)+a_2\tau _1(u)+a_3\tau _2(u)\right\} . \end{aligned}$$

Since $|\varphi (u,\varvec{\lambda }_0)|>0$ and since the functions $\tau _1(u)$, $\tau _2(u)$ and $\tau _3(u)$ are linearly independent, $\varvec{a}'\varvec{g}(u)=0$ for all u if and only if

$$\begin{aligned} a_2=a_3=0\quad \text{ and }\quad \varvec{a}_1'E\frac{1}{c_t}\frac{\partial c_t}{\partial \varvec{\theta }}(\varvec{\theta }_0)=0. \end{aligned}$$

This can obviously be achieved by choosing $\varvec{a}_1\ne \varvec{0}$. Note that the rank of $\varvec{G}$ is equal to 3. The singularity of $\varvec{V}$ is shown similarly. $\square$

Proof of Theorem 3.3

Assumption A6 and the consistency of $\widehat{\varvec{\vartheta }}_T$ entail that ${\partial \widetilde{\Delta }_T(\widehat{\varvec{\vartheta }}_T)}/{\partial \varvec{\vartheta }}=0$, at least for T large enough. In view of Lemma 8.5, (8.6) and Lemma 8.7, we thus have, a.s.

$$\begin{aligned} o(1)&=\sqrt{T}\frac{\partial \Delta _T(\widehat{\varvec{\vartheta }}_T)}{\partial \varvec{\vartheta }}\nonumber \\&=2 \int _{-\infty }^{+\infty }\frac{1}{\sqrt{T}}\sum _{t=1}^T\text{ Re }\left( \frac{1}{T}\sum _{s=1}^T\frac{\partial g_s(u,\widehat{\varvec{\vartheta }}_T)}{\partial \varvec{\vartheta }}\overline{g_t(u,\widehat{\varvec{\vartheta }}_T)}\right) W(u)du \nonumber \\&=2 \int _{-\infty }^{+\infty }\text{ Re }\left[ \left\{ \varvec{g}(u)+\varvec{R}_T(u)\right\} \frac{1}{\sqrt{T}}\sum _{t=1}^T\overline{g_t(u,\widehat{\varvec{\vartheta }}_T)}\right] W(u)du. \end{aligned}$$

(9.9)

A Taylor expansion shows that

$$\begin{aligned} \frac{1}{\sqrt{T}}\sum _{t=1}^T\text{ Re } g_t(u,\widehat{\varvec{\vartheta }}_T)= \frac{1}{\sqrt{T}}\sum _{t=1}^T\text{ Re } g_t(u,\varvec{\vartheta }_0)+\frac{1}{T}\sum _{t=1}^T\frac{\partial \text{ Re } g_t(u,\varvec{\vartheta }^*_T)}{\partial \varvec{\vartheta }'}\sqrt{T}(\widehat{\varvec{\vartheta }}_T-\varvec{\vartheta }_0) \end{aligned}$$

for some $\varvec{\vartheta }^*_T$ between $\widehat{\varvec{\vartheta }}_T$ and $\varvec{\vartheta }_0$. A similar expansion holds for the imaginary part. By Lemma 8.7, we thus have

$$\begin{aligned} \frac{1}{\sqrt{T}}\sum _{t=1}^T \overline{g_t(u,\widehat{\varvec{\vartheta }}_T)}= \frac{1}{\sqrt{T}}\sum _{t=1}^T \overline{g_t(u,\varvec{\vartheta }_0)}+\left\{ \overline{\varvec{g}}(u)+\overline{\varvec{R}_T(u)}\right\} '\sqrt{T}(\widehat{\varvec{\vartheta }}_T-\varvec{\vartheta }_0). \end{aligned}$$

(9.10)

Now, note that, by the central limit theorem

$$\begin{aligned} \int _{-\infty }^{+\infty }\text{ Re } \frac{1}{\sqrt{T}}\sum _{t=1}^T\overline{g_t(u,\varvec{\vartheta }_0)}W(u)du=O_P(1) \end{aligned}$$

and by A3(0) and A3(4)

$$\begin{aligned} \int _{-\infty }^{+\infty }\left\| \varvec{R}_T(u)\right\| ^2W(u)du=o_P(1). \end{aligned}$$

Theorem 4.1 in Billingsley (1968) thus entails

$$\begin{aligned} \int _{-\infty }^{+\infty }\text{ Re }\varvec{R}_T(u) \frac{1}{\sqrt{T}}\sum _{t=1}^T\overline{g_t(u, \varvec{\vartheta }_0)}W(u)d u=o_P(1). \end{aligned}$$

(9.11)

In view of (9.10) and (9.11), (9.9) implies

$$\begin{aligned} o_P(1)= \frac{1}{\sqrt{T}}\sum _{t=1}^T\varvec{\Upsilon }_t+ \varvec{G}_T\sqrt{T}(\widehat{\varvec{\vartheta }}_T-\varvec{\vartheta }_0), \end{aligned}$$

(9.12)

where

$$\begin{aligned} \varvec{G}_T=\int _{-\infty }^{+\infty }\text{ Re }\left[ \left\{ \varvec{g}(u)+\varvec{R}_T(u)\right\} \left\{ \overline{\varvec{g}(u)}+\overline{\varvec{R}_T(u)}\right\} '\right] W(u)du. \end{aligned}$$

Note that

$$\begin{aligned} \left\| \varvec{G}_T-\varvec{G}\right\| \le Kr_T \int _{-\infty }^{+\infty }(1+|u|+u^2)W(u)du+Kr^2_T \int _{-\infty }^{+\infty }(1+|u|+u^2)^2W(u)du, \end{aligned}$$

which tends almost surely to 0 as $T\rightarrow \infty$, under A3(0) and A3(4). The conclusion follows from the central limit theorem. $\square$

Proof of Theorem 3.4

In view of (9.5) and (9.8), we have

$$\begin{aligned} \varvec{g}(u)=\varvec{A}\varvec{\tau }(u),\qquad \varvec{\tau }(u)=\varphi (u,\varvec{\lambda }_0)\left( \begin{array}{c}\tau _3(u)\\ \tau _1(u)\\ \tau _2(u)\end{array}\right) . \end{aligned}$$

(9.13)

Since the functions $\tau _i(u)$, $i=1,2,3$, are linearly independent, the matrix

$$\begin{aligned} \varvec{B}:=\text{ Re }\int \overline{\varvec{\tau }(u)}\varvec{\tau }(u)'W(u)du \end{aligned}$$

(9.14)

is invertible (see Theorem 2 in Bryant and Paulson 1979), and we have

$$\begin{aligned} \varvec{G}=\text{ Re }\int \overline{\varvec{g}}(u)\varvec{g}'(u)W(u)du=\varvec{A}\varvec{B}\varvec{A}'. \end{aligned}$$

Letting $\varvec{A}_T'=\varvec{B}^{-1}(\varvec{A}'\varvec{A})^{-1}\varvec{A}'\varvec{G}_T$, we thus have $\varvec{A}_T\rightarrow \varvec{A}$ a.s. as $T\rightarrow \infty$. By Theorem 3.3 we then obtain $\sqrt{T}\varvec{A}_T'(\widehat{\varvec{\vartheta }}_T-\varvec{\vartheta }_0)\mathop {\rightarrow }\limits ^{d}\mathcal{N}\left( \varvec{0},\varvec{\Sigma }\right)$ with

$$\begin{aligned} \varvec{\Sigma }=\varvec{B}^{-1}(\varvec{A}'\varvec{A})^{-1}\varvec{A}'\varvec{V}\varvec{A}(\varvec{A}'\varvec{A})^{-1}\varvec{B}^{-1}. \end{aligned}$$

Since

$$\begin{aligned} \varvec{\Upsilon }_t=\varvec{A}\int _{-\infty }^{\infty }\text{ Re }\left\{ e^{\mathrm {i}u\epsilon _t(\varvec{\theta }_0)}\overline{\varvec{\tau }(u)}\right\} W(u)du- \varvec{A}\int _{-\infty }^{\infty }\text{ Re }\left\{ \varphi (u,\varvec{\lambda }_0)\overline{\varvec{\tau }(u)}\right\} W(u)du, \end{aligned}$$

a computation similar to that of (17) in Thornton and Paulson (1977) gives $\varvec{V}=\varvec{A}\varvec{U}\varvec{A}'$ with

$$\begin{aligned} \varvec{U}=&\frac{1}{2}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty } \text{ Re }\left\{ \varphi (u+v,\varvec{\lambda }_0)\overline{\varvec{\tau }(u)}\, \overline{\varvec{\tau }(v)}'\right\} W(u)W(v)dudv\nonumber \\&+\frac{1}{2}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty } \text{ Re }\left\{ \varphi (u-v,\varvec{\lambda }_0)\overline{\varvec{\tau }(u)}\, \varvec{\tau }(v)'\right\} W(u)W(v)dudv\nonumber \\&-\int _{-\infty }^{\infty }\text{ Re }\left\{ \varphi (u, \varvec{\lambda }_0)\overline{\varvec{\tau }(u)}\right\} W(u)du \left( \int _{-\infty }^{\infty }\text{ Re }\left\{ \varphi (v, \varvec{\lambda }_0)\overline{\varvec{\tau }(v)}\right\} W(v)dv\right) '.\nonumber \\ \end{aligned}$$

(9.15)

The conclusion follows. $\square$

Proof of Theorem 3.5

From (9.9), Lemmas 8.7–8.8 and already employed arguments, we have

$$\begin{aligned} o_P(1)&= \frac{1}{\sqrt{T}}\sum _{t=1}^T\varvec{\Upsilon }_t+ \varvec{G}_T\sqrt{T}(\widehat{\varvec{\vartheta }}_T-\varvec{\vartheta }_0)\nonumber \\&\quad + \int \varvec{g}(u)T^{1/4}(\widehat{\varvec{\vartheta }}_T-\varvec{\vartheta }_0)'\left\{ \varvec{H}(u)+\varvec{R}^{(2)}_T(u)\right\} T^{1/4}(\widehat{\varvec{\vartheta }}_T-\varvec{\vartheta }_0)W(u)du. \end{aligned}$$

(9.16)

In view of (9.12), we thus have

$$\begin{aligned} \int \varvec{g}(u)T^{1/4}(\widehat{\varvec{\vartheta }}_T-\varvec{\vartheta }_0)'\left\{ \varvec{H}(u)+\varvec{R}^{(2)}_T(u)\right\} T^{1/4}(\widehat{\varvec{\vartheta }}_T-\varvec{\vartheta }_0)W(u)du=o_P(1). \end{aligned}$$

(9.17)

Note that ${\partial ^2\overline{g_t(u,\varvec{\vartheta })}}/{\partial \varvec{\vartheta }\partial \varvec{\vartheta }'}$ is block-diagonal. Therefore $\varvec{H}(u)$ and $\varvec{R}^{(2)}_T(u)$ are also block-diagonal matrices. The block-diagonal terms of $\varvec{H}(u)$ are

$$\begin{aligned} \varvec{H}_{\varvec{\theta }_0}(u)=\frac{1}{2}E\frac{\partial ^2 e^{-\mathrm {i}u\varepsilon _1(\varvec{\theta }_0)}}{\partial \varvec{\theta }\partial \varvec{\theta }'} \quad \text{ and } \quad \varvec{H}_{\varvec{\lambda }_0}(u)=-\frac{1}{2}\frac{\partial ^2 \overline{\varphi (u, \varvec{\lambda }_0)}}{\partial \varvec{\lambda }\partial \varvec{\lambda }'}. \end{aligned}$$

Those of $\varvec{R}^{(2)}_T(u)$ are denoted by $\varvec{R}^{(2)}_{\varvec{\theta }_0,T}(u)$ and $\varvec{R}^{(2)}_{\varvec{\lambda }_0,T}(u)$. Since, under (3.3), Theorem 3.4 entails that $\sqrt{T}(\widehat{\varvec{\lambda }}_T-\varvec{\lambda }_0)=O_P(1)$, we have

$$\begin{aligned}&T^{1/4}(\widehat{\varvec{\vartheta }}_T-\varvec{\vartheta }_0)'\left\{ \varvec{H}(u)+\varvec{R}^{(2)}_T(u)\right\} T^{1/4}(\widehat{\varvec{\vartheta }}_T-\varvec{\vartheta }_0)\end{aligned}$$

(9.18)

$$\begin{aligned}&\quad =T^{1/4}(\widehat{\varvec{\theta }}_T-\varvec{\theta }_0)'\left\{ \varvec{H}_{\varvec{\theta }_0}(u)+\varvec{R}^{(2)}_{\varvec{\theta }_0,T}(u)\right\} T^{1/4}(\widehat{\varvec{\theta }}_T-\varvec{\theta }_0)+o_P(1). \end{aligned}$$

(9.19)

Noting also that $\varvec{g}(u)=\varvec{A}\varvec{\tau }(u)$ where $\varvec{A}$ has full rank, it follows from (9.17) that

$$\begin{aligned} \int \varvec{\tau }(u)T^{1/4}(\widehat{\varvec{\theta }}_T-\varvec{\theta }_0)'\left\{ \varvec{H}_{\varvec{\theta }_0}(u)+\varvec{R}^{(2)}_{\varvec{\theta }_0,T}(u)\right\} T^{1/4}(\widehat{\varvec{\theta }}_T-\varvec{\theta }_0)W(u)du= o_P(1). \end{aligned}$$

(9.20)

Under A3(0) and A3(4), Lemma 8.8 entails

$$\begin{aligned} \int T^{1/4}(\widehat{\varvec{\theta }}_T-\varvec{\theta }_0)'\varvec{R}^{(2)}_{\varvec{\theta }_0,T}(u) T^{1/4}(\widehat{\varvec{\theta }}_T-\varvec{\theta }_0)W(u)du= o_P\left( T^{1/2}\left\| \widehat{\varvec{\theta }}_T-\varvec{\theta }_0)\right\| ^2\right) . \end{aligned}$$

(9.21)

Similarly to (9.8), we obtain

$$\begin{aligned} \varvec{H}_{\varvec{\theta }_0}(u)=\tau _4(u)\varvec{H}^{(1)}_{\varvec{\theta }_0}+\tau _5(u)\varvec{H}^{(2)}_{\varvec{\theta }_0}, \end{aligned}$$

with

$$\begin{aligned} \varvec{H}^{(1)}_{\varvec{\theta }_0}=E\frac{1}{c_t^2(\varvec{\theta }_0)}\frac{\partial c_t(\varvec{\theta }_0)}{\partial \varvec{\theta }} \frac{\partial c_t(\varvec{\theta }_0)}{\partial \varvec{\theta }'},\qquad \varvec{H}^{(2)}_{\varvec{\theta }_0}=E\frac{1}{c_t(\varvec{\theta }_0)}\frac{\partial ^2 c_t(\varvec{\theta }_0)}{\partial \varvec{\theta }\partial \varvec{\theta }'} \end{aligned}$$

and

$$\begin{aligned} \tau _4(u)=-\frac{u^2}{2}E\epsilon _t^2(\varvec{\theta }_0)e^{-\mathrm {i}u\epsilon _t(\varvec{\theta }_0)}-2\tau _5(u),\quad \tau _5(u)=\frac{\mathrm {i}u}{2} E\epsilon _t(\varvec{\theta }_0)e^{-\mathrm {i}u\epsilon _t(\varvec{\theta }_0)}. \end{aligned}$$

In view of (9.20) and (9.21), we then have

$$\begin{aligned} \sum _{i=1}^2T^{1/4}(\widehat{\varvec{\theta }}_T-\varvec{\theta }_0)'\varvec{H}^{(i)}_{\varvec{\theta }_0}T^{1/4}(\widehat{\varvec{\theta }}_T-\varvec{\theta }_0)\int \varvec{\tau }(u)\tau _{3+i}(u)W(u)du= r_T. \end{aligned}$$

(9.22)

where $r_T=o_P(1)+o_P\left( T^{1/2}\left\| \widehat{\varvec{\theta }}_T-\varvec{\theta }_0)\right\| ^2\right)$ as $T\rightarrow \infty$. In view of (3.6) this entails that

$$\begin{aligned} T^{1/4}(\widehat{\varvec{\theta }}_T-\varvec{\theta }_0)'\varvec{H}^{(i)}_{\varvec{\theta }_0}T^{1/4}(\widehat{\varvec{\theta }}_T-\varvec{\theta }_0)\int \varvec{\tau }(u)\tau _{3+i}(u)W(u)du= r_T. \end{aligned}$$

(9.23)

for $i=1$ and $i=2$. Since $\varvec{H}^{(1)}_{\varvec{\theta }_0}$ is positive definite (see (ii) in the proof of Theorem 2.2 in HZ), the proof is complete. $\square$