Asymptotic normality of the MLE in the level-effect ARCH model

Abstract

We establish consistency and asymptotic normality of the maximum likelihood estimator in the level-effect ARCH model of Chan et al. (J Financ 47(3):1209–1227, 1992). Furthermore, it is shown by simulations that the asymptotic properties also apply in finite samples.

1 Introduction

After Engle (1982) initiated the literature on autoregressive conditional heteroskedasticity (ARCH) and the model proved itself to be very useful in empirical applications, an immense amount of research has been directed towards extending Engle’s original ideas empirically as well as theoretically (see for example Dias-Curto et al. (2009) as an example of the usefulness of GARCH models in practice). Chan, Karolyi, Longstaff and Sanders (1992, CKLS from now onwards) proposed to introduce the lagged level of the spot interest rate in the conditional variance equation, generating the so called level-effect ARCH model. This model has subsequently been successfully used and extended by Brenner et al. (1996), Andersen and Lund (1997), Ball and Torous (1999) among others. Recently, Maheu and Yang (2016) have estimated with Bayesian methods and using financial data both CKLS model and also the CKLS model with ARCH disturbances; while Bu et al. (2017) have estimated the CKLS in two different regimes also using financial data. However, despite of its empirical success, the asymptotic behavior of the quasi-maximum likelihood (QML) estimator associated with the level-effect ARCH model has, to the best of our knowledge, not been formally established yet. Most papers on conditional heteroskedastic time series, see, e.g., Berkes and Horváth (2004), Straumann and Mikosch (2006), Hamadeh and Zakoïan (2011) and Francq et al. (2018), do not allow for the introduction of the level of a series, such as the interest rate, in the conditional variance equation. The double autoregressive model of Ling (2004) is an exception. Triffi (2006) illustrates the convergence results for the Constant Elasticity of Variance (CEV)-ARCH model of Fornari and Mele (2006), but from the best of our knowledge, the asymptotic normality and consistency of the QML/ML estimator for the level-ARCH model is still unknown. In the following sections we will present the model and provide a simple proof of asymptotic normality and consistency of the ML estimator within the traditional level-effect ARCH(1) setting. The simulation section confirms our theoretical results and finally, we conclude.

2 The level-effect ARCH model

Consider the discrete-time approximation of the CKLS model

$$\begin{aligned} y_{t}^{*}= & {} \Delta y_{t}-\left( a+by_{t-1}\right) =\sigma _{t}\left| y_{t-1}\right| ^{\gamma }z_{t}, \end{aligned}$$

(1)

$$\begin{aligned} \sigma _{t}^{2}= & {} w+\alpha \left( \frac{y_{t-1}^{*}}{\left| y_{t-2}\right| ^{\gamma }}\right) ^{2}, \end{aligned}$$

(2)

for $t=1,\ldots ,T$ that we denote the level-effect ARCH.¹ Let us denote the parameter vector of interest by $\theta =\left( \gamma ,w,\alpha \right) {\acute{}}$ and let the true parameter values be given by $\theta _{0}=\left( \gamma _{0},w_{0},\alpha _{0}\right) ^{\prime }$. Further, in many applications $y_{t}^{*}$ is chosen as a transformation of current and lagged values of $y_{t},$ such as $y_{t}^{*}=y_{t}^{*}\left( y_{t},y_{t-1},y_{t-2},\ldots ;\delta \right)$ where $\delta$ is a vector of parameters. One such specification could be $y_{t}^{*}=\Delta y_{t}-\left( a+by_{t-1}\right) ,$ where $\delta =\left( a,b\right) ^{\prime },$ such as the one considered in Andersen and Lund (1997, p. 354) and in Broze et al (1995, Eq. 2). However, also note that model (1), (2) is not exactly that of Andersen and Lund (1997), as they use $\Delta y_{t}=\left( a+by_{t-1}\right) +\sigma _{t}y_{t-1}^{\gamma }z_{t}$ instead of $\Delta y_{t}=\left( a+by_{t-1}\right) +\sigma _{t}\left| y_{t-1}\right| ^{\gamma }z_{t}.$ The absolute value is needed, as otherwise the discrete-time model does not ensure that $y_{t}$ is non-negative. In practice, $\delta$ can be pre-estimated in a first stage. This pre-estimation approach is very common in empirical research, particularly, when modelling spot interest rates, see, for example, Ball and Torous (1999, p. 2349). In order to avoid additional complexity of the proofs, we assume throughout this paper that $\delta$ is known, see also Remark 2.² It should be noted that (1)–(2) is a generalization of Frydman (1994), who consider a discrete-time process, but where $\alpha _{0}=0.$ Broze, Scaillet and Zakoïan (1995, Eq. 2 in p. 202) have also analyzed the regular level effect model but without the ARCH component, but where they introduce also $\left| y_{t-1}\right| ^{\gamma }z_{t}$ in (1). Moreover, for the case where $\gamma _{0}$ is known, (1), (2) is the standard linear model for which a complete characterization of the estimation theory has been developed by Jensen and Rahbek (2004a, b) and in Kristensen and Rahbek (2005, 2008). The “quasi”-log likelihood function (conditional on past values of $y_{t}$) associated with (1), (2) is given as

$$\begin{aligned} L_{T}\left( \theta \right)= & {} \sum _{t=1}^{T}l_{t}\left( \theta \right) =-\frac{ 1}{2}\sum _{t=1}^{T}\ln \left[ y_{t-1}^{2\gamma }\left( w+\alpha \left( \frac{ y_{t-1}^{*}}{\left| y_{t-2}\right| ^{\gamma }}\right) ^{2}\right) \right] \nonumber \\&-\frac{1}{2}\sum _{t=1}^{T}\frac{\left( y_{t}^{*}\right) ^{2}}{y_{t-1}^{2\gamma }\left( w+\alpha \left( \frac{y_{t-1}^{*} }{\left| y_{t-2}\right| ^{\gamma }}\right) ^{2}\right) }. \end{aligned}$$

(3)

We proceed under the following set of maintained assumptions:

Assumption A

A1 :

$z_{t}\sim N.i.i.d.\left( 0,1\right) ,$

A2 :

$\infty>w_{0}>0,$ $\infty>\alpha _{0}>0,$

A3 :

$E\left[ \left( \frac{y_{t-1}^{*}}{\left| y_{t-2}\right| ^{i}}\right) ^{\varphi }\left| \ln \left( \left| y_{t-1}\right| \right) \right| ^{3}\right] <\infty ,$ $E\left[ \left( \frac{y_{t-1}^{*}}{\left| y_{t-2}\right| ^{i}}\right) ^{\varphi }\left( \ln \left( \left| y_{t-1}\right| \right) \right) ^{2}\left| \ln \left( \left| y_{t-2}\right| \right) \right| \right] <\infty ,$ $E\left[ \left( \frac{y_{t-1}^{*}}{\left| y_{t-2}\right| ^{i}} \right) ^{\varphi }\left( \ln \left( \left| y_{t-2}\right| \right) \right) ^{2}\left| \ln \left( \left| y_{t-1}\right| \right) \right| \right] <\infty ,$ for both $\varphi =0$ and for some $\varphi >0$ for $i=0,1.$

Assumption B

B1 :

$E\ln \left( \alpha _{0}z_{t}^{2}\right) <0$,

B2 :

$1\ge \gamma _{0}\ge 0,$ $\left| b+1\right| <1,$ $E\ln \left| b+1+\alpha _{0}z_{t}^{2}\right| <0.$

Assumptions A1, A2 and B1 are very common in the traditional ARCH literature (see e.g. Jensen and Rahbek (2004a, b), p. 1205). In A1 we need to impose Gaussianity as we make use in our proofs of the results in Broze et al. (1995). Also, A1 implies that $E\left( \left( 1-z_{t}^{2}\right) ^{2}\right) =\zeta =2.$ Note also, A3 -an assumption very specific for the level-effect ARCH model- explains the necessity to introduce $\left| y_{t-1}\right| ^{\gamma }$ in (1) as in Broze et al (1995, Eq. 2).³ In B2, we require $1\ge \gamma _{0}\ge 0$-see Broze et al (1995, Proposition 3) for more details-. The condition for stationarity of $\sigma _{t}^{2},$ $\left( y_{t}^{*}/y_{t-1}^{\gamma }\right) ,$ $y_{t}^{*}$ and $y_{t}$ is given by the following two Lemmas:

Lemma 1

Let Assumption A hold. A necessary and sufficient condition for strict stationarity of $\sigma _{t}^{2}$ and $\left( y_{t}^{*}/\left| y_{t-1}\right| ^{\gamma _{0}}\right)$ as generated by (1), (2) is given by

$$\begin{aligned} E\ln \left( \alpha _{0}z_{t}^{2}\right) <0. \end{aligned}$$

Proof of Lemma 1

Lemma 1 is not a new result in the literature since if $\left( y_{t}^{*}/\left| y_{t-1}\right| ^{\gamma _{0}}\right)$ is known, the level-effect ARCH model reduces to the ARCH(1) and the condition for strict stationarity is well known (see e.g. Jensen and Rahbek 2004a). $\square$

Lemma 2

Let Assumption A and B1 hold. A necessary and sufficient condition for ergodicity and second order stationarity of $y_{t}^{*}$ and $y_{t}$ as generated by (1), (2) is given by

$$\begin{aligned} 1>\gamma _{0}\ge 0,\left| b+1\right|<1,E\ln \left| b+1+\alpha _{0}z_{t}^{2}\right| <0. \end{aligned}$$

When $\gamma _{0}=1,$ the previous condition is a sufficient condition for second order stationarity and ergodicity.

Proof of Lemma 2

Given in the proof of Proposition 3 of Broze et al (1995, Appendix D) but replacing $\sigma _{0,h}$ in Broze et al. (1995) by the required strict stationarity condition of $\sigma _{t}^{2}$ that was proved in Lemma 1, and where the Gaussianity of $z_{t}$ is used in the definition of the transition density.

Next, the main result of the paper regarding the limiting distribution of the ML estimator in the level-effects ARCH model can be established. $\square$

Theorem 1

Define $u_{1t}\left( \theta _{0}\right) =\left( \ln \left| y_{t-1}\right| -w_{0}\left( \frac{1}{w_{0}}-\frac{1}{\sigma _{t}^{2}\left( \theta _{0}\right) }\right) \ln \left| y_{t-2}\right| \right)$, $u_{2t}\left( \theta _{0}\right) =\left( \frac{1}{\sigma _{t}^{2}\left( \theta _{0}\right) }\right)$ and $u_{3t}\left( \theta _{0}\right) =\left( \frac{w_{0}}{\alpha _{0}}\right) \left( \frac{1}{w_{0}}- \frac{1}{\sigma _{t}^{2}\left( \theta _{0}\right) }\right)$, let Assumptions A and B hold, $\theta _{0}$ is assumed to be an interior point of the parameter space and assume that $\delta =\left( a,b\right) ^{\prime }$ is known. Consider the log likelihood function given by (3). Then, there exists a fixed open neighborhood $U=U\left( \theta _{0}\right)$ of $\theta _{0}$ such that with probability tending to one as $T\longrightarrow \infty ,$ $L_{T}\left( \theta \right)$ has a unique maximum point ${\widehat{\theta }}$ in U. In addition, the ML estimator ${\widehat{\theta }}$ is consistent and asymptotically normal

$$\begin{aligned} \sqrt{T}\left[ {\widehat{\theta }}-\theta _{0}\right] {\acute{}} \overset{d}{\longrightarrow }N\left( 0,\left( \zeta ^{2}/4\right) \Lambda ^{-1}\right) , \end{aligned}$$

where

$$\begin{aligned} \Lambda =\zeta \left[ \begin{array}{ccc} {\overline{m}}_{11} &{} \frac{1}{2}{\overline{m}}_{12} &{} \frac{1}{2}{\overline{m}} _{13} \\ \frac{1}{2}{\overline{m}}_{12} &{} \frac{1}{4}{\overline{m}}_{22} &{} \frac{1}{4} {\overline{m}}_{23} \\ \frac{1}{2}{\overline{m}}_{13} &{} \frac{1}{4}{\overline{m}}_{23} &{} \frac{1}{4} {\overline{m}}_{33} \end{array} \right] >0, \end{aligned}$$

$\zeta =2$ and ${\overline{m}}_{ij}=E\left( u_{it}\left( \theta _{0}\right) u_{jt}\left( \theta _{0}\right) \right)$ for $i=1,2,3$ and $j=1,2,3$.⁴

Proof of Theorem 1

The proof of Theorem 1 is given in the Appendix.

Importantly, Theorem 1 applies to the MLE of the stationary level-effect ARCH(1) process. However, if $\gamma _{0}$ is known under Assumption A but $E\ln \left( \alpha _{0}z_{t}^{2}\right) >0$ then the asymptotics for the ML estimator of $\alpha$ can still be established. To see this, simply notice that the model (1), (2) in this case can be rewritten as

$$\begin{aligned} {\widetilde{y}}_{t}= & {} \sigma _{t}z_{t} \\ \sigma _{t}^{2}= & {} \omega +\alpha {\widetilde{y}}_{t-1}^{2} \end{aligned}$$

for ${\widetilde{y}}_{t}\equiv \frac{y_{t}^{*}}{\left| y_{t-1}\right| ^{\gamma _{0}}}.$ This representation of the process ${\widetilde{y}}_{t}$ is exactly identical to the model given by equation (1) in Jensen and Rahbek (2004a). Consequently, when $E\ln \left( \alpha _{0}z_{t-1}^{2}\right) >0$ (Assumption B fails) then ${\widetilde{y}}_{t}^{2} \overset{a.s.}{\longrightarrow }\infty$ from Lemma 1 as $\sigma _{t}^{2} \overset{a.s.}{\longrightarrow }\infty$ (see also Klüppelberg et al. (2004)) and the asymptotic results follows directly from Jensen and Rahbek (2004a, Lemmas 1–5).⁵ The case of $E\ln \left( \alpha _{0}z_{t-1}^{2}\right) =0$ implies, as shown in Klüppelberg et al. (2004) for an ARCH(1) (under suitable conditions), that $\sigma _{t}^{2}\overset{p}{\longrightarrow }\infty$ and therefore different arguments are required in this case (see Pedersen and Rahbek (2016)). Three remarks should be added: $\square$

Remark 1

It is well known, that the stationary level-effect ARCH model, can be estimated by non-parametric techniques, since the variance function is smooth and only depends on $y_{t-1}.$ In the nonstationary case, Han and Zhang (2009) consider ARCH models by applying the results of Wang and Phillips (2009a, b), although they do not allow for a level-effect.

Remark 2

In model (1), (2), we are assuming that $\delta =\left( a,b\right) ^{\prime }$ is known. In practice, $\delta$ can be pre-estimated in a first stage. If $\delta$ and $\theta$ are estimated jointly in mean and variance equation (contrary to Ball and Torous (1999)), then our proof will need to be extended to account for the joint estimation. At this stage we do not know if this would require stronger assumptions than those in Assumptions A and B.

Remark 3

The generalization of the asymptotic results when going from ARCH(1) to GARCH(1,1) can most likely be provided in a similar fashion as the extension from Jensen and Rahbek (2004a) to Jensen and Rahbek (2004b), with the added complexity in the proofs.

3 Simulations

In this section we evaluate and discuss, based on simulations, how well the asymptotic results of ${\widehat{\theta }}$ given by Theorem 1 can approximate the finite sample properties. We also report simulation results to check the consequences when some of the assumptions are violated. We set $\delta =\left( a,b\right) ^{\prime }=\left( 0,-1\right) ^{\prime }$ in all simulations and we do not estimate it.

In Panel A of Table 1, results are reported on point estimates, their associated biases and the root mean squared errors (RMSE’s) when Assumption B1 holds, i.e., the volatility process is stationary. In Panel A, three alternative data generating processes are considered: The first is characterized by having standard Gaussian distributed innovations whereas the two remaining processes have fatter tails as the innovations are standard t-distributed with 5 degrees of freedom. The results show that under all three data generating processes, the MLE is relatively accurate with small biases and small RMSE’s even at small sample sizes, i.e., $T=1000$ . As expected from Theorem 1, the biases and the RMSE’s are decreasing for all the estimators as sample sizes increase. Note that in Theorem 1, we show the asymptotic theory for the MLE, however in this simulation section we also show what happens under alternative distribution functions for the innovations.

In Panel B of Table 1, two data generating processes, both in violation with Assumption B1, are considered. As noted in Jensen and Rahbek (2004a, b) $\omega$ is not identified in this case hence is fixed at its true value in the population. The results of Panel B, Table 1, clearly illustrates that the consistency properties of the MLE of $\alpha$ still holds when $\sigma _{t}^{2}$ and $\left( y_{t}^{*}/y_{t-1}^{\gamma _{0}}\right)$ are nonstationary. The main theoretical results of this paper are silent about the asymptotic properties of $\gamma$ when B1 is violated. This is due to the complexity of the expressions of the first, second and third order derivatives of the loglikelihood with respect to $\gamma$, and we can only analyze the case when B1 holds. However the simulations seem to indicate strongly that consistency of the MLE of $\gamma$ is maintained also when Assumption B1 does not hold.

Table 1 Simulation results on the QMLE, i.e., ${\widehat{\theta }} = \left( {\widehat{\gamma }},{\widehat{\omega }},{\widehat{\alpha }} \right)$

In Table 2, consistency of the estimated standard errors of the MLE in finite samples is illustrated. The term SE $\left( {\widehat{\gamma }}\right)$ denotes the “feasible” standard error of ${\widehat{\gamma }}$ computed according to the expression for the asymptotic variance-covariance function derived in Theorem 1, but where all the population parameters are replaced by sample estimates (sample analogy estimation). ASE $\left( {\widehat{\gamma }}\right)$ denotes the asymptotic standard error of ${\widehat{\gamma }}$. It is defined similar to SE $\left( {\widehat{\gamma }}\right)$ but computed based on the true populations parameters.

We see that for all the models presented in Table 2, there is a very close correspondence between SE $\left( {\widehat{\gamma }}\right)$ and ASE $\left( {\widehat{\gamma }}\right)$. These results are very encouraging, implying that the asymptotic variance covariance matrix provides a good approximation of the parameter estimation uncertainty also in finite sample. It is also noticeable, that in the cases of relative fat tailed t-distributed innovations (when using the misspecified Gaussian quasi-likelihood) the standard errors of the MLE increase as expected, relative to the cases where innovations are normally distributed. But the observed increase in parameter estimation uncertainty seems to be only of limited magnitude. Similar results hold for SE $\left( \widehat{\omega }\right)$, SE $\left( {\widehat{\alpha }}\right)$, ASE $\left( {\widehat{\omega }}\right)$ and ASE $\left( {\widehat{\alpha }}\right)$.⁶

Table 2 Simulation results on the standard errors of the QMLE

4 Conclusion

In this paper we establish consistency and asymptotic normality of the ML estimator in the level-effect ARCH model. We also show in simulations that the asymptotic theory provides a good approximation in finite samples.

Appendix

The analytical expressions for the first, second and third order derivatives of the quasi log likelihood function are given in a Supplementary Appendix available upon request from any of the authors. We provide now three important propositions that we need in order to prove Theorem 1. The proof technique for the MLE utilizes the classic Cramér type conditions for consistency and asymptotic normality (central limit theorem for the score, convergence of the Hessian and uniformly bounded third-order derivatives); see e.g. Lehmann (1999).

Proposition 1

Let $u_{jt}\left( \theta _{0}\right)$ be defined as in Theorem 1. Under Assumptions A and B, the joint distribution of the score functions evaluated at $\theta =\theta _{0}$ are asymptotically Gaussian,

$$\begin{aligned} \frac{1}{\sqrt{T}}\frac{\partial }{\partial \theta }L_{T}\left( \theta _{0}\right) \overset{d}{\rightarrow }N\left( 0,\Lambda \right) , \end{aligned}$$

where

$$\begin{aligned} \Lambda =\zeta \left[ \begin{array}{ccc} {\overline{m}}_{11} &{} \frac{1}{2}{\overline{m}}_{12} &{} \frac{1}{2}{\overline{m}} _{13} \\ \frac{1}{2}{\overline{m}}_{12} &{} \frac{1}{4}{\overline{m}}_{22} &{} \frac{1}{4} {\overline{m}}_{23} \\ \frac{1}{2}{\overline{m}}_{13} &{} \frac{1}{4}{\overline{m}}_{23} &{} \frac{1}{4} {\overline{m}}_{33} \end{array} \right] >0, \end{aligned}$$

and ${\overline{m}}_{ij}=E\left( u_{it}\left( \theta _{0}\right) u_{jt}\left( \theta _{0}\right) \right)$ for $i=1,2,3$ and $j=1,2,3$.

Proof of Proposition 1

For the proof of Proposition 1, we need first the following 2 Lemmas. $\square$

Lemma A

Let Assumptions A and B hold and define $u_{1t}\left( \theta _{0}\right) =\left( \ln \left| y_{t-1}\right| -\right.$ $\left. w_{0}\left( \frac{1}{w_{0}}-\frac{ 1}{\sigma _{t}^{2}\left( \theta _{0}\right) }\right) \ln \left| y_{t-2}\right| \right) ,$

$u_{2t}\left( \theta _{0}\right) =\left( \frac{1}{\sigma _{t}^{2}\left( \theta _{0}\right) }\right)$ and $u_{3t}\left( \theta _{0}\right) =\left( \frac{w_{0}}{\alpha _{0}}\right) \left( \frac{1}{w_{0}}-\frac{1}{\sigma _{t}^{2}\left( \theta _{0}\right) }\right)$. Then $u_{it}\left( \theta _{0}\right)$ is a stationary and ergodic sequence. In addition $\frac{1}{T} \sum _{t=1}^{T}u_{it}\left( \theta _{0}\right) \overset{p}{\rightarrow } E\left( u_{it}\left( \theta _{0}\right) \right) \equiv {\overline{u}}_{i}$ and $\frac{1}{T}\sum _{t=1}^{T}u_{it}^{2}\left( \theta _{0}\right) \overset{p}{ \rightarrow }E\left( u_{it}^{2}\left( \theta _{0}\right) \right) \equiv {\overline{m}}_{ii}$ for $i=1,2,3.$

Proof of Lemma A

Define $I_{t}=\{y_{t},z_{t,}y_{t-1},z_{t-1,}y_{t-2},z_{t-2,}...\}.$ Note first that

$$\begin{aligned} \left| u_{1t}\left( \theta _{0}\right) \right| \le \left| \ln \left| y_{t-1}\right| \right| +w_{0}\left| \ln \left| y_{t-2}\right| \right| \left( \frac{1}{w_{0}}+\frac{1}{\sigma _{t}^{2}\left( \theta _{0}\right) }\right) \le \left| \ln \left| y_{t-1}\right| \right| +2\left| \ln \left| y_{t-2}\right| \right| , \end{aligned}$$

hence

$$\begin{aligned} E\left| u_{1t}\left( \theta _{0}\right) \right| \le 3E(\ln \left| y_{t}\right| )<\infty , \end{aligned}$$

where we have used assumptions A and B and where the last inequality follows from A3 where the first two moments of $\ln \left| y_{t}\right|$ are assumed to be bounded. Hence we can write

$$\begin{aligned} u_{1t}\left( \theta _{0}\right) \equiv g_{1}\left( y_{t-1},y_{t-2},\sigma _{t}^{2}\left( \theta _{0}\right) \right) , \end{aligned}$$

where $g_{1}$ is a $I_{t}$-measurable function and where all arguments $y_{t-1},y_{t-2}$ and $\sigma _{t}^{2}\left( \theta _{0}\right)$ are stationary and ergodic as a consequence of Lemmas 1 and 2. This implies that $u_{1t}\left( \theta _{0}\right)$ is stationary and ergodic by Theorem 3.35 in White (1984). Consequently $\frac{1}{T}\sum _{t=1}^{T}u_{1t}\left( \theta _{0}\right) \overset{p}{\rightarrow }E\left( u_{1t}\left( \theta _{0}\right) \right)$ follows by the Ergodic Theorem. Similarly, it follows straightforwardly that $E\left| u_{2t}\left( \theta _{0}\right) \right| \le \left( \frac{1}{w_{0}}\right)$ and $E\left| u_{3t}\left( \theta _{0}\right) \right| \le \left( \frac{2}{\alpha _{0}} \right) .$ We can write $u_{2t}\left( \theta _{0}\right) \equiv g_{2}\left( \sigma _{t}^{2}\left( \theta _{0}\right) \right)$ and $u_{3t}\left( \theta _{0}\right) \equiv g_{3}\left( \sigma _{t}^{2}\left( \theta _{0}\right) \right)$ and as above conclude that $(u_{2t}\left( \theta _{0}\right) ,u_{3t}\left( \theta _{0}\right) )$ is stationary and ergodic, and hence $\frac{1}{T}\sum _{t=1}^{T}u_{it}\left( \theta _{0}\right) \overset{p}{ \rightarrow }E\left( u_{it}\left( \theta _{0}\right) \right)$ for $i=2,3$. Second, notice that

$$\begin{aligned} \left| u_{1t}^{2}\left( \theta _{0}\right) \right|= & {} |\ln ^{2}\left| y_{t-1}\right| -2w_{0}\left( \frac{1}{w_{0}}-\frac{1}{ \sigma _{t}^{2}\left( \theta _{0}\right) }\right) \ln \left| y_{t-2}\right| \ln \left| y_{t-1}\right| \\&\quad +\left( \ln \left| y_{t-2}\right| \right) ^{2}w_{0}^{2}\left( \frac{1}{w_{0}}-\frac{1}{ \sigma _{t}^{2}\left( \theta _{0}\right) }\right) ^{2}| \\\le & {} \ln ^{2}\left| y_{t-1}\right| +\frac{4w_{0}^{2}}{\sigma _{t}^{4}\left( \theta _{0}\right) }\ln ^{2}\left| y_{t-2}\right| +4 \frac{w_{0}}{\sigma _{t}^{2}\left( \theta _{0}\right) }\left| \ln \left| y_{t-1}\right| \ln \left| y_{t-2}\right| \right| \\\le & {} \ln ^{2}\left| y_{t-1}\right| +4\ln ^{2}\left| y_{t-2}\right| +4\left| \ln \left| y_{t-1}\right| \ln \left| y_{t-2}\right| \right| , \end{aligned}$$

such that

$$\begin{aligned} E\left| u_{1t}^{2}\left( \theta _{0}\right) \right| \le 5E\left( (\ln \left| y_{t}\right| )^{2}\right) +4E\left| \ln \left| y_{t}\right| \ln \left| y_{t-1}\right| \right| <\infty . \end{aligned}$$

On the right hand side of the first inequality we have used Lemmas 1 and 2 and the second inequality follows from A3 (existence of second order moments). In addition, $E\left| u_{2t}^{2}\left( \theta _{0}\right) \right| \le \left( \frac{1}{w_{0}^{2}}\right)$ and $E\left| u_{3t}^{2}\left( \theta _{0}\right) \right| \le \left( \frac{4}{\alpha _{0}^{2}}\right) .$ We can therefore conclude, by Theorem 3.35 in White (1984), that since $u_{it}\left( \theta _{0}\right)$ is stationary and ergodic then so is $u_{it}^{2}\left( \theta _{0}\right)$ for $i=1,2,3.$ Furthermore as $E|u_{it}^{2}\left( \theta _{0}\right) |$ is bounded then $\frac{1}{T}\sum _{t=1}^{T}u_{1t}^{2}\left( \theta _{0}\right) \overset{p}{ \rightarrow }E\left( u_{1t}^{2}\left( \theta _{0}\right) \right)$ for $i=1,2,3$ follows from the ergodicity theorem. This completes the proof of Lemma A. $\square$

Lemma B

Under Assumptions A and B, the marginal distributions of the score functions given by Eqs. (9)–(11) evaluated at $\theta =\theta _{0}$ are asymptotically Gaussian,

$$\begin{aligned} \frac{1}{\sqrt{T}}\frac{\partial }{\partial \gamma }L_{T}\left( \theta _{0}\right)= & {} \frac{-1}{\sqrt{T}}\sum _{t=1}^{T}\left( 1-z_{t}^{2}\right) u_{1t}\left( \theta _{0}\right) \overset{d}{\rightarrow }N\left( 0,\zeta {\overline{m}}_{11}\right) , \end{aligned}$$

(4)

$$\begin{aligned} \frac{1}{\sqrt{T}}\frac{\partial }{\partial w}L_{T}\left( \theta _{0}\right)= & {} \frac{-1}{\sqrt{T}}\sum _{t=1}^{T}\frac{1}{2}\left( 1-z_{t}^{2}\right) u_{2t}\left( \theta _{0}\right) \overset{d}{\rightarrow }N\left( 0,\zeta {\overline{m}}_{22}\right) , \end{aligned}$$

(5)

$$\begin{aligned} \frac{1}{\sqrt{T}}\frac{\partial }{\partial \alpha }L_{T}\left( \theta _{0}\right)= & {} \frac{-1}{\sqrt{T}}\sum _{t=1}^{T}\frac{1}{2}\left( 1-z_{t}^{2}\right) u_{3t}\left( \theta _{0}\right) \overset{d}{\rightarrow } N\left( 0,\zeta {\overline{m}}_{33}\right) , \end{aligned}$$

(6)

where ${\overline{m}}_{ii},$ $i=1,2,3$ and $\zeta$ are defined by Lemma A and A3 respectively.

Proof of Lemma B

We will prove (4) in detail. The results in (5) and ( 6) hold by identical arguments. Define again $I_{t}= \{y_{t},z_{t,}y_{t-1},z_{t-1,}y_{t-2},z_{t-2,}...\}$ and recall from Result 1 that

$$\begin{aligned} s_{1t}\left( \theta _{0}\right) =-\left( 1-z_{t}^{2}\right) u_{1t}\left( \theta _{0}\right) . \end{aligned}$$

Consequently

$$\begin{aligned} E\left( s_{1t}|I_{t-1}\right)= & {} -E\left( \left( 1-z_{t}^{2}\right) u_{1t}\left( \theta _{0}\right) |I_{t-1}\right) =-E\left( \left( 1-z_{t}^{2}\right) \right) u_{1t}\left( \theta _{0}\right) \nonumber \\= & {} 0. \end{aligned}$$

(7)

Since $\{ s_{1t},I_{t}\}$ is an adapted stochastic sequence the result in (7) implies that $\{s_{1t},I_{t}\}$ is a martingale difference sequence according to Definition 3.75 in White (1984). Further, notice that

$$\begin{aligned} V_{1T}^{2}\left( \theta _{0}\right) =\sum _{t=1}^{T}E\left( s_{1t}^{2}\left( \theta _{0}\right) |I_{t-1}\right) =\sum _{t=1}^{T}E\left( \left( 1-z_{t}^{2}\right) ^{2}\right) u_{1t}^{2}\left( \theta _{0}\right) =\zeta \sum _{t=1}^{T}u_{1t}^{2}\left( \theta _{0}\right) . \end{aligned}$$

Hence,

$$\begin{aligned} E(V_{1T}^{2}\left( \theta _{0}\right) )=\zeta \sum _{t=1}^{T}E\left( u_{1t}^{2}\left( \theta _{0}\right) \right) =T\zeta {\overline{m}}_{11}. \end{aligned}$$

Furthermore, according to Lemma A we have that

$$\begin{aligned} \frac{1}{T}\sum _{t=1}^{T}u_{1t}^{2}\left( \theta _{0}\right) \overset{p}{ \rightarrow }{\overline{m}}_{11}, \end{aligned}$$

implying that

$$\begin{aligned} \frac{1}{T}V_{1T}^{2}\left( \theta _{0}\right) \overset{p}{\rightarrow } \zeta {\overline{m}}_{11}. \end{aligned}$$

From this we see that

$$\begin{aligned} \left( V_{1T}^{2}\left( \theta _{0}\right) \right) \left( E(V_{1T}^{2}\left( \theta _{0}\right) )\right) ^{-1}\overset{p}{\rightarrow }1. \end{aligned}$$

(8)

Importantly, the result given by equation (8) corresponds to Condition (1), p. 60 in Brown (1971).⁷

Finally, we need to prove that the Lindeberg type condition, which is Condition (2) in Brown (1971). In particular, we need to show that

$$\begin{aligned} \left( E(V_{1T}^{2}\left( \theta _{0}\right) )\right) ^{-1}\sum _{t=1}^{T}E\left( s_{1t}^{2}\left( \theta _{0}\right) 1\left\{ \left| s_{1t}\left( \theta _{0}\right) \right| >\epsilon \sqrt{ E(V_{1T}^{2}\left( \theta _{0}\right) )}\right\} \right) \overset{p}{ \rightarrow }0, \end{aligned}$$

for all $\epsilon >0.$ By inserting the expression for $s_{1t}^{2}$ and $E(V_{1T}^{2}\left( \theta _{0}\right) )$ we get

$$\begin{aligned}&\underset{T\rightarrow \infty }{\lim }\frac{1}{T\zeta {\overline{m}}_{11}} \sum _{t=1}^{T}E\left( s_{1t}^{2}\left( \theta _{0}\right) 1\left\{ \left| s_{1t}\left( \theta _{0}\right) \right|>\epsilon \sqrt{ T\zeta {\overline{m}}_{11}}\right\} \right) =\underset{T\rightarrow \infty }{\lim }\frac{1}{\zeta {\overline{m}}_{1}}\\&\quad E\left( \left( \left( 1-z_{t}^{2}\right) ^{2}u_{1t}^{2}\left( \theta _{0}\right) \right) 1\left\{ \left| \left( 1-z_{t}^{2}\right) ^{2}u_{1t}^{2}\left( \theta _{0}\right) \right| >\sqrt{T\zeta {\overline{m}}_{11}}\right\} \right) \rightarrow 0, \end{aligned}$$

for all $\zeta {\overline{m}}_{11}$ because, from Lemma A and A1, $u_{1t}^{2}\left( \theta _{0}\right)$ and $z_{t}^{2}$ have finite moments and are stationary and ergodic. Consequently, the Lindeberg condition holds.

According to Theorem 2, p. 60, in Brown (1971) we can therefore conclude that

$$\begin{aligned} \frac{1}{\sqrt{T\zeta {\overline{m}}_{11}}}\sum _{t=1}^{T}s_{1t}\left( \theta _{0}\right) \overset{d}{\rightarrow }N(0,1), \end{aligned}$$

which completes the proof. $\square$

Along the same lines

$$\begin{aligned} \frac{1}{T}\sum _{t=1}^{T}E\left( s_{2t}^{2}\mid I_{t-1}\right)= & {} \frac{1}{T} \sum _{t=1}^{T}\frac{\zeta }{4}\frac{1}{\left( w_{0}+\alpha _{0}\left( \frac{ y_{t-1}^{*}}{\left| y_{t-2}\right| ^{\gamma }}\right) ^{2}\right) }\overset{p}{\longrightarrow }\frac{\zeta }{4w_{0}^{2}}>0, \\ \frac{1}{T}\sum _{t=1}^{T}E\left( s_{3t}^{2}\mid I_{t-1}\right)= & {} \frac{1}{T} \sum _{t=1}^{T}\frac{\zeta }{4}\frac{\left( \frac{y_{t-1}^{*}}{\left| y_{t-2}\right| ^{\gamma }}\right) ^{2}}{\left( w_{0}+\alpha _{0}\left( \frac{y_{t-1}^{*}}{\left| y_{t-2}\right| ^{\gamma }}\right) ^{2}\right) }\overset{p}{\longrightarrow }\frac{\zeta }{4\alpha _{0}^{2}}>0. \end{aligned}$$

and

$$\begin{aligned}&\frac{1}{T}\sum _{t=1}^{T}E\left( s_{2t}^{2}1\left\{ \left| s_{2t}\right|>\sqrt{T}\delta \right\} \right) \\&\quad \le E\left( \left( \frac{ \left( 1-z_{t}^{2}\right) ^{2}}{4w_{0}^{2}}\right) 1\left\{ \left| \frac{ \left( 1-z_{t}^{2}\right) }{2w_{0}}\right|>\sqrt{T}\delta \right\} \right) \rightarrow 0, \\&\frac{1}{T}\sum _{t=1}^{T}E\left( s_{3t}^{2}1\left\{ \left| s_{3t}\right|>\sqrt{T}\delta \right\} \right) \\&\quad \le E\left( \left( \frac{ \left( 1-z_{t}^{2}\right) ^{2}}{4\alpha _{0}^{2}}\right) 1\left\{ \left| \frac{\left( 1-z_{t}^{2}\right) }{2\alpha _{0}}\right| >\sqrt{T}\delta \right\} \right) \rightarrow 0, \end{aligned}$$

for some $\delta >0$ and as T tends to $\infty$ . $\square$

Proof of Proposition 1

In order to fully characterize the asymptotic distribution we need to determine the off-diagonal elements of the variance covariance matrix of the score vectors given by $\Lambda .$ In particular, because $u_{1t}\left( \theta _{0}\right) ,u_{2t}\left( \theta _{0}\right)$ and $u_{3t}\left( \theta _{0}\right)$ are all stationary and ergodic with finite first moments (from Lemma A) it follows straightforwardly that

$$\begin{aligned} \frac{1}{T}\sum _{t=1}^{T}s_{1t}\left( \theta _{0}\right) s_{2t}\left( \theta _{0}\right)= & {} \frac{1}{T}\sum _{t=1}^{T}\left( 1-z_{t}^{2}\right) ^{2}u_{1t}\left( \theta _{0}\right) u_{2t}\left( \theta _{0}\right) \overset{ p}{\rightarrow }\frac{1}{2}\zeta {\overline{m}}_{12}, \\ \frac{1}{T}\sum _{t=1}^{T}s_{1t}\left( \theta _{0}\right) s_{3t}\left( \theta _{0}\right)= & {} \frac{1}{T}\sum _{t=1}^{T}\frac{1}{2}\left( 1-z_{t}^{2}\right) ^{2}u_{1t}\left( \theta _{0}\right) u_{3t}\left( \theta _{0}\right) \overset{ p}{\rightarrow }\frac{1}{2}\zeta {\overline{m}}_{13}, \\ \frac{1}{T}\sum _{t=1}^{T}s_{2t}\left( \theta _{0}\right) s_{3t}\left( \theta _{0}\right)= & {} \frac{1}{T}\sum _{t=1}^{T}\frac{1}{4}\left( 1-z_{t}^{2}\right) ^{2}u_{2t}\left( \theta _{0}\right) u_{3t}\left( \theta _{0}\right) \overset{ p}{\rightarrow }\frac{1}{4}\zeta {\overline{m}}_{23}. \end{aligned}$$

Since all the elements in the score vector are asymptotically normal (see Lemma B), the result follows directly from application of the Cramer-Wold device, see for example Proposition 5.1 in White (1984), which completes the proof. $\square$

Proposition 2

Let $u_{jt}\left( \theta _{0}\right)$ be defined as in Theorem 1. Under Assumptions A and B, the observed information evaluated at $\theta =\theta _{0}$ converges in probability, i.e.,

$$\begin{aligned} -\frac{1}{T}\frac{\partial ^{2}}{\partial \theta \partial \theta ^{\prime }} L_{T}\left( \theta _{0}\right) \overset{p}{\rightarrow }\Omega , \end{aligned}$$

where

$$\begin{aligned} \Omega =\left[ \begin{array}{ccc} 2{\overline{m}}_{11} &{} {\overline{m}}_{12} &{} {\overline{m}}_{13} \\ {\overline{m}}_{12} &{} \frac{1}{2}{\overline{m}}_{22} &{} \frac{1}{2}{\overline{m}} _{23} \\ {\overline{m}}_{13} &{} \frac{1}{2}{\overline{m}}_{23} &{} \frac{1}{2}{\overline{m}} _{33} \end{array} \right] >0, \end{aligned}$$

and ${\overline{m}}_{ij}=E\left( u_{it}\left( \theta _{0}\right) u_{jt}\left( \theta _{0}\right) \right)$ for $i=1,2,3$ and $j=1,2,3$.

Proof of Proposition 2

Recall from Result 2 (see Supplementary Appendix) that

$$\begin{aligned}&-\frac{1}{T}\frac{\partial ^{2}}{\partial \gamma ^{2}}L_{T}\left( \theta _{0}\right) =2\frac{1}{T}\sum _{t=1}^{T}z_{t}^{2}u_{1t}^{2}\left( \theta _{0}\right) -\frac{2}{T}\sum _{t=1}^{T}\left( 1-z_{t}^{2}\right) \nonumber \\&\quad \left( \ln \left| y_{t-2}\right| \right) ^{2}w_{0}^{2}\left( \frac{1}{w_{0}}- \frac{1}{\sigma _{t}^{2}\left( \theta _{0}\right) }\right) ^{2} \\&\qquad +\frac{2}{T}\sum _{t=1}^{T}\left( 1-z_{t}^{2}\right) w_{0}\left( \frac{1}{ w_{0}}-\frac{1}{\sigma _{t}^{2}\left( \theta _{0}\right) }\right) \left( \ln \left| y_{t-2}\right| \right) ^{2}. \end{aligned}$$

Since $z_{t}^{2}$ and $u_{1t}^{2}\left( \theta _{0}\right)$ are independent, the first term on the right hand side converges to $2{\overline{m}}_{11}$ by Lemma A. Furthermore, since $\left( \ln \left| y_{t-2}\right| \right) ^{2}w_{0}^{2}\left( \frac{1}{w_{0}}-\frac{1}{ \sigma _{t}^{2}\left( \theta _{0}\right) }\right) ^{2}$ and $w_{0}\left( \frac{1}{w_{0}}-\frac{1}{\sigma _{t}^{2}\left( \theta _{0}\right) }\right) \left( \ln \left| y_{t-2}\right| \right) ^{2}$ have bounded moments, they are ergodic and stationary and since $E\left( 1-z_{t}^{2}\right) =0,$ it follows from the ergodic theorem that the last term on the right hand side converges in probability to zero. Therefore, the result follows. Using identical arguments we find

$$\begin{aligned} -\frac{1}{T}\frac{\partial ^{2}}{\partial w^{2}}L_{T}\left( \theta _{0}\right)= & {} -\frac{1}{2}\frac{1}{T}\sum _{t=1}^{T}\left( 1-2z_{t}^{2}\right) u_{2t}^{2}\left( \theta _{0}\right) \overset{p}{\rightarrow }\frac{1}{2} {\overline{m}}_{22},-\frac{1}{T}\frac{\partial ^{2}}{\partial \alpha ^{2}} L_{T}\left( \theta _{0}\right) \\= & {} -\frac{1}{2}\frac{1}{T}\sum _{t=1}^{T}\left( 1-2z_{t}^{2}\right) u_{3t}^{2}\left( \theta _{0}\right) \overset{p}{\rightarrow }\frac{1}{2} {\overline{m}}_{33},-\frac{1}{T}\frac{\partial ^{2}}{\partial \gamma \partial w }L_{T}\left( \theta _{0}\right) \\= & {} \frac{1}{T}\sum _{t=1}^{T}z_{t}^{2}u_{1t}\left( \theta _{0}\right) u_{2t}\left( \theta _{0}\right) \\&+\,\frac{1}{T}\sum _{t=1}^{T}\left( 1-z_{t}^{2}\right) \left( \left( \ln \left| y_{t-2}\right| \right) \left( \frac{w_{0}}{\sigma _{t}^{2}\left( \theta _{0}\right) }\right) \left( \frac{1}{w_{0}}-\frac{1}{\sigma _{t}^{2}\left( \theta _{0}\right) }\right) \right) \overset{p}{\rightarrow }{\overline{m}}_{12}, \\&-\,\frac{1}{T}\frac{\partial ^{2}}{\partial \gamma \partial \alpha } L_{T}\left( \theta _{0}\right) =\frac{1}{T}\sum _{t=1}^{T}z_{t}^{2}u_{1t} \left( \theta _{0}\right) u_{3t}\left( \theta _{0}\right) \\&-\,\frac{1}{T} \sum _{t=1}^{T}\left( 1-z_{t}^{2}\right) w_{0}\left( \ln \left| y_{t-2}\right| \right) \left( \frac{w_{0}}{\alpha _{0}}\right) \left( \frac{1}{w_{0}}-\frac{1}{_{\sigma _{t}^{2}\left( \theta _{0}\right) }} \right) ^{2} \\&\overset{p}{\rightarrow }{\overline{m}}_{13},-\frac{1}{T}\frac{\partial ^{2}L_{T}\left( \theta _{0}\right) }{\partial w\partial \alpha }=-\frac{1}{2} \frac{1}{T}\sum _{t=1}^{T}\left( 1-2z_{t}^{2}\right) u_{2t}\left( \theta _{0}\right) u_{3t}\left( \theta _{0}\right) \overset{p}{\rightarrow }\frac{1 }{2}{\overline{m}}_{23}. \end{aligned}$$

We proceed now to show that $\Lambda$ is positive definite. $\Lambda$ will be positive definite if for any non-zero column vector z with entries a, b and c, we show that $z^{T}\Lambda z>0$. In our case

$$\begin{aligned} z^{T}\Lambda z= & {} \zeta \left( \begin{array}{c} aE\left( u_{1t}^{2}\right) +\frac{b}{2}E\left( u_{1t}u_{2t}\right) +\frac{c}{ 2}E\left( u_{1t}u_{3t}\right) \\ \frac{a}{2}E\left( u_{1t}u_{2t}\right) +\frac{b}{4}E\left( u_{2t}^{2}\right) +\frac{c}{4}E\left( u_{2t}u_{3t}\right) \\ \frac{a}{2}E\left( u_{1t}u_{3t}\right) +\frac{b}{4}E\left( u_{2t}u_{3t}\right) +\frac{c}{4}E\left( u_{3t}^{2}\right) \end{array} \right) ^{T}\left( \begin{array}{c} a \\ b \\ c \end{array} \right) \\= & {} \zeta \Bigg [ a^{2}E\left( u_{1t}^{2}\right) +\frac{ab}{2}E\left( u_{1t}u_{2t}\right) +\frac{ac}{2}E\left( u_{1t}u_{3t}\right) +\frac{ab}{2} E\left( u_{1t}u_{2t}\right) \\&+\,\frac{b^{2}}{4}E\left( u_{2t}^{2}\right) +\frac{ bc}{4}E\left( u_{2t}u_{3t}\right) \\&+\,\frac{ac}{2}E\left( u_{1t}u_{3t}\right) +\frac{bc}{4}E\left( u_{2t}u_{3t}\right) +\frac{c^{2}}{4}E\left( u_{3t}^{2}\right) \Bigg ] \\= & {} \zeta \Bigg [ a^{2}E\left( u_{1t}^{2}\right) +\frac{b^{2}}{4}E\left( u_{2t}^{2}\right) +\frac{c^{2}}{4}E\left( u_{3t}^{2}\right) \\&+\, abE\left( u_{1t}u_{2t}\right) +acE\left( u_{1t}u_{3t}\right) +\frac{bc}{2}E\left( u_{2t}u_{3t}\right) \Bigg ] \end{aligned}$$

where we have written $u_{it}\left( \theta _{0}\right) =u_{it}$ for simplicity reasons. Since $\zeta ,$ by Assumption A1, is always positive and larger than zero, and from Lemma A we have that

$$\begin{aligned} u_{3t}=\left( \frac{w_{0}}{\alpha _{0}}\right) \left( \frac{1}{w_{0}} -u_{2t}\right) , \end{aligned}$$

then, we need to show if the following term is strictly positive

$$\begin{aligned}&a^{2}E\left( u_{1t}^{2}\right) +\frac{b^{2}}{4}E\left( u_{2t}^{2}\right) +\left( \frac{cw_{0}}{2\alpha _{0}}\right) ^{2}E\left( \left( \frac{1}{w_{0}} -u_{2t}\right) ^{2}\right) \\&\qquad +abE\left( u_{1t}u_{2t}\right) +\frac{acw_{0}}{ \alpha _{0}}E\left( \frac{u_{1t}}{w_{0}}-u_{1t}u_{2t}\right) \\&\qquad +\frac{bcw_{0}}{2\alpha _{0}}E\left( \frac{u_{2t}}{w_{0}}-u_{2t}^{2}\right) =a^{2}E\left( u_{1t}^{2}\right) +\frac{1}{4}\left( b-\frac{cw_{0}}{\alpha _{0}}\right) ^{2}\\&E\left( u_{2t}^{2}\right) +\left( \frac{c}{2\alpha _{0}} \right) ^{2}+\frac{ac}{\alpha _{0}}E\left( u_{1t}\right) \\&\qquad +\frac{c}{2\alpha _{0}}\left( b-\frac{cw_{0}}{\alpha _{0}}\right) E\left( u_{2t}\right) +a\left( b-\frac{cw_{0}}{\alpha _{0}}\right) E\left( u_{1t}u_{2t}\right) \\&\quad =a^{2}E\left( u_{1t}^{2}\right) +\frac{1}{4}\left( b-\frac{cw_{0}}{\alpha _{0}}\right) ^{2}E\left( u_{2t}^{2}\right) +a\left( b-\frac{cw_{0}}{\alpha _{0}}\right) E\left( u_{1t}u_{2t}\right) \\&\qquad +\frac{c}{\alpha _{0}}\left[ aE\left( u_{1t}\right) +\frac{1}{2}\left( b- \frac{cw_{0}}{\alpha _{0}}\right) E\left( u_{2t}\right) \right] +\left( \frac{c}{2\alpha _{0}}\right) ^{2} \\&\quad =E\left[ a^{2}u_{1t}^{2}+\frac{1}{4}\left( b-\frac{cw_{0}}{\alpha _{0}} \right) ^{2}u_{2t}^{2}+a\left( b-\frac{cw_{0}}{\alpha _{0}}\right) u_{1t}u_{2t}\right. \\&\qquad \left. +\frac{c}{\alpha _{0}}\left[ au_{1t}+\frac{1}{2}\left( b-\frac{ cw_{0}}{\alpha _{0}}\right) u_{2t}\right] +\left( \frac{c}{2\alpha _{0}} \right) ^{2}\right] \\&\quad =E\left[ \left( au_{1t} +\frac{1}{2}\left( b-\frac{cw_{0}}{\alpha _{0}} \right) u_{2t}\right) ^{2}\right. \\&\left. \qquad +\frac{c}{\alpha _{0}}\left[ au_{1t} +\frac{1}{2} \left( b-\frac{cw_{0}}{\alpha _{0}}\right) u_{2t}\right] +\left( \frac{c}{ 2\alpha _{0}}\right) ^{2}\right] \\&\quad =E\left( au_{1t}+\frac{1}{2}\left( b-\frac{cw_{0}}{2\alpha _{0}}\right) u_{2t}+\frac{c}{2\alpha _{0}}\right) ^{2}>0. \end{aligned}$$

Finally notice that since $\Omega =2\Lambda \zeta ^{-1}=\Lambda ,$ then $\Omega >0.$ This completes the proof of Proposition 2. $\square$

Proposition 3

Define the lower and upper values for each parameter in $\theta _{0}$ as $\gamma _{L}<\gamma _{0}<\gamma _{U},w_{L}<w_{0}<w_{U},$ and $\alpha _{L}<\alpha _{0}<\alpha _{U},$ respectively and the neighborhood $N\left( \theta _{0}\right)$ around $\theta _{0}$ as

$$\begin{aligned} N\left( \theta _{0}\right) =\left\{ \theta \backslash \gamma _{L}\le \gamma \le \gamma _{U},w_{L}\le w\le w_{U},\text { and }\alpha _{L}\le \alpha \le \alpha _{U}\right\} . \end{aligned}$$

Under Assumptions A and B, there exists a neighborhood $N\left( \theta _{0}\right)$ for which for $i,j,k=1,2,3$

$$\begin{aligned} \sup _{\theta \in N\left( \theta _{0}\right) }\left| \frac{1}{T}\frac{ \partial ^{3}}{\partial \theta _{i}\partial \theta _{j}\partial \theta _{k}} L_{T}\left( \theta \right) \right| \le \frac{1}{T} \sum _{t=1}^{T}w_{ijkt}, \end{aligned}$$

where $w_{ijkt}$ is stationary. Furthermore $\frac{1}{T} \sum _{t=1}^{T}w_{ijkt}\overset{a.s.}{\longrightarrow }E\left( w_{ijkt}\right) <\infty$ for $\forall ijk.$

Proof of Proposition 3

Let us start from the components of $\left| \frac{1}{T}\frac{\partial ^{3}}{\partial \gamma ^{3}}L_{T}\left( \theta \right) \right|$ defined in Result 3 (see Supplementary Appendix). Part I (which is also defined in Result 3) can be written as

$$\begin{aligned}&\left| \frac{4}{T}\sum _{t=1}^{T}\frac{\alpha \left( \frac{y_{t-1}^{*} }{\left| y_{t-2}\right| ^{\gamma }}\right) ^{2}\left( \ln \left( \left| y_{t-2}\right| \right) \right) ^{3}}{\left( w+\alpha \left( \frac{y_{t-1}^{*}}{\left| y_{t-2}\right| ^{\gamma }}\right) ^{2}\right) }\left( 1-\frac{\left( y_{t}^{*}\right) ^{2}}{ y_{t-1}^{2\gamma }\left( w+\alpha \left( \frac{y_{t-1}^{*}}{\left| y_{t-2}\right| ^{\gamma }}\right) ^{2}\right) }\right) \right| \end{aligned}$$

$$\begin{aligned}&\quad \le \left| \frac{4}{T}\sum _{t=1}^{T}\frac{\left( \left( w+\alpha \left( \frac{y_{t-1}^{*}}{\left| y_{t-2}\right| ^{\gamma }} \right) ^{2}\right) -w\right) \left| \ln \left( \left| y_{t-2}\right| \right) \right| ^{3}}{\left( w+\alpha \left( \frac{ y_{t-1}^{*}}{\left| y_{t-2}\right| ^{\gamma }}\right) ^{2}\right) } \right. \\&\qquad \times \, \left. \left( \frac{\left( y_{t}^{*}\right) ^{2}}{y_{t-1}^{2\gamma }\left( w+\alpha \left( \frac{y_{t-1}^{*}}{\left| y_{t-2}\right| ^{\gamma }}\right) ^{2}\right) }+1\right) \right| \\&\quad \le \left| \frac{4}{T}\sum _{t=1}^{T}\left( \frac{\left( y_{t}^{*}\right) ^{2}}{y_{t-1}^{2\gamma } \left( w+\alpha \left( \frac{y_{t-1}^{*} }{\left| y_{t-2}\right| ^{\gamma }}\right) ^{2}\right) }+1\right) \left| \ln \left( \left| y_{t-1}\right| \right) \right| ^{3}\right| \\&\quad \le \left| \frac{4}{T}\sum _{t=1}^{T}\left( \left( \frac{ y_{t-1}^{2\gamma _{0}}\left( w_{0}+\alpha _{0}\left( \frac{y_{t-1}^{*}}{ \left| y_{t-2}\right| ^{\gamma _{0}}}\right) ^{2}\right) }{ y_{t-1}^{2\gamma }\left( w+\alpha \left( \frac{y_{t-1}^{*}}{\left| y_{t-2}\right| ^{\gamma }}\right) ^{2}\right) }\right) z_{t}^{2}+1\right) \left| \ln \left( \left| y_{t-1}\right| \right) \right| ^{3}\right| \\&\quad \le \left| \frac{4}{T}\sum _{t=1}^{T}\left( \left( \frac{w_{0}}{w} y_{t-1}^{2\left( \gamma _{0}-\gamma \right) }+\frac{\alpha _{0}}{\alpha } \left( \frac{y_{t-1}^{*}}{\left| y_{t-2}\right| }\right) ^{2\left( \gamma _{0}-\gamma \right) }\right) z_{t}^{2}+1\right) \left| \ln \left( \left| y_{t-1}\right| \right) \right| ^{3}\right| \\&\quad \le \left| \frac{4}{T}\sum _{t=1}^{T}\left( \left\{ \frac{w_{U}}{w_{L}} \Lambda _{t-1}+\frac{\alpha _{U}}{\alpha _{L}}\Lambda _{t-2}\right\} z_{t}^{2}+1\right) \left| \ln \left( \left| y_{t-1}\right| \right) \right| ^{3}\right| \\&\quad \le \left| \frac{4}{T}\sum _{t=1}^{T}\left( \left\{ \frac{w_{U}}{w_{L}} \Lambda _{t-1}+\frac{\alpha _{U}}{\alpha _{L}}\Lambda _{t-2}\right\} z_{t}^{2}+1\right) \left| \ln \left( \left| y_{t-1}\right| \right) \right| ^{3}\right| \\&\quad \le \left| \frac{4}{T}\sum _{t=1}^{T}\left( \left\{ \frac{w_{U}}{w_{L}} \Lambda _{t-1}+\frac{\alpha _{U}}{\alpha _{L}}\Lambda _{t-2}\right\} z_{t}^{2}+1\right) \left| \ln \left( \left| y_{t-1}\right| \right) \right| ^{3}\right| , \end{aligned}$$

where we can define the lower bound for all t,  $y_{L}\le \left| y_{t-1}\right| ,$ $y_{L}\le \left| y_{t-2}\right| ,$ $\Lambda _{t-1}=\max \left\{ y_{L}^{2\left| \gamma _{U}-\gamma _{L}\right| },y_{t-1}^{2\left| \gamma _{U}-\gamma _{L}\right| }\right\}$, $\Lambda _{t-2}=\max \left\{ 1,\left( \frac{y_{t-1}^{*}}{\left| y_{t-2}\right| }\right) ^{2\left| \gamma _{U}-\gamma _{L}\right| }\right\}$ and the result follows by setting $2\left| \gamma _{U}-\gamma _{L}\right| =\varphi$, Assumptions A and B and the law of large numbers (see Jensen and Rahbek (2004a), Lemma 5). Part II requires also assumption A3 since

$$\begin{aligned}&\left| \frac{8}{T}\sum _{t=1}^{T}\frac{\alpha ^{3}\left( \frac{ y_{t-1}^{*}}{\left| y_{t-2}\right| ^{\gamma }}\right) ^{6}\left( \ln \left( \left| y_{t-2}\right| \right) \right) ^{3}}{\left( w+\alpha \left( \frac{y_{t-1}^{*}}{\left| y_{t-2}\right| ^{\gamma }}\right) ^{2}\right) ^{3}}\left( 1-\frac{3\left( y_{t}^{*}\right) ^{2}}{y_{t-1}^{2\gamma }\left( w+\alpha \left( \frac{y_{t-1}^{*} }{\left| y_{t-2}\right| ^{\gamma }}\right) ^{2}\right) }\right) \right| \\&\quad \le \left| \frac{8}{T}\sum _{t=1}^{T}\left( \frac{3\left( y_{t}^{*}\right) ^{2}}{y_{t-1}^{2\gamma }\left( w+\alpha \left( \frac{y_{t-1}^{*} }{\left| y_{t-2}\right| ^{\gamma }}\right) ^{2}\right) }-1\right) \left| \ln \left( \left| y_{t-2}\right| \right) \right| ^{3}\right| \\&\quad \le \left| \frac{8}{T}\sum _{t=1}^{T}\left( 3\left\{ \frac{w_{U}}{w_{L} }\Lambda _{t-1}+\frac{\alpha _{U}}{\alpha _{L}}\Lambda _{t-2}\right\} z_{t}^{2}+1\right) \left| \ln \left( \left| y_{t-1}\right| \right) \right| ^{3}\right| . \end{aligned}$$

Parts III, IV, V and VI follow the same argument.

Along the same lines for $\left| \frac{1}{T}\frac{\partial ^{3}}{ \partial \alpha ^{3}}L_{T}\left( \theta \right) \right|$

$$\begin{aligned} \left| \frac{1}{T}\frac{\partial ^{3}}{\partial \alpha ^{3}}L_{T}\left( \theta \right) \right|= & {} \left| \frac{1}{T}\sum _{t=1}^{T}\left( 3 \frac{\left( y_{t}^{*}\right) ^{2}}{y_{t-1}^{2\gamma }\left( w+\alpha \left( \frac{y_{t-1}^{*}}{\left| y_{t-2}\right| ^{\gamma }} \right) ^{2}\right) }-1\right) \frac{\left( \frac{y_{t-1}^{*}}{ y_{t-2}^{\gamma }}\right) ^{6}}{\left( w+\alpha \left( \frac{y_{t-1}^{*} }{\left| y_{t-2}\right| ^{\gamma }}\right) ^{2}\right) ^{3}} \right| \\\le & {} \left| \frac{1}{T}\sum _{t=1}^{T}\left( 3\frac{\left( y_{t}^{*}\right) ^{2}}{y_{t-1}^{2\gamma }\left( w+\alpha \left( \frac{y_{t-1}^{*} }{\left| y_{t-2}\right| ^{\gamma }}\right) ^{2}\right) }-1\right) \right| \frac{1}{\alpha _{L}^{3}} \\\le & {} \frac{1}{T}\sum _{t=1}^{T}\left( 3\left\{ \frac{w_{U}}{w_{L}}\Lambda _{t-1}+\frac{\alpha _{U}}{\alpha _{L}}\Lambda _{t-2}\right\} z_{t}^{2}+1\right) \frac{1}{\alpha _{L}^{3}}. \end{aligned}$$

The rest of the cases follow directly using the same argument. This completes the proof of Proposition 3. $\square$

Proof of Theorem 1

Given the conditions provided by Propositions 1–3, Theorem 1 follows from Lumsdaine (1996, pp. 593–595, Theorem 3), the ergodic theorem and Lemma 1, p. 1206 in Jensen and Rahbek (2004b). $\square$