# Stationarity and goodness-of-fit tests for locally stationary time series

Jean-Marc Bardet¹

¹University Paris 1 Panthéon-Sorbonne, SAMM, France, e-mail: bardet@univ-paris1.fr

**Abstract:** This paper considers the trajectory of a time series with time-varying coefficients and proposes to test the adequacy of these parameters at a finite and fixed number of instants of the trajectory. For this purpose, a Wald test is constructed from point estimates of the parameters obtained by minimization of a kernel contrast. This can take the form of a localized near-maximum likelihood estimator for ARMA or GARCH processes, or a localized least squares estimator for a GLARCH process, but many other time-varying time series such as AR($\infty$), ARCH($\infty$), ARMA-GARCH, APARCH,..., could be considered. Above all, this allows the introduction of a new stationarity test for these processes, whose very good numerical performance has been demonstrated by numerical experiments.

MSC 2010 subject classifications: 62G05, 62G20, 62M05.

## 1. Introduction

In the classic study of an observed trajectory of a time series, one of the first steps is to test the stationarity of this trajectory. This is most often done using the well-known stationarity tests based on the presence or absence of a unit root in an ARMA model, such as the Augmented Dickey-Fuller (ADF), Phillips and Perron (PP) or Kwiatkowski, Phillips, Schmidt and Shin (KPSS) tests (Details of these well-known tests can be found, for example, in [20]).

It is clear that these tests are useful and effective for a time series that potentially has a trend or is clearly a stationary ARMA process. However, there are many cases where they are not appropriate at all. For example, we can mention the case of long memory processes (we highlighted this in [2]). Another, ultimately quite common, setting in which these stationarity tests are not relevant is that of time series with time-dependent coefficients. Consider, for example, a tv-ARMA(1, 1) process whose coefficients are functions following one of the cases (LS1) described in the section on numerical applications. The trajectory is thus that shown in Figure 1. And the result of the three tests ADF, KPSS and PP is clearly that the trajectory is stationary, whereas it clearly is not (visible on the plot). Whereas with the $\hat{S}_n$ stationarity test que nous construisons dans cet article (see its definition in 14, with $J = 3$), the p-value of the test is $2.10^{-16}$ since $\hat{S}_n \approx 222.09$ while the distribution under the stationary hypothesis is a $\chi^2(6)$ distribution: the stationarity hypothesis is clearly rejected.
<figure><img src="image_1.png" /><figcaption>Fig 1: A generated trajectory of a tvARMA(1, 1) whose coefficients are functions defined in case (LS1)</figcaption></figure>

To explain the steps leading to the establishment of this new stationarity test, let's start by going back to what locally stationary time series are, as defined by Dahlhaus in the seminal paper [5]. The chapter [9] by Dahlhaus gives an overview of the results obtained between 1992 and 2012 on this topic. There are numerous papers by Dahlhaus and his co-authors on the definitions and properties of locally stationary models (see e.g. [12], [10], and [21]). Others have been devoted to identifying and estimating these processes (see [6], [7], [12], [10], [8], and [21]).

But it is undoubtedly in the recent Dahlhaus paper et al. (2019) [11] that a very general approach to estimating the parameters or functions of these processes can be found. Another generalization was obtained in [3], with a different methodology, and the possibility of applying it to locally stationary processes with infinite memory, defined by a recursive equation:

$$X_t^{(n)} = F_{\theta^*(t/n)}((X_{t-k}^{(n)})_{k \in \mathbb{N}^*}; \xi_t), \quad 1 \le t \le n, n \in \mathbb{N}^*, \quad (1)$$

with a continuous function $\theta^*$: [0, 1] $\to \mathbb{R}^d$, a known real-valued function $F_\theta$ and $(\xi_t)$ a sequence of independent and identically distributed (i.i.d.) random variables (see more details in Section 2). Among the results obtained is a central limit theorem satisfied by a point estimator $\hat{\theta}_n(u)$ in all $u \in (0, 1)$, this estimator being obtained by minimizing a localized contrast by a kernel.

In Section 3, we propose a multivariate extension to the previously obtained central limit theorem, i.e. we prove the asymptotic normality of $(\hat{\theta}_n(u_1), \dots, \hat{\theta}_n(u_J))$, where $J \in \mathbb{N}^*$ and $0 < u_1 < \dots < u_J < 1$ are fixed. Using an approximation of the asymptotic covariance matrix, two Wald-type tests can be set up. The first tests whether $\theta^*(\cdot) = \theta_0(\cdot)$, where $\theta_0(\cdot)$ is a function of [0, 1] in $\mathbb{R}^d$ given. The second, more far-reaching, test is designed to ascertain whether $\theta^*(\cdot)$ is a constant function on [0, 1]. This is equivalent to testing whether the trajectory can
be described as that of a stationary process. The test is straightforward to im-
plement, computationally efficient, and the p-value can be obtained in relation
to a classic chi-squared distribution. We give detailed examples of tv-ARMA,
tv-GARCH or tv-GLARCH processes for which these tests naturally apply, but
it would also be possible to implement them for AR(∞), ARCH(∞), ARMA-
GARCH or APARCH processes.

There are many other stationarity tests. In the framework we have chosen, that
of potential local-stationarity, as we have said, the classic ADF, KPSS or PP
tests are of no relevance. But several other tests have been established for this
purpose: we can note in particular those set up in Dette et al. (2011) [13], Preuß
et al. (2013) [22] or more recently in Heinrich and (2021) [19]. However, these
tests are based on a distance to the spectral density of the process, which is per-
fectly appropriate for ARMA processes, but not at all for GARCH, LARCH or
APARCH processes, for example, whose spectral density is that of white noise
and in no way allows the process to be identified. We could also mention the test
developed in [25], but again this is a very specific context of a tv-ARCH process.
The advantage of the stationarity test we propose is that it can be applied to
all the most known locally stationary processes.

Numerical studies are also proposed. First, Monte Carlo experiments show
the expected convergence of the two tests in the case of tv-ARMA(1, 1), tv-
GARCH(1, 1) and tv-GLARCH(1, 1) processes, though it should be noted that
these are obtained for the adequacy test for sufficiently large trajectory sizes
(convergence being faster for the stationarity test). Second, an application to
financial data (log-returns of the S&P500 data from October 1990 to Octo-
ber 2020) shows that while it is not possible to use a stationary GARCH(1, 1)
process to model the entire trajectory, it does seem legitimate for the period
between the financial crisis of September 2008 and the COVID crisis of March
2020.

The upcoming Section 2 is devoted to the definition and existence of new non-
stationary models. In Section 3, the multidimensional asymptotic normality of
($\hat{\theta}_n(u_1), \dots, \hat{\theta}_n(u_J)$) and the definitions and convergences in law of the two
types of tests proposed are stated. Section 4 examines three important examples:
tv-ARMA, tv-GARCH and tv-GLARCH. Numerical experiments are proposed
in Section 5 and proofs are followed up in Sections 6.

## 2. Preliminaries

### 2.1. Notation

Some standard notation is used:

* The symbol $0$ denotes the null vector in every vector space;

* If $S$ is an arbitrary vector space then $S^\infty = \{(x_n)_{n \in \mathbb{N}} \in S^\mathbb{N}; \exists N \in \mathbb{N}, x_k = 0, \text{ for all } k > N\}$;
• The symbol $\| \cdot \|$ denotes the usual Euclidean norm of a vector or the associated norm of a matrix;

• For $p \ge 1$ and $Z$ a random vector in $\mathbb{R}^m$, denote: $\|Z\|_p = [\mathbb{E}(\|Z\|^p)]^{1/p}$.

• From now on $\Theta$ denotes a subset of $\mathbb{R}^d$, and $\mathring{\Theta}$ is the interior of $\Theta$. For a function $f: \Theta \to f(\Theta) \in \mathbb{R}^m$, we define $\|f\|_{\Theta} = \sup_{\theta \in \Theta} \|f(\theta)\|$.

• For $\theta \in \Theta$ and $\Psi_{\theta}: \mathbb{R}^{\infty} \to V$ a Borel function with values in a finite dimensional vector space $V$, $\partial_{\theta}^k \Psi_{\theta}(x)$ denotes respectively for $k = 0, 1, 2$, in case they exist, $\Psi_{\theta}(x)$, $\partial \Psi_{\theta}(x)/\partial \theta$ and $\partial^2 \Psi_{\theta}(x)/\partial \theta^2$ for $x \in \mathbb{R}^{\infty}$.

## 2.2. Time-varying infinite memory process

In all the sequel, we will consider a given real number $p \ge 2$.

If it exists, we define $(X_t)_{t \in \mathbb{Z}}$ as a solution of the recursion

$$X_t = F_{\theta^*}((X_{t-k})_{k \ge 1}, \xi_t), \quad t \in \mathbb{Z}, \qquad (2)$$

where

• $(\xi_t)_{t \in \mathbb{Z}}$ is a sequence of independent and identically distributed random variables (i.i.d.r.v.) satisfying $\mathbb{E}[\xi_0] = 0$, $\mathbb{E}[\xi_0^2] = \text{Var}(\xi_0) = 1$ and $\|\xi_0\|_p < \infty$;

• the parameter $\theta^*$ is such as $\theta^* \in \Theta \subset \mathbb{R}^d$;

• the function $\theta \in \Theta \subset \mathbb{R}^d \mapsto F_{\theta} \in \mathbb{R}$ is such that for any $\theta \in \Theta$, $\partial_{\theta}^k F_{\theta}$ exists on $\mathbb{R}^{\infty} \times \mathbb{R}$ for $k = 0, 1, 2$ and there exists a sequence $(b_j^{(k)}(\theta))_j$ of nonnegative numbers such that for all $x, y \in \mathbb{R}^{\infty}$

$$\bullet \ C_k(\Theta) = \sup_{\theta \in \Theta} \|\partial_{\theta}^k F_{\theta}(0, \xi_0)\|_p < \infty \qquad (3)$$

$$\bullet \quad \left\| \partial_{\theta}^{k} F_{\theta}(x, \xi_0) - \partial_{\theta}^{k} F_{\theta}(y, \xi_0) \right\|_{p} \le \sum_{j=1}^{\infty} b_{j}^{(k)}(\theta) \|x_j - y_j\|_{p}, \\ \text{with } \left\| \sum_{j=1}^{\infty} b_{j}^{(k)} \right\|_{\Theta} < 1 \text{ and } \left\| \sum_{t=1}^{\infty} t \log(t) b_{t}^{(0)} \right\|_{\Theta} < \infty. \qquad (4)$$

From [14], there exists a.s. a unique stationary stationary version $(X_t)_{t \in \mathbb{Z}}$ satisfying (2) and $\|X_0\|_p < \infty$. There are many examples of such series: the ARMA, AR($\infty$), ARFIMA, GARCH, ARCH($\infty$), APARCH, ARMA-GARCH processes... (see for example [4]).

In [3], a locally stationary time-varying version of such a series is defined and studied. To do this, the parameter $\theta^*$ is replaced by a function $u \in [0, 1] \mapsto \theta^*(u) \in \Theta \subset \mathbb{R}^d$ which is a $\rho$-Hölder function with $\rho \in (0, 1]$, that is for any $u \in [0, 1]$,

$$\|\theta^*(v) - \theta^*(u)\| \le K_{\theta} |v - u|^{\rho}, \quad \text{for every } n \in \mathbb{N}^* \text{ and } 1 \le t \le n. \qquad (5)$$
Then we can consider the following locally stationary triangular array $(X_t^{(n)})_{1\le t\le n, n\in\mathbb{N}^*}$ defined for any $n \in \mathbb{N}^*$ by:

$$X_t^{(n)} = F_{\theta^*(\frac{t}{n})}((X_{t-k}^{(n)})_{k \in \mathbb{N}^*}; \xi_t), \text{ for } 1 \le t \le n \text{ and } X_t^{(n)} = 0 \text{ for } t \ge 0, \quad (6)$$

It was proved in [3] that $\sup_{n \in \mathbb{N}^*, 0 \le s \le n} \|X_s^{(n)}\|_p < \infty$.

### 2.3. M-estimation of the function $\theta^*(\cdot)$

It is assumed that $(X_1^{(n)}, X_2^{(n)}, \dots, X_n^{(n)})$ is an observed trajectory, and the aim is a statistical study of this trajectory.

The first thing, already done in [3], concerns the estimation of the function $\theta^*(\cdot)$. To do this, consider a contrast function $\Phi: (x, \theta) \in \mathbb{R}^\infty \times \Theta \to \mathbb{R} \mapsto \Phi(x, \theta) \in \mathbb{R}$, which is a continuous function, such that for every $x \in \mathbb{R}^\infty$, $\theta \in \Theta \mapsto \Phi(x, \theta)$ is a $C^2(\Theta)$ function, and

* There exist two sequences $(\alpha_i(\Phi, \Theta))_{i \in \mathbb{N}}$ and $(\alpha_i(\partial_\theta \Phi, \Theta))_{i \in \mathbb{N}}$ of $[0, \infty)^\mathbb{N}$ such that $\sum_{s=1}^\infty s(\alpha_s(\Phi, \Theta) + \alpha_s(\partial_\theta \Phi, \Theta)) < \infty$ and a function $g: [0, \infty) \to [0, \infty)$ such as for all sequences $U = (U_i)_{i \in \mathbb{N}^*} \in (\mathbb{L}^p)^\infty$ and $V = (V_i)_{i \in \mathbb{N}^*} \in (\mathbb{L}^p)^\infty$ satisfying $\sup_{s \ge 1} \{\|U_s\|_p \vee \|V_s\|_p\} < \infty$, we have:

$$ \left\{ \begin{array}{l} \mathbb{E}\left[\sup_{\theta \in \Theta} \left|\Phi(0, \theta)\right| + \left|\partial_{\theta}\Phi(0, \theta)\right|\right] < \infty; \\[2ex] \mathbb{E}\left[\sup_{\theta \in \Theta} \left|\Phi(U, \theta) - \Phi(V, \theta)\right|\right] \le g\left(\sup_{s \ge 1} \{\|U_s\|_p \vee \|V_s\|_p\}\right) \sum_{s=1}^{\infty} \alpha_s(\Phi, \Theta) \|U_s - V_s\|_p \\[2ex] \mathbb{E}\left[\sup_{\theta \in \Theta} \left|\partial_{\theta}\Phi(U, \theta) - \partial_{\theta}\Phi(V, \theta)\right|\right] \le g\left(\sup_{s \ge 1} \{\|U_s\|_p \vee \|V_s\|_p\}\right) \sum_{s=1}^{\infty} \alpha_s(\partial_{\theta}\Phi, \Theta) \|U_s - V_s\|_p \end{array} \right. \quad (7) $$

* For $(X_t)_{t \in \mathbb{Z}}$ satisfying the infinite memory model (2) with parameter $\theta^* \in \mathring{\Theta}$ and with $\mathcal{F}_0 = \sigma((X_{-k})_{k \in \mathbb{N}})$,

$$ \theta^* = \underset{\theta \in \mathring{\Theta}}{\operatorname{argmin}} \left\{ \mathbb{E}[\Phi((X_{1-k})_{k \in \mathbb{N}}, \theta) | \mathcal{F}_0] \right\} \quad \text{a.s.} \quad (8) $$

As an example, consider the case of a causal AR(1) process defined $X_t = \theta^* X_{t-1} + \xi_t$ for $t \in \mathbb{Z}$ and the usual quadratic contrast $\Phi((X_{1-k})_{k \in \mathbb{N}}, \theta) = (X_1 - \theta X_0)^2$ for which the conditions (7) are clearly verified. Moreover, with $p=2$, $\sigma_\xi^2 = \mathbb{E}[\xi_0^2]$ and for $\Theta = [-r, r]$ where $|r| < 1$,

$$ \mathbb{E}[\Phi((X_{1-k})_{k \in \mathbb{N}}, \theta) | \mathcal{F}_0] = \mathbb{E}[(X_1 - \theta X_0)^2 | \mathcal{F}_0] = \sigma_\xi^2 + (\theta^* - \theta)^2 X_0^2, $$

which is almost surely minimized by $\theta = \theta^*$ in $\mathring{\Theta}$.
For $u \in [0, 1]$, the kernel-based estimator of $\theta^*(u)$ as

$$ \hat{\theta}(u) := \underset{\theta \in \Theta}{\operatorname{argmin}} \frac{1}{nh_n} \sum_{j=1}^{n} \Phi((X_{j-i}^{(n)})_{i \in \mathbb{N}}, \theta) K\left(\frac{\frac{j}{n} - u}{h_n}\right), \quad (9) $$

where:

* $(h_n)_{n \in \mathbb{N}}$ is a bandwidth sequence of positive real numbers such that

$$ nh_n^{1+2\rho} \xrightarrow{n \to \infty} 0 \quad \text{and} \quad nh_n \xrightarrow{n \to \infty} \infty; $$

* $K: \mathbb{R} \to \mathbb{R}$ is a piecewise differentiable kernel function with a compact support, i.e.

$$ \int_{\mathbb{R}} K(x)dx = 1 \quad \text{and there exists } c > 0 \text{ such as } K(x) = 0 \text{ for } |x| \ge c. $$

In Theorem 3.1 of [3], under all these assumptions, the pointwise consistency of $\hat{\theta}(\cdot)$ is established, i.e. $\hat{\theta}(u) \xrightarrow[n \to \infty]{\mathbb{P}} \theta^*(u)$ for any $u \in (0, 1)$.

Moreover, after further assuming, with the tangent process $(\tilde{X}_t(u))_t$ of $(X_t^{(n)})_t$ in $u$ defined by $\tilde{X}_t(u) = F_{\theta^*(u)}((\tilde{X}_{t-k}(u))_{k \ge 1}, \xi_t)$ for $t \in \mathbb{Z}$, that

* $\mathbb{E}[\|\partial_{\theta} \Phi((\tilde{X}_k(u))_{k \le 0}, \theta^*(u))\|^2] < \infty$ and $\Sigma(\theta^*(u))$ is a definite positive matrix with

$$ \Sigma(\theta^*(u)) = \left( \int_{\mathbb{R}} K^2(x)dx \right) \mathbb{E} \left[ \partial_{\theta} \Phi((\tilde{X}_{-k}(u))_{k \in \mathbb{N}}, \theta^*(u)) \left( \partial_{\theta} \Phi((\tilde{X}_{-k}(u))_{k \in \mathbb{N}}, \theta^*(u)) \right)^{\top} \right]; $$

* $\Gamma(\theta^*(u)) = \mathbb{E}[\partial_{\theta^2}^2 \Phi((\tilde{X}_{-k}(u))_{k \in \mathbb{N}}, \theta^*(u))]$ is a positive definite matrix,

it is shown in Theorem 3.3, that for every $u \in (0, 1)$,

$$ \sqrt{nh_n} (\hat{\theta}(u) - \theta^*(u)) \xrightarrow[n \to \infty]{\mathcal{L}} \mathcal{N}_d \left( 0, \Gamma^{-1}(\theta^*(u)) \Sigma(\theta^*(u)) \Gamma^{-1}(\theta^*(u)) \right). \quad (10) $$

# 3. Multivariate asymptotic normality and Wald tests

## 3.1. A multivariate extension of the asymptotic normality

A first step before setting up Wald tests for locally stationary time series satisfying (6) is to establish a multivariate version of the asymptotic normality theorem (10).

To do this, we start by choosing $0 < u_1 < u_2 < \dots < u_J < 1$ where $J$ is fixed and not dependent on $n$. We can then prove the following result:
**Theorem 3.1.** For any $(\lambda_1, \dots, \lambda_J) \in (\mathbb{R}^d)^J$, then

$$ \sqrt{nh_n} \left( \sum_{j=1}^{J} \lambda_j^\top (\hat{\theta}(u_j) - \boldsymbol{\theta}^*(u_j)) \right) \xrightarrow[n \to \infty]{\mathcal{L}} \mathcal{N}_d \left( 0, \sum_{j=1}^{J} \lambda_j^\top \Gamma^{-1}(\boldsymbol{\theta}^*(u_j)) \Sigma(\boldsymbol{\theta}^*(u_j)) \Gamma^{-1}(\boldsymbol{\theta}^*(u_j)) \lambda_j \right). \quad (11) $$

In particular, this theorem shows that if $u_i \neq u_j$ then the asymptotic distributions of $\sqrt{nh_n}(\hat{\theta}(u_i) - \boldsymbol{\theta}^*(u_i), \hat{\theta}(u_j) - \boldsymbol{\theta}^*(u_j))$ is that of 2 independent Gaussian vectors. In order to obtain Wald tests from (11), for $0 < u < 1$ one will use estimators following asymptotic covariance matrices:

$$ \hat{\Sigma}(u) = \frac{\int_{\mathbb{R}} K^2(x) dx}{nh_n} \sum_{t=1}^{n} K\left(\frac{\frac{t}{n} - u}{h_n}\right) \partial_{\boldsymbol{\theta}} \Phi\left(\left(X_{t-i}^{(n)}\right)_{i \in \mathbb{N}}, \hat{\theta}(u)\right) \partial_{\boldsymbol{\theta}} \Phi\left(\left(X_{t-i}^{(n)}\right)_{i \in \mathbb{N}}, \hat{\theta}(u)\right)^{\top} \\ \hat{\Gamma}(u) = \frac{1}{nh_n} \sum_{t=1}^{n} K\left(\frac{\frac{t}{n} - u}{h_n}\right) \partial_{\boldsymbol{\theta}^2}^2 \Phi\left(\left(X_{t-i}^{(n)}\right)_{i \in \mathbb{N}}, \hat{\theta}(u)\right). $$

Taking up what has already been done in [3], with the consistency of $\hat{\theta}(u_i)$ and Slutsky's theorem, we can transform the asymptotic normality (11) into the one below:

$$ \sqrt{nh_n} \left( \sum_{j=1}^{J} \lambda_j^\top (\hat{\Gamma}(u_j) (\hat{\Sigma}(u_j))^{-1} \hat{\Gamma}(u_j))^{1/2} (\hat{\theta}(u_j) - \boldsymbol{\theta}^*(u_j)) \right) \xrightarrow[n \to \infty]{\mathcal{L}} \mathcal{N}_d \left( 0, \sum_{j=1}^{J} \|\lambda_j\|^2 \right). \quad (12) $$

## 3.2. A Wald test for the function $\boldsymbol{\theta}^*(\cdot)$

A first use of the (12) theorem is to set up a Wald "goodness-of-fit" test for the function $\boldsymbol{\theta}^*(\cdot)$. If we consider a function fixed in advance $\boldsymbol{\theta}_0(\cdot)$, the test problem is:

$$ \left\{ \begin{array}{ll} \mathbf{H}_0: & \boldsymbol{\theta}^*(\cdot) = \boldsymbol{\theta}_0(\cdot); \\ & \text{vs} \\ \mathbf{H}_1: & \boldsymbol{\theta}^*(\cdot) \neq \boldsymbol{\theta}_0(\cdot). \end{array} \right. $$

Quite naturally, the Wald test statistic for this test problem is as follows:

$$ \hat{W}_n = nh_n \sum_{j=1}^{J} \left\| \left( \hat{\Gamma}(u_j) (\hat{\Sigma}(u_j))^{-1} \hat{\Gamma}(u_j) \right)^{1/2} (\hat{\theta}(u_j) - \boldsymbol{\theta}_0(u_j)) \right\|^2 \quad (13) $$

Using (12), we show:

**Proposition 3.1.** Under the assumption $\mathbf{H}_0$, then $\hat{W}_n \xrightarrow[n \to \infty]{\mathcal{L}} \chi^2(dJ)$.
### 3.3. A stationarity test for locally stationary time series

The asymptotic normality (12) will also enable us to test the possible stationarity of a time series. Under the $H_0$ hypothesis of stationarity, it is clear that the function $\theta^*(\cdot)$ is necessarily constant. Hence the test problem is:

$$ 
\left\{ 
\begin{array}{l}
\mathbf{H}_0: (X_t^{(n)})_t \text{ is a stationary time series} \\
\qquad \implies \exists \theta_0 \in \mathbb{R}^d \text{ such that } \theta^*(u) = \theta_0 \text{ for all } u \in [0, 1]; \\
\text{vs} \\
\mathbf{H}_1: (X_t^{(n)})_t \text{ is not a stationary time series} \\
\qquad \implies \neg \theta_0 \in \mathbb{R}^d \text{ such that } \theta^*(u) = \theta_0 \text{ for all } u \in [0, 1].
\end{array}
\right.
$$

Then, without knowing $\theta_0$, the following Wald test statistic can be considered:

$$ 
\begin{gathered}
\hat{S}_n = nh_n \sum_{j=1}^{J} \left\| (\hat{\Gamma}(u_j) (\hat{\Sigma}(u_j))^{-1} \hat{\Gamma}(u_j))^{1/2} \hat{\theta}(u_j) - \bar{\theta} \right\|^2 \\
\text{with } \bar{\theta} = \frac{1}{J} \sum_{j=1}^{J} (\hat{\Gamma}(u_j) (\hat{\Sigma}(u_j))^{-1} \hat{\Gamma}(u_j))^{1/2} \hat{\theta}(u_j). \quad (14)
\end{gathered}
$$

Using (12), we show:

**Proposition 3.2.** Under the assumption $H_0$, then $\hat{S}_n \xrightarrow{n \to \infty} \chi^2(d(J-1))$.

It is even possible to simplify the previous test a little: under the assumption $H_0$, the matrix $\Gamma(\theta^*(u_j))(\Sigma(\theta^*(u_j)))^{-1}\Gamma(\theta^*(u_j))$ is the same for all $u_j \in (0, 1)$. It is therefore more interesting to estimate this matrix only once, but over the whole trajectory $(X_t^{(n)})_{1 \le t \le n}$, an estimate we will note:

$$ \hat{\Gamma}_n \hat{\Sigma}_n^{-1} \hat{\Gamma}_n \quad \text{with} \quad \left\{ \begin{array}{ll} \hat{\Sigma}_n = \hat{\Sigma}(1/2) & \text{with } h_n = 1 \\ \hat{\Gamma}_n = \hat{\Gamma}(1/2) & \text{with } h_n = 1 \end{array} \right. $$

**Corollary 3.1.** Under the same assumptions as Proposition 3.2, the latter is still valid when we replace $\hat{\Gamma}(u_j)(\hat{\Sigma}(u_j))^{-1}\hat{\Gamma}(u_j)$ by $\hat{\Gamma}_n \hat{\Sigma}_n^{-1} \hat{\Gamma}_n$ in the expression of $\hat{S}_n$.

## 4. Examples

### 4.1. Time-varying causal affine processes and Gaussian QMLE

We consider a time-varying causal affine processes, that is the local stationary extension of causal affine processes defined in [4], and is defined by

$$ 
\left\{ 
\begin{array}{ll}
X_t^{(n)} = M_{\theta^*(\frac{t}{n})}((X_{t-i}^{(n)})_{1 \le i} \xi_t + f_{\theta^*(\frac{t}{n})}((X_{t-i}^{(n)})_{1 \le i}) & \text{for every } 1 \le t \le n \\
X_t^{(n)} = 0 & \text{for every } t \le 0,
\end{array}
\right.
\quad (15)
$$
for any $n \in \mathbb{N}^*$, with $\theta^* : [0, 1] \mapsto \Theta$ a compact set of $\mathbb{R}^d$ and for every $\theta \in \Theta$,
$f_{\theta}: \mathbb{R}^{\infty} \to \mathbb{R}$ and $M_{\theta}: \mathbb{R}^{\infty} \to [0, \infty)$ are $C^1(\mathbb{R}^{\infty})$ functions such that

$$
\sum_{j=1}^{\infty} \left( \sup_{x \in \mathbb{R}^{\infty}} |\partial_{x_j} f_{\theta}(x)| + \|\xi_0\|_4 \sup_{x \in \mathbb{R}^{\infty}} |\partial_{x_j} M_{\theta}(x)| \right) < 1 \quad \text{for any } \theta \in \Theta, \quad (16)
$$

where we suppose that $\|\xi_0\|_4 = (\mathbb{E}[\xi_0^4])^{1/4} < \infty$. From these assumptions,
for any $u \in [0,1], (\tilde{X}_t(u))_{t \in \mathbb{Z}}$ defined by $\tilde{X}_t(u) = M_{\theta^*}(u)((\tilde{X}_{t-i}(u))_{1 \le i}) \xi_t +$
$f_{\theta^*}(u)((\tilde{X}_{t-i}(u))_{1 \le i})$ for any $t \in \mathbb{Z}$ is a stationary and $\mathbb{L}^4$ process.

In such a framework this is interesting to consider $\Phi$ as $(-2)$ times the Gaussian
conditional log-density, inducing

$$
\Phi_G(x, \boldsymbol{\theta}) = \log (H_{\boldsymbol{\theta}}((x_i)_{i \ge 2})) + \frac{(x_1 - f_{\boldsymbol{\theta}}((x_i)_{i \ge 2}))^2}{H_{\boldsymbol{\theta}}((x_i)_{i \ge 2})} \quad \text{with } H_{\boldsymbol{\theta}}(\cdot) = M_{\boldsymbol{\theta}}^2(\cdot). \tag{17}
$$

The M-estimator resulting from this contrast is the Gaussian Quasi-Maximum
Likelihood estimator (QMLE), notably used for estimating the parameters of
GARCH processes, but also for ARMA, APARCH, ARMA-GARCH processes...
As in [3], we will add the following assumptions on the functions $f_{\theta}$ and $M_{\theta}$ :

1. There exists $\underline{M} > 0$ such as $M_{\theta} \ge \underline{M}$ for all $\theta \in \Theta$;

2. For all $\boldsymbol{\theta}, \boldsymbol{\theta}' \in \Theta$,

$$
(f_{\boldsymbol{\theta}} = f_{\boldsymbol{\theta}'} \text{ and } M_{\boldsymbol{\theta}} = M_{\boldsymbol{\theta}'}) \implies \boldsymbol{\theta} = \boldsymbol{\theta}'
$$

3. For every $u \in [0, 1]$, we have

$$
\left( \sum_{j=1}^{d} \mu_j \frac{\partial}{\partial \theta_j} f_{\theta^*(u)}((\tilde{X}_{-k}(u))_{k \in \mathbb{N}}) \right) = 0 \quad \text{a.s.} \implies \mu_j = 0, j = 1, \dots, d,
$$

$$
\text{or} \quad \left( \sum_{j=1}^{d} \mu_j \frac{\partial}{\partial \theta_j} H_{\theta^*(u)}((\tilde{X}_{-k}(u))_{k \in \mathbb{N}}) \right) = 0 \quad \text{a.s.} \implies \mu_j = 0, j = 1, \dots, d.
$$

4. The functions $f_{\theta}$ and $M_{\theta}$ are such that:

$$
\sum_{j=1}^{\infty} \sup_{x \in \mathbb{R}^{\infty}, \theta \in \Theta} \left\{ j \log j \left( |\partial_{x_j} f_{\theta}(x)| + |\partial_{x_j} H_{\theta}(x)| \right) + j \left( |\partial_{\theta, x_j}^2 f_{\theta}(x)| + |\partial_{\theta, x_j}^2 H_{\theta}(x)| \right) \right. \\
\left. \qquad + \left( |\partial_{\theta, x_j}^3 f_{\theta}(x)| + |\partial_{\theta, x_j}^3 H_{\theta}(x)| \right) \right\} < \infty.
$$

We can then use Proposition 4.1 of [3] and we have:
**Proposition 4.1.** Under all these assumptions, Theorem 3.1, Propositions 3.1 and 3.2 hold, with

$$ egin{cases} 
\Sigma(\boldsymbol{\theta}^*(u)) = \left(\int_{\mathbb{R}} K^2(x)dx\right) \mathbb{E} \left[ \frac{\partial_{\boldsymbol{\theta}} f_{\boldsymbol{\theta}^*(u)}^0 (\partial_{\boldsymbol{\theta}} f_{\boldsymbol{\theta}^*(u)}^0)^\top}{H_{\boldsymbol{\theta}^*(u)}^0} + (\mu_4 - 1) \frac{\partial_{\boldsymbol{\theta}} H_{\boldsymbol{\theta}^*(u)}^0 (\partial_{\boldsymbol{\theta}} H_{\boldsymbol{\theta}^*(u)}^0)^\top}{(H_{\boldsymbol{\theta}^*(u)}^0)^2} \right] \\ 
\Gamma(\boldsymbol{\theta}^*(u)) = \mathbb{E} \left[ \frac{\partial_{\boldsymbol{\theta}} f_{\boldsymbol{\theta}^*(u)}^0 (\partial_{\boldsymbol{\theta}} f_{\boldsymbol{\theta}^*(u)}^0)^\top}{H_{\boldsymbol{\theta}^*(u)}^0} + \frac{\partial_{\boldsymbol{\theta}} H_{\boldsymbol{\theta}^*(u)}^0 (\partial_{\boldsymbol{\theta}} H_{\boldsymbol{\theta}^*(u)}^0)^\top}{(H_{\boldsymbol{\theta}^*(u)}^0)^2} \right] 
\end{cases} $$

where $f_{\boldsymbol{\theta}^*(u)}^0 = f_{\boldsymbol{\theta}}((\tilde{X}_{-k}(u))_{k \in \mathbb{N}})$, $H_{\boldsymbol{\theta}}^0 = H_{\boldsymbol{\theta}}((\tilde{X}_{-k}(u))_{k \in \mathbb{N}})$ and $\mu_4 = \mathbb{E}[\xi_0^4]$.

## I. Example 1: Application to time-varying ARMA(p, q) processes

For any $n \in \mathbb{N}^*$, define a time-varying ARMA(p, q) process by

$$ egin{cases} 
X_t^{(n)} = \sum_{i=1}^{p} \phi_i \left(\frac{t}{n}\right) X_{t-i}^{(n)} + \sigma \left(\frac{t}{n}\right) \left(\xi_t + \sum_{j=1}^{q} \psi_j \left(\frac{t}{n}\right) \xi_{t-j}\right) & \text{for every } 1 \le t \le n \\ 
X_t^{(n)} = 0 & \text{for every } t \le 0 
\end{cases} 
\quad (18) $$

where $u \in [0, 1] \mapsto \boldsymbol{\theta}^*(u) = (\phi_1^*(u), \dots, \phi_p^*(u), \psi_1^*(u), \dots, \psi_q^*(u), (\sigma^*(u))^2) \in (-1, 1)^{p+q} \times (0, \infty)$. We assume that functions $\phi_1^*(\cdot), \dots, \phi_p^*(\cdot), \psi_1^*(\cdot), \dots, \psi_q^*(\cdot)$ and $(\sigma^*(\cdot))^2$ are $\rho$-Hölderian functions, and therefore $\boldsymbol{\theta}^*(\cdot)$ satisfies (5).

Note that Dahlhaus in [7] for Gaussian time-varying ARMA processes (using Whittle likelihood approximation) and by Azrak and Mélard in [1], obtained the asymptotic normality of point estimates of $\boldsymbol{\theta}^*(u)$. In [3], this asymptotic normality was obtained with an estimator based on the QMLE version, when

adding the following condition: $\sum_{i=1}^p |\phi_i^*(u)| + 2\sum_{j=1}^q |\psi_j^*(u)| < 1$ for any $u \in [0, 1]$.

Note that such a condition obtained in [3] is very restrictive in the $q \ge 1$ case, but close to being optimal for a pure AR(p) autoregressive model (i.e. in the $q = 0$ case). This leads to the following corollary:

**Corollary 4.1.** If $(X_t^{(n)})$ is a time-varying ARMA(p, q) process defined in (18), Theorem 3.1, Propositions 3.1 and 3.2 hold with

$$ \Sigma(\boldsymbol{\theta}^*(u)) = \frac{1}{(\sigma^*(u))^2} \left( \int_{\mathbb{R}} K^2(x)dx \right) \left( \begin{array}{cc} \mathbb{E}[\partial_{\boldsymbol{\theta}} f_{\boldsymbol{\theta}^*(u)}^0 (\partial_{\boldsymbol{\theta}} f_{\boldsymbol{\theta}^*(u)}^0)^\top] & 0 \\ 0 & \frac{\mu_4 - 1}{(\sigma^*(u))^2} \end{array} \right) $$

$$ \text{and } \Gamma(\boldsymbol{\theta}^*(u)) = \frac{1}{(\sigma^*(u))^2} \left( \begin{array}{cc} \mathbb{E}[\partial_{\boldsymbol{\theta}} f_{\boldsymbol{\theta}^*(u)}^0 (\partial_{\boldsymbol{\theta}} f_{\boldsymbol{\theta}^*(u)}^0)^\top] & 0 \\ 0 & \frac{1}{(\sigma^*(u))^2} \end{array} \right) $$
with $f_{\boldsymbol{\theta}^*(u)}^t = \sum_{k=1}^p \phi_k(u) \tilde{X}_{t-k}(u) + \sum_{\ell=1}^q \psi_\ell(u) (\tilde{X}_{t-\ell}(u) - f_{\boldsymbol{\theta}^*(u)}^{t-\ell})$ for any $t \in \mathbb{Z}$.

In the special case of an AR(p) time-varying process with no moving average part, then the matrices $\Sigma(\boldsymbol{\theta}^*(u))$ and $\Gamma(\boldsymbol{\theta}^*(u))$ simplify significantly and we obtain:

$$ \hat{\Gamma}(u) (\hat{\Sigma}(u))^{-1} \hat{\Gamma}(u) = \frac{\hat{\sigma}^{-2}(u)}{\int_{\mathbb{R}} K^2(x)dx} \begin{pmatrix} \hat{r}_0(u) & \hat{r}_1(u) & \cdots & \hat{r}_{p-1}(u) & 0 \\ \hat{r}_1(u) & \hat{r}_0(u) & \cdots & \hat{r}_{p-2}(u) & 0 \\ \hat{r}_2(u) & \hat{r}_1(u) & \cdots & \hat{r}_{p-3}(u) & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ \hat{r}_{p-1}(u) & \hat{r}_{p-2}(u) & \cdots & \hat{r}_0(u) & 0 \\ 0 & 0 & \cdots & 0 & \frac{\hat{\sigma}^{-2}(u)}{\hat{\mu}_4 - 1} \end{pmatrix} $$
with $$ \begin{cases} \hat{\mu}_4 = \frac{1}{nh_n} \sum_{t=1}^n (X_t^{(n)} - \sum_{j=1}^p \hat{\phi}_j(u) X_{t-j}^{(n)})^4 K(\frac{t}{n} - u) \\ \hat{r}_i(u) = \frac{1}{nh_n} \sum_{k=1}^{n-i} X_t^{(n)} X_{t+i}^{(n)} K(\frac{t}{n} - u) \quad \text{for } 0 \le i \le p-1 \end{cases} $$

In the case of ARMA(p, q) time-varying processes, it is also possible to implement numerical algorithms to obtain $\hat{\Gamma}(u) (\hat{\Sigma}(u))^{-1} \hat{\Gamma}(u)$. Indeed, the following computation by iterations can be done:

$$ \hat{f}^t(u) = \sum_{k=1}^{p} \hat{\phi}_k(u) X_{t-k}^{(n)} + \sum_{\ell=1}^{q} \hat{\psi}_{\ell}(u) (X_{t-\ell}^{(n)} - \hat{f}^{t-\ell}(u)) \quad \text{for } t \ge 1 \text{ and } \hat{f}^j(u) = 0 \text{ for } j \le 0. $$

Next, we use another iteration calculation to estimate $\partial_{\boldsymbol{\theta}} f_{\boldsymbol{\theta}^*(u)}^t$, i.e. for $1 \le t \le n$,

$$ \left\{ \begin{aligned} \partial_{\phi_i} \hat{f}^t(u) &= X_{t-i}^{(n)} - \sum_{\ell=1}^q \hat{\psi}_\ell(u) \partial_{\phi_i} \hat{f}^{t-\ell}(u) \\ &\quad \text{and } \partial_{\phi_i} \hat{f}^t(u) = 0 \quad \text{for } t \le 0 \quad (1 \le i \le p) \\ \partial_{\psi_i} \hat{f}^t(u) &= (X_{t-i}^{(n)} - \hat{f}^{t-i}(u)) - \sum_{\ell=1}^q \hat{\psi}_\ell(u) \partial_{\psi_i} \hat{f}^{t-\ell}(u) \\ &\quad \text{and } \partial_{\psi_i} \hat{f}^t(u) = 0 \quad \text{for } t \le 0 \quad (1 \le i \le q) \end{aligned} \right. $$

Then we define:

$$ \hat{\Gamma}(u) (\hat{\Sigma}(u))^{-1} \hat{\Gamma}(u) = \frac{1}{nh_n} \frac{1}{\hat{\sigma}^2(u)} \left( \int_{\mathbb{R}} K^2(x)dx \right)^{-1} \\ \times \sum_{t=1}^{n} K\left(\frac{t}{n} - u \middle| h_n\right) \begin{pmatrix} \partial_{\phi_1} \hat{f}^t(u) \\ \vdots \\ \partial_{\psi_q} \hat{f}^t(u) \\ 0 \end{pmatrix} \begin{pmatrix} 0 \\ \hat{\sigma}^{-2}(u) \\ \hat{\mu}_4 - 1 \end{pmatrix} $$
with

$$ 
\begin{cases}
\hat{\sigma}^2(u) = \frac{1}{nh_n} \sum_{t=1}^{n} (X_t^{(n)} - \hat{f}^t(u))^2 K\left(\frac{t}{n} - \frac{u}{h_n}\right) \\
\hat{\mu}_4 = \frac{1}{nh_n} \sum_{t=1}^{n} K\left(\frac{t}{n} - \frac{u}{h_n}\right) \frac{(X_t^{(n)} - \hat{f}^t(u))^4}{\hat{\sigma}^4(u)}
\end{cases}
$$

## II. Example 2: Application to time-varying GARCH(p, q) processes

As a particular case, we also define the time-varying GARCH(p, q) (see for instance [12]) such that for $n \in \mathbb{N}^*$,

$$ 
\left\{ 
\begin{array}{ll}
X_t^{(n)} = \sigma_t^{(n)} \xi_t & \text{for } 1 \le t \le n \\
(\sigma_t^{(n)})^2 = \omega(\frac{t}{n}) + \sum_{j=1}^{p} \alpha_j(\frac{t}{n}) (X_{t-j}^{(n)})^2 + \sum_{j=1}^{q} \beta_j(\frac{t}{n}) (\sigma_{t-j}^{(n)})^2 & \text{for } 1 \le t \le n \\
X_t^{(n)} = 0 \quad \text{and} \quad \sigma_t^{(n)} = 0 & \text{for } t \le 0
\end{array}
\right. 
\quad (19) 
$$

where $\theta(u) = (\omega(u), \alpha_1(u), \dots, \alpha_p(u), \beta_1(u), \dots, \beta_q(u))^\top$ is a vector of non-positive real numbers for every $u \in [0, 1]$ and $\omega(u) \ge \omega^* > 0$. We will assume that the functions $\phi_1^*(\cdot), \dots, \phi_p^*(\cdot), \psi_1^*(\cdot), \dots, \psi_q^*(\cdot)$ and $(\sigma^*(\cdot))^2$ are $\rho$-Hölderian functions, and therefore $\theta^*(\cdot)$ satisfies (5). We add another condition to guarantee stationarity and a moment of order 4 for the tangent process:

$$ \sum_{j=1}^{q} \beta_j(u) + \|\xi_0\|_4^2 \sum_{i=1}^{p} \alpha_i(u) < 1 \quad \text{for any } u \in [0, 1]. $$

**Corollary 4.2.** If $(X_t^{(n)})$ is a time-varying GARCH(p, q) process defined in (19), Theorem 3.1, Propositions 3.1 and 3.2 hold with

$$ 
\left\{ 
\begin{aligned}
\Sigma(\boldsymbol{\theta}^*(u)) &= (\mu_4 - 1) \left( \int_{\mathbb{R}} K^2(x)dx \right) \mathbb{E} \left[ \frac{\partial_{\boldsymbol{\theta}} H_{\boldsymbol{\theta}^*(u)}^0 (\partial_{\boldsymbol{\theta}} H_{\boldsymbol{\theta}^*(u)}^0)^\top}{(H_{\boldsymbol{\theta}^*(u)}^0)^2} \right] \\
\Gamma(\boldsymbol{\theta}^*(u)) &= \mathbb{E} \left[ \frac{\partial_{\boldsymbol{\theta}} H_{\boldsymbol{\theta}^*(u)}^0 (\partial_{\boldsymbol{\theta}} H_{\boldsymbol{\theta}^*(u)}^0)^\top}{(H_{\boldsymbol{\theta}^*(u)}^0)^2} \right] = \frac{1}{\mu_4 - 1} \left( \int_{\mathbb{R}} K^2(x)dx \right)^{-1} \Sigma(\boldsymbol{\theta}^*(u))
\end{aligned}
\right. 
$$

with $H_{\boldsymbol{\theta}^*(u)}^t = \omega(u) + \sum_{j=1}^{p} \alpha_j(u) (\tilde{X}_{t-j}(u))^2 + \sum_{j=1}^{q} \beta_j(u) H_{\boldsymbol{\theta}^*(u)}^{t-j}$ for any $t \in \mathbb{Z}$.

From a practical point of view, it is especially interesting to be able to calculate $\hat{\Gamma}(u) (\hat{\Sigma}(u))^{-1} \hat{\Gamma}(u)$ and we can already see that this will amount to calculating
$\frac{1}{\hat{\mu}_4 - 1} \left( \int_{\mathbb{R}} K^2(x) dx \right)^{-1} \hat{\Gamma}(u)$. To do this, we start by calculating by iteration:

$\hat{H}^t(u) = \hat{\omega}(u) + \sum_{j=1}^{p} \hat{\alpha}_j(u) (X_{t-j}^{(n)})^2 + \sum_{j=1}^{q} \hat{\beta}_j(u) \hat{H}^{t-j}(u) \text{ for } t \ge 1 \text{ and } \hat{H}^t(u) = 0 \text{ for } t \le 0$.

Next, we use another iteration calculation to estimate $\partial_{\theta} H_{\theta^*}^t(u)$, i.e. for $1 \le t \le n$, $1 \le i \le p$ and $1 \le j \le q$,

$$ \left\{ \begin{array}{ll} \partial_{\omega} \hat{H}^t(u) = 1 + \sum_{j=1}^q \hat{\beta}_j(u) \partial_{\omega} \hat{H}^{t-j}(u) & \text{and } \partial_{\omega} \hat{H}^t(u) = 0 \text{ for } t \le 0 \\ \partial_{\alpha_i} \hat{H}^t(u) = (X_{t-i}^{(n)})^2 + \sum_{j=1}^q \hat{\beta}_j(u) \partial_{\alpha_i} \hat{H}^{t-j}(u) & \text{and } \partial_{\alpha_i} \hat{H}^t(u) = 0 \text{ for } t \le 0 \\ \partial_{\beta_i} \hat{H}^t(u) = \hat{H}^{t-i}(u) + \sum_{j=1}^q \hat{\beta}_j(u) \partial_{\beta_i} \hat{H}^{t-j}(u) & \text{and } \partial_{\beta_i} \hat{H}^t(u) = 0 \text{ for } t \le 0 \end{array} \right. . $$

Then we define:

$$ \left\{ \begin{array}{l} \hat{\Gamma}(u) = \frac{1}{nh_n} \sum_{t=1}^{n} K\left(\frac{\frac{t}{n}-u}{h_n}\right) \frac{1}{(\hat{H}^t(u))^2} \partial_{\theta} \hat{H}^t(u)^{\top} \partial_{\theta} \hat{H}^t(u) \\ \\ \hat{\mu}_4 = \frac{1}{nh_n} \sum_{t=1}^{n} K\left(\frac{\frac{t}{n}-u}{h_n}\right) \frac{(X_t^{(n)})^4}{(\hat{H}^t(u))^2} , \end{array} \right. $$

$$ \text{with } \partial_{\theta} \hat{H}^t(u) = (\partial_{\omega} \hat{H}^t(u), \partial_{\alpha_1} \hat{H}^t(u), \dots, \partial_{\alpha_p} \hat{H}^t(u), \partial_{\beta_1} \hat{H}^t(u), \dots, \partial_{\beta_q} \hat{H}^t(u)) \\ \text{and thus } \hat{\Gamma}(u) (\hat{\Sigma}(u))^{-1} \hat{\Gamma}(u) = \frac{1}{\hat{\mu}_4 - 1} \left( \int_{\mathbb{R}} K^2(x) dx \right)^{-1} \hat{\Gamma}(u) \text{ can be computed.} $$

## 4.2. Time-varying GLARCH(p, q) processes and least square contrast

Here we will study the case of tv-GLARCH($p, q$) processes, which are special cases of tv-LARCH($\infty$) processes. Stationary LARCH($\infty$) process have been introduced by Robinson in [23] and studied intensively by Giraitis et al. in [17]. A tv-GLARCH($p, q$) process can be defined as for $n \in \mathbb{N}^*$,

$$ \left\{ \begin{array}{ll} X_t^{(n)} = \sigma_t^{(n)} \xi_t & \text{for every } 1 \le t \le n \\ \sigma_t^{(n)} = c_0 \left(\frac{t}{n}\right) + \sum_{i=1}^p c_i \left(\frac{t}{n}\right) X_{t-i}^{(n)} + \sum_{j=1}^q d_j \left(\frac{t}{n}\right) \sigma_{t-j}^{(n)} & \text{for every } 1 \le t \le n \\ X_t^{(n)} = 0 \quad \text{and} \quad \sigma_t^{(n)} = 0 & \text{for every } t \le 0 \end{array} \right. \quad (20) $$

where $\theta(u) = (c_0(u), c_1(u), \dots, c_p(u), d_1(u), \dots, d_q(u))^\top$ is a vector of non-positive real numbers for every $u \in [0, 1]$ and $c_0(u) \ge c_0 > 0$. We will assume that the functions $c_0(\cdot), c_1(\cdot), \dots, c_p(\cdot), d_1(\cdot), \dots, d_q(\cdot)$ are $\rho$-Hölderian functions, and therefore $\theta^*(\cdot)$ satisfies (5).

In the general context of LARCH($\infty$) processes, or more specifically GLARCH($p, q$) processes, the use of a quasi-maximum likelihood contrast is not appropriate, as the conditional variance $\sigma_t^2$ of the process may cancel out (on this question, see
the excellent work by Francq and Zakoïan in [16]). We would rather consider here the following ordinary LS-contrast of square values: for $x \in \mathbb{R}^{\infty}$, define

$$ \Phi_{LARCH}(x, \boldsymbol{\theta}) = \left( x_1^2 - \left( c_0 + \sum_{j=1}^{\infty} a_j(\boldsymbol{\theta}) x_{j+1} \right)^2 \right)^2, \quad (21) $$

where $(a_j)$ is defined by

$$ c_0 + \sum_{j=1}^{\infty} a_j(\boldsymbol{\theta}) z^j = (c_0 + \sum_{i=1}^{p} c_i z^i) (1 + \sum_{\ell=1}^{\infty} d_{\ell}^{(-1)} z^{\ell}) \\ \text{and } (1 - \sum_{i=1}^{q} d_i z^i) (1 + \sum_{\ell=1}^{\infty} d_{\ell}^{(-1)} z^{\ell}) = 1 \text{ for any } z \in (-1, 1). $$

**Corollary 4.3.** If $\|\xi_0\|_8 < \infty$ and the probability distribution of $\xi_0$ is absolutely continuous with respect to the Lebesgue measure, then if $(X_t^{(n)})$ is a time-varying GLARCH($p, q$) process defined in (20), Theorem 3.1, Propositions 3.1 and 3.2 hold with

$$ \left\{ \begin{array}{l} \Sigma(\boldsymbol{\theta}^*(u)) = 16 (\mu_4 - 1) \left( \int_{\mathbb{R}} K^2(x) dx \right) \mathbb{E} \left[ \left( (M_{\boldsymbol{\theta}}^0(u))^6 (\partial_{\boldsymbol{\theta}} M_{\boldsymbol{\theta}}^0(u)) \right) (\partial_{\boldsymbol{\theta}} M_{\boldsymbol{\theta}}^0(u))^\top \right] \\ \Gamma(\boldsymbol{\theta}^*(u)) = 8 \mathbb{E} \left[ \left( (M_{\boldsymbol{\theta}}^0(u))^2 (\partial_{\boldsymbol{\theta}} M_{\boldsymbol{\theta}}^0(u)) \right) (\partial_{\boldsymbol{\theta}} M_{\boldsymbol{\theta}}^0(u))^\top \right] \end{array} \right. $$

$$ \text{with } M_{\boldsymbol{\theta}^*(u)}^t = c_0(u) + \sum_{j=1}^{p} c_j(u) \tilde{X}_{t-j}(u) + \sum_{j=1}^{q} d_j(u) M_{\boldsymbol{\theta}^*(u)}^{t-j} \text{ for any } t \in \mathbb{Z}. $$

As in the previous cases, we need to propose a method for numerically obtaining these two asymptotic covariance matrices. So, as in the case of GARCH($p, q$) time-varying, we start by defining:

$$ \widehat{M}^t(u) = \hat{c}_0(u) + \sum_{j=1}^{p} \hat{c}_j(u) X_{t-j}^{(n)} + \sum_{j=1}^{q} \hat{d}_j(u) \widehat{M}^{t-j}(u) \quad \text{for } t \ge 1 \text{ and } \widehat{M}^j(u) = 0 \text{ for } j \le 0. $$

Next, we use another iteration calculation to estimate $\partial_{\boldsymbol{\theta}} M_{\boldsymbol{\theta}^*(u)}^t$: for $1 \le t \le n$, $1 \le i \le p$ and $1 \le j \le q$,

$$ \left\{ \begin{array}{ll} \partial_{c_0} \widehat{M}^t(u) = 1 + \sum_{k=1}^q \hat{d}_k(u) \partial_{c_0} \widehat{M}^{t-k}(u) & \text{and } \partial_{c_0} \widehat{M}^t(u) = 0 \quad \text{for } t \le 0 \\ \partial_{c_i} \widehat{M}^t(u) = X_{t-i}^{(n)} + \sum_{k=1}^q \hat{d}_k(u) \partial_{c_i} \widehat{M}^{t-k}(u) & \text{and } \partial_{c_i} \widehat{H}^t(u) = 0 \quad \text{for } t \le 0 \\ \partial_{d_j} \widehat{M}^t(u) = \widehat{M}^{t-j}(u) + \sum_{k=1}^q \hat{d}_k(u) \partial_{d_j} \widehat{M}^{t-k}(u) & \text{and } \partial_{d_i} \widehat{M}^t(u) = 0 \quad \text{for } t \le 0 \end{array} \right. $$

$$ \text{Then, with } \partial_{\boldsymbol{\theta}} \widehat{M}_{\boldsymbol{\theta}}^t(u) = (\partial_{c_0} \widehat{M}^t(u), \partial_{c_1} \widehat{M}^t(u), \dots, \partial_{c_p} \widehat{M}^t(u), \partial_{d_1} \widehat{M}^t(u), \dots, \partial_{d_q} \widehat{M}^t(u))^\top, $$
we define:

$$
\left\{
\begin{aligned}
\hat{\Sigma}(u) &= \frac{16(\hat{\mu}_4 - 1)}{nh_n} \left( \int_{\mathbb{R}} K^2(x) dx \right) \sum_{t=1}^{n} K\left(\frac{\frac{t}{n}-u}{h_n}\right) (\hat{M}_{\theta}^t(u))^6 (\partial_{\theta}\hat{M}_{\theta}^t(u)) (\partial_{\theta}\hat{M}_{\theta}^t(u))^{\top} \\
\hat{\Gamma}(u) &= 8 \frac{1}{nh_n} \sum_{t=1}^{n} K\left(\frac{\frac{t}{n}-u}{h_n}\right) (\hat{M}_{\theta}^t(u))^2 (\partial_{\theta}\hat{M}_{\theta}^t(u)) (\partial_{\theta}\hat{M}_{\theta}^t(u))^{\top} \\
\hat{\mu}_4 &= \frac{\frac{1}{nh_n} \sum_{t=1}^{n} K\left(\frac{\frac{t}{n}-u}{h_n}\right) (X_t^{(n)})^4}{\frac{1}{nh_n} \sum_{t=1}^{n} K\left(\frac{\frac{t}{n}-u}{h_n}\right) (\hat{M}^t(u))^4}
\end{aligned}
\right.
$$

**Remark 4.1.** We note a different definition for $\hat{\mu}_4$ in the case of tv-GLARCH processes than those proposed for tv-ARMA or tv-GARCH processes. Indeed, if we were to use an empirical mean of $(\frac{X_t^{(n)}}{M^t(u)})_t^4$ as before, we would come up against the fact that $\hat{M}^t(u)$ can cancel out in the GLARCH case and for any $u$, and this empirical mean may not converge to $\mu_4$ (on this subject, see again Francq and Zakoïan in [16]). On the other hand, we know that $\mathbb{E}[(\tilde{X}_0(u))^4] = \mathbb{E}[(M_\theta^0(u)\xi_0)^4] = \mathbb{E}[(M_\theta^0(u))^4]\mathbb{E}[\xi_0^4]$ since $(\tilde{X}_t(u))_t$ is a causal stationary process and we deduce that $\mathbb{E}[\xi_0^4] = \mathbb{E}[(M_\theta^0(u)\xi_0)^4]/\mathbb{E}[(M_\theta^0(u))^4]$. This explains the expression of $\hat{\mu}_4$ given just above.

**5. Numerical experiments**

In the sequel we are going to apply the two tests seen above in several different cases of (local) stationary processes: tv-ARMA(1, 1), tv-GARCH(1, 1) and tv- GLARCH(2) processes.

With regard to the choice of the kernel function and the selected window size $h_n$, we have chosen an Epanechnikov kernel such as:

$$K(x) = \frac{3}{4} (1 - x^2) \mathbb{I}_{x \in [-1, 1]} \quad \text{and} \quad h_n = 2 n^{-\lambda} \text{ with } \lambda = 0.35,$$

inducing $n h_n \xrightarrow{n \to \infty} \infty$ and $n h_n^3 \xrightarrow{n \to \infty} 0$, which is the consistency and the asymptotic normality condition required for $\rho = 1$-Hölder function $\theta^*$.

**5.1. Monte Carlo simulations**

Here we will consider three examples of locally stationary processes and 4 cases:

* The cases of a locally stationary (LS1), (LS2) and (LS3), and non-stationary process, i.e. such that $\theta^*(\cdot)$ is not a constant function. Note that from (LS1) to (LS3), the functions get closer and closer to the constant case, and non-stationarity becomes harder and harder to detect;
* The case of a stationary (S) process, i.e. such that $\theta^*(\cdot)$ is a constant function.

Note that we generate all those processes from a Gaussian white noise ($\xi_t$). As regards the choice of parameters, we have chosen:

* The size $n$ of the trajectory is such that

$$n = 1000, 2000, 5000, 10000, 20000, 50000;$$

* The number $J$ of points $u_j$ chosen from a uniform grid of (0, 1) is such that $u_j + n h_n < u_{j+1} - n h_n$, so that the supports of the kernels do not overlap, which led us to consider:

<table><tr><td>n</td><td>1000</td><td>2000</td><td>5000</td><td>10000</td><td>20000</td><td>50000</td></tr><tr><td>J</td><td>2</td><td>3</td><td>4</td><td>6</td><td>8</td><td>11</td></tr></table>

* We carried out 5000 independent replications of each case to ensure that our results were sufficiently precise.

Finally, the results of the simulations will be presented in tables using the empirical rejection rate for the with 0.05 type I error. With $L = 5000$ independent replications, and for a test $\hat{T}$ with an asymptotic $\chi^2(d)$ distribution, the empirical rejection rate $\hat{R}_L$ is defined by:

$$\hat{R}_L = \frac{1}{L} \sum_{j=1}^{L} \mathbb{I}_{\hat{T}_j > q_{\chi^2(d)}(0.95)},$$

where $q_{\chi^2(d)}(0.95)$ is the quantile of order 0.95 of the $\chi^2(d)$ distribution.

## 1. Framework of tv-ARMA(1, 1) processes.

Here we consider a tv-ARMA(1, 1) process that follows the equation (18), with several possible functions defining its coefficients:

<table><thead><tr><th>Functions</th><th>loc. stat. 1 (LS1)</th><th>loc. stat. 2 (LS2)</th><th>loc. stat. 3 (LS3)</th><th>stat. (S)</th></tr></thead><tbody><tr><td>$\phi_1^*(u)$</td><td>$0.1 + 0.5 |\sin(3u)|$</td><td>$0.1 + 0.05 |\sin(3u)|$</td><td>$0.1 + 0.02 |\sin(3u)|$</td><td>$0.2$</td></tr><tr><td>$\psi_1^*(u)$</td><td>$0.1 + 0.5 u$</td><td>$0.1 + 0.05 u$</td><td>$0.1 + 0.02 u$</td><td>$0.5$</td></tr><tr><td>$\sigma^*(u)$</td><td>$1 + 0.5 \sin(5u)$</td><td>$1 + 0.05 \sin(5u)$</td><td>$1 + 0.02 \sin(5u)$</td><td>$0.3$</td></tr></tbody></table>

In the framework of tv-ARMA(1, 1) processes we use the QML estimator obtained from the contrast defined in (17) and the estimated matrix $\hat{\Gamma}(u) (\hat{\Sigma}(u))^{-1} \hat{\Gamma}(u)$ obtained in Section 4, I.

We begin by testing the goodness-of-fit of the function $\theta$ using the Wald test $\hat{W}_n$ defined in (3.1). We will confine ourselves here to cases (S) and
<table><caption>TABLE 1
Empirical rejection rate of the Wald test with 0.05 type I error for tv-ARMA processes</caption><thead><tr><th>Function θ<sup>*</sup>(u)</th><th>Case (S)</th><th>Case (LS1)</th></tr></thead><tbody><tr><td>n = 1000</td><td>0.141</td><td>0.139</td></tr><tr><td>n = 2000</td><td>0.106</td><td>0.107</td></tr><tr><td>n = 5000</td><td>0.086</td><td>0.083</td></tr><tr><td>n = 10000</td><td>0.075</td><td>0.072</td></tr><tr><td>n = 20000</td><td>0.061</td><td>0.061</td></tr></tbody></table>

<table><caption>TABLE 2
Empirical rejection rate of the stationarity test with 0.05 type I error for tv-ARMA processes</caption><thead><tr><th>Function θ<sup>*</sup>(u)</th><th>Case (S)</th><th>Case (LS1)</th><th>Case (LS2)</th><th>Case (LS3)</th></tr></thead><tbody><tr><td>n = 1000</td><td>0.048</td><td>1.000</td><td>0.205</td><td>0.100</td></tr><tr><td>n = 2000</td><td>0.053</td><td>1.000</td><td>0.287</td><td>0.129</td></tr><tr><td>n = 5000</td><td>0.054</td><td>1.000</td><td>0.613</td><td>0.156</td></tr><tr><td>n = 10000</td><td>0.052</td><td>1.000</td><td>0.903</td><td>0.190</td></tr><tr><td>n = 20000</td><td>0.055</td><td>1.000</td><td>1.000</td><td>0.343</td></tr></tbody></table>

(LS1), since cases (LS2) and (LS3) give virtually the same results, which are shown in Table 1:

We can then test the stationarity of processes (S), which is indeed stationary (and we expect an empirical size close to 0.95), and processes (LS1), (LS2) and (LS3), which are not stationary (and we expect an empirical size close to 0):

## 2. Framework of tv-GARCH(1, 1) processes.

Here we consider a tv-GARCH(1, 1) process that follows the equation (19), with several possible functions defining its coefficients:

<table><thead><tr><th>Functions</th><th>loc. stat. 1 (LS1)</th><th>loc. stat. 2 (LS2)</th><th>stat. (S)</th></tr></thead><tbody><tr><td>ω<sup>*</sup>(u)</td><td>1 + 0.5 sin(3u)</td><td>0.5 + 0.1 sin(3u)</td><td>1</td></tr><tr><td>α<sub>1</sub><sup>*</sup>(u)</td><td>0.4 - 0.3 |sin(2u)|</td><td>0.4 - 0.1 |sin(2u)|</td><td>0.3</td></tr><tr><td>β<sub>1</sub><sup>*</sup>(u)</td><td>0.1 + 0.3u</td><td>0.1 + 0.1u</td><td>0.5</td></tr></tbody></table>

We begin by testing the goodness-of-fit of the function $\theta$ using the Wald test $\hat{W}_n$ defined in (3.1). We will confine ourselves here to cases (S) and (LS1), since case (LS2) give the same results, which are shown in Table 3:

In Table 4, we have tested the stationarity of processes (S), which is indeed stationary (and we expect an empirical rejection rate of to 0.05), and processes (LS1) and (LS2), which are not stationary (and we expect an
<table><caption>TABLE 3
Empirical rejection rate of the Wald test with 0.05 type I error for tv-GARCH processes</caption><thead><tr><th>Function θ<sup>*</sup>(u)</th><th>Case (S)</th><th>Case (LS1)</th></tr></thead><tbody><tr><td>n = 1000</td><td>0.162</td><td>0.312</td></tr><tr><td>n = 2000</td><td>0.136</td><td>0.282</td></tr><tr><td>n = 5000</td><td>0.102</td><td>0.256</td></tr><tr><td>n = 10000</td><td>0.077</td><td>0.201</td></tr><tr><td>n = 20000</td><td>0.064</td><td>0.180</td></tr><tr><td>n = 50000</td><td>0.053</td><td>0.123</td></tr></tbody></table>

<table><caption>TABLE 4
Empirical rejection rate of the stationarity test with 0.05 type I error for tv-GARCH processes</caption><thead><tr><th>Function θ<sup>*</sup>(u)</th><th>Case (S)</th><th>Case (LS1)</th><th>Case (LS2)</th></tr></thead><tbody><tr><td>n = 1000</td><td>0.142</td><td>0.367</td><td>0.078</td></tr><tr><td>n = 2000</td><td>0.116</td><td>0.990</td><td>0.114</td></tr><tr><td>n = 5000</td><td>0.096</td><td>1.000</td><td>0.183</td></tr><tr><td>n = 10000</td><td>0.071</td><td>1.000</td><td>0.357</td></tr><tr><td>n = 20000</td><td>0.064</td><td>1.000</td><td>0.636</td></tr><tr><td>n = 50000</td><td>0.047</td><td>1.000</td><td>0.973</td></tr></tbody></table>

empirical rejection rate close to 1):

3. Framework of tv-GLARCH(1, 1) processes. Here we consider a tv-GLARCH(1, 1) process that follows the equation (20), with several possible functions defining its coefficients:

<table><thead><tr><th>Functions</th><th>loc. stat. 1 (LS1)</th><th>loc. stat. 2 (LS2)</th><th>stat. (S)</th></tr></thead><tbody><tr><td>ω<sup>*</sup>(u)</td><td>1 + 0.5 sin(3u)</td><td>0.5 + 0.1 sin(3u)</td><td>1</td></tr><tr><td>α<sub>1</sub><sup>*</sup>(u)</td><td>0.4 - 0.3 |sin(2u)|</td><td>0.4 - 0.1 |sin(2u)|</td><td>0.3</td></tr><tr><td>β<sub>1</sub><sup>*</sup>(u)</td><td>0.1 + 0.3u</td><td>0.1 + 0.1u</td><td>0.5</td></tr></tbody></table>

We begin by testing the goodness-of-fit of the function $\theta$ using the Wald test $\tilde{W}_n$ defined in (3.1). We will confine ourselves here to cases (S) and (LS1), since case (LS2) give the same results, which are shown in Table 5:

<table><caption>TABLE 5
Empirical rejection rate of the Wald test with 0.05 type I error for tv-GLARCH processes</caption><thead><tr><th>Function θ<sup>*</sup>(u)</th><th>Case (S)</th><th>Case (LS1)</th></tr></thead><tbody><tr><td>n = 1000</td><td>0.171</td><td>0.199</td></tr><tr><td>n = 2000</td><td>0.150</td><td>0.182</td></tr><tr><td>n = 5000</td><td>0.114</td><td>0.162</td></tr><tr><td>n = 10000</td><td>0.073</td><td>0.119</td></tr><tr><td>n = 20000</td><td>0.062</td><td>0.097</td></tr><tr><td>n = 50000</td><td>0.050</td><td>0.072</td></tr></tbody></table>
<table><caption>TABLE 6
Empirical rejection rate of the stationarity test with 0.05 type I error for tv-GLARCH processes</caption><thead><tr><th>Function θ*(u)</th><th>Case (S)</th><th>Case (LS1)</th><th>Case (LS2)</th></tr></thead><tbody><tr><td>n = 1000</td><td>0.089</td><td>0.659</td><td>0.060</td></tr><tr><td>n = 2000</td><td>0.087</td><td>1.000</td><td>0.117</td></tr><tr><td>n = 5000</td><td>0.083</td><td>1.000</td><td>0.338</td></tr><tr><td>n = 10000</td><td>0.078</td><td>1.000</td><td>0.705</td></tr><tr><td>n = 20000</td><td>0.069</td><td>1.000</td><td>0.958</td></tr><tr><td>n = 50000</td><td>0.064</td><td>1.000</td><td>1.000</td></tr></tbody></table>

We can also test the stationarity of processes (S), which is indeed stationary (and we expect an empirical rejection rate of to 0.05), and processes (LS1) and (LS2), which are not stationary (and we expect an empirical rejection rate close to 1). The results are shown in Table 6:

**Conclusions from the Monte-Carlo experiments:** From all these simulations, we can deduce:

* Convergence of the Wald-type goodness-of-fit test is clear, but very slow for all 3 types of process, and being in the stationary frame accelerates convergence only slightly. The explanation lies mainly in the estimation of the matrices $\Gamma(\theta^*(u_j)) (\Sigma(\theta^*(u_j)))^{-1} \Gamma(\theta^*(u_j))$ by the matrices $\hat{\Gamma}(u_j) (\hat{\Sigma}(u_j))^{-1} \hat{\Gamma}(u_j)$, which is very slow: Numerical tests in the stationary case, taking the matrix estimate over a very large trajectory, show that this significantly improves test efficiency.

* On the other hand, convergence under $H_0$ of the stationarity test is much faster and more satisfactory, and this can also be explained by the fact that under $H_0$ we use the matrix $\hat{\Gamma}_n (\hat{\Sigma}_n)^{-1} \hat{\Gamma}_n$ calculated on the entire trajectory (particularly in the ARMA case). And under the different cases (LS1), (LS2) or (LS3) of $H_1$, the power of the test tends well towards 1 if the size of the trajectory is sufficiently large in relation to the amplitude of the fluctuations of the coefficients of the locally stationary processes.

## 5.2. Application to financial data

We apply the stationarity test to a trajectory of financial data, that of the log-returns of the daily closing values of S&P500 index between October 1990 and October 2020 (therefore $n = 7556$, see also Figure 2 for the graph of this trajectory).

Many studies have shown that the GARCH(1, 1) process is a relevant model for this type of data (We refer to the monograph of Francq and Zakoïan [15] for more details). However, the socio-economic and political environment can
<figure><img src="image_2.png" /><figcaption>Fig 2: log-returns of daily closing values of S&P500 index between October 1990 and October 2020</figcaption></figure>

influence the "normal" functioning of financial markets (think in particular of
the September 2008 crisis and of the spring 2020 COVID crisis which can be
seen on the figure). It can therefore be interesting to model such a financial se-
ries by a tv-GARCH(1,1) process and see how the parameters evolve over time.
And that is what we have already done in [3], highlighting this evolution (with
notably an overall growth in the α₁ coefficient from 0.01 in 1990 to 0.22 in 2020
and a decrease in the β₁ coefficient from 0.98 in 1990 to 0.73 in 2020).

So we are going to go a step further and answer a first question: can we consider
this trajectory as that of a stationary GARCH(1, 1)? And the answer is clear:
with $J = 5$, $h_n = 2n^{0.35}$, we obtain $\hat{S}_n \simeq 376.25$ and the assumption $H_0$ of
stationarity is clearly rejected.

We have then considered two periods between 1990 and 2020 during which we
can perhaps accept a stationary GARCH(1, 1) process model:

* The first is the period before the September 2008 crisis: $n = 4520$ observations between October 1990 and the end of August 2008. With $J = 4$ and $h_n = 2n^{0.35}$, we obtain $\hat{S}_n \simeq 223.82$ and the assumption $H_0$ of stationarity is clearly rejected.

* The second is the period after September 2008 crisis and before the covid crisis of March 2020: $n = 2740$ observations between October 2008 and the end of February 2020. With $J = 3$ and $h_n = 2n^{0.35}$, we obtain $\hat{S}_n \simeq 9.36$: the p.value of the test is 0.154 and the assumption $H_0$ of stationarity is not rejected and the modeling of these data by a stationary GARCH(1, 1) process is legitimate.

**6. Proofs**

Proof of Theorem 3.1. A/ First, using Theorem 3.1 of [3], we have for all $j = 1, \dots, J$, $J, \hat{\theta}(u_j) \xrightarrow{n \to \infty} \theta^*(u_j)$ and therefore $\sum_{j=1}^J \lambda_j^T \hat{\theta}(u_j) \xrightarrow{n \to \infty} \sum_{j=1}^J \lambda_j^T \theta^*(u_j)$ for all $(\lambda_j)_{1 \le j \le J} \in (\mathbb{R}^d)^J$.
B/ Now, consider the proof of Theorem 3.3 of [3]. Some of the points in this proof can be taken up directly, but the asymptotic normality (91) must be reviewed and extended.

First define the column vectors $\tilde{\theta} = (\tilde{\theta}_i)_{1\le i\le dJ}$, $\hat{\theta} = (\hat{\theta}(u_1)^\top, \dots, \hat{\theta}(u_J)^\top)^\top$ and $\tilde{\theta}^* = (\theta^*(u_1)^\top, \dots, \theta^*(u_J)^\top)^\top$ of $\mathbb{R}^{dJ}$.

Then we can write that:

$$ \hat{\theta} = (\hat{\theta}(u_1)^\top, \dots, \hat{\theta}(u_J)^\top)^\top = \underset{\tilde{\theta} \in \Theta^J}{\operatorname{argmin}} \hat{W}_n(\tilde{\theta}) $$

$$ \text{with } \hat{W}_n(\tilde{\theta}) := \sum_{j=1}^{J} \sum_{t=1}^{n} \Phi((X_{t-i}^{(n)})_{i \in \mathbb{N}}, (\tilde{\theta}_i)_{1+(j-1)d \le i \le jd}) K\left(\frac{\frac{t}{n} - u_j}{h_n}\right). \quad (22) $$

Therefore, due to the differentiability of the $\Phi$ function, we can already write that:

$$ \partial_{\tilde{\theta}} \hat{W}_n(\hat{\theta}) = 0. \quad (23) $$

The aim of this part is to establish a central limit theorem verified by $\partial_{\tilde{\theta}} \hat{W}_n(\tilde{\theta}^*)$. To do this, we will first establish a central limit theorem for $\partial_{\tilde{\theta}} W_n(\tilde{\theta}^*)$ with:

$$ W_n(\tilde{\theta}) := \sum_{j=1}^{J} \sum_{t=1}^{n} \Phi((\tilde{X}_{t-k}(u_j))_{k \in \mathbb{N}}, (\tilde{\theta}_i)_{1+(j-1)d \le i \le jd}) K\left(\frac{\frac{t}{n} - u_j}{h_n}\right) \text{ for any } \tilde{\theta} \in \Theta^J. \quad (24) $$

We begin by recalling that, based on the assumptions, for any $j \in \{1, \dots, J\}$, $\partial_{\tilde{\theta}} \mathbb{E}[\Phi((\tilde{X}_{t-i}(u_j))_{i \in \mathbb{N}}, \theta^*(u_j)) | \mathcal{F}_0] = 0$ as $\theta^*(u)$ is the unique minimizer of $\mathbb{E}[\Phi((\tilde{X}_{t-i}(u))_{i \in \mathbb{N}}, \theta) | \mathcal{F}_0]$ over the open set $\tilde{\Theta}^o$. As a consequence, for any $t \in \mathbb{Z}$,

$$ \partial_{\tilde{\theta}} \mathbb{E}[R^t(\tilde{\theta}^*) | \mathcal{F}_0] = \mathbb{E}[\partial_{\tilde{\theta}} R^t(\tilde{\theta}^*) | \mathcal{F}_0] = 0 $$

$$ \text{with } R^t(\tilde{\theta}) = \sum_{j=1}^{J} \alpha_j \Phi((\tilde{X}_{t-i}(u_j))_{i \in \mathbb{N}}, (\tilde{\theta}_i)_{(j-1)d+1 \le i \le jd}), \quad (25) $$

where $\alpha = (\alpha_j)_{1\le j\le J} \in \mathbb{R}^J$ and $\alpha \ne 0$. Moreover it is supposed that for any $u \in (0, 1)$, $\mathbb{E}[\|\partial_{\tilde{\theta}} \Phi((\tilde{X}_k(u))_{k \le 0}, \theta^*(u))\|^2] < \infty$. Then, we deduce that

$$ \mathbb{E}[\|\partial_{\tilde{\theta}} R^t(\tilde{\theta}^*)\|^2] < \infty. \quad (26) $$

From (25), the sequence $(\partial_{\tilde{\theta}} R^t(\tilde{\theta}^*))_{t \in \mathbb{Z}}$ constitutes a difference of martingale sequence. And from (26) we can apply the CLT for triangular arrays of stationary differences of martingale sequences from [18], Corollary 3.1, in order to obtain the following multidimensional asymptotic normality: for every sequences $(v_n)_{n \in \mathbb{N}}$ and $(w_n)_{n \in \mathbb{N}}$ of integer numbers such that $\min(v_n, w_n, w_n -$
$v_n) \underset{n \to \infty}{\longrightarrow} \infty,$

$$
\frac{1}{\sqrt{w_n - v_n}} \sum_{t=v_n}^{w_n} \partial_{\tilde{\theta}} R^t(\tilde{\theta}^*) \xrightarrow{n \to \infty} \mathcal{N}_{dJ} \left( 0, \alpha^\top \sum_{t \in \mathbb{Z}} \mathbb{E} \left[ \partial_{\tilde{\theta}} R^0(\tilde{\theta}^*) \partial_{\tilde{\theta}} R^t(\tilde{\theta}^*)^\top \right] \alpha \right).
$$

The asymptotic covariance matrix of size dJ can be specified. Indeed $(R^t(\tilde{\theta}^*))_t$ is an increment martingale process and therefore this implies:

$$
\frac{1}{\sqrt{w_n - v_n}} \sum_{t=v_n}^{w_n} \partial_{\tilde{\theta}} R^t (\tilde{\theta}^*) \xrightarrow{n \to \infty} \mathcal{N}_{dJ} \left( 0, \alpha^\top \mathbb{E} \left[ \partial_{\tilde{\theta}} R^0 (\tilde{\theta}^*) \partial_{\tilde{\theta}} R^0 (\tilde{\theta}^*)^\top \right] \alpha \right). \quad (27)
$$

Now for $1 \le i_0 \le dJ$, we are going to consider for any $\tilde{\theta} \in \mathbb{R}^{dJ}$, $\partial_{\tilde{\theta}_{i_0}} W_n(\tilde{\theta}^*)$.

Since the compact support of the kernel $K$ is included in $[-c, c]$, for all $u \in (0, 1)$ and $n$ large enough the sequences $(i_n(u))_n$ and $(j_n(u))_n$ can be defined as

$$
\begin{align}
1 \le i_n(u) &:= [n(u - ch_n)] < j_n(u) := [n(u + ch_n)] \le n \nonumber \\
&\implies K\left(\frac{t - nu}{nh_n}\right) = 0 \text{ for } t \notin [i_n(u), j_n(u)]. \tag{28}
\end{align}
$$

Now, since $0 < u_1 < \dots < u_J < 1$ do not depend on $n$, we will consider $n$ large enough so that for any $j \in \{1, \dots, J-1\}$, $j_n(u_j) < i_n(u_{j+1})$. Thus the intervals $[i_n(u_j), j_n(u_j)]_j$ are disjoint. Therefore, for $1 \le i_0 \le dJ$, we can write that:

$$
\partial_{\tilde{\theta}_{i_0}} W_n(\tilde{\theta}^*) = \sum_{t=i_n(u_{j_0})}^{j_n(u_{j_0})} \partial_{\theta_{i_0-(j_0-1)d}} \Phi((\tilde{X}_{t-i}(u_{j_0}))_{i \in \mathbb{N}}, \theta^*(u_{j_0})) K\left(\frac{t-nu_{j_0}}{nh_n}\right), \quad (29)
$$

with $j_0$ such as $(j_0-1)d+1 \le i_0 \le j_0d$. And this simplified writing of $\partial_{\tilde{\theta}_{i_0}} W_n(\tilde{\theta}^*)$
is also valid for any $i_0 \in \{(j_0-1)d+1, \dots, j_0d\}$. Then it is possible to use directly
the multidimensional result (91) obtained in [3] and this implies:

$$
\frac{1}{\sqrt{nh_n}} \left( \partial_{\tilde{\theta}_{i_0}} W_n(\tilde{\theta}^*) \right)_{(j_0-1)d+1 \le i_0 \le j_0 d} \xrightarrow{n \to \infty} \mathcal{N} \left( 0, \Sigma(\theta^*(u_{j_0})) \right) \quad (30)
$$

with

$$
\Sigma(\boldsymbol{\theta}^*(u)) = \left( \int_{\mathbb{R}} K^2(x)dx \right) \mathbb{E} \left[ \partial_{\boldsymbol{\theta}} \Phi((\tilde{X}_{-k}(u))_{k \in \mathbb{N}}, \boldsymbol{\theta}^*(u)) \left( \partial_{\boldsymbol{\theta}} \Phi((\tilde{X}_{-k}(u))_{k \in \mathbb{N}}, \boldsymbol{\theta}^*(u)) \right)^{\top} \right].
$$

Now we will consider the asymptotic behavior of $U_n(i_1, i_2)$ defined by

$$
U_n(i_1, i_2) := \beta_1 \partial_{\tilde{\theta}_{i_1}} W_n(\tilde{\theta}^*) + \beta_2 \partial_{\tilde{\theta}_{i_2}} W_n(\tilde{\theta}^*) \quad \text{where} \quad (\beta_1, \beta_2) \in \mathbb{R}^2.
$$

Let $1 \le j_1 < j_2 \le J$ be integers such that $(j_1 - 1)d + 1 \le i_1 \le j_1d$ and $(j_2 - 1)d + 1 \le i_2 \le j_2d$. This will be sufficiently representative to demonstrate,
with (30), the asymptotic behavior of $\sum_{i=1}^{dJ} \beta_i W_n(i)$, with $\beta_i \in \mathbb{R}$. First, we can write that for $n$ large enough and such that $i_n(u_{j_2}) - j_n(u_{j_1}) > 0$, we have

$$ U_n(i_1, i_2) = \beta_1 \sum_{t=i_n(u_{j_1})}^{j_n(u_{j_1})} \partial_{\theta_{i_1-(j_1-1)_d}} \Phi((\tilde{X}_{t-k}(u))_{k \in \mathbb{N}}, \theta^*(u_{j_1})) K\left(\frac{t-n u_{j_1}}{nh_n}\right) \\ + \beta_2 \sum_{t=i_n(u_{j_2})}^{j_n(u_{j_2})} \partial_{\theta_{i_2-(j_2-1)_d}} \Phi((\tilde{X}_{t-k}(u))_{k \in \mathbb{N}}, \theta^*(u_{j_2})) K\left(\frac{t-n u_{j_2}}{nh_n}\right), $$

We will prove that:

$$ \sqrt{nh_n} U_n(i_1, i_2) \xrightarrow[n \to \infty]{\mathcal{L}} \mathcal{N}\left[0, \left(\int_{\mathbb{R}} K^2(x) dx\right) \times \left(\beta_1^2 \left(\Sigma(\theta^*(u_{j_1}))\right)_{i_1 - (j_1-1)_d, i_1 - (j_1-1)_d} + \beta_2^2 \left(\Sigma(\theta^*(u_{j_2}))\right)_{i_2 - (j_2-1)_d, i_2 - (j_2-1)_d}\right)\right], \quad (31) $$

which means that the asymptotic covariance between $\sqrt{nh_n} \partial_{\tilde{\theta}_{i_1}} W_n(\tilde{\theta}^*)$ and $\sqrt{nh_n} \partial_{\tilde{\theta}_{i_2}} W_n(\tilde{\theta}^*)$ is zero.

Let $\ell \in \mathbb{N}^*$ and for $m \in \{1, \dots, \ell\}$, we denote $I_m^{(\ell)} = [-c + 2c \frac{m-1}{\ell}, -c + 2c \frac{m}{\ell}]$, $T_{m,j}^{(\ell)} = \{t \in \mathbb{N}, \frac{t-u_j n}{nh_n} \in I_m\}$ for $j \in \{1, \dots, d\}$, and let $K_\ell$ be a piecewise constant kernel function

$$ K_\ell(x) = \sum_{m=1}^{\ell} a_m^{(\ell)} \mathbb{I}_{x \in I_m^{(\ell)}} \quad \text{with } a_m^{(\ell)} \in \mathbb{R} \text{ for all } m. $$

Then define

$$ U_{n,m}^{(\ell)}(i_1, i_2) = \alpha_1 \sum_{t \in T_{m,j_1}^{(\ell)}} \partial_{\theta_{i_1-(j_1-1)_d}} \Phi((\tilde{X}_{t-k}(u))_{k \in \mathbb{N}}, \theta^*(u_{j_1})) \\ + \alpha_2 \sum_{t \in T_{m,j_2}^{(\ell)}} \partial_{\theta_{i_2-(j_2-1)_d}} \Phi((\tilde{X}_{t-k}(u))_{k \in \mathbb{N}}, \theta^*(u_{j_2})) \quad (32) $$

and

$$ \begin{aligned} U_n^{(\ell)}(i_1, i_2) &= \sum_{m=1}^{\ell} a_m^{(\ell)} U_{n,m}^{(\ell)}(i_1, i_2) \\ &= \alpha_1 \sum_{t=i_n(u_{j_1})}^{j_n(u_{j_1})} \partial_{\theta_{i_1-(j_1-1)_d}} \Phi((\tilde{X}_{t-k}(u))_{k \in \mathbb{N}}, \theta^*(u_{j_1})) K_\ell\left(\frac{t-n u_{j_1}}{nh_n}\right) \\ &\quad + \alpha_2 \sum_{t=i_n(u_{j_2})}^{j_n(u_{j_2})} \partial_{\theta_{i_2-(j_2-1)_d}} \Phi((\tilde{X}_{t-k}(u))_{k \in \mathbb{N}}, \theta^*(u_{j_2})) K_\ell\left(\frac{t-n u_{j_2}}{nh_n}\right) \end{aligned} \quad (33) $$
But as it was already established in (6), $(\alpha_1 \partial_{\theta_{i_1-(j_1-1)d}} \Phi((\tilde{X}_{t-k}(u))_{k \in \mathbb{N}}, \theta^*(u_{j_1})) + \alpha_2 \partial_{\theta_{i_2-(j_2-1)d}} \Phi((\tilde{X}_{t-k}(u))_{k \in \mathbb{N}}, \theta^*(u_{j_2})))_t$ is a martingale difference process with asymptotic normality and therefore, since $\text{Card}(T_{m,j}^{(\ell)}) \sim 2 \frac{c}{\ell} nh_n \xrightarrow{n \to \infty} \infty$ for every $m \in \{1, \dots, \ell\}$ and $j \in \{1, \dots, J\}$,

$$ \sqrt{\frac{\ell}{2c nh_n}} U_{n,m}^{(\ell)}(i_1, i_2) \xrightarrow[n \to \infty]{\mathcal{L}} \mathcal{N}\left(0, \alpha_1^2 \left(\Sigma(\theta^*(u_{j_1}))\right)_{\substack{i_1 - (j_1-1)d, \\ i_1 - (j_1-1)d}} + \alpha_2^2 \left(\Sigma(\theta^*(u_{j_2}))\right)_{\substack{i_2 - (j_2-1)d, \\ i_2 - (j_2-1)d}}\right). \quad (34) $$

Indeed

$$ \text{Cov}(\partial_{\theta_{i_1-(j_1-1)d}} \Phi((\tilde{X}_{t_1-k}(u))_{k \in \mathbb{N}}, \theta^*(u_{j_1})), \partial_{\theta_{i_2-(j_2-1)d}} \Phi((\tilde{X}_{t_2-k}(u))_{k \in \mathbb{N}}, \theta^*(u_{j_2})) = 0 $$

for any $t_1 \in T_{m,j_1}^{(\ell)}$ and $t_2 \in T_{m,j_2}^{(\ell)}$ when $j_1 \neq j_2$. Hence $U_n^{(\ell)}(i_1, i_2)$ defined in (33), which is linear combinations of $U_{n,m}^{(\ell)}(i_1, i_2)$ also satisfies the following central limit theorem

$$ \sqrt{\frac{\ell}{2c nh_n}} U_n^{(\ell)}(i_1, i_2) \xrightarrow[n \to \infty]{\mathcal{L}} \mathcal{N}\left(0, \left(\int_{\mathbb{R}} K_{\ell}^2(x) dx \times \left(\alpha_1^2 \left(\Sigma(\theta^*(u_{j_1}))\right)_{\substack{i_1 - (j_1-1)d, \\ i_1 - (j_1-1)d}} + \alpha_2^2 \left(\Sigma(\theta^*(u_{j_2}))\right)_{\substack{i_2 - (j_2-1)d, \\ i_2 - (j_2-1)d}}\right)\right), $$

since for piecewise kernel we have $\int_{\mathbb{R}} K_{\ell}^2(x) dx = \frac{2c}{\ell} \sum_{m=1}^{\ell} (a_m^{(\ell)})^2$.

For the same central limit theorem for a general piecewise differentiable kernel $K$, consider a piecewise constant $K_{\ell}$ satisfying $\int_{\mathbb{R}} K_{\ell}^2(x) dx \xrightarrow{\ell \to \infty} \int_{\mathbb{R}} K^2(x) dx$. The end of the proof of (31) mimics the end of the proof of Lemma 7.1 3. in [3]. Therefore

$$ \frac{1}{\sqrt{nh_n}} \partial_{\tilde{\theta}} W_n(\tilde{\theta}^*) \xrightarrow[n \to \infty]{\mathcal{L}} \mathcal{N}_{dJ}\left(0, \tilde{\Sigma}(u_1, \dots, u_j)\right) \\ \text{with } \tilde{\Sigma}(u_1, \dots, u_j) = \begin{pmatrix} \Sigma(\theta^*(u_1)) & 0 & \dots & 0 \\ 0 & \Sigma(\theta^*(u_2)) & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \Sigma(\theta^*(u_J)) \end{pmatrix} \quad (35) $$

$$ \text{with } \Sigma(\theta^*(u)) = \left( \int_{\mathbb{R}} K^2(x) dx \right) \mathbb{E} \left[ \partial_{\theta} \Phi((\tilde{X}_{-k}(u))_{k \in \mathbb{N}}, \theta^*(u)) \left( \partial_{\theta} \Phi((\tilde{X}_{-k}(u))_{k \in \mathbb{N}}, \theta^*(u)) \right)^{\top} \right]. $$

C/ We use a Taylor-Lagrange expansion for establishing

$$ \frac{1}{\sqrt{nh_n}} \partial_{\tilde{\theta}} W_n(\tilde{\theta}) = \frac{1}{\sqrt{nh_n}} \partial_{\tilde{\theta}} W_n(\tilde{\theta}^*) + \sqrt{nh_n} \left( \frac{1}{nh_n} \partial_{\tilde{\theta}^2}^2 W_n(\tilde{\theta}) \right) (\tilde{\theta} - \tilde{\theta}^*), \quad (36) $$
where $\bar{\theta}$ belongs to the segment with extremities $\tilde{\theta}^*$ and $\hat{\theta}$. Then using (87) and (93) of [3], which can be applied because we only consider a finite number $J$ of $(u_1, \dots, u_J)$, and with the point A/ of consistency of $\hat{\theta}$, we have:

$$ \frac{1}{nh_n} \partial_{\tilde{\theta}^2}^2 W_n(\bar{\theta}) \xrightarrow[n \to \infty]{\mathbb{P}} \tilde{\Gamma}(u_1, \dots, u_j) = \left( \int_{\mathbb{R}} K^2(x)dx \right) \mathbb{E}[\partial_{\tilde{\theta}^2}^2 W_n(\tilde{\theta}^*)] $$
$$ \text{with } \tilde{\Gamma}(u_1, \dots, u_j) = \begin{pmatrix} \Gamma(\theta^*(u_1)) & 0 & \dots & 0 \\ 0 & \Gamma(\theta^*(u_2)) & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \Gamma(\theta^*(u_J)) \end{pmatrix} \quad (37) $$

$$ \text{with } \Gamma(\theta^*(u)) = \left( \int_{\mathbb{R}} K^2(x)dx \right) \mathbb{E}[\partial_{\tilde{\theta}^2}^2 \Phi((\tilde{X}_{-k}(u))_{k \in \mathbb{N}}, \theta^*(u))] $$

Moreover, using (95) of [3], we also have:

$$ \left\| \frac{1}{\sqrt{nh_n}} \partial_{\hat{\theta}} W_n(\hat{\theta}) \right\|_1 \le C \left( \frac{1}{\sqrt{nh_n}} + h_n^\rho \sqrt{nh_n} \right) \implies \frac{1}{\sqrt{nh_n}} \partial_{\hat{\theta}} W_n(\hat{\theta}) \xrightarrow[n \to \infty]{\mathbb{P}} 0, \quad (38) $$

$$ \text{by using the condition } nh_n^{1+2\rho} \xrightarrow{n \to \infty} 0. $$

Finally, using (36), from (37) and (38), we obtain:

$$ \sqrt{nh_n}(\hat{\theta} - \tilde{\theta}^*) = \left(\frac{1}{nh_n} \partial_{\tilde{\theta}^2}^2 W_n(\bar{\theta})\right)^{-1} \left(\frac{1}{\sqrt{nh_n}} \partial_{\tilde{\theta}} W_n(\hat{\theta}) - \frac{1}{\sqrt{nh_n}} \partial_{\tilde{\theta}} W_n(\tilde{\theta}^*)\right) \\ \xrightarrow[n \to \infty]{\mathcal{L}} \mathcal{N}_{dJ} \left(0, \tilde{\Gamma}(u_1, \dots, u_j)^{-1} \tilde{\Sigma}(u_1, \dots, u_j) \tilde{\Gamma}(u_1, \dots, u_j)^{-1}\right). \quad (39) $$

The end of the proof is obtained from:

$$ \tilde{\Gamma}(u_1, \dots, u_j)^{-1} \tilde{\Sigma}(u_1, \dots, u_j) \tilde{\Gamma}(u_1, \dots, u_j)^{-1} = \begin{pmatrix} \Gamma^{-1}(\theta^*(u_1)) \Sigma(\theta^*(u_1)) \Gamma^{-1}(\theta^*(u_1)) & 0 & 0 \\ 0 & \Gamma^{-1}(\theta^*(u_2)) \Sigma(\theta^*(u_2)) \Gamma^{-1}(\theta^*(u_2)) & 0 \\ \vdots & \vdots & \vdots \\ 0 & 0 & 0 \\ \vdots & 0 & 0 \\ \vdots & \vdots & \vdots \\ \vdots & \vdots & \Gamma^{-1}(\theta^*(u_J)) \Sigma(\theta^*(u_J)) \Gamma^{-1}(\theta^*(u_J)) \end{pmatrix}. \quad \square $$

Proof of Proposition 3.1. From (12), we can still write that under $\mathbf{H}_0$:

$$ \sqrt{nh_n}(\tilde{\theta} - \tilde{\theta}_0) \xrightarrow[n \to \infty]{\mathcal{L}} \mathcal{N}_{dJ}(0, I_{dJ}) \quad (40) $$
by considering the column vectors of $\mathbb{R}^{dJ}$:

$$ \tilde{\theta} = (\hat{\theta}(u_1)^{\top}, \dots, \hat{\theta}(u_J)^{\top})^{\top} \quad \text{and} \quad \tilde{\theta}_0 = (\theta_0(u_1)^{\top}, \dots, \theta_0(u_J)^{\top})^{\top}. $$

Then the convergence in law of $\hat{W}_n$ is obtained by applying the application $V \in \mathbb{R}^{dJ} \to \|V\|^2$ on both sides of (40) due to the continuity of this application. □

**Proof of Proposition 3.2.** Under $\mathbf{H}_0$, the asymptotic normality (40) holds with $\tilde{\theta}_0 = (\theta_0^\top, \dots, \theta_0^\top)^\top$. Let $V_1, \dots, V_d$ be the $d$ column vectors of $\mathbb{R}^{dJ}$ such that

$$ V_1 = ((1, 0, \dots, 0), \dots, (1, 0, \dots, 0))^\top, $$

$$ V_2 = ((0, 1, 0, \dots, 0), \dots, (0, 1, 0, \dots, 0))^\top, $$

$$ \vdots \vdots \vdots $$

$$ V_d = ((0, 0, \dots, 0, 1), \dots, (0, 0, \dots, 0, 1))^\top. $$

Then, always with $\tilde{\theta} = (\hat{\theta}(u_1)^{\top}, \dots, \hat{\theta}(u_J)^{\top})^{\top}$, we obtain:

$$ \sum_{j=1}^{J} \left\| \left( \hat{\Gamma}(u_j) (\hat{\Sigma}(u_j))^{-1} \hat{\Gamma}(u_j) \right)^{1/2} \hat{\theta}(u_j) - \bar{\theta} \right\|^2 = \| \tilde{\theta} - P_{[V]} \tilde{\theta} \|^2 $$

where $P_{[V]}$ is the orthogonal projection matrix on the vector subspace $[V]$ of $\mathbb{R}^{dJ}$ generated by $V_1, \dots, V_d$. But $\|\tilde{\theta} - P_{[V]}\tilde{\theta}\|^2 = \|P_{[V]}\tilde{\theta}\|^2$ and using Cochran's Theorem, we know that $\|P_{[V]}\perp Z\|^2 \stackrel{\mathcal{L}}{\sim} \chi^2(d(J-1))$ since $\dim([V]^\perp) = dJ-d = d(J-1)$ when $Z \stackrel{\mathcal{L}}{\sim} \mathcal{N}_{dK}(0, I_{dJ})$. Finally, since $V \in \mathbb{R}^{dJ} \to \|V\|^2$ is a continuous application, we obtain $\hat{S}_n \xrightarrow{n \to \infty} \chi^2(d(J-1))$. □

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