Explosive Episodes and Time-Varying Volatility: A New MARMA–GARCH Model Applied to Cryptocurrencies

Abstract

Financial assets often exhibit explosive price surges followed by abrupt collapses, alongside persistent volatility clustering. Motivated by these features, we introduce a mixed causal–noncausal invertible–noninvertible autoregressive moving average generalized autoregressive conditional heteroskedasticity (MARMA–GARCH) model. Unlike standard ARMA processes, our model admits roots inside the unit disk, capturing bubble-like episodes and speculative feedback, while the GARCH component explains time-varying volatility. We propose two estimation approaches: (i) Whittle-based frequency-domain methods, which are asymptotically equivalent to Gaussian likelihood under stationarity and finite variance, and (ii) time-domain maximum likelihood, which proves to be more robust to heavy tails and skewness—common in financial returns. To identify causal vs. noncausal structures, we develop a higher-order diagnostics procedure using spectral densities and residual-based tests. Simulation results reveal that overlooking noncausality biases GARCH parameters, downplaying short-run volatility reactions to news ($\alpha$) while overstating volatility persistence ($\beta$). Our empirical application to Bitcoin and Ethereum enhances these insights: we find significant noncausal dynamics in the mean, paired with pronounced GARCH effects in the variance. Imposing a purely causal ARMA specification leads to systematically misspecified volatility estimates, potentially underestimating market risks. Our results emphasize the importance of relaxing the usual causality and invertibility assumption for assets prone to extreme price movements, ultimately improving risk metrics and expanding our understanding of financial market dynamics.

1. Introduction

Financial returns often exhibit relatively weak autocorrelation in levels but strong and persistent volatility, a pattern that poses notable challenges for conventional linear time series models. Although ARCH-type approaches capture time-varying volatility, standard ARMA processes assume that all autoregressive and moving average roots lie strictly outside the unit circle, thereby excluding the possibility of forward-looking or explosive dynamics. Recent research has emphasized that certain economic or financial time series—particularly those prone to speculative bubbles or local explosions—may benefit from noncausal autoregressive components, wherein some roots reside inside the unit disk (Aguirre & Lobato, 2024; Blasques et al., 2024; Giancaterini & Hecq, 2022; Gouriéroux & Zakoïan, 2017; Hall & Jasiak, 2024; Lanne & Saikkonen, 2011; Lobato & Velasco, 2022; Lof & Nyberg, 2017). In response, the mixed causal–noncausal invertible–noninvertible autoregressive moving average (MARMA) model extends traditional ARMA by allowing both causal and noncausal (as well as invertible and noninvertible) factors in the same framework, thus permitting local explosions consistent with bubble-like phenomena.

A distinctive feature of the MARMA model is that it admits multiple ARMA representations with identical second-order properties. Under Gaussianity, this multiplicity renders the true placement of roots (inside vs. outside the unit disk) unidentifiable through second moments alone, as the resulting ARMA forms share the same variance–covariance structure and spectral density (Breidt et al., 2001). Identification only becomes possible by exploiting higher-order probability structures, including autocorrelations of squared residuals or third- and fourth-order spectral densities. In this sense, noncausality reveals itself through richer nonlinear dynamics that standard causal ARMA models cannot replicate.

Although MARMA models capture nonlinear behavior arising from noncausal feedback, they do not, by themselves, generate the well-known volatility clustering phenomenon commonly observed in financial markets (Breidt et al., 2001; Engle, 1982). For instance, the magnitude of asset returns often displays strong autocorrelation, so that large shocks tend to be followed by further volatility. These dynamics are typically addressed by GARCH processes (Bollerslev, 1986), which model conditional variances as functions of past shocks and past conditional variances. Accordingly, a combined MARMA–GARCH structure can better reproduce the main stylized facts in financial returns: weak autocorrelation in levels, local explosiveness when bubbles arise, and persistent conditional volatility.

Yet introducing a GARCH component into the MARMA framework raises theoretical and practical challenges. Much of the existing MARMA literature assumes i.i.d. errors, while GARCH fundamentally presumes conditional heteroskedasticity in these errors. Although Meitz and Saikkonen (2013) study noninvertible ARMA models with ARCH errors, more general MARMA–GARCH processes with potentially noncausal terms remain relatively unexplored. In this paper, we fill that gap by detailing estimation and identification procedures for MARMA–GARCH processes—including cases where the process is noninvertible, noncausal, or both.

We compare two estimation approaches. The first is Whittle estimation in the frequency domain, which is asymptotically equivalent to Gaussian maximum likelihood for stationary processes with finite variance. The second is time-domain maximum likelihood, which can be more flexible in heavy-tailed situations. While both methods provide consistent and asymptotically normal estimators of the MARMA parameters under stationarity, they differ in their requirements for the GARCH component. Maximum likelihood only demands finite second moments, whereas Whittle estimation requires the existence of up to eight moments—a significant constraint in the presence of high kurtosis or skewness. Our Monte Carlo evidence confirms that maximum likelihood outperforms Whittle estimation for GARCH parameters when faced with heavy-tailed or skewed innovations.

An additional insight from our analysis is that erroneously imposing a purely causal (or purely invertible) representation for the mean may distort GARCH parameter estimates. Specifically, forcing roots outside the unit disk when the true process is partly noncausal can lead to a lower estimated $\alpha$, implying an understated immediate reaction to new information, and a higher estimated $\beta$, suggesting an exaggerated persistence of volatility. Such parameter biases compromise both the fit of the model and the reliability of derived risk measures, such as Value at Risk (VaR).

To illustrate these concepts, we apply the MARMA–GARCH model to daily prices of two major cryptocurrencies, Bitcoin and Ethereum. After detrending the data (via cubic splines or Fourier filtering), we find that noncausal specifications dominate purely causal alternatives and yield more plausible estimates of conditional volatility during bubble-like episodes. In particular, the MARMA($0,1,0,0$) structure accommodates future-dependent feedback, while the GARCH(1,1) component captures persistent volatility clustering. By contrast, imposing a causal AR(1) in the mean tends to inflate $\beta$ and attenuate $\alpha$, underestimating the susceptibility of volatility to immediate shocks.

The remainder of this paper proceeds as follows: Section 2 develops the MARMA–GARCH model, describes our proposed estimation and identification strategies, and addresses practical issues surrounding data detrending. Section 3 presents a Monte Carlo study and an empirical application to cryptocurrencies, highlighting how noncausal modeling affects GARCH parameter estimates and risk metrics. Finally, Section 4 discusses the main findings and concludes.

2. Materials and Methods

2.1. MARMA($r,s,r^{\prime },s^{\prime }$)–GARCH($p,q$) Processes

Let $\{Y_t:t\in \mathbb{Z}\}$ be a strictly stationary univariate process defined on a probability space $(\mathsf{\Omega },\mathcal{F},\mathbb{P})$. We denote it as a mixed causal–noncausal invertible–noninvertible autoregressive moving average generalized autoregressive conditional heteroskedasticity process, abbreviated as

$$\mathrm{MARMA}(r,s,r^{\prime },s^{\prime })–\mathrm{GARCH}(p,q).$$

Specifically, it satisfies

$${\varphi }^{+}\left(L\right){\varphi }^{*}\left(L^{-1}\right)Y_t={\theta }^{+}\left(L\right){\theta }^{*}\left(L^{-1}\right){\epsilon }_t,$$

(1)

where

$${\epsilon }_t={\eta }_t{\sigma }_t,{\eta }_t\stackrel{\mathrm{i}.\mathrm{i}.\mathrm{d}.}{\sim }\mathfrak{F}\left(\zeta \right),\mathrm{and}{\sigma }_t^2=\omega +\alpha \left(L\right){\epsilon }_t^2+\beta \left(L\right){\sigma }_t^2.$$

(2)

The innovations $\left\{{\eta }_t\right\}$ are assumed i.i.d. with

$$\mathbb{E}\left[{\eta }_t\right]=0,\mathbb{E}\left[{\eta }_t^2\right]=1,\mathbb{E}\left[{\eta }_t^4\right]<\infty .$$

Hence, $\left\{{\epsilon }_t\right\}$ is a martingale difference sequence satisfying

$$\mathbb{E}\left[{\epsilon }_t\right]=0,\mathbb{E}\left[{\epsilon }_t^2\right]={\sigma }_t^2,\mathbb{E}\left[{\epsilon }_t^3\right]={\kappa }_3,\mathbb{E}\left[{\epsilon }_t^4\right]={\mu }_4,{\kappa }_4={\mu }_4-3{\sigma }_t^4.$$

Throughout, L denotes the backward shift operator, i.e., $L^k{Y}_t=Y_{t-k}$ for $k\in \{0,\pm 1,\pm 2,\dots \}$. Define the following lag polynomials:

$$\begin{array}{cc}\hfill {\varphi }^{+}\left(L\right)& =1-\sum _{j=1}^r{\varphi }_j^{+}{L}^j,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\theta }^{+}\left(L\right)& =1+\sum _{i=1}^{r^{\prime }}{\theta }_i^{+}{L}^i,\hfill \end{array}$$

(4)

and their corresponding lead polynomials,

$$\begin{array}{cc}\hfill {\varphi }^{*}\left(L^{-1}\right)& =1-\sum _{k=1}^s{\varphi }_k^{*}{L}^{-k},\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill {\theta }^{*}\left(L^{-1}\right)& =1+\sum _{m=1}^{s^{\prime }}{\theta }_m^{*}{L}^{-m}.\hfill \end{array}$$

(6)

We collect all parameters in the vector

$$\mathit{\vartheta }=\left({\varphi }_1^{+},\dots ,{\varphi }_r^{+};{\varphi }_1^{*},\dots ,{\varphi }_s^{*};{\theta }_1^{+},\dots ,{\theta }_{r^{\prime }}^{+};{\theta }_1^{*},\dots ,{\theta }_{s^{\prime }}^{*}\right)\in {\mathbb{C}}^{r+s+r^{\prime }+s^{\prime }}.$$

If $r^{\prime }=0$ and $s^{\prime }=0$, then the above reduces to a classical causal–invertible ARMA($r,s$) model.

From (1), an equivalent rearrangement is

$$\begin{array}{c}(1-\sum _{j=1}^r{\varphi }_j^{+}{L}^j-\sum _{k=1}^s{\varphi }_k^{*}{L}^{-k}+\sum _{j=1}^r\sum _{k=1}^s{\varphi }_j^{+}{\varphi }_k^{*}{L}^{j-k})Y_t\hfill \\ =\left(1+\sum _{i=1}^{r^{\prime }}{\theta }_i^{+}{L}^i+\sum _{m=1}^{s^{\prime }}{\theta }_m^{*}{L}^{-m}+\sum _{i=1}^{r^{\prime }}\sum _{m=1}^{s^{\prime }}{\theta }_i^{+}{\theta }_m^{*}{L}^{i-m}\right){\epsilon }_t.\hfill \end{array}$$

(7)

We assume any common roots between ${\varphi }^{+}\left(z\right){\varphi }^{*}\left(z\right)$ and ${\theta }^{+}\left(z\right){\theta }^{*}\left(z\right)$ are canceled. In the complex domain (substituting $L\mapsto z$), these polynomials become

$$\begin{array}{cccc}\hfill {\varphi }^{+}\left(z\right)& =1-\sum _{j=1}^r{\varphi }_j^{+}{z}^j,\hfill & \hfill {\varphi }^{*}\left(z\right)& =1-\sum _{k=1}^s{\varphi }_k^{*}{z}^{-k},\hfill \end{array}$$

(8)

$$\begin{array}{cccc}\hfill {\theta }^{+}\left(z\right)& =1+\sum _{i=1}^{r^{\prime }}{\theta }_i^{+}{z}^i,\hfill & \hfill {\theta }^{*}\left(z\right)& =1+\sum _{m=1}^{s^{\prime }}{\theta }_m^{*}{z}^{-m}.\hfill \end{array}$$

(9)

All these factors are assumed to have zeros outside the unit disk. They may be factorized as

$$\begin{array}{cc}\hfill {\varphi }^{+}\left(z\right)& =\prod _{\ell =1}^r\left(1-m_{\ell }z\right),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\varphi }^{*}\left(z\right)& =\prod _{\ell =r+1}^{r+s}\left(1-m_{\ell }z\right),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\theta }^{+}\left(z\right)& =\prod _{h=1}^{r^{\prime }}\left(1-n_{h}z\right),\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\theta }^{*}\left(z\right)& =\prod _{h=r^{\prime }+1}^{r^{\prime }+s^{\prime }}\left(1-n_{h}z\right).\hfill \end{array}$$

(13)

Observe that any root $m_{\ell }$ (or $n_h$) lying outside the unit circle can equivalently be represented by its reciprocal $m_{\ell }^{-1}$ (or $n_h^{-1}$) inside the unit circle. Flipping a root from outside to inside the unit disk effectively switches a causal factor to a noncausal one (or an invertible factor to a noninvertible one), but does not affect second-order properties such as the autocorrelation function or spectrum. Hence, for Gaussian processes, there exist $2^{r+s+r^{\prime }+s^{\prime }}$ alternative factorizations corresponding to the same second-order structure, making them observationally indistinguishable via second-order statistics alone. Under non-Gaussianity, however, higher-order moments and cumulants break this equivalence, thus providing identifiability of the particular causal–noncausal and invertible–noninvertible roots (Hecq & Velasquez-Gaviria, 2024).

• GARCH Component.

For the GARCH($p,q$) part, let

$$\begin{array}{ccc}\hfill \alpha \left(L\right)& =\sum _{u=1}^p{\alpha }_u{L}^u,\hfill & \hfill p\ge 0,\end{array}$$

(14)

$$\begin{array}{ccc}\hfill \beta \left(L\right)& =\sum _{v=1}^q{\beta }_v{L}^v,\hfill & \hfill q>0,\end{array}$$

(15)

with $\omega >0$ and ${\alpha }_u,{\beta }_v>0$. These restrictions ensure ${\sigma }_t^2\ge 0$. Hence, (2) may be written more explicitly as

$${\sigma }_t^2=\omega +\sum _{u=1}^p{\alpha }_u{\epsilon }_{t-u}^2+\sum _{v=1}^q{\beta }_v{\sigma }_{t-v}^2.$$

(16)

As shown in Engle (1982) and Bollerslev (1986), a squared GARCH($p,q$) sequence can be rewritten as an ARMA(max$(p,q),q$) model. From (16), we obtain

$${\epsilon }_t^2-\sum _{\ell =1}^{max(p,q)}\left({\alpha }_{\ell }+{\beta }_{\ell }\right){\epsilon }_{t-\ell }^2=\omega +v_t-\sum _{\ell =1}^q{\beta }_{\ell }{v}_{t-\ell },$$

(17)

where

$$v_t={\epsilon }_t^2-{\sigma }_t^2={\sigma }_t^2\left({\eta }_t^2-1\right).$$

Since $\left\{v_t\right\}$ is a martingale difference (assuming $\mathbb{E}\left[{\epsilon }_t^2\right]<\infty$), it follows that $\left\{{\epsilon }_t^2\right\}$ satisfies an ARMA(max$(p,q),q$) representation:

$$\left(\alpha +\beta \right)\left(L\right){\epsilon }_t^2=\omega +\beta \left(L\right)v_t.$$

(18)

Though $\left\{v_t\right\}$ is uncorrelated, it is not independent. This equivalence allows GARCH parameters to be estimated via standard ARMA-based techniques.

2.1.1. Causal and Invertible Representation

A MARMA($r,s,r^{\prime },s^{\prime }$) process typically admits multiple ARMA($p,q$) representations (with $p=r+s$ and $q=r^{\prime }+s^{\prime }$) sharing the same second-order structure (Hecq & Velasquez-Gaviria, 2024). A convenient representation for some purposes is the causal and invertible (often termed “pseudo-causal and invertible”) form, in which all roots lie strictly outside the unit disk. Even if the true process is noncausal or noninvertible, this representation yields errors that are uncorrelated (though not necessarily independent) (Breidt et al., 2001).

A common causal–invertible representation is

$$\left(1-\sum _{j=1}^{r+s}{\varphi }_j{L}^j\right)Y_t=\left(1+\sum _{i=1}^{r^{\prime }+s^{\prime }}{\theta }_i{L}^i\right){\epsilon }_t^{*},$$

(19)

with parameter vector

$$\mathit{\beta }=\left({\varphi }_1,\dots ,{\varphi }_{r+s},{\theta }_1,\dots ,{\theta }_{r^{\prime }+s^{\prime }}\right)\in {\mathbb{R}}^{r+s+r^{\prime }+s^{\prime }}.$$

Causality means there exists an absolutely summable filter $\left\{{\pi }_j\right\}$ such that

$$Y_t=\sum _{j=0}^{\infty }{\pi }_j{\epsilon }_{t-j}^{*},$$

and invertibility means there is a similar filter $\left\{{\pi }_j^{*}\right\}$ such that

$${\epsilon }_t^{*}=\sum _{j=0}^{\infty }{\pi }_j^{*}{Y}_{t-j}.$$

In practice, fitting this causal–invertible form (often all-pass under genuine noncausality) can simplify the subsequent extraction of the correct factors and identification of their roots (Hecq & Velasquez-Gaviria, 2024). Misallocating the roots, however, may yield residuals (while uncorrelated) whose squares exhibit dependence differing from the correct GARCH specifications.

2.1.2. Spectral Representations

The spectral representation of a MARMA($r,s,r^{\prime },s^{\prime }$)–GARCH($p,q$) process splits naturally into MARMA and GARCH components. The MARMA part has second-order properties equivalent to those of the corresponding ARMA($p,q$) (Hecq & Velasquez-Gaviria, 2024), while the GARCH component corresponds to an ARMA(max$(p,q),q$) model for $\left\{{\epsilon }_t^2\right\}$ (Bollerslev, 1986).

Let $\left\{Y_t\right\}$ be the MARMA($r,s,r^{\prime },s^{\prime }$)–GARCH($p,q$) process satisfying (1). Define $z=e^{-i\omega }$. The transfer function for the conditional mean is

$$\psi \left(\mathit{\vartheta },\omega \right)={\displaystyle \frac{{\theta }^{+}\left(z\right){\theta }^{*}\left(z\right)}{{\varphi }^{+}\left(z\right){\varphi }^{*}\left(z\right)}}={\displaystyle \frac{1+{\sum }_{i=1}^{r^{\prime }}{\theta }_i^{+}{z}^i+{\sum }_{m=1}^{s^{\prime }}{\theta }_m^{*}{z}^{-m}+{\sum }_{i=1}^{r^{\prime }}{\sum }_{m=1}^{s^{\prime }}{\theta }_i^{+}{\theta }_m^{*}{z}^{i-m}}{1-{\sum }_{j=1}^r{\varphi }_j^{+}{z}^j-{\sum }_{k=1}^s{\varphi }_k^{*}{z}^{-k}+{\sum }_{j=1}^r{\sum }_{k=1}^s{\varphi }_j^{+}{\varphi }_k^{*}{z}^{j-k}}}.$$

(20)

The k-th order spectral density is given by

$$S_k\left(\mathit{\vartheta },{\kappa }_k;{\omega }_1,\dots ,{\omega }_{k-1}\right)={\displaystyle \frac{{\kappa }_k}{{\left(2\pi \right)}^{k-1}}}\left(\prod _{\ell =1}^{k-1}\psi (\mathit{\vartheta },{\omega }_{\ell })\right)\psi \left(\mathit{\vartheta },-\sum _{\ell =1}^{k-1}{\omega }_{\ell }\right),$$

(21)

where ${\kappa }_k$ is the k-th order cumulant of $\left\{Y_t\right\}$, and typically ${\omega }_{\ell }=2\pi \ell /T$.

Likewise, the transfer function for the conditional variance component (i.e., the ARMA form of $\left\{{\epsilon }_t^2\right\}$) is

$$g\left(\mathit{\xi },\omega \right)={\displaystyle \frac{\beta \left(z\right)}{(\alpha +\beta )\left(z\right)}}={\displaystyle \frac{1-{\sum }_{v=1}^q{\beta }_v{z}^v}{1-{\sum }_{u=1}^{max(p,q)}({\alpha }_u+{\beta }_u)z^u}}.$$

Its k-th order spectral density is

$$S_k^v\left(\mathit{\xi },{\kappa }_k^v;{\omega }_1,\dots ,{\omega }_{k-1}\right)={\displaystyle \frac{{\kappa }_k^v}{{\left(2\pi \right)}^{k-1}}}\left(\prod _{\ell =1}^{k-1}g(\mathit{\xi },{\omega }_{\ell })\right)g\left(\mathit{\xi },-\sum _{\ell =1}^{k-1}{\omega }_{\ell }\right),$$

(22)

where ${\kappa }_k^v$ denotes the k-th order cumulant of $\left\{v_t\right\}=\{{\epsilon }_t^2-{\sigma }_t^2\}$.

• Higher-Order Periodogram.

A sample analog of the k-th order spectral density is the k-th order periodogram:

$$I_k({\omega }_1,\dots ,{\omega }_{k-1})={\displaystyle \frac{1}{{\left(2\pi \right)}^{k-1}T}}\left(\prod _{\ell =1}^{k-1}{d}_T\left({\omega }_{\ell }\right)\right)d_T\left(-\sum _{\ell =1}^{k-1}{\omega }_{\ell }\right),$$

(23)

where

$$d_T\left(\omega \right)=\sum _{t=1}^T{Y}_t{e}^{-i\omega t}$$

is the discrete Fourier transform of $\left\{Y_t\right\}$. For $k=2$, (23) reduces to the ordinary periodogram.

2.2. Parameter Estimation and Identification

We now outline a two-stage procedure for estimating and identifying the MARMA$(r,s,r^{\prime },s^{\prime })$ structure and the associated GARCH$(p,q)$ parameters. In the first stage, we fit the mean equation using a causal–invertible ARMA$(p,q)$ representation of the MARMA process, factorize its polynomials, and select among the resulting $2^{r+s+r^{\prime }+s^{\prime }}$ candidate representations via higher-order diagnostics. In the second stage, we fit the GARCH$(p,q)$ dynamics, comparing Whittle-based and maximum likelihood approaches.

2.2.1. Two-Step Approach

• First Step (MARMA Identification).
  • Fit a Causal–Invertible ARMA$(p,q)$. Estimate the orders p and q (where $p=r+s$ and $q=r^{\prime }+s^{\prime }$) using information criteria (AIC, BIC) and significance tests. Treat the process as if it were causal and invertible, which allows for a standard ARMA($p,q$) fit in the frequency or time domain (e.g., Whittle or Gaussian maximum likelihood).
  • Redistribute Roots via Factorization. Once the ARMA($p,q$) fit is obtained, factorize its polynomials and systematically explore all $2^{r+s+r^{\prime }+s^{\prime }}$ possible root allocations, corresponding to distinct causal/noncausal and invertible/noninvertible configurations. In practice, this step involves flipping certain roots inside/outside the unit disk (see Section 2.1).
  • Select Final Representation. For each candidate, compute residual diagnostics such as higher-order spectral measures (Hecq & Velasquez-Gaviria, 2022, 2024; Velasco & Lobato, 2018) or a portmanteau-type test on $\left\{{\epsilon }_t\right\}$ and $\left\{{\epsilon }_t^2\right\}$. The representation with the weakest remaining serial dependence (in both levels and squares) is chosen as the final MARMA$(r,s,r^{\prime },s^{\prime })$ specification.
• Second Step (GARCH(p,q) Estimation).
  • Estimate GARCH$(p,q)$. Using the final residual series $\left\{{\epsilon }_t\right\}$ from the identified MARMA model, estimate the parameters $\{\omega ,{\alpha }_u,{\beta }_v\}$ of a GARCH$(p,q)$ process (commonly GARCH$(1,1)$ in many financial applications).
  • Whittle vs. MLE. We compare frequency-domain Whittle estimation (which typically requires higher-order moment assumptions) against time-domain maximum likelihood methods (which may hold under weaker moment conditions). Our Monte Carlo experiments suggest that MLE often exhibits smaller bias and variance, although Whittle remains a practical alternative if the required moments exist.

2.2.2. Estimation and Identification of the MARMA($r,s,r^{\prime },s^{\prime }$) Process

We begin with the causal–invertible ARMA($p,q$) representation that shares second-order properties with the MARMA$(r,s,r^{\prime },s^{\prime })$ process; see Section 2.1 and Fries and Zakoïan (2019). Suppose we have observations ${\left\{Y_t\right\}}_{t=1}^T$. Under mild regularity conditions (stationarity, finite fourth moments), Whittle estimation of the ARMA parameters is asymptotically equivalent to maximizing the Gaussian likelihood, because (i) the discrete Fourier transforms of $\left\{Y_t\right\}$ converge in distribution to complex normal variables, and (ii) the periodogram converges in probability to the underlying second-order spectral density.

• Whittle Estimation.

Let

$${\mathit{\beta }}_0=\left({\varphi }_{0,1},\dots ,{\varphi }_{0,r+s},{\theta }_{0,1},\dots ,{\theta }_{0,r^{\prime }+s^{\prime }}\right)$$

denote the true parameter vector in the causal–invertible ARMA($p,q$) representation. Recall from (21) that $S_2(\mathit{\beta },{\kappa }_2,\omega )$ denotes the second-order (power) spectral density of $\left\{Y_t\right\}$. The Whittle criterion is

$$W\left(\mathit{\beta }\right)=\sum _{\ell =1}^{T/2}\left\{log\left[S_2(\mathit{\beta },{\kappa }_2;{\omega }_{\ell })\right]+{\displaystyle \frac{I_2\left({\omega }_{\ell }\right)}{S_2(\mathit{\beta },{\kappa }_2;{\omega }_{\ell })}}\right\},$$

(24)

where $I_2\left({\omega }_{\ell }\right)$ is the usual periodogram at frequency ${\omega }_{\ell }$. The Whittle estimator is

$$\widehat{\mathit{\beta }}=arg\underset{\mathit{\beta }\in \mathit{B}}{min}W\left(\mathit{\beta }\right).$$

(25)

Provided $\mathbb{E}\left[Y_t^4\right]<\infty$, standard theorems (e.g., Hannan, 1973) imply

$$\widehat{\mathit{\beta }}\stackrel{p}{\to }{\mathit{\beta }}_0,\mathrm{and}\sqrt{T}\left(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0\right)\stackrel{d}{\to }\mathcal{N}\left(\mathbf{0},\mathit{V}\left({\mathit{\beta }}_0\right)\right),$$

where the asymptotic covariance matrix can be approximated by the inverse of

$$\mathit{I}\left({\mathit{\beta }}_0\right)={\displaystyle \frac{1}{4\pi }}{\int }_{-\pi }^{\pi }{\displaystyle \frac{1}{{\left[S_2({\mathit{\beta }}_0,{\kappa }_2,\omega )\right]}^2}}\left({\displaystyle \frac{\partial S_2({\mathit{\beta }}_0,{\kappa }_2,\omega )}{\partial \mathit{\beta }}}\right){\left({\displaystyle \frac{\partial S_2({\mathit{\beta }}_0,{\kappa }_2,\omega )}{\partial \mathit{\beta }}}\right)}^{\prime }d\omega .$$

(26)

• Redistributing Roots.

After obtaining $\widehat{\mathit{\beta }}$, we factorize the fitted polynomials and move each root inside or outside the unit disk, as described in (10)–(13), thereby generating all $2^{r+s+r^{\prime }+s^{\prime }}$ candidates consistent with $\widehat{\mathit{\beta }}$. We select the final model (i.e., the correct allocation of causal/noncausal and invertible/noninvertible factors) by finding the root configuration that most reduces higher-order correlation in the residuals (see Section 2.1 and, e.g., Velasco and Lobato (2018) for third/fourth-order diagnostics). Formally, one may minimize $W\left(\mathit{\beta }\right)$ over this discrete set of admissible factorizations or rely on higher-order tests outlined below.

2.2.3. Spectral Identification Function

To incorporate higher-order dynamics, Velasco and Lobato (2018) propose combining the third- and fourth-order spectral densities into a single loss function. For $k=3$ or 4, define

$$L_{kT}\left(\mathit{\vartheta }\right)={\displaystyle \frac{{\left(2\pi \right)}^{k-2}}{2k{T}^{k-1}}}\sum _{\mathit{j}=1}^{T-1}{\displaystyle \frac{{\left|I_k({\omega }_1,\dots ,{\omega }_{k-1})-S_k\left(\mathit{\vartheta },{\kappa }_k\left(\mathit{\vartheta }\right);{\omega }_1,\dots ,{\omega }_{k-1}\right)\right|}^2}{{\left|\left({\prod }_{\ell =1}^{k-1}\psi (\mathit{\beta },{\omega }_{\ell })\right)\psi \left(\mathit{\beta },-{\sum }_{\ell =1}^{k-1}{\omega }_{\ell }\right)\right|}^2}},$$

(27)

where $\mathit{j}=(j_1,\dots ,j_{k-1})$ indexes all relevant frequency combinations. Let

$$L_T\left(\mathit{\vartheta }\right)=L_{3T}\left(\mathit{\vartheta }\right)+L_{4T}\left(\mathit{\vartheta }\right).$$

Minimizing $L_T\left(\mathit{\vartheta }\right)$ over $\mathit{\vartheta }$ helps detect non-Gaussian structures (e.g., nonlinearity, noncausality) not captured by second-order spectral analysis. In the Gaussian case ($k=2$), one recovers the usual Whittle function.

2.2.4. Portmanteau-Type Tests

We also employ a simplified joint portmanteau test (cf. Dalla et al., 2020; Jasiak & Neyazi, 2023) that targets both linear and nonlinear dependencies:

$${\rho }_{\epsilon }\left(k\right)=0\mathrm{and}{\rho }_{{\epsilon }^2}\left(k\right)=0(\forall k\ge 1),$$

where ${\rho }_{\epsilon }\left(k\right)$ is the autocorrelation of the residuals $\left\{{\epsilon }_t\right\}$ and ${\rho }_{{\epsilon }^2}\left(k\right)$ is the autocorrelation of their squares. Define

$$J_{\epsilon ,{\epsilon }^2,k}={\displaystyle \frac{T^2}{T-k}}\left({\widehat{\rho }}_{\epsilon }{\left(k\right)}^2+{\widehat{\rho }}_{{\epsilon }^2}{\left(k\right)}^2\right),$$

and the cumulative statistic

$$C_{\epsilon ,{\epsilon }^2,m}=\sum _{k=1}^m{J}_{\epsilon ,{\epsilon }^2,k}.$$

Under the null hypothesis of i.i.d. residuals, $J_{\epsilon ,{\epsilon }^2,k}$ converges in distribution to ${\chi }_2^2$, and $C_{\epsilon ,{\epsilon }^2,m}$ to ${\chi }_{2m}^2$. The preferred model is the one for which the null is not rejected at typical significance levels (or, among those rejected, yields the smallest value of $C_{\epsilon ,{\epsilon }^2,m}$).

2.2.5. Whittle Estimation for the GARCH($p,q$) Model

Whittle estimation of GARCH models was introduced by Giraitis et al. (2000), leveraging the ARMA(max$(p,q),q$) representation in (18). One key distinction is that Whittle methods for the GARCH process typically require higher-order moments: $\mathbb{E}\left[Y_t^8\right]<\infty$ in the original treatment. Later, Mikosch and Straumann (2002) showed that for squared GARCH processes, existence of the fourth moment suffices, though convergence is often slower than under maximum likelihood.

• Whittle Criterion for GARCH.

Let

$${\mathit{\xi }}_0=\left({\alpha }_{0,1},\dots ,{\alpha }_{0,p},{\beta }_{0,1},\dots ,{\beta }_{0,q}\right)$$

be the true GARCH($p,q$) parameter vector (excluding $\omega$). From (22), denote by $S_2^v(\mathit{\xi },{\kappa }_2^v,\omega )$ the power spectrum of $\left\{{\epsilon }_t^2\right\}$. The Whittle function becomes

$$W\left(\mathit{\xi }\right)=\sum _{\ell =1}^{T/2}\left\{log\left[S_2^v(\mathit{\xi },{\kappa }_2^v;{\omega }_{\ell })\right]+{\displaystyle \frac{I_2\left({\omega }_{\ell }\right)}{S_2^v(\mathit{\xi },{\kappa }_2^v;{\omega }_{\ell })}}\right\},$$

(28)

where $I_2\left({\omega }_{\ell }\right)$ is now the periodogram of $\left\{{\epsilon }_t^2\right\}$. The Whittle estimator is

$$\widehat{\mathit{\xi }}=arg\underset{\mathit{\xi }\in \mathsf{\Xi }}{min}W\left(\mathit{\xi }\right).$$

(29)

Because this method uses only $\left\{{\epsilon }_t^2\right\}$, the latent $\left\{{\sigma }_t^2\right\}$ sequence need not be explicitly computed, and no initial values for ${\sigma }_t^2$ are required. In practice, $\omega$ can be estimated via the sample variance:

$$var\left(Y_t\right)={\displaystyle \frac{\omega }{1-{\sum }_{u=1}^{max(p,q)}({\alpha }_u+{\beta }_u)}}.$$

However, if the mean $\left\{Y_t\right\}$ is noncausal or noninvertible, it remains an open question whether the classic Whittle approach to the GARCH component is fully robust, since the conditional variance depends only on past shocks, whereas the MARMA mean may incorporate lead terms.

2.2.6. Maximum Likelihood Estimation for the GARCH($p,q$) Model

Maximum likelihood (or quasi-maximum likelihood) estimation is the standard for GARCH models and requires weaker moment conditions than the Whittle approach (Francq & Zakoian, 2019). In particular, normal QMLE imposes only the existence of the second moment, whereas Whittle typically requires up to the fourth moment (Mikosch & Straumann, 2002).

• Likelihood Function.

Let

$$\mathit{\xi }=\left({\alpha }_1,\dots ,{\alpha }_p,{\beta }_1,\dots ,{\beta }_q\right)\in {\mathbb{R}}^{p+q},$$

excluding $\omega$. Conditional on initial values, the log-likelihood (assuming Gaussian errors) is

$$log\left({\mathrm{Lik}}_T\left(\mathit{\xi }\right)\right)=\sum _{t=1}^T\left[-\frac{1}{2}log\left(2\pi \right)-\frac{1}{2}log\left({\tilde{\sigma }}_t^2\left(\mathit{\xi }\right)\right)-\frac{{\epsilon }_t^2}{2{\tilde{\sigma }}_t^2\left(\mathit{\xi }\right)}\right],$$

(30)

where

$${\tilde{\sigma }}_t^2\left(\mathit{\xi }\right)=\omega +\sum _{u=1}^p{\alpha }_u{\epsilon }_{t-u}^2+\sum _{v=1}^q{\beta }_v{\tilde{\sigma }}_{t-v}^2.$$

The MLE is

$${\widehat{\mathit{\xi }}}_T=arg\underset{\mathit{\xi }\in \mathsf{\Xi }}{max}log\left({\mathrm{Lik}}_T\left(\mathit{\xi }\right)\right).$$

Under stationarity and finite variance, ${\widehat{\mathit{\xi }}}_T$ is consistent and asymptotically normal, with asymptotic covariance given by the inverse Fisher information. Empirical work often finds that MLE (or QMLE) outperforms Whittle if the moment requirements of the latter are not comfortably satisfied.

Because many financial time series (e.g., equity prices) appear nonstationary, one typically detrends or works with returns rather than levels. In the following section, we outline standard approaches for differencing or transforming $\left\{Y_t\right\}$ to achieve approximate stationarity before MARMA–GARCH modeling.

2.3. Detrending Methods

Many economic and financial time series exhibit trends that must be removed before modeling. Whether a trend is stochastic or deterministic has important implications for subsequent inference, and different detrending methods can alter the resulting dynamics (Aguirre & Lobato, 2024; Fries & Zakoïan, 2019; Hencic & Gouriéroux, 2015). In practice, formal unit-root tests (e.g., augmented Dickey–Fuller or Phillips–Perron) guide this decision, but their conclusions can be sensitive to lag-length choices and might not always reflect the true data-generating process.

When a series is believed to have a stochastic trend, simple differencing or a log-transformation is commonly applied. However, excessive differencing can remove meaningful low-frequency information and potentially introduce artificial serial dependence. Conversely, for a deterministic trend, one typically estimates a low-order polynomial (linear, quadratic, cubic) or employs more flexible smoothing (e.g., spline fitting). Each approach influences the residual dynamics that remain for MARMA–GARCH estimation.

The literature offers many possibilities: for instance, Hencic and Gouriéroux (2015) used a cubic polynomial trend, Fries and Zakoïan (2019) and Aguirre and Lobato (2024) relied on linear trends or simple mean-centering, while Hecq et al. (2017); Hecq and Voisin (2023) used seasonal filters (X–11 or Hodrick–Prescott). More recently, Blasques et al. (2023) proposed filtering that separately models a random-walk (fundamental) component and a bubble component.

• Our Implementation.

Here, we discuss two flexible methods for detrending: cubic splines and Fourier detrending. Each seeks to remove low-frequency components, thus leaving a (approximately) stationary series suitable for MARMA–GARCH modeling. In both cases, the knot placements (for splines) or the number of frequencies removed (in Fourier detrending) are ultimately user choices and can influence the estimated dynamics. Consequently, we treat these methods as a preprocessing step before specifying and fitting the MARMA–GARCH model.

2.3.1. Cubic Splines

Cubic splines provide a smooth, piecewise polynomial fit to the observed data. Let ${\left\{Y_t\right\}}_{t=1}^T$ be the original series. A cubic spline $S(·)$ is a function defined by separate cubic polynomials on each sub-interval (or “knot”):

$$S\left(Y\right)=a_t+b_t\left(Y-Y_t\right)+c_t{\left(Y-Y_t\right)}^2+d_t{\left(Y-Y_t\right)}^3,Y\in [Y_t,Y_{t+1}],$$

for $t=1,\dots ,T-1$. The coefficients $\{a_t,b_t,c_t,d_t\}$ are chosen so that the spline and its first two derivatives are continuous at each knot. Common boundary conditions are $S^{″}\left(Y_1\right)=0$ and $S^{″}\left(Y_T\right)=0$.

If one partitions $\left\{Y_t\right\}$ into n knots, the spline coefficients can be estimated by solving a tridiagonal system:

$${\displaystyle \frac{h_{t-1}}{6}}{M}_{t-1}+{\displaystyle \frac{h_{t-1}+h_t}{3}}{M}_t+{\displaystyle \frac{h_t}{6}}{M}_{t+1}={\displaystyle \frac{Y_{t+1}-Y_t}{h_t}}-{\displaystyle \frac{Y_t-Y_{t-1}}{h_{t-1}}},$$

where $M_t=S^{″}\left(Y_t\right)$ and $h_t=Y_{t+1}-Y_t$. Once $\left\{M_t\right\}$ are found, the polynomial coefficients follow straightforwardly:

$$\begin{array}{cccc}\hfill a_t& =Y_t,\hfill & \hfill b_t& ={\displaystyle \frac{Y_{t+1}-Y_t}{h_t}}-{\displaystyle \frac{h_t}{3}}\left(2M_t+M_{t+1}\right),\hfill \\ \hfill c_t& =M_t,\hfill & \hfill d_t& ={\displaystyle \frac{M_{t+1}-M_t}{3h_t}}.\hfill \end{array}$$

The fitted spline defines a smooth trend $t\left(Y\right)$. The detrended observations are then

$$Y_t^{\left(\mathrm{detrend}\right)}=Y_t-t\left(Y_t\right).$$

In our numerical examples, we follow Hall and Jasiak (2024) by using a second-order spline with one knot every 30 observations, but in practice, the choice of knot frequency can be refined using graphical diagnostics or information criteria.

2.3.2. Fourier Detrending

Fourier detrending filters out low-frequency components of the discrete Fourier transform (DFT). Let

$$d_T\left(\omega \right)=\sum _{t=1}^T{Y}_t{e}^{-it\omega }$$

be the DFT of $\left\{Y_t\right\}$, with frequencies ${\omega }_{\ell }=\frac{2\pi \ell }{T}$ for $\ell =0,1,\dots ,T-1$. Suppose one identifies $\ell =1,\dots ,D$ (and by symmetry, $\ell =T-D+1,\dots ,T$) as the “lowest” and “highest” frequencies capturing long-run or trend variations. Following Lobato and Velasco (2022), we zero out these components:

$$d_T\left({\omega }_{\ell }\right)=0\mathrm{for}\ell =1,\dots ,D\mathrm{and}\ell =T-D+1,\dots ,T.$$

Then, an inverse DFT yields the detrended series:

$$Y_t^{\left(\mathrm{detrend}\right)}=\Re \left({\displaystyle \frac{1}{T}}\sum _{\ell =0}^{T-1}{d}_T\left({\omega }_{\ell }\right)e^{i{\omega }_{\ell }t}\right),$$

where $\Re (·)$ denotes the real part. The integer D controls how many frequencies are removed, and hence determines the amount of low-frequency content retained in $\left\{Y_t^{\left(\mathrm{detrend}\right)}\right\}$. Typically, one inspects the periodogram or employs domain knowledge to choose D.

Both methods—cubic splines and Fourier filtering—serve to remove gradual trends or low-frequency cycles. In subsequent sections, we apply these detrended data to the MARMA–GARCH estimation procedures described above (Section 2.1 and Section 2.2). As with any filtering, the choice of knot frequency or frequency cutoffs can subtly alter the remaining dynamics; hence, we emphasize that careful empirical checking, including residual diagnostics, is essential to ensure an appropriate trend-removal strategy for a given application.

3. Results

3.1. Simulation Study

In this section, we investigate and compare frequency-domain (Whittle) and time-domain (maximum likelihood) estimation methods for the GARCH component of a MARMA$(r,s,r^{\prime },s^{\prime })$–GARCH$(p,q)$ model. While the properties of Whittle estimation for MARMA processes have been studied by Hecq and Velasquez-Gaviria (2024), our focus here is on the GARCH parameters.

• Simulation Design.

To generate a sample ${\left\{Y_t\right\}}_{t=1}^T$ from a MARMA$(r,s,r^{\prime },s^{\prime })$–GARCH$(p,q)$ process, we follow these steps:

• Generate Innovations: Draw an i.i.d. sequence $\left\{{\eta }_t\right\}$ of length T from a chosen distribution $\mathfrak{F}\left(\zeta \right)$ (e.g., Student’s t, skew-t, or exponential), with $\mathbb{E}\left[{\eta }_t\right]=0$ and $\mathrm{Var}\left({\eta }_t\right)=1$.
• Simulate GARCH($p,q$) Variance: Initialize ${\sigma }_1^2>0$. Recursively compute

  $${\sigma }_t^2=\omega +\sum _{u=1}^p{\alpha }_u{\eta }_{t-u}^2{\sigma }_{t-u}^2+\sum _{v=1}^q{\beta }_v{\sigma }_{t-v}^2,t=2,\dots ,T,$$

  ensuring $\left\{{\sigma }_t^2\right\}$ remains strictly positive.
• Form GARCH Innovations:

  $${\epsilon }_t={\eta }_t{\sigma }_t,t=1,\dots ,T.$$
• Fourier Representation of Residuals: Compute the DFT of $\left\{{\epsilon }_t\right\}$:

  $$d_{\epsilon }\left({\omega }_j\right)=\sum _{t=1}^T{\epsilon }_t{e}^{-it{\omega }_j},{\omega }_j=\frac{2\pi j}{T},j=0,1,\dots ,T-1.$$
• MARMA Transfer Function: Select parameters $\mathit{\vartheta }$ for the MARMA$(r,s,r^{\prime },s^{\prime })$ part. The frequency-domain transfer function is

  $$\psi (\mathit{\vartheta },\omega )={\displaystyle \frac{{\theta }^{+}\left(z\right){\theta }^{*}\left(z\right)}{{\varphi }^{+}\left(z\right){\varphi }^{*}\left(z\right)}},z=e^{-i\omega }.$$
• Construct the MARMA Process: Multiply the DFT of the innovations by the transfer function:

  $$d_T\left({\omega }_j\right)=d_{\epsilon }\left({\omega }_j\right)\times \psi (\mathit{\vartheta },{\omega }_j).$$
• Inverse DFT (Real Part):

  $$Y_t=\Re \left({\displaystyle \frac{1}{T}}\sum _{j=0}^{T-1}{d}_T\left({\omega }_j\right)e^{it{\omega }_j}\right),t=1,\dots ,T.$$
  This final step yields $\left\{Y_t\right\}$, a realization from the specified MARMA$(r,s,r^{\prime },s^{\prime })$–GARCH$(p,q$) process, provided stationarity and positivity conditions are met.

Figure 1 displays a single simulated trajectory of a MARMA(1,1,0,0)–GARCH(1,1) process, including its conditional variance $\left\{{\sigma }_t^2\right\}$ and the resulting time series $\left\{Y_t\right\}$. The conditional variance in the top panel exhibits clusters of high volatility and strong responses to shocks. The bottom panel reveals nonlinear dynamics in $\left\{Y_t\right\}$, such as localized explosions and volatility bursts, reminiscent of real-world financial markets.

3.1.1. Whittle vs. Maximum Likelihood for Pure GARCH($p,q$)

We first assess the estimation accuracy for a martingale difference GARCH(1,1) process (i.e., no mean structure).

Table 1. Bias and RMSE for GARCH(1,1) Parameter Estimates Under Various Distributions. True parameters: $\omega =0.1, \alpha =0.15, \beta =0.85$. Sample sizes: $T=1000, 2000, 5000$. Results use Whittle and Maximum Likelihood estimation. Cell shading reflects absolute magnitude of each estimate (white = near 0, red = larger bias or RMSE).

	Bias						RMSE
	Whittle Estimation			Maximum Likelihood			Whittle Estimation			Maximum Likelihood
T	${\displaystyle \mathit{\omega }}$	${\displaystyle \mathit{\alpha }}$	${\displaystyle \mathit{\beta }}$	${\displaystyle \mathit{\omega }}$	${\displaystyle \mathit{\alpha }}$	${\displaystyle \mathit{\beta }}$	${\displaystyle \mathit{\omega }}$	${\displaystyle \mathit{\alpha }}$	${\displaystyle \mathit{\beta }}$	${\displaystyle \mathit{\omega }}$	${\displaystyle \mathit{\alpha }}$	${\displaystyle \mathit{\beta }}$
Exp(1) Innovations
1000	0.089	−0.030	−0.030	0.020	0.006	−0.020	0.288	0.073	0.182	0.067	0.063	0.081
2000	0.076	−0.028	−0.016	0.010	0.006	−0.011	0.163	0.049	0.100	0.030	0.028	0.035
5000	0.073	−0.024	−0.016	0.002	0.000	−0.002	0.115	0.032	0.075	0.010	0.011	0.013
Student’s t (8) Innovations
1000	0.055	−0.018	−0.015	0.014	−0.001	−0.009	0.136	0.039	0.092	0.034	0.026	0.036
2000	0.043	−0.011	−0.013	0.007	0.001	−0.005	0.064	0.030	0.050	0.015	0.012	0.015
5000	0.039	−0.008	−0.012	0.001	−0.001	0.000	0.042	0.016	0.031	0.006	0.005	0.006
Student’s t (4.5) Innovations
1000	0.103	−0.023	−0.039	0.020	0.003	−0.017	0.248	0.054	0.152	0.046	0.039	0.051
2000	0.079	−0.024	−0.020	0.012	0.004	−0.012	0.120	0.033	0.072	0.022	0.022	0.026
5000	0.054	−0.024	−0.008	0.003	0.000	−0.003	0.063	0.019	0.045	0.007	0.008	0.009
skew-t (4.5,1.5) Innovations
1000	0.128	−0.022	−0.051	0.021	0.007	−0.022	0.327	0.064	0.166	0.049	0.050	0.062
2000	0.107	−0.021	−0.033	0.008	0.007	−0.011	0.197	0.034	0.095	0.021	0.024	0.027
5000	0.103	−0.019	−0.037	0.004	0.004	−0.005	0.097	0.025	0.065	0.009	0.011	0.012

reports the bias and RMSE of the Whittle and maximum likelihood (ML) estimators for parameters $\omega ,\alpha ,\beta$. We consider several distributions (exponential, Student’s t with different degrees of freedom, and skew-t) and three sample sizes.

The ML estimator achieves lower bias and RMSE than the Whittle estimator in all cases, and this gap becomes more pronounced for heavier-tailed distributions (Student’s t with lower degrees of freedom or skew-t). Whittle’s heavier moment requirements appear to render it more sensitive to large outliers. Overall, ML estimation delivers more reliable GARCH parameter estimates, a finding that aligns with common practices in financial econometrics.

3.1.2. Estimating MARMA($r,s,r^{\prime },s^{\prime }$)–GARCH($p,q$)

We next compare how parameter estimates differ when fitting a causal ARMA representation (e.g., AR(2) for a MARMA(1,1,0,0)) versus the true noncausal specification. In particular,

Table 2. Parameter Estimates for the MARMA(1,1,0,0)–GARCH(1,1) process. True parameters: ${\varphi }^{+}=0.2,{\varphi }^{*}=0.7,\omega =0.01,\alpha =0.15,\beta =0.85$. Each row gives mean estimates (top row) and standard deviations of estimates (bottom row) across repeated simulations for different sample sizes (T) and innovation distributions.

	AR(2)–GARCH(1,1) Fit				MARMA(1,1,0,0)–GARCH(1,1) Fit
T	${\displaystyle {\mathit{\varphi }}_{\mathbf{1}}}$	${\displaystyle {\mathit{\varphi }}_{\mathbf{2}}}$	${\displaystyle \mathit{\omega }}$	${\displaystyle \mathit{\alpha },\mathit{\beta }}$	${\displaystyle {\mathit{\varphi }}^{+}}$	${\displaystyle {\mathit{\varphi }}^{*}}$	${\displaystyle \mathit{\omega }}$	${\displaystyle \mathit{\alpha },\mathit{\beta }}$
Exp(1) Innovations
1000	0.893	−0.143	0.023	0.358, 0.542	0.675	0.218	0.012	0.108, 0.828
	(0.048)	(0.047)	(0.009)	(0.087, 0.116)	(0.064)	(0.089)	(0.009)	(0.060, 0.093)
2000	0.896	−0.142	0.022	0.359, 0.549	0.686	0.211	0.011	0.106, 0.836
	(0.036)	(0.036)	(0.006)	(0.060, 0.077)	(0.048)	(0.065)	(0.005)	(0.041, 0.054)
5000	0.899	−0.141	0.021	0.359, 0.558	0.695	0.204	0.011	0.104, 0.843
	(0.025)	(0.024)	(0.004)	(0.040, 0.049)	(0.030)	(0.041)	(0.003)	(0.028, 0.033)
Student’s t (8) Innovations
1000	0.896	−0.141	0.013	0.115, 0.820	0.685	0.211	0.012	0.102, 0.836
	(0.040)	(0.039)	(0.006)	(0.032, 0.055)	(0.059)	(0.076)	(0.006)	(0.031, 0.054)
2000	0.900	−0.141	0.011	0.115, 0.828	0.694	0.206	0.011	0.100, 0.844
	(0.032)	(0.030)	(0.004)	(0.021, 0.034)	(0.038)	(0.053)	(0.004)	(0.021, 0.033)
5000	0.899	−0.140	0.011	0.115, 0.830	0.698	0.201	0.010	0.101, 0.846
	(0.020)	(0.019)	(0.002)	(0.013, 0.021)	(0.024)	(0.033)	(0.002)	(0.013, 0.020)
Student’s t (4.5) Innovations
1000	0.897	−0.140	0.016	0.180, 0.736	0.685	0.212	0.013	0.111, 0.825
	(0.047)	(0.047)	(0.009)	(0.050, 0.088)	(0.066)	(0.089)	(0.008)	(0.053, 0.078)
2000	0.898	−0.142	0.014	0.177, 0.751	0.689	0.209	0.012	0.108, 0.834
	(0.037)	(0.037)	(0.005)	(0.036, 0.053)	(0.049)	(0.068)	(0.005)	(0.040, 0.050)
5000	0.901	−0.142	0.013	0.175, 0.756	0.695	0.205	0.011	0.104, 0.842
	(0.026)	(0.025)	(0.003)	(0.021, 0.032)	(0.032)	(0.044)	(0.003)	(0.024, 0.032)
skew-t (4.5,1.5) Innovations
1000	0.897	−0.144	0.017	0.222, 0.683	0.678	0.219	0.012	0.111, 0.825
	(0.050)	(0.048)	(0.009)	(0.063, 0.103)	(0.067)	(0.090)	(0.010)	(0.065, 0.094)
2000	0.897	−0.143	0.016	0.217, 0.699	0.684	0.213	0.012	0.109, 0.832
	(0.039)	(0.038)	(0.005)	(0.040, 0.063)	(0.051)	(0.070)	(0.006)	(0.046, 0.061)
5000	0.899	−0.140	0.015	0.214, 0.708	0.695	0.204	0.011	0.104, 0.843
	(0.028)	(0.027)	(0.003)	(0.025, 0.037)	(0.036)	(0.049)	(0.003)	(0.028, 0.036)

considers a data-generating process (DGP) of MARMA(1,1,0,0)–GARCH(1,1), but estimates parameters under both (i) AR(2)–GARCH(1,1) and (ii) MARMA(1,1,0,0)–GARCH(1,1). Although both mean specifications generate uncorrelated residuals, higher-order statistics differ, affecting the estimated GARCH parameters.

When the mean is incorrectly constrained to a causal AR(2) form (left panel), $\left\{{\epsilon }_t\right\}$ are uncorrelated but contain higher-order dependencies that invalidate the correct GARCH structure (Hecq & Velasquez-Gaviria, 2024). This mismatch translates into biased estimates of $\omega ,\alpha ,\beta$. By contrast, the correctly specified MARMA(1,1,0,0) representation (right panel) leads to parameter estimates that converge accurately on the true values.

Figure 2 illustrates this difference more clearly. Estimates from the AR(2)–GARCH(1,1) representation (left boxplots) systematically overestimate $\omega$ and $\alpha$ while underestimating $\beta$. In financial terms, this translates to higher perceived long-term volatility and an exaggerated immediate reaction to shocks, but less persistence. In contrast, the correct MARMA(1,1,0,0)–GARCH(1,1) fit (right boxplots) recovers nearly unbiased estimates and narrower spreads with a growing sample size.

These Monte Carlo experiments produce two key findings:

• Whittle vs. MLE for GARCH: Maximum likelihood estimation more accurately recovers the GARCH parameters, particularly in heavy-tailed environments, while Whittle estimation exhibits larger bias and RMSE.
• Importance of Correct MARMA Specification: Using a purely causal ARMA representation when the data are partially noncausal can distort the higher-order structure and lead to biased GARCH estimates. Correctly assigning the roots to causal/noncausal polynomials (as in a MARMA model) mitigates this issue.

These points are particularly relevant in financial applications, where heavy tails and bubbles or local explosiveness can invalidate causal ARMA assumptions. In such contexts, careful attention to model specification and estimation methods is crucial for reliable volatility forecasts and risk assessments.

3.2. Empirical Applications in Cryptocurrencies

We next illustrate the proposed MARMA–GARCH framework using daily data for two prominent cryptocurrencies, Bitcoin (BTC) and Ethereum (ETH), over the sample period January 2021–July 2024. These assets have been noted for their extreme volatility (Cheah & Fry, 2015; Corbet et al., 2017; Hafner, 2018), heavy tails and clustering in returns (Ardia et al., 2019), and bubble-like local explosions (Hall & Jasiak, 2024). As a result, cryptocurrencies provide a natural testing ground for models that allow for noncausal (forward-looking) dynamics.

Figure 3 plots the daily USD prices of Bitcoin and Ethereum. Both assets experienced significant peaks during 2021—reaching approximately USD 60,000 (BTC) and USD 4000 (ETH)—fueled by institutional adoption. Regulatory constraints, including a direct ban in China, contributed to a sharp drawdown by mid-2021, reinforcing the perception that these markets can be subject to speculative bubbles (Bazán-Palomino, 2022). Further volatility ensued in 2022 amid macroeconomic uncertainties (notably inflationary pressures and geopolitical tensions), with partial recoveries in late 2023 tied to central bank policy signals and a growing array of crypto-derivative products.

Although changes in regulation, macro news, and investor sentiment have been widely cited as drivers of cryptocurrency volatility, these markets also appear to exhibit persistent departures from pure random-walk behavior. Recurrent episodes of rapid price surges followed by abrupt declines suggest potential noncausal features—i.e., future expectations or bubble-like behavior—making this domain suitable for our MARMA–GARCH methodology.

• Detrending.

Both BTC and ETH prices are nonstationary, exhibiting long-run growth punctuated by sharp local explosions. As detailed in Section 2.3, we use two detrending techniques to remove low-frequency components while preserving short-run dynamics:

• Cubic Splines: Following Hall and Jasiak (2024), we employ a piecewise cubic spline with knots every 30 days. Additional experiments (e.g., quadratic vs. cubic, varying knot frequencies) suggested that a 30-day cubic spline balances flexibility with parsimony for these data.
• Fourier Filtering: We remove the lowest and highest 32 frequency components in the discrete Fourier transform of $\left\{Y_t\right\}$, corresponding to low-frequency trends and ultra-long cycles (Lobato & Velasco, 2022). This approach effectively zeroes out the first (and last) 32 harmonics before an inverse transform.

Figure 4 and Figure 5 depict the resulting detrended series for Bitcoin and Ethereum under both spline and Fourier filtering.

Table 3. Descriptive Statistics of Original and Detrended Prices (Bitcoin and Ethereum).

	Mean	Std. Dev.	Min	Max	Skewness	Kurtosis
Bitcoin
Price	38,384.65	15,015.49	15,787.28	73,083.50	0.473	2.188
Cubic Splines	0.000	2190.83	−8347.69	8712.28	0.034	4.210
Fourier	0.000	2177.26	−19,449.88	12,595.56	−0.874	13.362
Ethereum
Price	2324.60	888.75	730.37	4812.09	0.680	2.439
Cubic Splines	0.000	172.25	−858.58	1003.89	0.193	6.543
Fourier	0.000	175.12	−1433.18	1162.61	−0.434	13.595

contrasts descriptive statistics of the raw and detrended data. While both methods yield mean-zero series, the Fourier-detrended versions manifest higher kurtosis, indicating that subtracting low-frequency oscillations may amplify outlier effects at shorter horizons. Notably, both detrending approaches leave intact the marked skewness and heavy tails often reported in cryptocurrency markets (Ardia et al., 2019).

• Fitting MARMA–GARCH.

Once detrended, each series displays weak autocorrelation at the first or second lag. For both Bitcoin and Ethereum, AIC/BIC selection suggests an AR(1) model for the conditional mean if one imposes a purely causal structure. However, to allow for potential noncausality, we invert the root(s) of the AR polynomial and compare all possible causal–noncausal configurations (see Section 2.1 and Section 2.2).

In all cases, the data favor a fully noncausal representation, $\mathrm{MARMA}(0,1,0,0)$. Concretely, the estimated parameter ${\widehat{\varphi }}^{*}\approx 0.825$ indicates strong feedback from future expectations or bubble-like dynamics (Hall & Jasiak, 2024). The residuals from this MARMA(0,1,0,0) fit (in both frequency and time domains) pass higher-order spectral diagnostics (Velasco & Lobato, 2018) and a portmanteau-type test on raw and squared residuals.

• GARCH(1,1) Estimation.

We then fit a GARCH(1,1) model to the residuals $\left\{{\widehat{\epsilon }}_t\right\}$, comparing the following:

$$\mathrm{MARMA}(0,1,0,0)–\mathrm{GARCH}(1,1)\mathrm{versus}\mathrm{AR}\left(1\right)–\mathrm{GARCH}(1,1).$$

Table 4. Estimated parameters for the noncausal model (MARMA(0,1,0,0)–GARCH(1,1)) and the causal model (AR(1)–GARCH(1,1)) for Bitcoin and Ethereum. The table also reports the Ljung–Box test p-values for the residuals and the McLeod–Li test p-values for the standardized residuals, under two different detrending methods (Cubic Splines and Fourier).

	Bitcoin				Ethereum
	Cubic Splines		Fourier		Cubic Splines		Fourier
	Noncausal	Causal	Noncausal	Causal	Noncausal	Causal	Noncausal	Causal
Estimated Parameters
${\varphi }^{+},{\varphi }^{*}$	0.8115 (0.0197)	0.8115 (0.0197)	0.8341 (0.0157)	0.8341 (0.0157)	0.8243 (0.1580)	0.8243 (0.1580)	0.8395 (0.0154)	0.8395 (0.0154)
$\omega$	0.0048 (0.0017)	0.0029 (0.0010)	0.0035 (0.0016)	0.0031 (0.0011)	0.0053 (0.0021)	0.0056 (0.0017)	0.0065 (0.0020)	0.0057 (0.0018)
$\alpha$	0.0952 (0.0179)	0.0564 (0.0106)	0.0717 (0.0215)	0.0593 (0.0117)	0.0937 (0.0195)	0.0828 (0.0147)	0.1142 (0.0208)	0.0856 (0.0156)
$\beta$	0.9037 (0.0171)	0.9406 (0.0102)	0.9274 (0.0210)	0.9382 (0.0111)	0.9053 (0.0187)	0.9137 (0.0141)	0.8847 (0.0297)	0.9116 (0.0149)
Residuals: Ljung–Box Test p-values
Lag(1)	0.7320	0.6234	0.3571	0.6033	0.5967	0.6533	0.4724	0.8152
Lag(2)	0.3611	0.2300	0.3124	0.1920	0.3602	0.1375	0.2636	0.1378
Lag(3)	0.2495	0.1177	0.1752	0.1663	0.1543	0.1506	0.0768	0.0861
Lag(4)	0.1799	0.1193	0.1608	0.1393	0.4288	0.1074	0.2514	0.1522
Lag(5)	0.2680	0.1801	0.2524	0.2210	0.1006	0.2217	0.1296	0.2971
Standardized Residuals: McLeod–Li Test p-values
Lag(1)	0.5715	0.4675	0.2073	0.3690	0.7695	0.7769	0.6491	0.9418
Lag(2)	0.7797	0.4635	0.4348	0.4882	0.8757	0.8125	0.8982	0.9142
Lag(3)	0.3872	0.2529	0.3059	0.3532	0.7137	0.5604	0.5430	0.5474
Lag(4)	0.5366	0.3933	0.4488	0.5099	0.8478	0.6703	0.6943	0.6602
Lag(5)	0.6540	0.3478	0.5191	0.5179	0.6248	0.4927	0.7015	0.4562

summarizes the estimation results under both spline and Fourier detrending. Three primary findings emerge:

• Long-Run Volatility (ω): The noncausal MARMA–GARCH model consistently yields higher estimates of $\omega$ than the causal AR(1)–GARCH(1,1). Hence, ignoring noncausality may understate the baseline variance level.
• Reactivity to Shocks (α): The noncausal specification produces larger $\alpha$, suggesting that these cryptocurrencies exhibit more immediate responses to new information when future expectations (or bubble-like feedback) are accounted for.
• Volatility Persistence (β): Reflecting the stronger short-run reactivity, $\beta$ is lower in the MARMA–GARCH model, implying reduced persistence once the noncausal feedback is captured.

Figure 6 compares the conditional volatility estimates for BTC and ETH. The MARMA(0,1,0,0)–GARCH(1,1) model (red lines) reacts more sharply during volatile episodes (2021 and 2022), yet tends to be lower in tranquil periods. By contrast, the AR(1)–GARCH(1,1) model (black lines) smooths out these extremes, potentially understating tail risks in real-time risk management contexts.

• Diagnostic Checks.

Table 4 also reports Ljung–Box test p-values (on raw residuals) and McLeod–Li test p-values (on squared residuals) for both the MARMA–GARCH and AR–GARCH fits. Across multiple lags, neither model shows significant serial correlation in the final residuals, indicating that both representations adequately capture second-order dependencies. However, only the MARMA–GARCH specification is consistent with the higher-order spectral tests, reinforcing the conclusion that noncausal dynamics are present.

Although we obtain broadly consistent results under both the spline and Fourier methods, our analysis reveals that detrending can alter higher-order moments and thus affect the identification of noncausal roots. In practice, it remains crucial to verify that the chosen method for removing low-frequency components does not inadvertently remove important cyclical or bubble-like behaviors. For the Bitcoin and Ethereum data examined here, a 30-day spline and a low-frequency cutoff of 32 harmonics yield quite similar outcomes.

These empirical findings highlight that allowing for noncausal (forward-looking) dynamics in cryptocurrency prices can substantially affect the estimated volatility process. Specifically, in MARMA(0,1,0,0)–GARCH(1,1) models, shocks are met with higher immediate volatility responses and a somewhat lower persistence of volatility clustering, relative to a standard AR(1)–GARCH(1,1). This may be interpreted as evidence of speculative or bubble elements, where future price expectations significantly amplify immediate volatility reactions. From a risk-management perspective, such models potentially offer more realistic scenarios of rapid volatility escalation in cryptocurrency markets—an important consideration for both institutional investors and regulators.

Allowing for noncausal (forward-looking) components in cryptocurrency price dynamics leads to a larger estimated immediate impact of shocks ($\alpha$) and higher overall volatility levels ($\omega$), while reducing persistence ($\beta$). In practical terms, a MARMA–GARCH approach may provide more realistic volatility forecasts during bubble-like expansions and subsequent crashes, helping financial practitioners better gauge tail risk. Of course, model identification remains sensitive to the detrending method and the assumption of stationarity after trend removal; thus, careful robustness checks are recommended.

Our analysis suggests that both Bitcoin and Ethereum exhibit strong noncausal features and that GARCH estimates can be substantially biased if one imposes a purely causal ARMA structure. By contrast, MARMA–GARCH captures higher-order dynamics and more pronounced reactivity to new information. The results confirm the practical relevance of noncausal modeling in markets prone to speculative cycles, with repercussions for volatility forecasting and risk management.

4. Discussion and Conclusions

This paper has introduced a new MARMA–GARCH model that accounts for bubble-like (explosive) episodes in financial time series together with time-varying conditional volatility. The central motivation is to allow for noncausal (forward-looking) components in the mean dynamics while simultaneously capturing ARCH-type effects in the variance. This joint specification becomes particularly relevant during speculative booms and busts, when prices can exhibit abrupt run-ups and subsequent collapses.

A key contribution lies in showing how to factorize the AR and MA polynomials in a manner that places one or more roots inside the unit disk, thereby capturing the noncausal dimension. By relaxing the typical ARMA restriction that all roots lie outside the unit circle, the MARMA model can generate local explosiveness or feedback from future expectations. This capacity to accommodate bubble-like behavior is rare in standard time series approaches, which generally assume causality. In addition, the GARCH component models time-varying volatility in a way that remains valid under noncausal mean dynamics.

With regard to estimation, two broad approaches were proposed. The first is a frequency-domain method based on the Whittle likelihood, which is asymptotically equivalent to maximizing the Gaussian likelihood in the presence of finite variance. The second is time-domain maximum likelihood, which can be particularly advantageous for heavy-tailed or skewed return distributions. While both methods deliver broadly consistent results for the MARMA portion of the model (assuming stationarity and finite variance), the simulation evidence indicates that time-domain maximum likelihood provides more robust estimation of the GARCH parameters in the presence of high kurtosis or significant skewness. This finding reflects the greater flexibility of maximum likelihood estimation in accommodating non-Gaussian features—a salient feature in many financial contexts.

An important insight is that a misspecified mean equation can bias the estimated parameters of the GARCH process. In particular, forcing the mean to be a causal AR(1) when it is, in fact, partially or fully noncausal leads to underestimating the immediate response to shocks (the $\alpha$ parameter) and overstating the degree of volatility persistence (the $\beta$ parameter). Consequently, the model may fail to capture swift volatility reactions to new information and systematically underestimate tail risk. Such biases have material consequences for practitioners and regulators who rely on these models for real-time risk management, portfolio hedging, and broader policy decisions.

Applying the MARMA–GARCH framework to two major cryptocurrencies, Bitcoin and Ethereum, highlights its empirical relevance. After detrending the data using both cubic splines and Fourier filters, we observe that noncausal specifications consistently outperform purely causal representations. The noncausal character, embodied in a MARMA($0,1,0,0$) structure, captures bubble-like episodes that standard ARMA models treat as anomalous volatility clusters. The associated GARCH estimates reveal stronger shock responsiveness (higher $\alpha$) and lower persistence ($\beta$) under the noncausal model, suggesting that bubble dynamics alter the interplay between incoming news and volatility. Such findings resonate with the high levels of speculation and price explosiveness often witnessed in crypto markets.

Notwithstanding these strengths, several caveats merit attention. First, both Whittle and Gaussian likelihood methods require at least finite second (and ideally higher) moments. Markets with extremely heavy tails could violate such assumptions. Second, the detection of noncausal roots hinges on the presumption that the data become stationary post detrending. Different detrending choices (e.g., polynomial, spline, or frequency-based) can shift the low-frequency components and affect the subsequent root factorization. Third, the method could be extended to more complex scenarios, including structural breaks, vector autoregressions, or regime-switching volatility, all of which may be essential in certain market environments.

Looking ahead, the MARMA–GARCH approach invites further research on the role of anticipatory or bubble-related behavior in a broader class of assets, including equities, commodities, and housing markets. Its capacity to depict forward-looking price formation can potentially elucidate herding phenomena, investor sentiment cycles, and other behavioral dynamics that conventional ARMA–GARCH models may fail to capture. From a methodological standpoint, future investigations might explore distributional assumptions beyond standard parametric families, such as robust filters or mixture distributions, to accommodate the pronounced skewness and kurtosis frequently encountered in high-volatility markets.

• Practical and Societal Implications.

The findings in this study have notable implications for financial practitioners, regulatory authorities, and market participants more generally. By explicitly modeling the noncausal behavior, the MARMA–GARCH framework enhances the detection of bubble-like episodes and sharp downturns, thus improving the short-run forecasts of volatility during turmoil periods. This refinement is crucial in risk management applications, where underestimation of volatility can lead to inadequate capital reserves and margin requirements. Regulators may find value in an approach that isolates forward-looking or speculative episodes, thereby enabling more effective macroprudential oversight. Finally, a better grasp of bubble mechanisms and extreme volatility also bears on societal concerns: overly leveraged investors, pension funds, and retail traders can be severely impacted by rapid price collapses, and the systemic repercussions of such events can extend to broader economic stability. By offering a more nuanced model of how future expectations amplify shock impacts, the MARMA–GARCH framework contributes to a deeper, more informed discourse on the interplay between speculation, volatility, and financial market integrity.