M-estimation in GARCH models without higher order moments

1 Introduction

The Generalized autoregressive conditional heteroscedastic (GARCH) models have been used extensively to analyze the volatility or the instantaneous variability of a financial time series

{X_{t}; 1 \leq t \leq n}

. A series

{X_{t}; t \in Z}

is said to follow a GARCH

(p, q)

model if

X_{t} = σ_{t} ϵ_{t},

(1.1)

where ${ϵ_{t}; t \in Z}$ are unobservable i.i.d. errors with symmetric distribution around zero and

σ_{t} = (ω_{0} + p \sum i = 1 α_{0 i} X_{t - i}^{2} + q \sum j = 1 β_{0 j} σ_{t - j}^{2})^{1 / 2}, t \in Z,

(1.2)

with $ω_{0}$ , $α_{0 i}, β_{0 j} > 0$ , $\forall i, j$ . Mukherjee (2008) proposed a class of M-estimators for estimating the GARCH parameter

\boldmathθ0=(ω0,α01,…,α0p,β01,…,β0q)′

(1.3)

based on observations ${X_{t}; 1 \leq t \leq n}$ . The M-estimators are asymptotically normal under some moment assumptions on the error distribution and are more robust than the commonly-used quasi maximum likelihood estimator (QMLE). Mukherjee (2020) considered a class of weighted bootstrap methods to approximate the distributions of these estimators and established the asymptotic validity of such bootstrap. In this paper, we apply an iteratively re-weighted algorithm to compute the M-estimates and the corresponding bootstrap estimates with specific attention to Huber’s, $μ$ - and Cauchy-estimates which were not considered in the literature in details. The iteratively re-weighted algorithm turns out to be particularly useful in computing bootstrap replicates since it avoids the re-computation of some core quantities for new bootstrap samples.

The class of M-estimators includes the QMLE. The asymptotic normality and the asymptotic validity of bootstrapping the QMLE were derived under the finite fourth moment assumption on the error distribution. However, there are other M-estimators such as the $μ$ -estimator and Cauchy-estimator which are asymptotic normal under mild assumption on the finiteness of lower order moments. Since heavy-tailed error distributions without higher order moments are common in the GARCH modeling of many real financial time series, it becomes worthwhile to use these estimators for such series but unfortunately they have not been investigated in the literature. One of the contributions of this paper is to reveal precisely the importance of such alternative M-estimators to analyze financial data instead of using the QMLE.

In an earlier work, Muler and Yohai (2008) analyzed the Electric Fuel Corporation (EFCX) time series and fitted a GARCH (1, 1) model. Using exploratory analysis, they detected presence of outliers and considered estimation of parameters based on robust methods. It turned out that estimates based on different methods vary widely and so it is difficult to assess which method should be relied on in similar situations. In this paper, we use M-estimates with mild assumptions on error moments to analyze the EFCX series.

Francq and Zakoïan (2009) underscored the importance of using higher order GARCH models such as GARCH (2, 1) for some real financial time series but the computation and simulation results for such models are not available widely in the literature. We investigate the role of M-estimators for the GARCH (2, 1) model through extensive simulations and real data analysis. We also provide simulation results and analysis for the GARCH (1, 2) model.

The paper is organized as follows. Sections 2 and 3 set the background. In particular, we discuss the class of M-estimators and give examples in Section 2. Section 3 contains bootstrap formulation and the statement on the asymptotic validity of the bootstrap. Section 4 discusses computational aspects of M-estimators and its bootstrapped replicates. Section 5 reports simulation results for various M-estimators. Section 6 compares bootstrap approximation with the asymptotic normal approximation to distributions of M-estimators through simulation. Section 7 analyzes three real financial time series.

2 M-estimators of the GARCH parameters

Throughout this paper, for a function $g$ , we use $˙ g$ to denote its derivative whenever it exists. Also, $sign (x) := I (x > 0) - I (x < 0)$ . For $x > 0$ , ${log}^{+} (x) := I (x > 1) log (x)$ . Moreover, $ϵ$ will denote a generic r.v. having same distribution as errors ${ϵ_{t}}$ of (1.1).

Let $ψ : R \to R$

be an odd function which is differentiable at all but finite number of points. Let

D \subset R

denote the set of points where

ψ

is differentiable and let

¯ D

denote its complement. Let

H (x) := x ψ (x)

x \in R

so that

H

is symmetric. The function

H

is called the score function of the M-estimation in the scale model. Examples are as follows.

Example 1. QMLE score: Let $ψ (x) = x$ . Then $H (x) = x^{2}$ .

Example 2. LAD score: Let $ψ (x) = sign (x)$ . Then $¯ D = {0}$ and $H (x) = | x |$ .

Example 3. Huber’s $k$ score: Let $ψ (x) = x I (| x | \leq k) + k sign (x) I (| x | > k)$ , where $k > 0$ is a known constant. Then $¯ D = {- k, k}$ and $H (x) = x^{2} I (| x | \leq k) + k | x | I (| x | > k)$ .

Example 4. Score function for the maximum likelihood estimation (MLE): Let $ψ (x) = - ˙ f (x) / f (x)$ , where $f$ is the true density of $ϵ$ , assumed to be known. Then $H (x) = x {- ˙ f (x) / f (x)}$ .

Example 5. $μ$ score: Let $ψ (x) = μ sign (x) / (1 + | x |)$ , where $μ > 1$ is a known constant. Then $¯ D = {0}$ and $H (x) = μ | x | / (1 + | x |)$ is bounded.

Example 6. Cauchy score: Let $ψ (x) = 2 x / (1 + x^{2})$ . Then $H (x) = 2 x^{2} / (1 + x^{2})$ is bounded.

Example 7. Score function for the exponential pseudo-maximum likelihood estimation: Let $ψ (x) = δ_{1} | x |^{δ_{2} - 1} sign (x)$ , where $δ_{1} > 0$ and $1 < δ_{2} \leq 2$ are known constants. Here $¯ D = {0}$ and $H (x) = δ_{1} | x |^{δ_{2}}$ .

Assume that for some $κ_{1} \geq 2$ and $κ_{2} > 0$ ,

E [| ϵ |^{κ_{1}}] < \infty and lim t \to 0 P [ϵ^{2} < t] / t^{κ_{2}} = 0.

(2.1)

Then $σ_{t}^{2}$ of (1.2) has the following unique almost sure representation:

σ_{t}^{2} = c_{0} + \infty \sum j = 1 c_{j} X_{t - j}^{2}, t \in Z,

(2.2)

where ${c_{j}; j \geq 0}$ are defined in (2.9)-(2.16) of Berkes et al. (2003).

Let $Θ$ be a compact subset of $(0, \infty)^{1 + p} \times (0, 1)^{q}$ . A typical element in $Θ$ is denoted by $\boldmathθ=(ω,α1,…,αp,β1,…,βq)′$

. Define the variance function on

Θ

vt(\boldmathθ)=c0(\boldmathθ)+∞∑j=1cj(\boldmathθ)X2t−j,\boldmathθ∈\boldmathΘ,t∈Z,

(2.3)

where the coefficients ${cj(\boldmathθ);j≥0}$ are given in Berkes et al. (2003) (Section 3, and display (3.1)) with the property

cj(\boldmathθ0)=cj,∀j≥0.

(2.4)

Hence the variance functions satisfy $vt(\boldmathθ0)=σ2t$ , $t \in Z$ . Using (2.4), (1.1) can be rewritten as

Xt={vt(\boldmathθ0)}1/2ϵt,1≤t≤n.

(2.5)

Consider observable approximation ${^vt(\boldmathθ)}$ of the process ${vt(\boldmathθ)}$ of (2.3) defined by

^vt(\boldmathθ)=c0(\boldmathθ)+I(2≤t)t−1∑j=1cj(\boldmathθ)X2t−j,% \boldmathθ∈\boldmathΘ,1≤t≤n.

(2.6)

Then an M-estimator $^\boldmathθn$ is defined as the solution of $ˆ\boldmathMn,H(\boldmathθ)=\boldmath% 0$ , where

ˆ\boldmathMn,H(\boldmathθ):=n∑t=1{1−H{Xt/^v1/2t(\boldmathθ)}}{˙^\boldmathvt(\boldmathθ)/^vt(\boldmathθ)}.

(2.7)

Next we describe the iterative relation of ${cj(\boldmathθ)}$ that is used to write computer program for their numerical evaluation. The computation is discussed in Section 4.

Example 1. GARCH $(1, 1)$ model: With $\boldmathθ=(ω,α,β)′$ ,

c_{0} (ω, α, β) = ω / (1 - β), c_{j} (ω, α, β) = α β^{j - 1}, j \geq 1.

Example 2. GARCH $(2, 1)$ model: With $\boldmathθ=(ω,α1,α2,β)′$ ,

c0(\boldmathθ)=ω/(1−β),c1(\boldmathθ% )=α1,c2(\boldmathθ)=α2+βc1(% \boldmathθ)=α2+βα1

and

cj(\boldmathθ)=βcj−1(\boldmathθ),j≥3.

Example 3. GARCH $(1, 2)$ model: With $\boldmathθ=(ω,α,β1,β2)′$ ,

c0(\boldmathθ)=ω/(1−β1−β2),c1(% \boldmathθ)=α,c2(\boldmathθ)=β1c1(\boldmathθ)=β1α,

and

cj(\boldmathθ)=β1cj−1(\boldmathθ)+β2cj−2(\boldmathθ),j≥3.

Example 4. GARCH $(2, 2)$ model: With $\boldmathθ=(ω,α1,α2,β1,β2)′$ ,

c0(\boldmathθ)=ω/(1−β1−β2),c1(% \boldmathθ)=α1,c2(\boldmathθ)=α2+β1α1

and

cj(\boldmathθ)=β1cj−1(\boldmathθ)+β2cj−2(\boldmathθ),j≥3.

2.1 Asymptotic distribution of $^\boldmathθn$

The asymptotic distribution of $^\boldmathθn$ is derived under the following assumptions.

Model assumptions: The parameter space $Θ$ is a compact set and its interior $\boldmathΘ0$ contains both $\boldmathθ0$ and $\boldmathθ0H$ of (1.3) and (2.10), respectively. Moreover, (2.1), (2.3) and (2.5) hold and ${X_{t}}$ is stationary and ergodic.

Conditions on the score function:

Identifiability condition: Corresponding to the score function $H$ , there exists a unique number $c_{H} > 0$ satisfying

E [H (ϵ / c_{H}^{1 / 2})] = 1.

(2.8)

Moment conditions:

E [H (ϵ / c_{H}^{1 / 2})]^{2} < \infty and 0 < E {(ϵ / c_{H}^{1 / 2}) ˙ H (ϵ / c_{H}^{1 / 2})} < \infty .

(2.9)

Also various Smoothness conditions on $H$ as in Mukherjee (2008) are assumed which are satisfied in all examples of $H$ considered above. Define the score function factor

σ^{2} (H) := 4 Var {H (ϵ / c_{H}^{1 / 2})} / [E {(ϵ / c_{H}^{1 / 2}) ˙ H (ϵ / c_{H}^{1 / 2})}]^{2},

the matrix

\boldmathG:=E{˙\boldmathv1(% \boldmathθ0H)˙\boldmathv′1(\boldmath% θ0H)/v21(\boldmathθ0H)}

and the transformed parameter

\boldmathθ0H=(cHω0,cHα01,…,cHα0p,β01,…,β0q)′.

(2.10)

Theorem 2.1.

Suppose that the model assumptions, identifiability condition, moment conditions and smoothness conditions hold. Then

n1/2(^\boldmathθn−\boldmathθ0H)→N(0,σ2(H)\boldmathG−1).

(2.11)

Note that $c_{H}$ used in above formulas are given by (i) $c_{H} = E (ϵ^{2})$ for the QMLE, (ii) $c_{H} = {(E | ϵ |)}^{2}$ for the LAD while for the Huber, $μ$ -estimator, Cauchy and other scores, $c_{H}$ does not have closed-form expression. For such score functions, $c_{H}$ is calculated using (2.8) as follows. We fix a large positive integer $I$ and generate ${ϵ_{i}; 1 \leq i \leq I}$ from the error distribution considered for the simulation. Then, using the bisection method on $c > 0$ , we solve the equation

(1 / I) I \sum i = 1 {H (ϵ_{i} / c^{1 / 2})} - 1 = 0.

Values of $c_{H}$ computed in this way were provided in Mukherjee (2008, page 1541) for some error distributions and score functions. In Table 1 we provide $c_{H}$ for few more error distributions and score functions such as Huber’s $k$ -score and $μ$ -estimator with $k = 1.5$ and $μ = 3$ which are used in simulations and data analysis of later sections. In the sequel, Double exponential is abbreviated as DE.

	Huber’s	$μ$ -estimator	Cauchy
Normal	0.825	1.692	0.377
DE	0.677	1.045	0.207
Logistic	0.781	1.487	0.316
$t (3)$	0.533	0.850	0.172
$t (2.2)$	0.204	0.274	0.053

Table 1: Values of

c_{H}

for M-estimators (Huber,

μ

-, Cauchy) under various error distributions.

3 Bootstrapping M-estimators

Let ${w_{n t}; 1 \leq t \leq n, n \geq 1}$ be a triangular array of r.v.’s such that for each $n \geq 1$ , ${w_{n t}; 1 \leq t \leq n}$ are exchangeable and independent of the data ${X_{t}; t \geq 1}$ and errors ${ϵ_{t}; t \geq 1}$ . Also, $\forall t \geq 1$ , $w_{n t} \geq 0$ and $E (w_{n t}) = 1$ .

Based on these weights, bootstrap estimate $^\boldmathθ∗n$ is defined as the solution of $ˆ\boldmathM∗n,H(\boldmathθ)=% \boldmath0$ , where

ˆ\boldmathM∗n,H(\boldmathθ):=n∑t=1wnt{1−H{Xt/^v1/2t(\boldmathθ)}}{˙^\boldmathvt(\boldmathθ)/^vt(\boldmathθ)}.

(3.1)

Examples. From many different choices of bootstrap weights, we consider the following three schemes for comparison.

(i) Scheme M. The sequence of weights ${w_{n 1}, \dots, w_{n n}}$ has a multinomial $(n, 1 / n, \dots, 1 / n)$ distribution, which is essentially the classical paired bootstrap.

(ii) Scheme E. When $w_{n t} = (n E_{t}) / \sum_{i = 1}^{n} E_{i}$ , where ${E_{t}}$ are i.i.d. exponential r.v. with mean $1$ . Under scheme E, $^\boldmathθ∗n$ is a weighted M-estimator with weights proportional to $E_{t}$ , $1 \leq i \leq n$ .

(iii) Scheme U. When $w_{n t} = (n U_{t}) / \sum_{i = 1}^{n} U_{i}$ , where ${U_{t}}$ are i.i.d. uniform r.v. on $(0.5, 1.5)$ . Under scheme U, $^\boldmathθ∗n$ is a weighted M-estimator with weights proportional to $U_{t}$ , $1 \leq i \leq n$ .

A host of other bootstrap methods in the literature are special cases of the above bootstrap formulation. Such general formulation of weighted bootstrap offers a unified way of studying several bootstrap schemes simultaneously. See, for example, Chatterjee and Bose (2005) for details in other contexts.

We assume that the weights satisfy the following basic conditions (Conditions BW of Chatterjee and Bose (2005)) where $σ_{n}^{2} = V a r (w_{n i})$ and $k_{3} > 0$ is a constant.

E (w_{n 1}) = 1, 0 < k_{3} < σ_{n}^{2} = o (n) and % Corr (w_{n 1}, w_{n 2}) = O (1 / n) .

(3.2)

Under (3.2) and some additional smoothness and moment conditions in Mukherjee (2020), weighted bootstrap is asymptotic valid.

Theorem 3.1.

For almost all data, as $n \to \infty$ ,

σ−1nn1/2(^\boldmathθ∗n−^% \boldmathθn)→N(0,σ2(H)\boldmathG% −1).

(3.3)

We remark that since $0 < 1 / σ_{n} < 1 / \sqrt{k_{3}}$

, the rate of convergence of the bootstrap estimator is the same as that of the original estimator. The standard deviation of the weights

{σ_{n}}

at the denominator of the scaling reflects the contribution of the corresponding weights.

The distributional result of (3.3

) is useful for constructing the confidence interval of the GARCH parameters as follows. Let

B

denote the number of bootstrap replicates,

γ_{0}

denote a generic parameter (one of

ω_{0}

α_{0 i}

β_{0 j}

) and let

{^γ}_{n}

and

{^γ}_{* n b}

denote its M-estimator and

b

-th bootstrap estimator (

1 \leq b \leq B

), respectively. Let

γ_{0 H}

be one of

c_{H} ω_{0}

c_{H} α_{0 i}

β_{0 j}

, as appropriate, which has a known value for a simulation experiment. Using the approximation of

\sqrt{n} ({^γ}_{n} - γ_{0 H})

σ_{n}^{- 1} n^{1 / 2} ({^γ}_{* n} - {^γ}_{n})

, the bootstrap confidence interval of

γ_{0 H}

is of the form

[^γn−n−1/2{σ−1nn1/2(^γ∗n,α/2−^γn)},^γn+n−1/2{σ−1nn1/2(^γ∗n,1−α/2−^γn)}]

(3.4)

where ${^γ}_{* n, α / 2}$ is the $α / 2$

-th quantile of the numbers

{^γ∗nb,1≤b≤B}

. Consequently, the bootstrap coverage probability is computed by the proportion of the above set of

B

confidence intervals containing

γ_{0 H}

Similarly, using (2.11) of Theorem 3.1, we can obtain the confidence interval of $γ_{0 H}$ based on the asymptotic normality of ${^γ}_{n}$ , and this will be called the normal confidence interval. Specifically, in view of Proposition 3.1 of Mukherjee (2008) on the estimation of the variance-covariance matrix $σ2(H)\boldmathG−1$ , we can obtain the asymptotic confidence interval of $γ_{0 H}$ as

[{^γ}_{n} - n^{- 1 / 2}^d z_{1 - α / 2}, {^γ}_{n} + n^{- 1 / 2}^d z_{1 - α / 2}],

(3.5)

where $(^d)^{2}$ is the estimated variance of ${^γ}_{n}$ obtained from the appropriate diagonal entry of the estimator of $σ2(H)\boldmathG−1$ and $z_{1 - α / 2}$ is the $1 - α / 2$

-th quantile of the standard normal distribution.

In the following Section 6, we will compare the accuracy of the confidence intervals constructed by the bootstrap and asymptotic approximations.

4 Algorithm

We discuss the implementation of an iteratively re-weighted algorithm proposed in Mukherjee (2020) for computing M-estimates. In particular, we highlight $μ$ -estimate and Cauchy-estimate of the GARCH parameters in this paper as their asymptotic distributions are derived under mild moment assumptions. We also consider the bootstrap estimators based on the corresponding score functions.

4.1 Computation of M-estimates

For the convenience of writing, let $α (c) = E [H (c ϵ)]$ for $c > 0$ . Using a Taylor expansion of $ˆ\boldmathMn,H$ , we obtain the following recursive equation for computing the updated estimate $~\boldmathθ$ of $^\boldmathθn$ from the current estimate $θ$ of $ˆ\boldmathMn,H(\boldmathθ)=\boldmath% 0$ :

~\boldmathθ=\boldmathθ+{˙α(1)/2}−1[n∑t=1˙^\boldmathvt(% \boldmathθ)˙^\boldmathvt(\boldmathθ% )′/^v2t(\boldmathθ)]−1n∑t=1{H{Xt/^v1/2t(\boldmathθ)}−1}{˙^\boldmathvt(\boldmathθ)/^vt(\boldmathθ)},

(4.1)

where $˙ α (1) = E {ϵ ˙ H (ϵ)}$ under smoothness conditions on $H$ . Since the GARCH residuals ${Xt/^v1/2t(^\boldmathθn)}$ estimate only ${ϵ_{t} / c_{H}^{1 / 2}}$ , in general, we cannot estimate $˙ α (1)$ from the data. Therefore, we use ad hoc techniques such as simulating ${{~ ϵ}_{t}; 1 \leq t \leq n}$ from $N (0, 1)$ or standardized DE distribution and then use $n^{- 1} \sum_{t = 1}^{n} ~ ϵ ˙ H (~ ϵ)$ to carry out the iteration. Note that if the iteration in (4.1) converges then $~\boldmathθ≈\boldmathθ$ . Therefore in this case from (4.1), $ˆ\boldmathMn,H(\boldmathθ)≈% \boldmath0$ and hence $~\boldmathθ$ is the desired $^\boldmathθn$ . Based on our extensive simulation study and real data analysis, the algorithm is robust enough to converge to the same value of $^\boldmathθn$ irrespective of different values of the unknown factor $˙ α (1)$ used in computation.

In the following examples, we discuss (4.1) when specialized to the M-estimators computed in this paper.

QMLE: Here $H (x) = x^{2}$ and $α (c) = c^{2} E (ϵ^{2})$ . Hence $˙ α (1) / 2 = E (ϵ^{2})$ and

~\boldmathθ

=

With

Wt=1/^v2t(\boldmathθ),xt=˙^% \boldmathvt(\boldmathθ),yt=X2t−^vt(\boldmathθ),

$~\boldmathθ$ can be computed iteratively as

~\boldmathθ(r+1)=~\boldmathθ(r)+{E(ϵ2)}−1{∑tWtxtx′t}−1{∑tWtxtyt}.

Note that when $E (ϵ^{2}) = 1$ , this is same as the formula obtained through the BHHH algorithm proposed by Berndt et al. (1974).

LAD: Here $H (x) = | x |$ and $α (c) = c E | ϵ |$ . Hence $˙ α (1) = E | ϵ |$ and

$~\boldmathθ$	$=$	$\boldmathθ+{2/E\|ϵ\|}[n∑t=1{˙^\boldmathvt(\boldmathθ)˙^\boldmathvt(\boldmathθ)′/^v2t(\boldmathθ)}]−1n∑t=1[\|Xt\|/^v1/2t(\boldmathθ)−1]{˙^\boldmathvt(\boldmathθ)/^vt(% \boldmathθ)}$
	$=$	$\boldmathθ+{2/E\|ϵ\|}[n∑t=1{˙^\boldmathvt(\boldmathθ)˙^\boldmathvt(\boldmathθ)′/^v2t(\boldmathθ)}]−1n∑t=1{\|Xt\|−^v1/2t(\boldmathθ)}{˙^\boldmathvt(\boldmathθ)/^v3/2t(% \boldmathθ)}$
	$=$	$\boldmathθ+{2/E\|ϵ\|}[n∑t=1{˙^\boldmathvt(\boldmathθ)˙^\boldmathvt(\boldmathθ)′/^v2t(\boldmathθ)}]−1n∑t=1{^v1/2t(\boldmathθ)(\|Xt\|−^v1/2t(\boldmathθ))}{˙^\boldmathvt(% \boldmathθ)/^v2t(\boldmathθ)}.$

With

Wt=1/^v2t(\boldmathθ),xt=˙^% \boldmathvt(\boldmathθ),yt=^v1/2t(% \boldmathθ)(|Xt|−^v1/2t(\boldmathθ)),

$~\boldmathθ$ can be computed iteratively as

~\boldmathθ(r+1)=~\boldmathθ(r)+{2/E|ϵ|}{∑tWtxtx′t}−1{∑tWtxtyt}.

Huber: Here $H (x) = x^{2} I (| x | \leq k) + k | x | I (| x | > k)$ and

α (c) = E [(c ϵ)^{2} I (| c ϵ | \leq k) + k | c ϵ | I (| c ϵ | > k)] .

Hence

˙ α (1) = E [2 ϵ^{2} I (| ϵ | \leq k) + k | ϵ | I (| ϵ | > k)]

and

	$~\boldmathθ$	$=$	$\boldmathθ−{˙α(1)/2}−1[n∑t=1{˙^\boldmathvt(% \boldmathθ)˙^\boldmathvt(\boldmathθ% )′^v2t(\boldmathθ)}]−1$
			$×n∑t=1⎡⎣1−X2t^vt(% \boldmathθ)I⎛⎝\|Xt\|^v1/2t(\boldmathθ)≤k⎞⎠−k\|Xt\|^v1/2t(\boldmathθ)I⎛⎝\|Xt\|^v1/2t(\boldmathθ)>k⎞⎠⎤⎦⎧⎪⎨⎪⎩˙^\boldmathvt(\boldmath% θ)^vt(\boldmathθ)⎫⎪⎬⎪⎭.$

With

Wt=1/^v2t(\boldmathθ),xt=˙^% \boldmathvt(\boldmathθ)

and

yt=X2tI(|Xt|/^v1/2t(\boldmathθ)≤k)+k|Xt|^v1/2t(\boldmathθ)I(|Xt|/^v1/2t(\boldmathθ)>k)−^vt(\boldmathθ),

$~\boldmathθ$ can be computed iteratively as

~\boldmathθ(r+1)=~\boldmathθ(r)+{˙α(1)/2}−1{∑tWtxtx′t}−1{∑tWtxtyt}.

$μ$ -estimator: Here $H (x) = μ | x | / (1 + | x |)$ and $α (c) = μ - μ E [1 / (1 + | c ϵ |)]$ . Hence

˙ α (1) = μ E [| ϵ | / (1 + | ϵ |)^{2}]

and

~\boldmathθ=\boldmathθ+{μ2E[|ϵ|(1+|ϵ|)2]}−1[n∑t=1{˙^\boldmathvt(% \boldmathθ)˙^\boldmathvt(\boldmathθ% )′^v2t(\boldmathθ)}]−1n∑t=1⎡⎣μ|Xt|^v1/2t(\boldmathθ)+|Xt|−1⎤⎦⎧⎪⎨⎪⎩˙^\boldmathvt(\boldmathθ)^vt(\boldmathθ)⎫⎪⎬⎪⎭.

With

Wt=1/^v2t(\boldmathθ),xt=˙^% \boldmathvt(\boldmathθ),yt=μ|Xt|^vt(\boldmathθ)^v1/2t(\boldmathθ)+|Xt|−^vt(\boldmathθ),

$~\boldmathθ$ can be computed iteratively as

Cauchy-estimator: Here $H (x) = 2 x^{2} / (1 + x^{2})$ and $α (c) = E [2 c^{2} ϵ^{2} / (1 + c^{2} ϵ^{2})]$ . Hence

˙ α (1) = E [4 ϵ^{2} / (1 + ϵ^{2})^{2}]

and

~\boldmathθ=\boldmathθ−{2E[ϵ2(1+ϵ2)2]}−1[n∑t=1{˙^\boldmathvt(\boldmathθ)˙^\boldmathvt(\boldmathθ)′^v2t(\boldmathθ)}]−1n∑t=1[1−2X2t^vt(\boldmathθ)+X2t]⎧⎪⎨⎪⎩˙^\boldmathvt(\boldmathθ)^vt(\boldmathθ)⎫⎪⎬⎪⎭.

With

Wt=1/^v2t(\boldmathθ),xt=˙^% \boldmathvt(\boldmathθ),yt=2X2t^vt(\boldmathθ)^vt(\boldmathθ)+X2t−^vt(\boldmathθ),

$~\boldmathθ$ can be computed iteratively as

~\boldmathθ(r+1)=~\boldmathθ(r)+{2E[ϵ2(1+ϵ2)2]}−1{∑tWtxtx′t}−1{∑tWtxtyt}.

4.2 Computation of bootstrap M-estimates

Here the relevant function is $ˆ\boldmathM∗n,H(\boldmathθ)$ defined in (3.1) and the bootstrap estimate $^\boldmathθ∗n$ can be computed using the updating equation

~\boldmathθ∗

=

\boldmathθ−{2/˙α(1)}[n∑t=1 wnt{˙^\boldmathvt(\boldmathθ)˙^\boldmathvt(\boldmathθ)′/^v2t(\boldmathθ)

M-estimation in GARCH models without higher order moments

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1 Introduction

2 M-estimators of the GARCH parameters

2.1 Asymptotic distribution of $^\boldmathθn$

Theorem 2.1.

3 Bootstrapping M-estimators

Theorem 3.1.

4 Algorithm

4.1 Computation of M-estimates

4.2 Computation of bootstrap M-estimates

M-estimation in GARCH models without higher order moments

Authors

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This week in AI

1 Introduction

2 M-estimators of the GARCH parameters

2.1 Asymptotic distribution of ^\boldmathθn

Theorem 2.1.

3 Bootstrapping M-estimators

Theorem 3.1.

4 Algorithm

4.1 Computation of M-estimates

4.2 Computation of bootstrap M-estimates

2.1 Asymptotic distribution of $^\boldmathθn$