The simplest estimation method for ARCH models is that of ordinary least squares (OLSs). This estimation procedure has the advantage of being numerically simple, but has two drawbacks: (i) the OLS estimator is not efficient and is outperformed by methods based on the likelihood or on the quasi‐likelihood that will be presented in the next chapters; (ii) in order to provide asymptotically normal estimators, the method requires moments of order 8 for the observed process. An extension of the OLS method, the feasible generalised least squares (FGLS) method, suppresses the first drawback and attenuates the second by providing estimators that are asymptotically as accurate as the quasi‐maximum likelihood under the assumption that moments of order 4 exist. Note that the least‐squares methods are of interest in practice because they provide initial estimators for the optimisation procedure that is used in the quasi‐maximum likelihood method.

We begin with the unconstrained OLS and FGLS estimators. Then, in Section 6.3, we will see how to take into account positivity constraints on the parameters.

6.1 Estimation of ARCH( q ) models by Ordinary Least Squares

In this section, we consider the OLS estimator of the ARCH( q ) model:

6.1

The OLS method uses the AR representation on the squares of the observed process. No assumption is made on the law of η _t .

The true value of the vector of the parameters is denoted by θ ₀ = (ω ₀, α ₀₁,…,α _0q )^′ , and we denote by θ a generic value of the parameter.

From (6.1), we obtain the AR (q) representation

6.2

where . The sequence (u _t , ℱ _t ) _t constitutes a martingale difference when , denoting by ℱ _t the σ ‐field generated by {ε _s  : s ≤ t}.

Assume that we observe ε₁,…,ε _n , a realisation of length n of the process (ε _t ), and let ε₀,…,ε_1 − q be initial values. For instance, the initial values can be chosen equal to zero. Introducing the vector

in view of (6.2), we obtain the system

6.3

which can be written as

with the n × (q + 1) matrix

and the n × 1 vectors

Assume that the matrix X ^′ X is invertible, or equivalently that X has full column rank (we will see that this is always the case asymptotically, and thus for n large enough). The OLS estimator of θ ₀ follows:

6.4

Under assumptions OLS1 and OLS2 below, the variance of u _t exists and is constant. The OLS estimator of is

For the asymptotic normality of the OLS estimator, we need the following additional assumption.

• OLS4: .

Consider the (q + 1) × (q + 1) matrices

The invertibility of A was established in the proof of Theorem 6.1, and the invertibility of B is shown by the same argument, noting that if and only if c ^′ Z _t − 1 = 0 because a.s. The following result establishes the asymptotic normality of the OLS estimator.

6.2 Estimation of ARCH( q ) Models by Feasible Generalised Least Squares

In a linear regression model when, conditionally on the exogenous variables, the errors are heteroscedastic, the FGLS estimator is asymptotically more accurate than the OLS estimator. Note that in ( 6.3), the errors u _t are, conditionally on Z _t − 1 , heteroscedastic with conditional variance .

For all θ = (ω, α ₁,…,α _q )^′ , let

The FGLS estimator is defined by

The moment condition required for the asymptotic normality of the FGLS estimator is . For the OLS estimator, we had the more restrictive condition . Moreover, when this eighth‐order moment exists, the following result shows that the OLS estimator is asymptotically less accurate than the FGLS estimator.

We will see in Chapter 7 that the asymptotic variance of the FGLS estimator coincides with that of the quasi‐maximum likelihood estimator (but the asymptotic normality of the latter is obtained without moment conditions). This result explains why quasi‐maximum likelihood is preferred to OLS (and even to FGLS) for the estimation of ARCH (and GARCH) models. Note, however, that the OLS estimator often provides a good initial value for the optimisation algorithm required for the quasi‐maximum likelihood method.

6.3 Estimation by Constrained Ordinary Least Squares

Negative components are not precluded in the OLS estimator defined by ( 6.4) (see Exercise 6.3). When the estimate has negative components, predictions of the volatility can be negative. In order to avoid this problem, we consider the constrained OLS estimator defined by

The existence of is guaranteed by the continuity of the function Q _n and the fact that

as ‖θ‖ → ∞ and θ ≥ 0, whenever X has non‐zero columns. Note that the latter condition is satisfied at least for n is large enough (see Exercise 6.5).

6.3.2 Computation of the Constrained OLS Estimator

We now give an explicit way to obtain the constrained estimator. We have already seen that if all the components of the unconstrained estimator are positive, we have . Now suppose that one component of is negative, for instance the last one. Let

and

Note that in general (see Exercise 6.11).

6.4 Bibliographical Notes

The OLS method was proposed by Engle (1982) for ARCH models. The asymptotic properties of the OLS estimator were established by Weiss (1984, 1986), in the ARMA‐GARCH framework, under eighth‐order moments assumptions. Pantula (1989) also studied the asymptotic properties of the OLS method in the AR(1)‐ARCH(q) case, and he gave an explicit form for the asymptotic variance. The FGLS method was developed, in the ARCH case, by Bose and Mukherjee (2003) (see also Gouriéroux 1997). Aknouche (2012) developed a multistage weighted least squares estimate which has the same asymptotic properties as the FGLS but avoids the moment conditions of the OLS. The convexity results used for the study of the constrained estimator can be found, for instance, in Moulin and Fogelman‐Soulié (1979).