In the previous chapter, we saw that the asymptotic normality of the QMLE of a GARCH model holds true under general conditions, in particular without any moment assumption on the observed process. An important application of this result concerns testing problems. In particular, we are able to test the IGARCH assumption, or more generally a given GARCH model with infinite variance. This problem is the subject of Section 8.1.

The main aim of this chapter is to derive tests for the nullity of coefficients. These tests are complex in the GARCH case because of the constraints that are imposed on the estimates of the coefficients to guarantee that the estimated conditional variance is positive. Without these constraints, it is impossible to compute the Gaussian log‐likelihood of the GARCH model. Moreover, asymptotic normality of the QMLE has been established assuming that the parameter belongs to the interior of the parameter space (assumption A5 in Chapter 7). When some coefficients, α _i or β _j , are null, Theorem 7.2 does not apply. It is easy to see that, in such a situation, the asymptotic distribution of cannot be Gaussian. Indeed, the components, of , are constrained to be positive or null. If, for instance θ _0i  = 0, then for all n and the asymptotic distribution of this variable cannot be Gaussian.

Before considering the significance tests, we shall, therefore, establish in Section 8.2 the asymptotic distribution of the QMLE without assumption A5, at the cost of a moment assumption on the observed process. In Section 8.3, we present the main tests (Wald, score and likelihood ratio) used for testing the nullity of some coefficients. The asymptotic distribution obtained for the QMLE will lead to modification of the standard critical regions. Two cases of particular interest will be examined in detail: the test of nullity of only one coefficient and the test of conditional homoscedasticity, which corresponds to the nullity of all the coefficients α _i and β _j . Section 8.4 is devoted to testing the adequacy of a particular GARCH(p, q) model, using portmanteau tests. The chapter also contains a numerical application in which the pre‐eminence of the GARCH(1, 1) model is questioned.

8.1 Test of the Second‐Order Stationarity Assumption

For the GARCH(p, q) model defined by Eq. (7.1), testing for second‐order stationarity involves testing

Introducing the vector c = (0, 1, …, 1)^′ ∈ ℝ ^{p + q + 1} , the testing problem is

8.1

In view of Theorem 7.2, the QMLE of θ ₀ satisfies

under assumptions which are compatible with H ₀ and H ₁ . In particular, if c ^′ θ ₀ = 1 we have

It is thus natural to consider the Wald statistic

where and are consistent estimators in probability of κ _η and J . The following result follows immediately from the convergence of T _n to when c ^′ θ ₀ = 1.

Note that for most real series (see, for instance Table 7.4), the sum of the estimated coefficients and is strictly less than 1: second‐order stationarity thus cannot be rejected, for any reasonable asymptotic level (when T _n  < 0, the p ‐value of the test is greater than 1/2). Of course, the non‐rejection of H ₀ does not mean that the stationarity is proved. It is interesting to test the reverse assumption that the data‐generating process is an IGARCH, or more generally that it does not have moments of order 2. We thus consider the problem

8.2

As an application, we take up the data sets of Table 7.4 again, and we give the p ‐values of the previous test for the 11 series of daily returns. For the FTSE (DAX, Nasdaq, and S&P 500), the assumption of infinite variance cannot be rejected at the 5% (3%, 2%, and 1%) level (see Table 8.1). The other series can be considered as second‐order stationary (if one believes in the GARCH(1, 1) model, of course).

Index		p ‐Value
CAC	0.983 (0.007)	0.0089
DAX	0.981 (0.011)	0.0385
DJA	0.982 (0.007)	0.0039
DJI	0.986 (0.006)	0.0061
DJT	0.983 (0.009)	0.0023
DJU	0.983 (0.007)	0.0060
FTSE	0.990 (0.006)	0.0525
Nasdaq	0.993 (0.003)	0.0296
Nikkei	0.980 (0.007)	0.0017
SMI	0.962 (0.015)	0.0050
S&P 500	0.989 (0.005)	0.0157

Estimated standard deviations are in parentheses.

8.2 Asymptotic Distribution of the QML When θ ₀ is at the Boundary

In view of the form of the parameter space (7.3), the QMLE is constrained to have a strictly positive first component, while the other components are constrained to be positive or null. A general technique for determining the distribution of a constrained estimator involves expressing it as a function of the unconstrained estimator (see Gouriéroux and Monfort 1995). For the QMLE of a GARCH, this technique does not work because the objective function

cannot be computed outside Θ (for an ARCH(1), it may happen that is negative when α ₁ < 0). It is thus impossible to define .

The technique that we will utilise here (see, in particular, Andrews 1999), involves writing with the aid of the normalised score vector, evaluated at θ ₀ :

8.3

with

where the components of ∂ l _n (θ ₀)/∂θ and of J _n are right derivatives (see (a) in the proof of Theorem 8.1 below).

In the proof of Theorem 7.2, we showed that

8.4

For any value of θ ₀ ∈ Θ (even when ), it will be shown that the vector Z _n is well defined and satisfies

8.5

provided J exists. By contrast, when

Result (8.4) is no longer valid. However, we will show that the asymptotic distribution of is well approximated by that of the vector n ^1/2(θ − θ ₀) which is located at the minimal distance of Z _n , under the constraint θ ∈ Θ. Consider thus a random vector (which is not an estimator, of course) solving the minimisation problem

8.6

It will be shown that J _n converges to the positive definite matrix J . For n large enough, we thus have

where is a distance between two points x and y of ℝ ^{p + q + 1} .

We allow θ ₀ to have null components, but we do not consider the (less interesting) case where θ ₀ reaches another boundary of Θ. More precisely, we assume that

where and . In this case and belong to the ‘local parameter space’

8.7

where Λ₁ = ℝ, for i = 2, …, p + q + 1, Λ _i  = ℝ if θ _0i  ≠ 0 and Λ _i  = [0, ∞) if θ _0i  = 0. With the notation

we thus have, with probability 1,

8.8

The vector is the projection of Z _n on Λ, with respect to the norm (see Figure 8.1). Since Λ is closed and convex, such a projection is unique. We will show that

8.9

Since (Z _n , J _n ) tends in law to (Z, J) and is a function of (Z _n , J _n ) which is continuous everywhere except at the points where J _n is singular (that is, almost everywhere with respect to the distribution of (Z, J) because J is invertible), we have , where λ ^Λ is the solution of limiting problem

8.10

In addition to B1, we retain most of the assumptions of Theorem 7.2:

1. B2: θ ₀ ∈ Θ and Θ is a compact set.
2. B3: γ(A ₀) < 0 and for all θ ∈ Θ, .
3. B4: has a non‐degenerate distribution with .
4. B5: If p > 0, , and do not have common roots, , and α _0q  + β _0p  ≠ 0.
5. B6: .

We also need the following moment assumption:

1. B7: .

When , we can show the existence of the information matrix

without moment assumptions similar to B7. The following example shows that, in the ARCH case, this is no longer possible when we allow θ ₀ ∈ ∂Θ.

We then have the following result.

8.2.1 Computation of the Asymptotic Distribution

In this section, we will show how to compute the solutions of the optimization problem ( 8.10). Switching the components of θ , if necessary, it can be assumed without loss of generality that the vector of the first d ₁ components of θ ₀ has strictly positive elements and that the vector of the last d ₂ = p + q + 1 − d ₁ components of θ ₀ is null. This can be written as

8.12

More generally, it will be useful to consider all the subsets of these constraints. Let

be the set of the matrices obtained by deleting no, one, or several (but not all) rows of K. Note that the solution of the constrained minimisation problem ( 8.10) is the unconstrained solution λ = Z when the latter satisfies the constraint, that is, when

When Z ∉ Λ, the solution λ ^Λ coincides with that of an equality constrained problem of the form

An important difference, compared to the initial minimisation program ( 8.10), is that the minimisation is done here on a vectorial space. The solution is given by a projection (non‐orthogonal when J is not the identity matrix). We thus obtain (see Exercise 8.1)

8.13

is the projection matrix (orthogonal for the metric defined by J) on the orthogonal subspace of the space generated by the rows of K _i . Note that does not necessarily belong to Λ because K _i λ = 0 does not imply that Kλ ≥ 0. Let be the class of the admissible solutions. It follows that the solution that we are looking for is

This formula can be used in practice to obtain realisations of λ ^Λ from realisations of Z . The can be obtained by writing

8.14

Another expression (of theoretical interest) for λ ^Λ is

where and the form a partition of ℝ ^{p + q + 1} . Indeed, according to the zone to which Z belongs, a solution is obtained. We will make explicit these formulas in a few examples. Let d = p + q + 1, z ⁺ = z _{(0, + ∞)}(z) and z ⁻ = z _{(− ∞ , 0)}(z).

8.3 Significance of the GARCH Coefficients

We make the assumptions of Theorem 8.1 and use the notation of Section 8.2. Assume , and consider the testing problem

8.19

Recall that under H ₀ , we have

where the distribution of λ ^Λ is defined by

8.20

with .

8.3.1 Tests and Rejection Regions

For parametric assumptions of the form (8.19), the most popular tests are the Wald, score, and likelihood ratio tests.

Wald Statistic

The Wald test looks at whether is close to 0. The usual Wald statistic is defined by

where is a consistent estimator of Σ = (κ _η  − 1)J ⁻¹.

Score (or Lagrange Multiplier or Rao) Statistic

Let

denote the QMLE of θ constrained by θ ⁽²⁾ = 0. The score test aims to determine whether is not too far from 0, using a statistic of the form

where and denote consistent estimators of κ _η and J .

Likelihood Ratio Statistic

The likelihood ratio test is based on the fact that under H ₀ : θ ⁽²⁾ = 0, the constrained (quasi) log‐likelihood should not be much smaller than the unconstrained log‐likelihood . The test employs the statistic

Usual Rejection Regions

From the practical viewpoint, the score statistic presents the advantage of only requiring constrained estimation, which is sometimes much simpler than the unconstrained estimation required by the two other tests. The likelihood ratio statistic does not require estimation of the information matrix J , nor the kurtosis coefficient κ _η . For each test, it is clear that the null hypothesis must be rejected for large values of the statistic. For standard statistical problems, the three statistics asymptotically follow the same distribution under the null. At the asymptotic level α , the standard rejection regions are thus

where is the (1 − α)‐quantile of the χ ² distribution with d ₂ degrees of freedom. In the case d ₂ = 1, for testing the significance of only one coefficient, the most widely used test is Student's t test, defined by the rejection region

8.21

where . This test is equivalent to the standard Wald test because ( t _n being here always positive or null, because of the positivity constraints of the QML estimates) and

Our testing problem is not standard because, by Theorem 8.1, the asymptotic distribution of is not normal. We will see that, among the previous rejection regions, only that of the score test asymptotically has the level α .

8.3.2 Modification of the Standard Tests

The following proposition shows that for the Wald and likelihood ratio tests, the asymptotic distribution is not the usual under the null hypothesis. The proposition also shows that the asymptotic distribution of the score test remains the distribution. The asymptotic distribution of R _n is not affected by the fact that, under the null hypothesis, the parameter is at the boundary of the parameter space. These results are not very surprising. Take the example of an ARCH(1) with the hypothesis H ₀ : α ₀ = 0 of absence of ARCH effect. As illustrated by Figure 8.2, there is a non‐zero probability that be at the boundary, that is, that . Consequently admits a mass at 0 and does not follow, even asymptotically, the law. The same conclusion can be drawn for the likelihood ratio test. On the contrary, the score n ^1/2 ∂ l _n (θ ₀)/∂θ can take as well positive or negative values, and does not seem to have a specific behaviour when θ ₀ is at the boundary.

8.3.3 Test for the Nullity of One Coefficient

Consider the case d ₂ = 1, which is perhaps the most interesting case and corresponds to testing the nullity of only one coefficient. In view of relations (8.15), the last component of λ ^Λ is equal to . We thus have

where . Using the symmetry of the Gaussian distribution, and the independence between Z ^*2 and when Z ^* follows the real normal law, we obtain

Testing

can thus be achieved by using the critical region at the asymptotic level α ≤ 1/2. In view of Remark 8.3, we can define a modified likelihood ratio test of critical region . Note that the standard Wald test has the asymptotic level α/2, and that the asymptotic level of the standard likelihood ratio test is much larger than α when the kurtosis coefficient κ _η is large. A modified version of the Student t test is defined by the rejection region

8.25

We observe that commercial software – such as GAUSS, R, RATS, SAS, and SPSS – do not use the modified version (8.25), but the standard version (8.21). This standard test is not of asymptotic level α but only α/2. To obtain a t test of asymptotic level α , it then suffices to use a test of nominal level 2α .

8.3.4 Conditional Homoscedasticity Tests with ARCH Models

Another interesting case is that obtained with d ₁ = 1, θ ⁽¹⁾ = ω , p = 0, and d ₂ = q . This case corresponds to the test of the conditional homoscedasticity null hypothesis

8.26

in an ARCH(q) model

8.27

We will see that for testing problem (8.26), there exist very simple forms of the Wald and score statistics.

Simplified Form of the Wald Statistic

Using Exercise 8.6, we have

Since KΣK ^′ = I _q , we obtain a very simple form for the Wald statistic:

8.28

Asymptotic Distribution W and L

A trivial extension of Example 8.3 yields

8.29

The asymptotic distribution of is thus that of

wherethe Z _i 's are independent and -distributed. Thus, in the case where an ARCH(q) is fitted to a white noise we have

8.30

This asymptotic distribution is tabulated and the critical values are given in Table 8.2. In view of Remark 8.3, Table  8.2 also yields the asymptotic critical values of the modified likelihood ratio statistic . Table 8.3 shows that the use of the standard ‐based critical values of the Wald test would lead to large discrepancies between the asymptotic levels and the nominal level α .

q	α (%)
	0.1	1	2.5	5	10	15
1	9.5493	5.4119	3.8414	2.7055	1.6424	1.0742
2	11.7625	7.2895	5.5369	4.2306	2.9524	2.2260
3	13.4740	8.7464	6.8610	5.4345	4.0102	3.1802
4	14.9619	10.0186	8.0230	6.4979	4.9553	4.0428
5	16.3168	11.1828	9.0906	7.4797	5.8351	4.8519

q	α (%)
	0.1	1	2.5	5	10	15
1	0.05	0.5	1.25	2.5	5	7.5
2	0.04	0.4	0.96	1.97	4.09	6.32
3	0.02	0.28	0.75	1.57	3.36	5.29
4	0.02	0.22	0.59	1.28	2.79	4.47
5	0.01	0.17	0.48	1.05	2.34	3.81

Score Test

For the hypothesis (8.26) that all the α coefficients of an ARCH(q) model are equal to zero, the score statistic R _n can be simplified. To work within the linear regression framework, write

where Y is the vector of length n of the ‘dependent’ variable , where X is the n × (q + 1) matrix of the constant (in the first column) and of the ‘explanatory’ variables (in column i + 1, with the convention ε _t  = 0 for t ≤ 0), and . Estimating J(θ ₀) by n ⁻¹ X ^′ X , and κ _η  − 1 by n ⁻¹ Y ^′ Y , we obtain

and one recognizes n times the coefficient of determination in the linear regression of Y on the columns of X . Since this coefficient is not changed by linear transformation of the variables (see Exercise 5.11), we simply have R _n  = nR ² , where R ² is the coefficient of determination in the regression of on a constant and q lagged values . Under the null hypothesis of conditional homoscedasticity, R _n asymptotically follows the law.

The previous simple forms of the Wald and score tests are obtained with estimators of J which exploit the particular form of the matrix under the null. Note that there exist other versions of these tests, obtained with other consistent estimators of J . The different versions are equivalent under the null, but can have different asymptotic behaviours under the alternative.

8.3.5 Asymptotic Comparison of the Tests

The Wald and score tests that we have just defined are in general consistent, that is, their powers converge to 1 when they are applied to a wide class of conditionally heteroscedastic processes. An asymptotic study will be conducted via two different approaches: Bahadur's approach compares the rates of convergence to zero of the p ‐values under fixed alternatives, whereas Pitman's approach compares the asymptotic powers under a sequence of local alternatives, that is, a sequence of alternatives tending to the null as the sample size increases.

Bahadur's Approach

Let S _W (t) = ℙ(W > t) and S _R (t) = ℙ(R > t) be the asymptotic survival functions of the two test statistics, under the null hypothesis H ₀ defined by ( 8.26). Consider, for instance, the Wald test. Under the alternative of an ARCH(q) which does not satisfy H ₀ , the p ‐value of the Wald test S _W (W _n ) converges almost surely to zero as n → ∞ because

The p‐value of a test is typically equivalent to exp{−nc/2}, where c is a positive constant called the Bahadur slope. Using the fact that

8.31

and that , the (approximate ¹ ) Bahadur slope of the Wald test is thus

To compute the Bahadur slope of the score test, note that we have the linear regression model is the linear innovation of . We then have

The previous limit is thus equal to the Bahadur slope of the score test. The comparison of the two slopes favours the score test over the Wald test.

Bahadur's approach is sometimes criticised for not taking account of the critical value of test, and thus for not really comparing the powers. This approach only takes into account the (asymptotic) distribution of the statistic under the null and the rate of divergence of the statistic under the alternative. It is unable to distinguish a two‐sided test from its one‐sided counterpart (see Exercise 8.8). In this sense, the result of Proposition 8.4 must be put into perspective.

Pitman's Approach

In the ARCH(1) case, consider a sequence of local alternatives . We can show that under this sequence of alternatives,

Consequently, the local asymptotic power of the Wald test is

8.32

The score test has the local asymptotic power

8.33

Note that the probability in (8.32) is the power of the test of the assumption H ₀ : θ = 0 against the assumption H ₁ : θ = τ > 0, based on the rejection region of {X > c ₁} with only one observation . The power (8.33) is that of the two‐sided test {∣X ∣  > c ₂}. The tests {X > c ₁} and {∣X ∣  > c ₂} have the same level, but the first test is uniformly more powerful than the second (by the Neyman–Pearson lemma, {X > c ₁} is even uniformly more powerful than any test of level less than or equal to α , for any one‐sided alternative of the form H ₁ ). The local asymptotic power of the Wald test is thus uniformly strictly greater than that of Rao's test for testing for conditional homoscedasticity in an ARCH(1) model.

Consider the ARCH(2) case, and a sequence of local alternatives . Under this sequence of alternatives

with Let c ₁ be the critical value of the Wald test of level α . The local asymptotic power of the Wald test is

Let c ₂ be the critical value of the Rao test of level α . The local asymptotic power of the Rao test is

where (U ₁ + τ ₁)² + (U ₂ + τ ₂)² follows a non‐central χ ² distribution, with two degrees of freedom and non‐centrality parameter . Figure 8.5 compares the powers of the two tests when τ ₁ = τ ₂ .

Thus, the comparison of the local asymptotic powers clearly favours the Wald test over the score test, counter‐balancing the result of Proposition 8.4.

8.4 Diagnostic Checking with Portmanteau Tests

To check the adequacy of a given time series model, for instance an ARMA(p, q) model, it is common practice to test the significance of the residual autocorrelations. In the GARCH framework this approach is not relevant because the process is always a white noise (more precisely a martingale difference) even when the volatility is misspecified, that is, when with . To check the adequacy of a volatility model, for instance a GARCH( p, q) of the form (7.1), it is much more fruitful to look at the squared residual autocovariances

where ∣h ∣  < n , , is defined by Eq. (7.4) and is the QMLE given by Eq. (7.9).

For any fixed integer m , 1 ≤ m < n , consider the statistic . Let and be weakly consistent estimators of κ _η and J . For instance, one can take

Define also the m × (p + q + 1) matrix whose (h, k)th element, for 1 ≤ h ≤ m and 1 ≤ k ≤ p + q + 1, is given by

8.5 Application: Is the GARCH(1,1) Model Overrepresented?

The GARCH(1,1) model is by far the most widely used by practitioners who wish to estimate the volatility of daily returns. In general, this model is chosen a priori, without implementing any statistical identification procedure. This practice is motivated by the common belief that the GARCH(1,1) (or its simplest asymmetric extensions) is sufficient to capture the properties of financial series and that higher‐order models may be unnecessarily complicated.

We will show that, for a large number of series, this practice is not always statistically justified. We consider daily and weekly series of 11 returns (CAC, DAX, DJA, DJI, DJT, DJU, FTSE, Nasdaq, Nikkei, SMI and S&P 500) and five exchange rates. The observations cover the period from 2 January 1990 to 22 January 2009 for the daily returns and exchange rates, and from 2 January 1990 to 20 January 2009 for the weekly returns (except for the indices for which the first observations are after 1990). We begin with the portmanteau tests defined in Section 8.4 . Table 8.4 shows that the ARCH models (even with large order q) are generally rejected, whereas the GARCH(1,1) is only occasionally rejected. This table only concerns the daily returns, but similar conclusions hold for the weekly returns and exchange rates. The portmanteau tests are known to be omnibus tests, powerful for a broad spectrum of alternatives. As we will now see, for the specific alternatives for which they are built, the tests defined in Section 8.3 (Wald, score, and likelihood ratio) may be much more powerful.

The GARCH(1,1) model is chosen as the benchmark model and is successively tested against the GARCH(1,2), GARCH(1,3), GARCH(1,4), and GARCH(2,1) models. In each case, the three tests (Wald, score, and likelihood ratio) are applied. The empirical p‐values are displayed in Table 8.5. This table shows that (i) the results of the tests strongly depend on the alternative; (ii) the p‐values of the three tests can be quite different; (iii) for most of the series, the GARCH(1,1) model is clearly rejected. Point (ii) is not surprising because the asymptotic equivalence between the three tests is only shown under the null hypothesis or under local alternatives. Moreover, because of the positivity constraints, it is possible (see, for instance, the DJU) that the estimated GARCH(1,2) model satisfies with . In this case, when the estimators lie at the boundary of the parameter space and the score is strongly positive, the Wald and LR tests do not reject the GARCH(1,1) model, whereas the score does reject it. In other situations, the Wald or LR test rejects the GARCH(1,1), whereas the score does not (see, for instance the DAX for the GARCH(1,4) alternative). This study shows that it is often relevant to employ several tests and several alternatives. The conservative approach of Bonferroni (rejecting if the minimal p‐value multiplied by the number of tests is less than a given level α ), leads to rejection of the GARCH(1,1) model for 16 out of the 24 series in Table  8.5. Other procedures, less conservative than that of Bonferroni, could also be applied (see Wright 1992) without changing the general conclusion.

In conclusion, this study shows that the GARCH(1,1) model is certainly overrepresented in empirical studies. The tests presented in this chapter are easily implemented and lead to selection of GARCH models that are more elaborate than the GARCH(1,1).

8.6 Proofs of the Main Results^*

8.7 Bibliographical Notes

It is well known that when the parameter is at the boundary of the parameter space, the maximum likelihood estimator does not necessarily satisfy the first‐order conditions and, in general, does not admit a limiting normal distribution. The technique, employed in particular by Chernoff (1954) and Andrews (1997) in a general framework, involves approximating the quasi‐likelihood by a quadratic function, and defining the asymptotic distribution of the QML as that of the projection of a Gaussian vector on a convex cone. Particular GARCH models are considered by Andrews (1997, 1999) and Jordan (2003). The general GARCH(p, q) case is considered by Francq and Zakoïan (2007). A proof of Theorem 8.1, when the moment assumption B7 is replaced by assumption B7 ^′ of Remark 8.2, can be found in the latter reference. When the nullity of GARCH coefficients is tested, the parameter is at the boundary of the parameter space under the null, and the alternative is one‐sided. Numerous works deal with testing problems where, under the null hypothesis, the parameter is at the boundary of the parameter space. Such problems have been considered by Chernoff (1954), Bartholomew (1959), Perlman (1969), and Gouriéroux, Holly, and Monfort (1982), among many others. General one‐sided tests have been studied by, for instance Rogers (1986), Wolak (1989), Silvapulle and Silvapulle (1995), and King and Wu (1997). Papers dealing more specifically with ARCH and GARCH models are Lee and King (1993), Hong (1997), Demos and Sentana (1998), Andrews (2001), Hong and Lee (2001), Dufour et al. (2004), Francq and Zakoïan (2009b) and Pedersen and Rahbek (2018).

The portmanteau tests based on the squared residual autocovariances were proposed by McLeod and Li (1983), Li and Mak (1994), and Ling and Li (1997). The results presented here closely follow Berkes, Horváth, and Kokoszka (2003a). Problems of interest that are not studied in this book are the tests on the distribution of the iid process (see Horváth, Kokoszka, and Teyssiére 2004; Horváth and Zitikis 2006).

Concerning the overrepresentation of the GARCH(1, 1) model in financial studies, we mention Stărică (2006). This paper highlights, on a very long S&P 500 series, the poor performance of the GARCH(1, 1) in terms of prediction and modelling, and suggests a non‐stationary dynamics of the returns.