2. A NEW DIAGNOSTIC TEST

Hong (1999) proposes a generalized spectrum as an analytic tool for linear and nonlinear time series. Suppose the time series {e_t} is strictly stationary. The basic idea of Hong (1999) is to consider the spectrum of the transformed series {e^iue_t}, where

. Define

the covariance between e^iue_t and e^ive_t−j. Straightforward algebra yields

where φ_j(u,v) = E [e^{i(ue_t+ve_t−j)}] and φ(u) = E(e^iue_t) are the joint and marginal characteristic functions of (e_t,e_t−j).

if and only if e_t and e_t−j are independent. Suppose

, which holds when, for example, {e_t} is a stationary α-mixing process with the mixing coefficients {α(j)} satisfying

for some ν > 1. Then the Fourier transform of σ_j(u,v) exists:

No moment condition on {e_t} is required. When var(e_t) exists, however, the negative partial derivative of f (ω,u,v) with respect to (u,v) at (0,0) yields the conventional spectral density:

where R(j) = cov(e_t,e_t−j). For this reason f (ω,u,v) is called in Hong (1999) a “generalized spectral density” of {e_t}. The introduction of parameters (u,v) offers much flexibility in capturing serial dependence in {e_t}. The generalized spectrum f (ω,u,v) can capture any type of pairwise dependence across various lags in {e_t}, including those with zero autocorrelations. Searching over the domain of (u,v), for example, one can find the “maximal dependence” of {e_t} at each frequency ω, as given by

where |·| is the Euclidean norm. This maximal spectral dependence may be contributed from linear or nonlinear serial dependence in {e_t}. A generalized spectral peak at some frequency will indicate a cycle, seasonality, or periodicity due to nonlinear dependence (e.g., volatility clustering) when {e_t} is a white noise.⁶

The generalized spectrum f (ω,u,v) differs from the well-known higher order spectra, which are the Fourier transforms of higher order cumulants (cf. Brillinger and Rosenblatt, 1967a, 1967b; Subba Rao and Gabr, 1980, 1984; Terdik, 1999). It does not require any moment condition on e_t. This is appealing because, for example, it has been argued that many high-frequency economic and financial time series have infinite variances (e.g., Fama and Roll, 1968; Pagan and Schwert, 1990). It can effectively capture any pairwise serially dependent processes, including ARCH with zero third cumulants. For such ARCH processes, the bispectrum—the Fourier transform of third-order cumulants—will miss them. We note, however, that f (ω,u,v) cannot capture dependent processes that are pairwise serially independent (i.e., e_t and e_t−j are independent for any nonzero j but {e_t} is serially dependent), which may or may not be captured by the bispectrum. It would be interesting to compare the generalized spectrum and the bispectrum thoroughly, but this is beyond the scope of this paper and should be pursued elsewhere.

When {e_t} is i.i.d., f (ω,u,v) becomes a flat generalized spectrum

Any deviation of f (ω,u,v) from f₀(ω,u,v) is evidence of serial dependence of {e_t}. To test the i.i.d. hypothesis for {e_t}, Hong (1999) suggests that one compare two consistent estimators of f (ω,u,v) and f₀(ω,u,v) via an L₂-norm. Define

where

Note that

. A kernel estimator for f (ω,u,v) can be defined as

where

is a symmetric kernel and p ≡ p_n is a bandwidth (or lag order) such that p → ∞, p/n → 0 as n → ∞. Examples of k(·) include the Bartlett, Daniell, quadratic-spectral, and truncated kernels (e.g., Priestley, 1981, p. 441). The factor (1 − | j|/n)^1/2 is a finite-sample correction factor that delivers a better approximation to the finite-sample distribution. We also have a consistent estimator for f₀(ω,u,v):

Let

be a nondecreasing function such that W′(u) = w(v) exists and is symmetric about 0, with ∫ dW(u) = ∫w(u) du < ∞. Examples of W(·) are the cumulative distribution functions of N(0,1), double exponential, and uniform distributions.

where the second equality follows from Parseval's identity,

, symmetry of weighting functions k(·) and w(·). Moreover, the centering and scaling factors have the following values:

Throughout, unspecified integrals are taken over the entire Euclidean space of proper dimension. The test statistic

involves one- and two-dimensional numerical integrations with respect to (u,v), which can be implemented using, e.g., Gauss–Legendre quadratures. Note that

involves no numerical integration over frequency ω, which has been integrated out as a result of the use of the L₂-norm. Divergence measures rather than the L₂-norm could be used, but they would generally involve numerical integrations over ω and also over (u,v), and the distribution theory might be different also. A GAUSS code for computing

with p chosen via a data-driven method is available from the authors.

3. ASYMPTOTIC DISTRIBUTION

We now derive the null limit distribution of

and establish its “nuisance-parameter-free” property under a wide class of time series models. Following are regularity conditions.

Assumption A.1. {Y_t} is a strictly stationary α-mixing process with

for some ν > 1.

Assumption A.2.

.

Assumption A.3. Let I_t be the pseudo information set from time t to the infinite past and let Θ₀ be a small convex neighborhood of θ₀. The functions g(I_t,·) and h(I_t,·) are twice continuously differentiable with respect to θ ∈ Θ₀ a.s., with E sup_θ∈Θ0∥h⁻¹(I_t,θ)(∂/∂θ)g(I_t,θ)∥⁴, E sup_θ∈Θ0∥h⁻¹(I_t,θ)(∂/∂θ)h(I_t,θ)∥⁴, E sup_θ∈Θ0∥h⁻¹(I_t,θ)(∂²/∂θ²)g(I_t,θ)∥², E sup_θ∈Θ0∥h⁻¹(I_t,θ)(∂²/∂θ²)h(I_t,θ)∥², and E sup_θ∈Θ0[e_t⁴(θ)] all bounded by some constant C ∈ (0,∞), where e_t(θ) = [Y_t − g(I_t−1,θ)]/h(I_t−1,θ).

Assumption A.4. Let

be the observed information set available at time t that may involve certain initial values. Then

Assumption A.5.

is symmetric about 0 and is continuous at 0 and all points in

except a finite number of points, with k(0) = 1,

as z → ∞ for some b > ½ and C ∈ (0,∞).

Assumption A.6.

is nondecreasing such that the derivative W′(u) = w(u) exists and is symmetric about 0, with

and

u⁴ dW(u) < ∞.

Assumption A.7. D₀ ≡ [∫|σ₀(u,v)|² dW(u) dW(v)]² > 0.

These are conditions on the data generating process (DGP) {Y_t}, model parameter estimator

, initial value conditions, models g(·,θ) and h(·,θ), and weight functions k(·) and W(·). In Assumption A.1, we permit but do not require var(Y_t) < ∞. An example with var(Y_t) = ∞ is the integrated GARCH(1,1) process (Engle and Bollerslev, 1986). In Assumption A.2, we permit but do not require

to be a quasi–maximum likelihood estimator (Lee and Hansen, 1994; Lumsdaine, 1996). Any n^1/2-consistent estimator

suffices. Assumption A.3 is a standard condition on the conditional mean and conditional variance models. We require that the fourth moment of the standardized error e_t exist.

Assumption A.4 is a start-up value condition. It ensures that the impact of initial values (if any) assumed in

is asymptotically negligible. This condition easily holds for many time series models. To illustrate this, we first consider an invertible MA(1) model Y_t = αu_t−1 + u_t, where u_t = σe_t,{e_t} is i.i.d.(0,1), and |α| < 1. Here, we have g(I_t−1,θ) = αu_t−1 and h(I_t−1,θ) = σ, where θ = (α,σ)′. Furthermore, we have I_t−1 = {Y_t−1,…,Y₁,Y₀,…} and

, where

is some assumed value (e.g.,

) for u₀. The condition on h(·,·) holds trivially, so we focus on the condition on g(·,·). By recursive substitution, we obtain

It follows that

provided |α| < 1, 0 < σ < ∞, E|e_t| < ∞, and

.

Next, we consider a GARCH(1,1) model: Y_t = e_t h_t, where h_t² = ω + αh_t−1 + βY_t−1². Here, we have I_t−1 = {Y_t−1,…,Y₁,Y₀,…} and

, where

is an assumed value (e.g.,

) for initial variance h₀². In this case, g(I_t−1,θ) = 0 so the condition on g(·,·) holds trivially. By recursive substitution, we have

It follows that

provided ω > 0, 0 < α, β < 1, α + β < 1, and E(h₀²) < ∞.

In Assumption A.5, the constant b governs the rate at which the kernel k(z) → 0 as z → ∞. For kernels with bounded support (e.g., the Bartlett, Parzen, Tukey, and truncated kernels), b = ∞. For the Daniell kernel and quadratic-spectral kernel, b = 1 and 2, respectively. Assumption A.7 ensures that the choice of W(·) does not lead to a degenerate test statistic.

THEOREM 1. Suppose Assumptions A.1–A.7 hold and p = cn^λ for λ ∈ (0,1) and c ∈ (0,∞). Then if {e_t} is i.i.d.,

.

Throughout, all the proofs are collected in the Appendix. In the proof of Theorem 1, we find that the use of any n^1/2-consistent estimator

rather than θ₀ has no impact on the limit distribution of

. Thus,

enjoys the same “nuisance-parameter-free” property as the BDS test but under a wider class of time series models—the “nuisance-parameter-free” property holds under ARCH models for

but not for BDS(m,d).

4. CONSISTENCY

Next, we establish the consistency of

under the alternative to the i.i.d. hypothesis.

THEOREM 2. Suppose Assumptions A.1–A.7 hold and p = cn^λ for λ ∈ (0,1) and c ∈ (0,∞). Then as n → ∞,

Suppose e_t and e_t−j are not independent at some lag j > 0. Then ∫|σ_j(u,v)|² dW(u) dW(v) > 0 for any weighting function W(·) that is positive, monotonically increasing, and continuous with unbounded support on

. Therefore,

is consistent against any type of pairwise dependence for any W(·) satisfying the aforementioned conditions. The examples of W(·) include the cumulative distribution functions of N(0,1), double exponential, and Student's t_ν distribution with ν ≥ 5. Thus, we expect that

has relatively omnibus power against a wide variety of alternatives. Because the L₂-norm in (4.1) is positive whenever there exists pairwise serial dependence at any nonzero lag,

is an asymptotically one-sided N(0,1) test. Upper-tailed asymptotic critical values (e.g., 1.645 at the 5% level) should be used.

The choice of W(·) for

may not be as important as the choice of distance parameter d for BDS(m,d), because the latter can render BDS(m,d) inconsistent against some alternatives. In contrast, any W(·) that is positive, monotonically increasing, and continuous with unbounded support on

always ensures consistency of

against any type of pairwise dependence across various lags in {e_t}. Nevertheless, the choice of W(·) might have impact on the power of

in finite samples. We investigate this in our simulation that follows. Our results show that a variety of choices of W(·) have little impact on the level and power of

, whereas the choice of d has significant impact on the level and power of BDS(m,d).

5. ASYMPTOTIC LOCAL POWER

Local power analysis is insightful for the power property of a test. As noted by Tjøstheim (1996), it is rather difficult to do asymptotic local power analysis in the context of nonparametric testing for serial dependence. For simplicity, we consider a class of local alternatives for which there exists only first-order serial dependence in {e_t} and the joint probability density of (e_t,e_t−1) is

where f₀(·) is a marginal probability density, r_n(·,·) is a remainder that may rise from the asymptotic expansion of f_1n(·,·), and a_n → 0 as n → ∞ is the rate at which

converges to the i.i.d. hypothesis. To ensure that f_1n(·,·) is a valid joint density, we make the following assumption.

Assumption A.8. (i) 1 + a_n g(x,y) + r_n(x,y) ≥ 0 for all

and all n ≥ 1; (ii) ∫g(x,y) f₀(x) f₀(y) dx dy = 0 and ∫r_n(x,y) f₀(x) f₀(y) dx dy = 0 for all n ≥ 1; and (iii) ∫g⁴(x,y) f₀(x) f₀(y) dx dy < ∞ and ∫r_n⁴(x,y) × f₀(x) f₀(y) dx dy = o(a_n⁴).

Note that f₀(·) is the marginal density of {e_t} when {e_t} is i.i.d. and g(·,·) characterizes the type of serial dependence in {e_t}. The condition on r_n(·,·) ensures that the remainder r_n(·,·) has no impact on the limit distribution of

under

. Two examples of

are an MA(1) process:

and an MA conditional heteroskedastic (MACH(1); Yang and Bewley, 1995) process:

where ε_t is i.i.d. N(0,1). Here, g(x,y) = xy for (5.2) and g(x,y) = (x² − 1) × (y² − 1) for (5.3).

THEOREM 3. Suppose Assumptions A.1–A.8 hold and p = cn^λ for λ ∈ (0,½) and c ∈ (0,∞). Then

under

as n → ∞, where the noncentrality

Whenever g(x,y) ≠ 0, we have μ > 0 provided W(u) is positive, monotonically increasing, and continuous with unbounded support on

. Consequently,

has nontrivial power against

. The rate p^1/4/n^1/2 is slower than n^−1/2, because p → ∞ as n → ∞. This is the price one has to pay to achieve consistency against any type of pairwise serial dependence in {e_t}. However, it is faster than n^−1/4 given p/n → 0. If p ∝ log(n), then p^1/4/n^1/2 ∝ n^−1/2 log^1/4(n), which is nearly the same as n^−1/2. If p ∝ n^1/5, as is the case with the data-driven method described subsequently for some commonly used kernels, p^1/4/n^1/2 ∝ n^−1/2+1/20, which is only slightly slower than n^−1/2. We note that the use of

rather than {e_t}_t=1ⁿ has no impact on the asymptotic local power of

(see Theorem A.3 in the Appendix), so the conclusion of Theorem 3 also applies to the tests considered in Hong (1999), where no local power analysis was given.

It is of interest to compare the asymptotic local power of

and BDS(m,d). For simplicity, we consider BDS(2,d) under a subclass of

where g(x,y) = g₁(x)g₂(y) for some functions

such that ∫g_l(x) f₀(x) dx = 0, l = 1,2. We find that BDS(2,d) has nontrivial power under

if the limit noncentrality

Straightforward algebra shows that under

with g(x,y) = g₁(x)g₂(y), we have

where f_0n(·) denotes the marginal density of e_t under

, which may not be the same as f₀(·), the marginal density of e_t when {e_t} is i.i.d. If the first term in (5.5) is identically 0 for all n, the asymptotic local power of BDS(2,d) will depend on the second term, which renders BDS(2,d) only able to detect

. This occurs, e.g., when the marginal density f₀(·) is uniform.

Then the first term in (5.5) is identically 0 for all n, because the integral

where the third equality follows from changes of variable and the last one follows from (5.6). Note that the MA(1) process in (5.2), where g₁(x) = x and g₂(y) = y, satisfies condition (5.6) and the MACH(1) process in (5.3), where g₁(x) = x² − 1 and g₂(y) = y² − 1, does not satisfy condition (5.6). Thus, BDS(2,d) can detect MACH(1) with rate n^−1/2 but MA(1) with rate n^−1/4 only. This explains why it is often found in practice that BDS(m,d) has excellent power against ARCH (e.g., Brock et al., 1991). Note that a local MACH(1) is equivalent to a local ARCH(1).

6. CHOICE OF DATA-DRIVEN BANDWIDTH

Both BDS(m,d) and

involve the choice of lag order m − 1 or p. Brock et al. (1991), based on simulation experiments, recommend some simple rule of thumb that m be small for finite sample sizes. Our generalized spectral smoothing provides a data-driven method to choose p, which, to some extent, lets data themselves speak for a proper p for

. Before discussing specific data-driven methods, we first justify the use of a data-driven lag order

. We impose a Lipschitz continuity condition on k(·), which rules out the truncated kernel k(z) = 1(|z| ≤ 1) but it includes most commonly used kernels.

Assumption A.9. For any

, for some constant C ∈ (0,∞).

THEOREM 4. Suppose Assumptions A.1–A.7 and A.9 hold and

is a data-driven bandwidth such that

for some β > (2b − ½)/(2b − 1), where b is as in Assumption A.5, p = cn^λ with λ ∈ (0,1), and c ∈ (0,∞). Then if {e_t} is i.i.d.,

as n → ∞.

Thus, as long as

converges to p sufficiently fast, the use of

rather than p has no impact on the limit distribution of

, an additional “nuisance-parameter-free” property. This extends the results of Hong (1999) to the estimated standardized residuals of model (1.2).

Theorem 4 allows for a wide range of admissible rates for

. One plausible choice of

is the plug-in method considered in Hong (1999), which minimizes an asymptotic integrated mean square error (IMSE) criterion for the estimator

f_n(·,·,·) in (2.7). This method is described as follows. Consider the “pilot” estimators based on a preliminary bandwidth p:

where

is a kernel not necessarily the same as the kernel k(·) used in (2.7). For example, k(·) can be the Bartlett kernel whereas k(·) is the Daniell kernel. Note that f_n(·,·,·) is an estimator for f (·,·,·) and f_n^(q)(·,·,·) is an estimator for the generalized spectral derivative

The plug-in bandwidth is then defined as

where

is the tuning parameter estimator given by

Note that the second equality in (6.5) follows from Parseval's identity.

The data-driven

still involves the choice of a preliminary bandwidth p, which either can be fixed or can grow with the sample size n. If p is fixed,

still grows at rate n^1/(2q+1) in general, but

does not converge to the optimal tuning constant. This is analogous in spirit to a parametric plug-in method. Hong (1999) shows that when p grows with n properly, the data-driven bandwidth

in (6.4) minimizes an asymptotic IMSE of

.

From a theoretical point of view, the choice of

based on the IMSE criterion may not maximize the power of the test. A more sensible alternative is to develop a data-driven

using a suitable power criterion or a criterion that trades off level distortion and power. This will necessitate higher order asymptotic analysis and is beyond the scope of this paper. We are content with the IMSE criterion here. Our simulation study shows that the power of the proposed test is relatively flat in the neighborhood of the optimal lag order that maximizes the power, and the data-driven

based on IMSE performs reasonably well.

is real-valued. One can take its integer part, and the impact of integer-clipping is expected to be negligible. The choice of p is somewhat arbitrary, but we expect that the choice of p is of secondary importance and may have no significant impact on

. This is confirmed in our subsequent simulation.

7. MONTE CARLO EVIDENCE

We now compare the level and power of

, BDS(m,d), the Box–Pierce–Ljung test BPL(p), the McLeod–Li test ML(p), and the Li–Mak test LM(p,r), in finite samples. We check adequacy of two basic time series models—AR(r) and ARCH(r)—for r = 1,4, respectively. With the null AR(1) model, we examine the level of the tests, their power against a variety of neglected dynamics and nonlinearities in conditional mean, and their power to distinguish AR(1) from a chaotic alternative that has the same autocorrelation structure as AR(1). With the null ARCH(1) model, we examine the level of the tests, their power against misspecification in conditional variance, their power to distinguish ARCH(1) from nonlinearities in mean that result in apparent ARCH structures, and their power to distinguish ARCH(1) from a chaotic process that behaves like a white noise but has similar autocorrelations in squares to ARCH(1). Finally, the null models AR(4) and ARCH(4) allow us to examine the impact of parameter estimation uncertainty on the level of the proposed tests when the parameter dimension increases. Such impact is asymptotically negligible but might be significant in finite samples.

7.1. Testing Conditional Mean Model

We first examine the adequacy of an AR(1) model:

Model A:

under each of the following DGP:

DGP A.0 (AR(1))

DGP A.1 (AR(2))

DGP A.2 (ARMA(1,1))

DGP A.3 (Bilinear)

DGP A.4 (Nonlinear MA)

DGP A.5 (Threshold AR, TAR)

DGP A.6 (Markov Regime-Switching)

where S_t is a latent state variable that follows a two-state Markov chain with transition probabilities P(S_t = 1|S_t−1 = 0) = P(S_t = 0|S_t−1 = 1) = 0.3.

DGP A.7 (Sign Autoregressive, SIGN)

where sign(x) = 1(x > 0) − 1(x < 0).

DGP A.8 (Tent Map)

where α = 0.49999 and Y₀ is generated from the uniform distribution on [0 1].

In DGPs A.0–A.7, {ε_t} is i.i.d.(0,1). DGPs A.1 and A.2 are used to check the power of tests against neglected dynamics in mean, and DGPs A.3–A.6 are used to check against various neglected nonlinearities in mean. DGPs A.3 and A.4 are not invertible but they are second-order stationary, as shown in Granger and Andersen (1978, pp. 90–91). The SIGN model examined in Granger and Teräsvirta (1999), DGP A.7, is a first-order nonlinear autoregressive process but has the same autocorrelation function as an AR(1) process: ρ(j) = (1 − 2q)^{| j|}, where q = P(σε_t < −1) = P(σε_t > 1) when ε_t is symmetric. Following Granger and Teräsvirta (1999), we choose σ = 0.43 so that q = 0.01 if ε_t is N(0,1). The tent map, DGP A.8, is a deterministic chaotic process, but it resembles in autocorrelation an AR(1) process with the AR coefficient 2α − 1 (see Sakai and Tokumaru, 1980). The DGPs A.7 and A.8 allow us to examine how a test can distinguish an AR(1) model from nonlinear stochastic and chaotic processes that behave like a linear process in terms of autocorrelation.

7.2. Testing Conditional Variance Model

Next, we examine the adequacy of an ARCH(1) model:

Model B:

when Y_t is generated from the following generating processes:

DGP B.0 (ARCH(1))

DGP B.1 (ARCH(2))

DGP B.2 (GARCH(1,1))

DGP B.3 (EGARCH(1,1))

DGP B.4 (Stochastic Volatility)

DGP B.5 (Bilinear)

DGP B.6 (TAR)

DGP B.7 (Nonlinear MA)

DGP B.8 (Logistic Map)

where Y₀ is generated from the uniform distribution on [0, 1].

In DGPs B.0–B.7, {ε_t} is i.i.d.(0,1). The DGPs B.1–B.4 are used to examine the power of the tests against misspecification in conditional variance. In DGP B.4, parameter value φ = 0.98 is empirically relevant; Harvey, Ruiz, and Shephard (1994) obtain estimates of φ in range of 0.9575–0.9948 for four different daily foreign exchange rates. The DGPs B.5–B.7 allow us to examine the power to distinguish ARCH from a variety of nonlinearities in mean that result in apparent ARCH structures. Such distinction has important implications in practice (Weiss, 1986; Bera and Higgins, 1997; Diebold, 1986). DGP B.8, the logistic map, behaves like a white noise but has similar autocorrelations in squares to ARCH(1) (e.g., Granger and Teräsvirta, 1993, p. 34). It is used to examine the power of a test to distinguish ARCH from a chaotic process with similar autocorrelations in squares. From the results of He and Teräsvirta (1999), we note that {Y_t} does not have finite fourth moments under DGP B.1 even when {ε_t} is i.i.d.N(0,1). However, {Y_t} has all finite moments under DGP B.3 when {ε_t} is i.i.d.N(0,1), but var(Y_t) is infinite when {ε_t} is i.i.d.t₅ (see Nelson, 1991). We conjecture that var(Y_t) is also infinite when {ε_t} is an i.i.d. sequence of mixed normals. Note that strict stationarity holds under both DGPs B.1 and B.3. On the other hand, DGPs B.5 and B.7 are not invertible, but they are second-order stationary (see Granger and Andersen, 1978, pp. 90–91).

7.3. Testing Higher Order Models

We now examine the impact of parameter estimation uncertainty on the level of the proposed tests. We examine how the level of the tests is affected by increasing the number of estimated parameters. Such impact is asymptotically negligible but might be significant in finite samples. We consider the following two higher order models under the null hypothesis.

First, we consider the adequacy of an AR(4) model for the conditional mean:

Model C:

under the following DGP:

DGP C.0 (AR(4))

Next, we consider the adequacy of an ARCH(4) model for the conditional variance:

Model D:

when Y_t is generated from the following DGP:

DGP D.0 (ARCH(4))

For all the DGPs except the chaotic processes A.8 and B.8, we use the GAUSS Windows version random number generator to generate i.i.d. innovations {ε_t} from four distributions: (i) N(0,1); (ii) exponential; (iii) mixed normal, P[ε_t ∼ N(−3,1)] = P[ε_t ∼ N(3,1)] = 0.5; and (iv) Student's t₅. All the ε_t have been rescaled to have mean 0 and variance 1. We generate n + 1,000 observations for {ε_t} under each of the distributions (i)–(iv) and then discard the first 1,000 to alleviate the impact of using some initial values. We report the levels of the tests under all four error distributions, but for space we report the power under the normal error only.

7.4. Monte Carlo Evidence

To compute the test statistic

, BDS(m,d), BPL(p), ML(p), and LM(p,r), we use the usual residual series

from Model A and

from Model C, estimated by the ordinary least squares method, and the standardized residual series

where

from Model B and

from Model D, estimated by the quasi–maximum likelihood method with a Gaussian likelihood function.

For the generalized spectral test

, we use Daniell kernel k(z) = sin(πz)/πz, which enjoys the optimal power property over a class of kernels (Hong, 1999, Theorem 5).

¹⁰

We have also tried the Bartlett, Parzen and quadratic-spectral kernels; they give level and power very similar to that of the Daniell kernel.

To examine the impact of the choice of preliminary bandwidths p on the level and power of

, we consider p = 1–10. To investigate the impact of the choice of weight function W(·) on the level and power of

, we consider the three distribution functions: (i) N(0,1), (ii) double exponential, and (iii) t₅-distribution. They are all scaled to have mean 0 and variance 1.

For BDS(m,d), Brock et al. (1991) recommend using d in range 0.5σ–1.5σ and m in range 2–5, for n = 500–1,000, where σ² = var(Y_t). To examine the impact of the choice of embedding dimension m on the level and power of BDS(m,d), we use m = 2–11, which is equivalent to the choice of a lag order p from 1 to 10. As some DGP may have no finite variance, we consider three choices of distance parameter: d = 0.5, 0.25, 0.125, in the unit of data range. For normal random samples, these choices roughly correspond to 2σ, σ, and 0.5σ, respectively.

For the Box–Pierce–Ljung test, BPL(p), we use p = 2–10 for Model A (p = 1 cannot be chosen because of the adjustment of the degree of freedoms for its asymptotic distribution) and p = 1–10 for Model B.

For the McLeod–Li test ML(p), which is suitable to test AR(r) models, we use p = 1–10 for Model A. For the Li–Mak test LM(p,r), which is suitable to test ARCH(r) models, p = 2–10 for Model B (similarly to BPL(p) for Model A, p = 1 cannot be chosen here for LM(p,r) when r = 1).

To examine the levels of the tests under the null AR(1) model and under the null ARCH(1) model, we estimate Model A and Model B under DGP A.0 and DGP B.0, respectively. We consider the empirical level at the 10%, 5%, and 1% significance levels for n = 100 and 200, using asymptotic critical values and 1,000 Monte Carlo iterations. To conserve space, Figures 1 and 2 only report the levels of the tests at the 5% level for n = 100. Similarly, to examine the levels of the tests under the null AR(4) model and under the null ARCH(4) model, we estimate Model C and Model D under DGP C.0 and DGP D.0, respectively, and report the results in Figures 1 and 2 also.¹¹

The results at the 10% and 1% levels and the results for n = 200 are available from the authors. The results at the 10% and 1% levels have similar patterns, and the levels with n = 200 are very similar to those with n = 100.

Size of testing conditional mean at 5% level—(a) DGP A.0: AR(1)–N(0,1), (b) DGP A.0: AR(1)–exponential, (c) DGP A.0: AR(1)–mixed normal, (d) DGP A.0: AR(1)–t5.

Size of testing conditional mean at 5% level—(e) DGP C.0: AR(4)–N(0,1), (f) DGP C.0: AR(4)–exponential, (g) DGP C.0: AR(4)–mixed normal, (h) DGP C.0: AR(4)–t5.

Size of testing conditional variance at 5% level—(a) DGP B.0: ARCH(1)–N(0,1), (b) DGP B.0: ARCH(1)–exponential, (c) DGP B.0: ARCH(1)–mixed normal, (d) DGP B.0: ARCH(1)–t5.

Size of testing conditional variance at 5% level—(e) DGP D.0: ARCH(4)–N(0,1), (f) DGP D.0: ARCH(4)–exponential, (g) DGP D.0: ARCH(4)–mixed normal, (h) DGP D.0: ARCH(4)–t5.

To examine the powers of the tests against various misspecifications of the AR(1) model, we report in Figure 3 the powers of the tests under DGPs A.1–A.8, each of which is fitted by an AR(1) model. To examine the powers of the tests against various misspecifications of the ARCH(1) model, we report in Figure 4 the powers of the tests under DGPs B.1–B.8, each of which is fitted by an ARCH(1) model. The power is level-adjusted by using the empirical critical values obtained under DGPs A.0 and B.0, respectively, which provide a fair comparison among the tests under study. We only report the power at the 5% level, for n = 100 and normal errors {ε_t}, using 1,000 replications.

Size-corrected power of testing conditional mean at 5% level—(a) DGP A.1: AR(2)–N(0,1), (b) DGP A.2: ARMA(1,1)–N(0,1), (c) DGP A.3: bilinear–N(0,1), (d) DGP A.4: nonlinear MA–N(0,1).

Size-corrected power of testing conditional mean at 5% level—(e) DGP A.5: TAR–N(0,1), (f) DGP A.6: Markov switching–N(0,1), (g) DGP A.7: SIGN–N(0,1), (h) DGP A.8: tent map.

Size-corrected power of testing conditional variance at 5% level—(a) DGP B.1: ARCH(2)–N(0,1), (b) DGP B.2: GARCH(1,1)–N(0,1), (c) DGP B.3: EGARCH–N(0,1), (d) DGP B.4: stochastic volatility–N(0,1).

Size-corrected power of testing conditional variance at 5% level—(e) DGP B.5: bilinear–N(0,1), (f) DGP B.6: TAR–N(0,1), (g) DGP B.7: nonlinear MA–N(0,1), (h) DGP B.8: logistic map.

In Figures 1, 2, 3, and 4, the levels or powers of

, BDS(m,d), BPL(p), ML(p), and LM(p,r) are plotted as functions of p, m − 1, p, p, and p in the horizontal axis, respectively. In each graph, there are three plots for

in solid lines, denoted as M_l (l = 1,2,3), that correspond to three weight functions W(·)—the distribution functions of N(0,1), double exponential, and t₅. There are also three plots for BDS(m,d) in dashed lines, denoted as BDS_l (l = 1,2,3), that correspond to three distance parameter values—d = 0.5^l. Moreover, ML(p) or LM(p,r) is plotted in dotted lines, and BPL(p) is plotted with more closely spaced dots. The test ML(p) is reported in Figures 1 and 3, where the usual estimated residuals are used, and LM(p,r) is reported in Figures 2 and 4, where the standardized estimated residuals are used.

We first examine the levels in Figures 1 and 2. We observe the following patterns.

1. Overall, the levels of the tests

, BPL(p), and ML(p) under the null AR(1) model and the levels of the tests

, BPL(p), and LM(p,r) under the null ARCH(1) model are more or less reasonable, whereas the level of BDS(m,d) appears not very satisfactory. The unsatisfactory level performance of BDS(m,d) under the null ARCH(1) model may be due to its violation of the “nuisance-parameter-free” property under the ARCH(1) model.

2. The level of

is robust to the choice of weight function W(·) and preliminary bandwidth p. The levels of BPL(p) and ML(p)/LM(p,r) are excellent and robust to the choice of lag order p. On the other hand, the level of BDS(m,d) is sensitive to the choices of distance parameter d and embedding dimension m. The fact that BDS(m,d) is sensitive to m whereas

is not sensitive to p indicates the practical merit of the data-driven choice of lag order

.

3.

displays some (not excessive) underrejection under the null AR(1) with normal errors or some (not excessive) overrejection under the ARCH(1) model with exponential and mixed normal errors. On the other hand, the level distortion of BDS(m,d) is quite large, especially under the mixed normal errors, which is consistent with the findings of Brock et al. (1991, p. 50).

4. The level patterns of each test under the null AR(1) model and the null ARCH(1) model are more or less similar.

5. The preceding four observed level patterns for the null models of AR(1) and ARCH(1) carry over to the higher order null models of AR(4) and ARCH(4). This indicates that model parameter estimation uncertainty does not affect the level of the tests at least for the models and sample sizes considered.

In Figures 1 and 2 we consider the levels of the tests under an AR coefficient of 0.6 for DGP A.0 and the levels of the tests with an ARCH coefficient of 0.1 for DGP B.0. We have also experimented (not reported) with a variety of coefficient values: 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9 in both cases. We note that the fourth moment of Y_t does not exist when the ARCH(1) coefficient value is larger than or equal to

for DGP B.0. In most scenarios, the levels of

, BDS(m,d), BPL(p), and ML(p)/LM(p,r) are generally robust to the values of the AR(1) and ARCH(1) coefficients in DGP A.0 and DGP B.0. One exception is BPL(p), which tends to overreject under the null AR(1) model when the AR coefficient is close to 1 (say, 0.9) in DGP A.0 and lag order p is small, but its level becomes reasonable for larger lag orders p, say, p > 5.

We now examine the powers of the tests. To be fair in comparison and to take into account the BDS(m,d)'s violation of the “nuisance-parameter-free” property under the ARCH(1) model, we use empirical critical values. We first examine the powers of the tests against various misspecifications of an AR(1) model, as reported in Figure 3. We observe the following patterns.

1. The power of

is generally not sensitive to the choice of the preliminary bandwidth (or lag order) p. The power of BDS(m,d) seems sensitive to the choice of the embedding dimension m, which is equivalent to the choice of a lag order. The tests BPL(p) and ML(p) are also sensitive to the choice of lag order p in some cases.

2. The power of

is robust to the choices of weight function W(·), whereas the power of BDS(m,d) is sensitive to the choice of distance parameter d.

3. (a) The autocorrelation test BPL(p) has excellent power against AR(2), ARMA(1,1), and SIGN alternatives to the AR(1) model. Nevertheless, as expected, BPL(p) cannot detect the nonlinear alternatives—bilinear, nonlinear MA, TAR, and Markov regime-switching. It also cannot distinguish AR(1) from the tent map, which resembles an AR(1) process in autocorrelation but is completely deterministic.

 (b) The correlation in squares test, ML(p), has good power against bilinear, nonlinear MA, Markov regime-switching, and tent map alternatives to the AR(1) model. However, it has low power against AR(2), ARMA(1,1), TAR, and SIGN alternatives.

 (c) BDS(m,d) has good power against bilinear, nonlinear MA, and tent map alternatives to the AR(1) model. However, it has low power against ARMA(1,1), TAR, and SIGN alternatives.

 (d) The generalized spectral test

has excellent power against AR(2), ARMA(1,1), bilinear, TAR, SIGN, and tent map alternatives to the AR(1) model. And it has moderate power for Markov regime-switching and low power for nonlinear MA alternatives.

4. Overall,

has reasonable omnibus power against all linear and nonlinear dependent alternatives except for nonlinear MA. Moreover, it is more powerful than the other tests in many cases.

Next, we examine the powers of the tests against various misspecifications of the ARCH(1) model, as reported in Figure 4. We observe the following patterns.

1. As in testing misspecifications in conditional mean, the power of

is robust to the choice of the preliminary bandwidth (or lag order) p in most cases. The power of BDS(m,d) is sensitive to the choice of the embedding dimension m. The tests BPL(p) and LM(p,r) are also sensitive to the choice of lag order p in some cases.

2. The power of

is robust to the choices of weight function W(·), whereas the power of BDS(m,d) is sensitive to the choice of distance parameter d.

3. (a) The correlation-based test BPL(p) has low power against ARCH(2), GARCH(1,1), EGARCH(1,1), stochastic volatility, bilinear, nonlinear MA, and logistic map alternatives to the ARCH(1) models. These alternatives are either martingale difference sequences or serially uncorrelated processes. However, BPL(p) has good power to distinguish TAR from ARCH(1).

 (b) The correlation-in-squares test (the Li–Mak test), LM(p,r), is most powerful against ARCH(2), for which it has good power by its design. Nevertheless, LM(p,r) has low power against other forms of conditional heteroskedastic alternatives to ARCH(1), such as GARCH(1,1), EGARCH(1,1), and stochastic volatility models. Moreover, it cannot distinguish ARCH(1) from bilinear, TAR, nonlinear MA, and logistic map processes. Many of these nonlinear conditional mean models have similar moment structures to ARCH(1). In particular, the logistic map behaves like a white noise but has similar autocorrelations in squares to ARCH(1) (cf. Granger and Teräsvirta, 1993, p. 34).

 (c) BDS(m,d) has poor power against bilinear, TAR, and nonlinear MA alternatives to the null ARCH(1) model. This finding is consistent with the findings of Brooks and Heravi (1999), who document that BDS(m,d) is a fairly poor discriminator of bilinear and TAR processes from ARCH processes. Such distinctions have important implications in terms of predictability in economics and finance (e.g., Bera and Higgins, 1997; Weiss, 1986).

 (d)

has high power against EGARCH(1,1) and stochastic volatility alternatives to the ARCH(1) model. It has high power to distinguish ARCH(1) from bilinear, TAR, nonlinear MA, and logistic map processes. Interestingly,

has better power than BDS(m,d) against the logistic map alternative.

(4) Overall, the generalized spectral test

has omnibus power against all alternatives except for GARCH(1,1). For the GARCH(1,1) alternative to ARCH(1), all the tests have low or little power. In most cases,

is the most powerful.

8. EMPIRICAL APPLICATION

To further highlight the merits of our generalized spectral test, we now apply it to evaluate an empirical financial time series model. As is well known, practical model-based financial decision making such as hedging, risk management, and option pricing will be satisfactory only if it builds on reasonable specification of the underlying asset price processes. In an important contribution, Andersen, Benzoni, and Lund (2002) use efficient method of moments (EMM) of Gallant and Tauchen (1996) to evaluate the adequacy of a variety of continuous-time parametric models for the daily S&P 500 equity index and the impact of different specifications on option pricing. The EMM method is based on the expectation of the score function of a discrete-time auxiliary semi-nonparametric model that adequately approximates the conditional distribution of the discretely sampled data in an asymptotic sense. An attractive feature of EMM is that it is generally applicable and achieves the same efficiency as the maximum likelihood estimator when the score function of the auxiliary model asymptotically spans the score function of the true conditional distribution. Moreover, the EMM criterion function can be used to construct an overall test of the overidentifying restrictions on the parametric model to be tested, and the fit of individual scores can be used to gauge how well the parametric model captures particular features of data.

In their EMM applications to the S&P 500 index, Andersen et al. (2002), based on some model selection criteria, choose the following auxiliary model for the daily S&P 500 price changes:

where the density of {e_t} is approximated by Hermite polynomials. The i.i.d. property for {e_t} is obtained as a consequence of using some model selection criteria. This model is referred to as MA(1)-EGARCH(p,q), as in the work of Andersen et al. (2002), who estimate the model in two steps: as the first-order autocorrelation in {Y_t} is largely “artificial,” induced by nonsynchronous trading effects, they filter out this effect before estimating the conditional variance.¹²

Estimates of MA(1)-EGARCH models for S&P 500 index changes

Andersen et al. (2002) find that an MA(1)-EGARCH(1,1) model can adequately capture the full serial dependence in the daily S&P 500 equity index changes from 01/02/1953 to 12/31/1996 (with n = 11,076), and an MA(1)-EGARCH(2,1) model can adequately capture the full serial dependence in daily changes of the S&P 500 index from 01/02/1980 to 12/31/1996 (with n = 4,298). In both cases, the density of the i.i.d. error e_t is approximated by a Hermite polynomial expansion.

The adequacy of the full dynamics of the auxiliary semi-nonparametric model for {Y_t} is crucial for the efficiency of the EMM estimator and the validity of the related EMM diagnostic tests. Although a semi-nonparametric model is asymptotically free of model misspecification from a theoretical point of view, the use of some model selection criteria might lead to a misspecified model in practice. It is therefore important to check if the aforementioned MA(1)-EGARCH models indeed completely capture the full serial dependence in the daily S&P 500 index data. Andersen et al. (2002) note that the standardized estimated residuals of these models pass Box–Pierce–Ljung type tests. We now check the adequacy of these MA(1)-EGARCH models by using our new procedure. For comparison, we also report the results for BPL(p), LM(p,r), and BDS(m,d) tests. Our results confirm that these models do pass the BLP(p), LM(p,r), and BDS(m,d) tests, but our new test finds very strong evidence of model misspecification missed by the BPL(p), LM(p,r), and BDS(m,d) tests.

Figure 5 reports the p-values of the proposed test

, BPL(p), LM(p,r), and BDS(m,d) tests. We report

only with the normal cumulative distribution function (c.d.f.) as the weighting function, denoted as M₁ in Figure 5, because

is robust to the choice of the weight functions. However, we still report BDS(m,d) with three choices of distance parameters d = 0.5, 0.25, and 0.125 in unit of data range, denoted as BDS₁, BDS₂, and BDS₃, respectively. As LM(p,r) is computed for the standardized residuals with

an estimated EGARCH model (not ARCH(r)), its asymptotic distribution is not χ_p−r² and is unknown to us. Thus, we report the bootstrap p-values in addition to the asymptotic p-values.

Let T be one of the statistics (

, BPL(p), LM(p,r), BDS(m,d)) based on the observed sample {Y_t}_t=1ⁿ. The bootstrap p-value of T is computed as follows: (i) Estimate the MA(1)-EGARCH(p,q) model and get the estimated standardized residuals

. (ii) Resample {ê_t}_t=1ⁿ with replacement to get

. (iii) Generate the bootstrap samples {Y_t^b}_t=1ⁿ recursively from the estimated MA(1)-EGARCH(p,q) model using

. (iv) Estimate the MA(1)-EGARCH(p,q) model using the bootstrapped data {Y_t^b}_t=1ⁿ to get the standardized residuals

. (v) Compute the statistic T^b using

. Repeat the steps (ii)–(v) B times (i.e., b = 1,…,B), where B is the number of bootstrap replications. We use B = 500. The bootstrap p-value of the statistic T is then obtained from

.

Empirical application—(a) Asymptotic p-values: 1953–1996, (b) bootstrap p-values: 1953–1996, (c) asymptotic p-values: 1980–1996, (d) bootstrap p-values: 1980–1996.

We first consider the results for the whole sample from 1953 to 1996 (see Figures 5a and 5b). We report asymptotic and bootstrap p-values for M₁, BPL(p), and LM(p,r). We do not report BDS(m,d) for the whole sample, because we use the program, accompanied by the book by Brock et al. (1991), that can handle only up to 7,500 observations. The M₁ test has zero asymptotic and bootstrap p-values. In fact, the M₁ statistic values range from 13.3 to 14.5 when the preliminary lag order p changes from 21 to 50 and the Parzen kernel is used, and they range from 10.5 to 14.3 when p changes from 21 to 50 and the Bartlett kernel is used. The BPL(p) test has large asymptotic and bootstrap p-values, ranged approximately from 10% to 25%, well above the conventional 5% level. The LM(p,r) test statistic is clearly insignificant in terms of both asymptotic and bootstrap p-values. Because the asymptotic distribution of LM(p,r) is not χ_p−r² when the conditional variance model is not ARCH(r), we rely on the bootstrap p-values in this case. The bootstrap p-values reported in Figure 5 are based on r = 0 for LM(p,r), but other values of r give similar bootstrap p-values (not reported). Using BPL(p) and LM(p,r), which have been commonly used in practice as diagnostic tools, we fail to detect any model inadequacy. In contrast, our proposed test indicates that the model does not adequately capture the full serial dependence in the S&P 500 index changes.

Next, we consider the results for the subsample (see Figures 5c and 5d). We now report BDS(m,d) also because the BDS program we use can handle this subsample (n = 4,298). Again, the M₁ test is the most powerful test: it has zero p-values in terms of both asymptotic and bootstrap p-values. The M₁ statistic values range from 5.9 to 6.6 when the preliminary lag order p changes from 21 to 50 and the Parzen kernel is used, and it ranges from 5.2 to 6.5 when p changes from 21 to 50 and the Bartlett kernel is used. The BPL test has both asymptotic and bootstrap p-values well above 10%, consistent with the results of Andersen et al. (2002). The LM(p,r) test has bootstrap p-values above 20%. The BDS(m,d) tests are sensitive to the choice of distance parameter d. Nevertheless, all three BDS tests fail to reject the MA(1)-EGARCH(2,1) model. For d = 0.125, BDS₃ has large p-values close to one in terms of both asymptotic and bootstrap p-values. When d = 0.25, BDS₂ is very sensitive to m and has asymptotic and bootstrap p-values in the range 0.35–1.00. When d = 0.5, BDS₁ has bootstrap p-values about 0.50.

In summary, the proposed test

firmly rejects the adequacy of MA(1)-EGARCH models for daily S&P 500 index changes, whereas BDS(m,d), BPL(p), and LM(p,r) tests fail to detect any model inadequacy. This implies that care should be taken in interpreting the statistical significance of the EMM test statistics reported in Andersen et al. (2002), because an inadequate auxiliary model affects the efficiency of the EMM estimator and the validity of the EMM diagnostic tests that have used efficient scores to estimate the asymptotic variance-covariance matrices. But we emphasize that the consistency of the EMM estimator is not affected.