Tests for conditional heteroscedasticity of functional data

Functional data objects derived from high‐frequency financial data often exhibit volatility clustering. Versions of functional generalized autoregressive conditionally heteroscedastic (FGARCH) models have recently been proposed to describe such data, however so far basic diagnostic tests for these models are not available. We propose two portmanteau type tests to measure conditional heteroscedasticity in the squares of asset return curves. A complete asymptotic theory is provided for each test. We also show how such tests can be adapted and applied to model residuals to evaluate adequacy, and inform order selection, of FGARCH models. Simulation results show that both tests have good size and power to detect conditional heteroscedasticity and model mis‐specification in finite samples. In an application, the tests show that intra‐day asset return curves exhibit conditional heteroscedasticity. This conditional heteroscedasticity cannot be explained by the magnitude of inter‐daily returns alone, but it can be adequately modeled by an FGARCH(1,1) model.


INTRODUCTION
Since the seminal work of Engle (1982) and Bollerslev (1986), generalized autoregressive conditionally heteroscedastic (GARCH) models and their numerous generalizations have become a cornerstone of financial time series modeling, and are frequently used as a model for the volatility of financial asset returns. As the name suggests, the main feature that these models account for is conditional heteroscedasticity, which for an uncorrelated financial time series can be detected by checking for the presence of serial correlation in the series of squared returns of the asset. This basic observation leads to several ways of testing for the presence of conditional heteroscedasticity in a given time series or series of model residuals by applying portmanteau tests to the squared series. Such tests have been developed by McLeod and Li (1983) and Li and Mak (1994) to test for conditional heteroscedasticity and perform model selection for GARCH models as well as autoregressive moving average models with GARCH errors. Diagnostic tests of this type are summarized in Li (2003), Shumway and Stoffer (2017), and with a special focus on GARCH models in Francq and Zakoïan (2010). Many of these methods have also been extended to multivariate time series of a relatively small dimension; see also Francq and Zakoïan (2010), Tse and Tsui (1999), Tse (2002), Duchesne and Lalancette (2003), Kroner and Ng (1998), Bauwens et al. (2006), and Catani et al. (2017).
In many applications, dense intra-day price data of financial assets are available in addition to the daily asset returns. One way to view such data is as daily observations of high dimensional vectors (consisting of hundreds or thousands of coordinates) that may be thought of as discrete observations of an underlying noisy intra-day price curve or function. We illustrate with the data that motivate our work and will be further studied below.

TESTS FOR FUNCTIONAL CONDITIONAL HETEROSCEDASTICITY
Consider a stretch of a functional time series of length N, X 1 (t), … , X N (t), which is assumed to have been observed from a strictly stationary sequence {X i (t), i ∈ ℤ, t ∈ [0, 1]} of stochastic processes with sample paths in L 2 [0, 1]. For example, in the application below X i (t) denotes the intra-day cumulative log returns derived from densely observed stock prices on day i at intraday time t, where t is normalized to be in the unit interval.
As described in Ding and Engle (2001), conditional heteroscedasticity or the presence of 'GARCH effects' in a multivariate time series is generally characterized by serial correlation in the squares of the component series, or lagged cross-correlation between the squared component series. This leads one to consider the following definition of conditional heteroscedasticity for functional observations: Definition 2.1 (Functional conditional heteroscedasticity). We say that a sequence {X i } is conditionally heteroscedastic in L 2 [0, 1] if it is strictly stationary, E[X i (t)| i−1 ] = 0, and cov(X 2 i (t), X 2 i+h (s)) ≠ 0, for some h ≥ 1, where the equality above is understood to be in the L 2 [0, 1] 2 sense.

G. RICE ET AL.
 0 : The sequence {X i } is i.i.d., vs.  A : The sequence of {X i } is conditionally heteroscedastic.
Clearly it is not the case in general that rejecting  0 would directly lead to  A , because {X i } might instead be serially correlated but not conditionally heteroscedastic. This concern can be alleviated though if we test serial correlation in the sequence of squared curves as described in Definition 2.1.
In particular, we might then test  0 vs.  A by measuring the serial correlation in the time series ‖X 1 ‖ 2 , … , ‖X N ‖ 2 , or in the sequence of curves X 2 1 (t), … , X 2 N (t). Testing for serial correlation in the time series ‖X i ‖ 2 can be viewed as measuring to what extent large in magnitude curves increase/decrease the likelihood of subsequent curves being large in magnitude, whereas testing for serial correlation in the curves X 2 i (t) aims to more directly evaluate whether the data follow Definition 2.1. For some positive integer K, we then consider portmanteau statistics of the form wherêh is the sample autocorrelation of the time series ‖X 1 ‖ 2 , … , ‖X N ‖ 2 , and̂h(t, s) ∈ L 2 [0, 1] 2 is the estimated autocovariance kernel of the functional time series X 2 i (t) at lag h , defined aŝ The test statistic V N,K is essentially the Box-Ljung-Pierce test statistic (Ljung and Box, 1978) derived from the scalar time series of squared norms, whereas the test statistic M N,K is the same as the portmanteau statistic defined in Kokoszka et al. (2017) applied to the squared functions.
Under  A , we expect the statistics V N,K and M N,K to be large, and hence a consistent test can be obtained by rejecting  0 whenever they exceed a threshold calibrated according to their limiting distributions under the null hypothesis. To establish the asymptotic distributions of each portmanteau statistic under  0 , we impose the following moment condition.
Under this assumption, the asymptotic distribution of M N,K depends on the eigenvalues i , i ≥ 1 of the kernel integral operator with kernel cov(X 2 i (t), X 2 i (s)), namely Theorem 2.1. If  0 and Assumption 2.1 are satisfied, then we have 5) and Theorem 2.1 shows that an approximate test of H 0 of size q is to reject if V N,K > 2 K,1−q or if M N,K exceeds the qth quantile of the distribution on the right-hand side of (2.6). The latter can be approximated in several ways, and in Section 4 we describe a Welch-Satterthwaite style 2 approximation to achieve this.
The eighth-order moment condition in Assumption 2.1 needed for this result is evidently quite strong, although this is a typical assumption in the literature on this topic to ensure consistency of the required higher-order moment estimates. Consequently one may wish to consider robust versions of such portmanteau tests. A nice discussion of this issue in the scalar case is given Aguilar and Hill (2015), and some approaches that may be used are to consider other transformations of the data to evaluate for volatility, for example measuring for serial dependence in the series |X i (t)| rather than X 2 i (t), or trimming the large in norm observations by discarding those that exceed a specified, high sample quantile of the observed norms.
The value of the parameter K must be chosen by the practitioner. Intuitively small values of K will increase the power of the test to detect GARCH effects that occur at small lags, but may miss effects occurring at longer lags, and taking a larger value of K may detect such effects at long lags, but decreases the power for detecting effects at short lags. In general we recommend applying the test for a range of values of K, as a default between 1 and 20, as is recommended when applying the Ljung-Box portmanteau test to scalar series in Shumway and Stoffer (2017), see p. 150. We demonstrate this in an application to functional time series derived from asset price data.

Consistency of the Proposed Tests
We now turn to the consistency of each test under  A . In particular, we consider the asymptotic behavior of V N,K and M N,K for sequences {X i } such that either: (i) they form a general weakly dependent sequences in L 2 [0, 1] that are conditionally heteroscedastic as described by Definition 2.1, or (ii) they follow a FARCH(1) model as described in (2.2). We use the notion of L p -m-approximability defined in Hörmann and Kokoszka (2010) to describe general weakly dependent sequences. A time series {X i } i∈ℤ in L 2 [0, 1] is called L p -m-approximable for some p > 0 if (i) There exists a measurable function g ∶ S ∞ → L 2 ([0, 1]), where S is a measurable space, and i.i.d. innovations Under suitable moment and stationarity conditions, the solutions {X i } to functional GARCH models are L p -m-approximable; see Hörmann et al. (2013) (2.10) The right-hand side of (2.10) is guaranteed to be strictly positive if ∫∫ (t, s)E (t)( 2 0 (t)−1) (s)( 2 0 (s)−1)dtds ≠ 0.
Remark 2.1. Theorem 2.3 shows that under an FARCH(1) model, the rate of divergence of V N,K and M N,K depend essentially on the size of the function (t, s) as well as how this kernel projects onto the intercept term in the conditional variance (t) and the covariance of the squared error 2 0 (t). If for example ∫∫ (t, s)E( 2 0 (t) − 1)( 2 0 (s) − 1)dtds = 0, then we do not expect the tests to be consistent.

DIAGNOSTIC CHECKING FOR FUNCTIONAL GARCH MODELS
The conditional heteroscedasticity tests proposed above can also be used to test for the adequacy of the estimated functional ARCH and GARCH models, and can aid in the order selection of these models. We introduce this approach in the context of testing the adequacy of the FGARCH(1,1) model, although one could more generally consider the same procedure applied to the FGARCH(p, q) models using the estimation procedures in Cerovecki et al. (2019). To this end, suppose that X i (t), 1 ≤ i ≤ N follows an FGARCH(1,1) model. To estimate (t), and the kernel functions (t, s) and (t, s), following Aue et al. (2017) and Cerovecki et al. (2019), we suppose that they have finite L-dimensional representations determined by a set of basis functions (3.1) least squares estimation, as is typically employed in multivariate GARCH models. To see this, under (3.1) we can re-express the FGARCH(1,1) model in terms of the coefficients as and the coefficient matrices A and B are ℝ L×L with (j, j ′ ) entries defined by a j,j ′ and b j,j ′ respectively. To estimate the vector of parameters 0 = (D ⊤ , vec(A) ⊤ , vec(B) ⊤ ) ⊤ , Aue et al. (2017) propose a least squares type estimator satisfyinĝN where Θ is a compact subset of ℝ L+2L 2 . Under certain regularity conditions, detailed at the beginning of Appendix B, it can be shown that̂N is a consistent estimator of 0 , and in fact √ N(̂N − 0 ) satisfies the central limit theorem. This yields estimated parameter functions given bŷ The functions j can be chosen in a number of ways, including using a deterministic basis system such as polynomials, b-splines, or the Fourier basis, as well as using a functional principal component basis; see for example, Chapter 6 of Ramsay and Silverman (2006). Cerovecki et al. (2019) and Aue et al. (2017) suggest using the principal component basis determined by the squared processes X 2 i (t), which we also consider below. Given these function estimates, we can estimatê2 i (t) recursively, see (B4) in Appendix B for specific details.
To test the adequacy of the FGARCH(1,1) model, we utilize the fact that if the model is well specified then the sequence of model residualŝi(t), 1 ≤ i ≤ N, should be approximately i.i.d., wherê (3.3) This suggests that we consider the portmanteau statistics constructed from the residuals wherê, h is the sample autocorrelation of the scalar time series ‖̂1‖ 2 , … , ‖̂N‖ 2 , and exceeds the 1 − qth quantile of the distribution on the right-hand side of (2.6), where again this distribution must be estimated from the squared residualŝ2 i (t). We abbreviate these tests below as being based on V heuristic N,K, and M heuristic N,K, , since even under the assumption that that the model is correctly specified the residualŝi are evidently not i.i.d. due to their common dependence on the estimated parameterŝN.

Accounting for the Effect of Parameter Estimation
The approximate goodness-of-fit tests proposed above provide a heuristic method to evaluate the model fit of a specified functional GARCH type model, however we now aim at more precisely describing how the asymptotic distribution of M N,K, based on the model residualŝi(t) depends on the joint asymptotics of the innovation process and the estimated parameterŝN. In this subsection, we focus on quantifying this effect for the fully functional statistic M N,K, , since this statistic showed generally better finite sample performance relative to V N,K, in Section 4, and also because this statistic is more amenable to such an asymptotic analysis due to the fact that it is based directly on the norms of the autocovariance kernels. Furthermore, we assume that the parameter estimatêN is obtained by the least squares method proposed in Aue et al. (2017), although this could easily be adapted to the QMLE parameter estimate as well.
Towards this, we define the terms J 0 , H 0 , and Q 0 respectively as

Given the regularity conditions stated Appendix B, it follows that
where  p (0, Σ) denotes a p dimensional normal random vector with mean zero and covariance matrix Σ. We use the notation 2 i (t, ) and 2 i ( ) to indicate how each of these terms depends on the vector of parameters defined in (3.6) We further define the covariance kernels where 2 1 (i), i ≥ 1 are i.i.d. 2 random variables with one degree of freedom. The coefficients ( , ) i,K are the eigenvalues of a covariance operator Ψ ( , ) K , defined in (B1), that is constructed from kernels of the form s, u, v) (3.8) wileyonlinelibrary.com/journal/jtsa © 2020 The Authors. Theorem 3.1 precisely describes the asymptotics for M N,K, , which in this case depend jointly on the autocovariance of the FGARCH innovations as well as the parameter estimates. The specific definition of the covariance operator Ψ ( , ) K along with the necessary assumptions on the FGARCH model are detailed in Appendix B. These assumptions basically imply that (3.5) holds, and that the solution {X i } of the FGARCH equations exists in C[0, 1] with sufficient moments. The proof relies on a functional delta method for partial sums of random variables taking values in C[0, 1] and depending on a vector of parameters that might be of independent interest. These results may also be easily generalized to FGARCH models of other orders, for instance, the FARCH(1) model, which we study in the simulation section below and also detail in Appendix B.

IMPLEMENTATION OF THE TESTS AND A SIMULATION STUDY
This section gives details on implementation of the proposed tests and evaluates the performance of the proposed tests in finite samples. Several synthetic data examples are considered for this purpose. A simulation study on diagnostic checking for the FGARCH model is also provided in the last subsection.

Computation of Test Statistics and Asymptotic Critical Values
In practice we only observe each functional data object X i (t) at a discrete collection of time points. Often in financial applications these time points can be taken to be regularly spaced and represented as  J = {t j = j∕J, j = 1, … , J} ⊂ (0, 1]. Given the observations of the function X i (t j ), t j ∈  J , we can estimate, for example, the squared norm ‖X i ‖ 2 by a simple Riemann sum, Other norms arising in the definitions of V N,K and M N,K can be approximated similarly. For data observed at different frequencies, such as tick-by-tick, the norms and inner-products can be estimated with Riemann sums or alternate integration methods as the data allows. In all of the simulations below we generate functional observations on J = 50 equally spaced points in the interval [0, 1].
The critical values of the null limiting distribution of V N,K can easily be obtained, but estimating the limiting null distribution of M N,K defined in (2.6) requires a further elaboration. To achieve this one could implement a covariance block bootstrap approach as described in Zhang (2016) and Pilavakis et al. (2019), but for the sake of computational speed, and due to its satisfactory performance, we instead propose to estimate the limiting distribution directly. This could be done by estimating the eigenvalues of the kernel integral operator with kernel cov(X 2 i (t), X 2 i (s)) via estimates of the kernel, or alternatively using a Welch-Satterthwaite style approximation, the later of which we pursue; see for example, Zhang (2013). The basic idea of this method is to approximate the limiting distribution in (2.6) by a random variable R K ∼ 2 , where and are estimated so that the distribution of R K has the same first two moments as the limiting distribution on the right-hand side of (2.6). If M K denotes the random variable on the right-hand side of (2.6), K = E(M K ), and 2 K = var(M K ), then in order that the first two moments of R K match those of M K we take (4.1) We verify below that These can be consistently estimated bŷ Similarly, to estimate the asymptotic critical values of M N,K, under the FGARCH model adequacy described in Theorem 3.1, we obtain the parameters and of approximated distribution by estimating, We can consistently estimate these terms using estimators of the form, wherê( , ) K,h,g are consistent estimators of the kernels ( , ) K,h,g in (3.8), which we define in the last subsection of Appendix B.
Calculating and storing such kernels, which can be thought of as four-dimensional tensors, is computationally intractable if J is large, which is commonly the case when considering high-frequency financial data. For example, J = 390 when using 1-minute resolution US stock market data. To solve this problem, we use a Monte Carlo integration to calculate the integrals above based on a randomly sparsified sample, with the sparse J * points determined by drawing from a uniform distribution on [0, 1]. Below we use J * = 20 points to estimate these integrals, which seems to work well in practice.

Simulation Study of FGARCH Goodness-of-Fit Tests
We have conducted numerous simulation experiments with several different data generating processes (DGPs) to evaluate the finite sample performance of the functional conditional heteroscedasticity tests introduced in Section 2. The results of these showed generally that those tests performed well in terms of empirical size and power, and to shorten the exposition we have relegated the presentation of those results to an online supplement to this article.
We consider a simulation study of the proposed test statistics applied to diagnostic checking of FGARCH models as described in Section 3. In particular, we generate data from the following three DGPs: the FARCH(1), FARCH(2), and FGARCH(1,1). Specifically, with {W i (t), t ∈ [0, 1], i ∈ ℤ} denoting i.i.d. sequences of standard Brownian motions defined on the unit interval, we consider the DGPs = 0.01 (a constant function), and i (t) follows (4.4).
For each simulated sample we then test for the model adequacy of the FARCH(1) model. When the data follows the FARCH(1) specification, we expect the test to reject the adequacy of the FARCH(1) model at only the specified significance level, while we expect that the adequacy of the FARCH(1) model is rejected at a high rate for data generated according to the FARCH(2) and FGARCH(1,1) models. To estimate these models, we set L = 1 in (3.1). In practice the number L can be selected in several ways, including using the 'total variance explained' approach common in principal component analysis, see Cerovecki et al. (2019), or using the tests that we propose to evaluate whether a given choice of L suitably whitens the model residuals. We take L = 1 here since the kernels defining each FGARCH process are rank one, and to more easily study the effect of estimating the basis used to approximate these kernels.
Also, to investigate whether using the estimates of the principal components of the squared process affects the rejection rates, we adopt two types of basis functions:̂1(t) is derived from the empirical principal components, or the 'oracle' basis function is used. Using the 'oracle' function to reduce the dimension of the operators to be estimated is ideal in our setting since for the processes that we consider the operators defining them are rank one with a range spanned by 1 .
Panel A of Table I  test, the asymptotic M N,K, test exhibits improved size when K = 1 and 5 under  0 , and displays a slightly lower power under  A , and this was in accordance with our expectation given that the asymptotic results are sharper for the latter statistic.
Another observation worthy of a remark is that the rejection rates of the adequacy of the FARCH(1) model tend to be low for all DGPs when K = 1. This is evidently because fitting an FARCH(1) model effectively removes serial correlation from the squared process at lag one. Hence it is advisable when using this test for the purpose of model diagnostic checking to incorporate several lags beyond the order of the applied model.
The rejection rates of the adequacy of each model with the modification of using the 'oracle' function are displayed in Panel B of Table I, which shows that both the size and the power are in general improved for all tests. This suggests in part that the discrepancy in the empirical size for the M N,K, test for large N can be attributed to poor estimation of the dimension reduction basis. This also suggests that we can improve the estimation of the FGARCH models by changing the basis used for dimension reduction, although it is in general not clear how to improve on the FPCA method; doing so is beyond the scope of the current article, and is something we hope to investigate further in future work.

APPLICATION TO DENSE INTRA-DAY ASSET PRICE DATA
A natural example of functional financial time series data are those derived from densely recorded asset price data, such as intraday stock price data. Recently there has been a great deal of research focusing on analyzing the information contained in the curves constructed from such data. Price curves associated with popular companies are routinely displayed for public review. The objectives of this section are to (i) test whether functional financial time series derived from the dense intraday price data exhibit conditional heteroscedasticity, and (ii) evaluate the adequacy of FGARCH models for such series.
The specific data that we consider consists of 5 minute resolution closing prices of S&P 500 market index, so that there are J = 78 observations of the closing price each day. For the purpose of applying a Monte Carlo integration to the asymptotic diagnostic test M N,K, , we employ a sparse grid of J * = 39 out of the 78 points. Then, we let P i (t) denote the price of either asset on day i at intraday time t, where t is normalized to the unit interval. We consider time series of curves from these data of length N = 502 taken from the dates between 31/December/2015 to 02/January/2018. There are several ways to define curves that are approximately stationary based on the raw price curves P i (t). We consider the following two cases: 2. Overnight cumulative intra-day log returns (OCIDRs) Curves of the former type have been studied extensively in the literature, see for example Kokoszka and Reimherr (2013) and Kokoszka et al. (2014), as their trajectories encode the cumulative asset price changes over the course of the day. The OCIDR cures have exactly the same shape as the CIDR curves, but also capture the overnight price change. A similar overnight return has been used in Koopman et al. (2005). Figure 1 shows these two types of intra-day curves across seven days. We used 30 cubic B-spline functions to interpolate the raw price information and estimate functional principal components of the squared process, as implemented in the fda package in R, see Ramsay et al. (2009). We recomputed the analysis below for several different values of the number of splines, and the results reported below appeared to be stable to this choice. The stationarity of both return curves was examined by using the stationarity test proposed by Horváth et al. (2014). The results suggest that all intra-day return series are reasonably stationary.
We begin by testing for functional conditional heteroscedasticity in the functional time series of each curve type. The results of these tests are given in Table II, and are not particularly surprising in that they suggest that each sample of curves exhibit strong conditional heteroscedasticity.
A natural next step is to posit and evaluate models to capture this conditional heteroscedasticity. For this we consider two models: standard scalar GARCH models and FGARCH models. The motivation for considering standard scalar GARCH models for this purpose is that we might at first expect that the volatility in each of these curves can be adequately accounted for by scaling each curve by the conditional standard deviation estimated by a scalar GARCH model fitted to the end-of-day returns, that is, a large magnitude of the return on the previous day spells high volatility for the entire intraday price on the following day. We compute the daily log returns as x i = log(P i (1))−log(P i−1 (1)), 1 ≤ i ≤ 500, to which we fit a scalar GARCH(p,q) model by using a quasi maximum likelihood estimation approach. The orders {p, q} are selected as the minimum orders for which the estimated residualŝi = x i ∕̂i are plausibly a strong white noise as measured by the Li-Mak test; see Li and Mak (1994), resulting in the selection of a GARCH(1,1) model, as shown in Panel A in Table III.
We then apply the proposed tests for conditional heteroscedasticity to the fitted residuals functions of intra-day returns̃i (t) = X i (t)∕̂i.  Table III. Heteroscedasticity tests of de-volatized daily return̂i and intra-day curves̃i(t) from an optimal selected GARCH(p,q) model, using Li-Mak test for scalar observations and V N,K and M N,K for functional observations respectively  The results of these tests are given in Panel B in Table III, which show that these curves still exhibit a substantial amount of conditional heteroscedasticity. Next, we consider the FARCH(1) and FGARCH(1,1) models for these curves. We fit each model with L = 1 in (3.1) to be consistent with the methods studied in the simulation section, and evaluate the adequacy of each model as proposed above. Figure 2 shows plots of w(t) and wire-frame plots of the kernels (t, s) and (t, s) for the FGARCH(1,1) model for both type of intra-day return curves. We then estimatêi(t) recursively with the initial values ofŵ(t), and the de-volatized intra-day return̂i(t) is fitted per 3.3. Figure 3 exhibits the de-volatized intra-day returns over seven days by using the FGARCH(1,1) model. Table IV reports the P-values from the diagnostic checks of the FGARCH(1,1) and FARCH(1) models applied to the de-volatized intra-day returns. All of the three diagnostic tests show consistent results at the specified significance levels. The FARCH(1) model is generally deemed to be inadequate for both types, although this model performs as we expected to adequately model conditional heteroscedasticity at lag 1. By contrast, the P values in Panel B of Table IV suggest that the FGARCH(1,1) model is acceptable for modeling the conditional heteroscedasticity of both curve types. Figure 4 shows the P values of these tests as a function of K in the case of the OCIDR curves (the results for the CIDR curves were similar), which show that this conclusion is apparently independent of the choice of K.
In conjunction with the above results showing that these curves cannot be adequately de-volatized simply by scaling with the conditional standard deviation estimates from GARCH models for the scalar returns, we draw the following tentative conclusions from this analysis: (i) the magnitude of the return cannot fully explain the volatility of intraday prices observed on subsequent days; instead we should consider the entire path of the price curve on previous days to adequately model future intra-day conditional heteroscedasticity, and (ii) the FGARCH class of models seems to be effective for modeling intra-day conditional heteroscedasticity.

CONCLUSION
We proposed two portmanteau-type conditional heteroscedasticity tests for functional time series. By applying the test statistics to model residuals from the fitted functional GARCH models, our tests also provide two heuristic and one asymptotically valid goodness-of-fit test for such models. Simulation results presented in this article show that both tests have good size and power to detect conditional heteroscedasticity in functional financial time series and assess the goodness-of-fit of the FGARCH models in finite samples. In an application to the dense intraday price 748 G. RICE ET AL.  data, we investigated the conditional heteroscedasticity of three types of the intra-day return curves, including the overnight cumulative intra-day returns, the cumulative intra-day returns and the intra-day log returns from two assets. Our results suggested that these curves exhibit substantial evidence of conditional heteroscedasticity that cannot be accounted for simply by rescaling the curves by using measurements of the conditional standard deviation based on the magnitude of the scalar returns. However, the functional conditional volatility models often appeared to be adequate for modeling this observed functional conditional heteroscedasticity in financial data.
Chapter 6.2 of Riesz and Nagy (1990), that Ψ K defines a nonnegative and decreasing sequence of eigenvalues and a corresponding orthonormal basis of eigenfunctions i,K (t, s), 1 ≤ i < ∞, satisfying With this notation, we now defineΓ N,  s) is a Gaussian process with covariance operator Ψ K , and ( 1 ) → denotes weak convergence in  1 . It now follows from the Karhunen-Loéve representation and continuous mapping theorem that A simple calculation based on (A1) shows that the eigenvalues of Ψ K are products of the eigenvalues defined by (2.4), { i j , 1 ≤ i, j < ∞}, with each eigenvalue having multiplicity K, giving the form of the limit distribution in (2.6).
Justification of (4.2). Using proposition 5.10.16 of Bogachev (1998), we have that and var(‖Γ K ‖ 2 ,1 ) = 2 tr(Ψ 2 K ) = 2K ( ∫ ∫ cov(X 2 0 (t), X 2 0 (s))dtds Proof of Theorem 2.2. We only show (2.7) as (2.8) follows similarly from it. Let C h (t, s) = cov(X 2 i (t), X 2 i+h (s)) ≠ 0. It follows from the assumed L 8 -m-approximability of X i that X 2 i is L 4 -m-approximable, from which we can show that ‖̂h(t, , and N‖C h ‖ 2 diverges to positive infinity at rate N, yielding the desired result. Proof of Theorem 2.3. Again we only prove (2.10) as (2.9) follows from it by a similar argument. By squaring both sides of of (2.1) and iterating (2.2), we obtain that where the series on the right-hand side of the above equation converges in L 2 [0, 1] with probability one, and (t) = ∞ Therefore, X 2 i (t) is a linear process in L 2 [0, 1] with mean (t) generated by the weak functional white noise innovations Y i as defined in Bosq (2000). It now follows from Assumption 2.1 and the ergodic theorem that It follows from this and the reverse triangle inequality that We first develop some notation and detail the assumptions that we use to establish Theorem 3.1. Recall from (3.2) that the FGARCH equations along with (3.1) imply that 2 and the coefficient matrices A and B are ℝ L×L with (j, j ′ ) entries by a j,j ′ and b j,j ′ respectively. Let Γ 0 (t, s) = (t, s) 2 0 (s) + (t, s). We make the following assumptions:  Aue et al. (2017), and imply both that there exists a strictly stationary and causal solution to the FGARCH equations in C[0, 1], and that̂N is a strongly consistent estimator of 0 that also satisfies the central limit theorem. Assumption B.5 is a somewhat stronger assumption than those of Theorem 3.2 of Aue et al (2017). It is used in the proofs below mainly to establish uniform integrability of terms of the form ‖X i ∕ i ‖ ∞ . Assumption B.6 is implied by the conditions of Cerovecki et al. (2019) that the functions i are strictly positive and that D ∈ Θ D ⊂ (0, ∞) L , where Θ D is compact, but also may hold under more general conditions. where Γ , is a mean zero Gaussian process in  1 with covariance operators Ψ ( , ) K defined by s, u, v) is a matrix valued kernel defined by (3.8). In addition, where i,K i ≥ 1 are the eigenvalues of Ψ ( , ) K .
Combining these results with (B12), we see that where ‖R 4,N ‖ = o P (1). We note that ( 2 i (t) − 1)( 2 i+h (s) − 1), depends solely on the error process: in particular it is √ N times the estimated autocovariance of the squared error processes that was considered in Appendix A. Let We now aim at establishing the weak limit of Γ ( ) N,K + Γ ( ) N,K in  1 . Γ ( ) N,K is tight in  1 as was established in Appendix A, and Γ ( ) N,K is tight in  1 since ℝ L+2L 2 is sigma-compact, hence Γ ( ) N,K + Γ ( ) N,K is tight in  1 . According to the proof of Theorem 4 on p. 19 of Aue et al. (2017), in particular their equations (5.15)-(5.22), we have under Assumptions B.1-B.6 that Therefore if z ∈  1 , where ⟨G h , z (h) ⟩ * is used to denote that the inner-product is carried out coordinate-wise, so that ⟨G h , z (h) ⟩ * ∈ ℝ L+2L 2 . Noting that 1) 2 i is  i−1 measurable, and 2) E[x i − i ( 0 )| i−1 ] = 0, we see that i (z) form a martingale difference