New Approaches to Robust Inference on Market (Non-)Efficiency, Volatility Clustering and Nonlinear Dependence
NNew Approaches to Robust Inference on Market (Non-)Efficiency,Volatility Clustering and Nonlinear Dependence Rustam Ibragimov
Imperial College Business School and St. Petersburg State University
Rasmus Søndergaard Pedersen
Department of Economics, the University of Copenhagen
Anton Skrobotov
Russian Presidential Academy of National Economy and Public Administration (RANEPA)and St. Petersburg State University
Abstract
Many key variables in finance, economics and risk management, including financial returns and foreignexchange rates, exhibit nonlinear dependence, heterogeneity and heavy-tailedness of some usually largelyunknown type.The presence of non-linear dependence (usually modelled using GARCH-type dynamics) and heavy-tailednessmay make problematic the analysis of (non-)efficiency, volatility clustering and predictive regressions in eco-nomic and financial markets using traditional approaches that appeal to asymptotic normality of sampleautocorrelation functions (ACFs) of returns and their squares.The paper presents several new approaches to deal with the above problems. We provide the results thatmotivate the use of measures of market (non-)efficiency, volatility clustering and nonlinear dependence basedon (small) powers of absolute returns and their signed versions. The paper provides asymptotic theory forsample analogues of the above measures in the case of general time series, including GARCH-type processes.It further develops new approaches to robust inference on them in the case of general GARCH-type processesexhibiting heavy-tailedness properties typical for real-world financial markets. The approaches are based onrobust inference methods exploiting conservativeness properties of t − statistics (Ibragimov and M¨uller, 2010,2016) and several new results on their applicability in the settings considered. In the approaches, estimatesof parameters of interest (e.g., measures of nonlinear dependence given by sample autocorrelations of powersof the returns’ absolute values) are computed for groups of data and the inference is based on t -statisticsin resulting group estimates. This results in valid robust inference under a wide range of heterogeneityand dependence assumptions satisfied in financial and economic markets. Numerical results and empiricalapplications confirm advantages of the new approaches over existing ones and their wide applicability inthe study of market (non-)efficiency, volatility clustering, nonlinear dependence, and other areas. The authors thank W. Distaso, J.-M. Dufour, L. Giraitis, P. Kattuman, A. Min, T. Mikosch, U. K. M¨uller, A.Prokhorov, A. Rahbek, L. Trapani and Y. Zu for helpful comments and suggestions. Research of R. Ibragimov andA. Skrobotov was supported in part by a grant from the Russian Foundation for Basic Research (RFBR, Project No.20-010-00960). R. S. Pedersen gratefully acknowledges financial support from the Carlsberg Foundation (CF-0909,”Robust methods for volatility modelling”). Pedersen is a research fellow at the Danish Finance Institute. a r X i v : . [ ec on . E M ] J un EL Classification:
C12, C22, C46, C51
Key words and phrases: robust inference, t − test, autocorrelations, financial markets, stylized facts, effi-ciency, volatility clustering, nonlinear dependence, GARCH Many studies have confirmed that financial returns time series exhibit several common statisticalproperties, often referred to as stylized facts (see Ch. 2 in Campbell et al. (1997), Chs. 1-3 inTsay (1997), Cont (2001), Chs. 1-2 in Taylor (2008), Ch. 1 in Christoffersen (2012), Ch. 3 inMcNeil et al. (2015), and references therein). The following are three most important stylized factsof financial returns time series, ( R t ), that much of the empirical literature agrees upon, togetherwith the standard mean-zero property ( E ( R t ) = 0) that implies the absence of systematic gains orlosses:(i) ear dependence and linear autocorrelations that provides the support for the weak efficientmarket hypothesis , that is, for the martingale difference property of financial returns: Corr ( R t , R t − h ) ≈ , (1.1)even for small lags h = 1 , , ..., (ii) The presence of nonlinear dependence and volatility clustering, captured by significant positiveautocorrelation in simple nonlinear functions of the returns and different measures of volatility,such as squared returns: Corr ( R t , R t − h ) >> , (1.2)even for large lags h > . This property implies, in particular, that financial returns are noti.i.d. and thus the strong market efficiency hypothesis does not hold.(iii) Heavy tails: The (unconditional) returns distributions are non-normal and exhibit power-lawor Pareto-like tails, P ( | R t | > x ) ∼ C/x ζ , (1.3)2or large positive x (cid:48) s, with C > ζ > The analysis and modelling of properties (i)-(ii) has been central to the development of modernfinancial theory and financial econometrics, including the development of market efficiency hypothe-ses by E. Fama and (G)ARCH time series by R. Engle and C. Granger - the contributions recognisedwith the Nobel Memorial Prize in Economic Sciences. E.g., (G)ARCH time series have been intro-duced to the literature to model the absence of linear autocorrelations and the presence of volatilityclustering - stylized facts (i)-(ii) - that they capture by the very definition (see, among others, thereviews in Ch. 12 in Campbell et al. (1997), Ch. 4 in Christoffersen (2012) and Ch. 4 in McNeilet al. (2015)).At the same time, much of the literature in finance, economics, econometrics, risk management andrelated fields has also focused on the analysis and modelling of heavy tails property (1.3) for keyfinancial and economic variables, including financial returns (see the reviews in Embrechts et al.(1997), Gabaix (2009), Ibragimov et al. (2015), and references therein). This interest is motivatedby the fact that heavy-tailed power laws (1.3) produce, as is confirmed by many empirical studies,a good fit to the distribution of financial returns and other important variables in finance andeconomics. Heavy-tailed distributions provide a convenient framework for modelling and quantifying (by theirkey parameter - the tail index ζ ) the likelihood of large downfalls, large fluctuations and crises infinancial and economic markets. In (1.3), the smaller values of the tail index parameter ζ correspondto a larger likelihood of crises, large downfalls and large fluctuations affecting the financial returnstime series ( R t ) , and vice versa (see the discussion in Ch. 1 in Ibragimov et al. (2015)). The tailindex parameter ζ is further important as it governs existence of moments of R t , with, for instance,the variance of R t being defined and finite: V ar ( R t ) < ∞ if and only if ζ > , and, more generally,the p th moment E | R t | p , p > , being finite: E | R t | p < ∞ if and only if ζ > p. The most ofthe empirical literature on heavy-tailed distributions agrees that, in the case of developed financialmarkets, the returns’ tail indices ζ belong to the interval (2 , , thus implying finite variances andinfinite fourth moments (op. cit.). Gabaix et al. (2003, 2006) review the empirical results that Following the standard definition and notation, for two positive functions f, g , f ( x ) ∼ g ( x ), if f ( x ) /g ( x ) → x → ∞ . The above properties and other stylized facts have been established for different frequencies, including, e.g.,weekly, monthly and high-frequency returns (see Cont (2001)); in this paper, we focus on daily returns for simplicityof presentation. See also, among others, the examples in Ch. 2 in Stock and Watson (2015) and the introduction in Ibragimovet al. (2015) that illustrate inappropriateness of Gaussian distributions as models for financial returns based on theirbehavior during the Black Monday crisis. Naturally, therefore, the standard OLS regression methods and autocorrelation-based time series analysis methodsare in principle inapplicable directly and need to be modified in the case of heavy-tailed time series with tail indices ζ smaller than two and infinite (or undefined) variances. Similar estimates are also obtained for developed country foreign exchange rates (see Ibragimov et al. (2013)). The property that the financial returns’ tail indices are smaller than 4 and their fourth moments are infiniteimplies that the use of the common measure of heavy-tailedness, the kurtosis, is inappropriate: E.g., under ζ ∈ (2 , , its estimate given by the sample kurtosis diverges to infinity in probability as the sample size grows. Thus, the sample It is important that, in addition to properties (i)-(ii), GARCH models also capture the heavy tailsstylized fact - property (iii) - as any stationary solution to a GARCH equation, even with thin-tailed- normal - innovations, has power-law tails (1.3) with the tail index ζ that depends on the parameters(and, in the general case, on the distribution of the innovations) in the GARCH model via Kesten’sequation (see Davis and Mikosch (1998); Mikosch and St˘aric˘a (2000) and Section 3). The standard widely used approach to testing properties (i)-(ii) relies on full-sample estimates (sam-ple autocorrelations) of population autocorrelation coefficients of the returns and their squares andappealing to the central limit theorem for them (see, e.g., Ch. 2 in Campbell et al. (1997) and Chs.1 and 4 in Christoffersen (2012)).However, as, typically, the tail index ζ <
Corr ( R t , R t − h ) of squared returns are not even defined,making condition (1.2) meaningless. Their sample analogues, the sample autocorrelations of R t , arenot consistent, implying random fluctuations even in large samples. Motivated by the analysis of long memory in financial returns and, in part, by the above problemswith undefined autocorrelations
Corr ( R t , R t − h ) of squared returns, several works in financial econo-metrics have focused on defining and measuring nonlinear dependence and volatility clustering in kurtosis of financial returns is expected to take on increasingly larger values as the sample size increases. In the model, heavy-tailedness of financial returns is implied by trading actions of largest market participants(mutual funds and other institutional investors) who have a size distribution with tail indices ζ = 1 (Zipfs law). The empirical results in Ibragimov et al. (2013); Gu and Ibragimov (2018); Chen and Ibragimov (2019) implythat the “Cubic Law of the Stock Returns” does not hold in the case of emerging country financial returns and foreignexchange rates as most of them have tail indices smaller than 3, and even tail indices smaller than 2 and infinitevariances are not uncommon. This implies that the model in Gabaix et al. (2003, 2006) may need to be modifiedin the case of emerging and developing markets, e.g., with possible deviations of the distribution of sizes of marketparticipants from Zipf’s law due to the governments’ regulatory interventions. Conceptually, this property is important as it implies that crises in the financial markets arise even in absence oflarge external shocks (that is, in absence of heavy-tailedness in the distribution of GARCH innovations) and thus canbe generated by the returns’ nonlinear dependence alone, that is, so to speak, within the system - by the interactionof market participants (e.g., by trades of large market participants - mutual funds and other institutional investors -followed by the rest of the market, see Gabaix et al. (2003, 2006), and by the investors’ herding behavior, see Contand Bouchaud (2000)) that leads to volatility clustering in the financial returns’ time series. Similarly, the population linear autocorrelations
Corr ( R t , R t − h ) are not defined in the case of tail indices ζ smallerthan 2: ζ < Corr ( | R t | p , | R t − h | p ) , p > , with most of the studies and empirical applications focusing on the case p = 1, corresponding to thereturns’ absolute values (see Ding et al. (1993); Ding and Granger (1996), the review and discussionin Section 5.2 in Cont (2001) and Chs. 1-2 in Taylor (2008), and references therein). For instance,Ding et al. (1993); Granger and Ding (1995); Ding and Granger (1996), and Cont (2001) indicatethat, for a given lag h, the autocorrelation Corr ( | R t | p , | R t − h | p ) appears to be maximised for p = 1thus implying that absolute returns tend to be more predictable than other powers of returns. Importantly, to our knowledge, the analysis of autocorrelations of powers of absolute returns inthe literature did not rely on econometrically justified or robust inference accounting for nonlineardependence and heavy-tailedness, which is the main focus in this paper.In view of undefined autocorrelations
Corr ( R t , R t − h ) and inconsistency of their sample analoguesin the case of financial returns time series ( R t ) with infinite fourth moments as in the real-worldfinancial markets, the above contributions make it natural to consider the modification of volatilityclustering stylized fact (ii) in terms of autocorrelations of absolute returns:(ii’) Corr ( | R t | , | R t − h | ) >> , even for large lags h > . Unfortunately, this does not resolve the problems with inference on the stylized facts of financialmarkets. The problem is that, in the case of GARCH models for financial returns time series ( R t ) , asymptotic normality of sample linear autocorrelations of R t and the absolute values | R t | holds onlyin the case of tail indices ζ greater than 4: ζ > In contrast, in the case of GARCH models ( R t ) with tail indices 2 < ζ <
4, as is typical for financialreturns in developed markets, the sample autocovariances and autocorrelations of R t and | R t | , albeitconsistent, converge in distribution to non-normal limits given by stable random variables (r.v.’s)or their ratios (op. cit.). The non-Gaussian stable limits are practically intractable and depend onthe unknown stability index determined by the unknown tail index value ζ as well as on locationand dispersion parameters. In addition, the rate of convergence of the sample autocovariances andautocorrelations depends on the unknown tail index value ζ and is slower than √ T , where T denotesthe sample length. Hence, the true confidence bands for the autocovariances and autocorrelationsof R t and | R t | are wider than those implied by the central limit theorem, and, importantly, itis essentially impossible to use these weak convergence results for testing for the stylized facts of Following Granger and Ding (1995), this property is often referred to as ’Taylor effect’; it strengthens the propertythat the autocorrelations of absolute returns tend to be larger than those of squared returns (see Chs. 1-2 in Taylor(2008)). The intuition for this is provided, for instance, by the fact that the variance
V ar ( R t R t − ) of the summands R t R t − that appears in sample first-order autocovariances and autocorrelations of a returns time series ( R t ) thatfollows GARCH(1, 1) process (3.4)-(3.5) discussed in the next section is given by V ar ( R t R t − ) = E ( R t R t − ) = E ( σ t σ t − ) = E (( ω + αR t − + βσ t − ) σ t − ) = ωE ( σ t − ) + ( α + β ) E ( σ t − ) = ωV ar ( R t ) + ( α + β ) E ( R t ) /E ( z t ) and isthus finite if and only if E ( R t ) < ∞ . See Davis and Mikosch (1998); Mikosch and St˘aric˘a (2000), the discussion in Section 5.3 in Cont (2001), andSection 3 in this paper).
The above conclusions emphasise the necessity in applications of econometrically justified definitionsand measures of market (non-)efficiency, nonlinear dependence and volatility clustering - analoguesof definitions and measures in (i), (ii) and (ii’) - that can be used in the case of GARCH-type modelswith the parameters and the implied volatility persistence and heavy-tailedness properties that aretypical for financial markets, e.g., for GARCH processes with tail indices ζ < The results further indicate that the standard conclusions in finance, economics and financial econo-metrics on absence of linear autocorrelations - stylized fact (i) - and the presence of nonlineardependence and volatility clustering in financial returns time series - stylized fact (ii) - may need tobe re-examined. They also emphasise the necessity of using econometrically justified and robust ap-proaches to the analysis of these and other statistical properties of real-world financial markets.The remainder of the paper is organized as follows. In Section 2 we discuss the contributions andthe main results of the paper. In Section 3 we introduce new measures for serial dependence andnon-linearity in time series, and propose a robust approach for carrying out reliable inference onsuch measures. In Section 4 we investigate the finite-sample properties of our proposed methods,and in Section 5 we provide an empirical illustration where we demonstrate the applicability of ourmethods in relation to testing for non-efficiency and non-linearity in returns on major stock marketindexes. Section 6 concludes the paper, and the Appendix contains additional technical details andnumerical results.
The analysis in this paper provides a support for the use of autocovariances
Cov ( | R t | p , | R t − h | p ) ,p > , of small order powers p of financial returns (with p < . R t ) with thetail index ζ > p < .
25 forthe returns with the tail index ζ > Similar conclusions hold for the weak convergence of sample autocovariances and autocorrelations of squares R t of GARCH time series ( R t ). Their asymptotic normality requires ζ > E | R t | < ∞ . In the case 4 < ζ < , the sample autocovariances and autocorrelations of R t are consistent but converge to non-normal limits given bystable r.v.’s or their ratios, with the rate of convergence that is slower than √ T .
Importantly, the stable limits andthe rate of convergence depend on the unknown tail index value ζ. Similar to the case of linear autocorrelations
Corr ( R t , R t − h ) , the autocorrelations Corr ( | R t | , | R t − h | ) are notdefined and definition (ii’) becomes meaningless for GARCH processes ( R t ) with tail indices ζ < Heavy-tailedness properties of emerging and developing markets further motivate the analysis in the case ofGARCH-type processes with even smaller tail indices, including the case ζ <
Corr ( | R t | p , | R t − h | p ) >> , even for large lags h > . In the general case of - for instance - GARCH-type processes with the tail index ζ, the results inthis paper justify the use of the above autocovariance measures with p < ζ/ . We further propose - to our knowledge, for the first time in the literature - the autocorrelations
Corr ( R t , | R t − h | s sign ( R t − h )) , s > , of ‘signed’ powers of absolute returns as measures of market(non-)efficiency in analogues of property (i) appropriate for the analysis of real-world financial mar-kets. These measures lead to formulation of natural analogues of stylized fact (i) that can be usedin the case of financial returns time series exhibiting heavy-tailedness and GARCH-type nonlineardependence as in the case of real-world financial markets data. These analogues of property (i) havethe form(i’) Corr ( R t , | R t − h | s sign ( R t − h )) ≈ , even for small lags h = 1 , , ... (property (i’) coincides with uncorrelatedness property (i) in the case s = 1).In (i’), as in the case of nonlinear dependence and volatility clustering measures in (ii”), the choiceof small powers s is justified under empirically observed heavy-tailedness in financial markets, withthe appropriate choice of power values s being s < ζ/ − ζ ∈ (2 ,
4) as is typical for real-world financial markets. E.g., in the case ζ = 3 as in the “CubicLaw of the Stock Returns” in Gabaix et al. (2003, 2006) (see Section 1.1), the appropriate choicesof powers are p < .
75 in property (ii”) and s < . t − statistics (Ibragimov and M¨uller, 2010, 2016) and several new re-sults on their applicability in the settings considered. In the approaches, estimates of parameters ofinterest (e.g., measures of volatility clustering and nonlinear dependence given by sample autocorre-lations of powers of the returns’ absolute values) are computed for groups of data and the inferenceis based on t -statistics in resulting group estimates (op. cit.; see also Section 3.3 in Ibragimov et al.(2015) and the review in Appendix A in this paper). This results in valid robust inference underheavy-tailedness and GARCH-type nonlinear dependence properties that are typically observed infinancial and economic markets. Numerical results and empirical applications confirm advantagesof the new approaches over existing ones and their wide applicability in the study of market (non-7efficiency, volatility clustering, nonlinear dependence, and other areas.The results in the paper emphasise that the appropriate choice of measures of market (non-)efficiencyand volatility clustering (and asymptotic validity of inference on them) depends on the degree ofnonlinear dependence (and volatility persistence) in GARCH-type models and the implied degreeof their heavy-tailedness, as measured by the tail index ζ. In particular, the focus on small powersof absolute returns in the measures used in the analysis of (non-)efficiency, nonlinear dependenceand volatility clustering is in contrast with the previous studies on the topic that mostly dealt withthe traditional choice p = 2 corresponding to autocorrelations of squared returns and p = 1 thatcorresponds to linear autocorrelations of returns and their absolute values. Most of the studies thatused other powers p of absolute returns in the analysis have dealt with the case of relatively largervalues equal to or around 1, and their comparisons with the case p = 2 , with some of empiricalresults presented for the case p = 0 . , . , . Importantly, as indicatedin the introduction, the previous studies on the topic did not rely on econometrically justified orrobust inference, which is the main focus in this paper.The results in Ibragimov and M¨uller (2010, 2016) (see also Section 3.3 in Ibragimov et al. (2015)and the review in Appendix A in this paper) imply that the t − statistic robust approaches resultin asymptotically valid inference under the assumption that the group estimators are asymptoti-cally normal and asymptotically independent. More generally (op. cit.), asymptotic validity of the t − statistic robust inference approaches holds in the case of scale mixture of normals, e.g., symmet-ric stable, limits of group estimators of the parameter of interest. As discussed in Appendix A,asymptotic normality of group estimators or their weak convergence to scale mixture of normalstypically follows from the same arguments and holds under the same conditions as in the case ofcorresponding full-sample estimators. The condition of asymptotic independence is a condition onthe degree of (weak) dependence in the data (e.g., returns’ time series) dealt with; typically, it isweaker than the conditions required for consistent estimation of limiting variances of (asymptoticallynormal) full-sample estimators using, e.g., heteroskedasticity and autocorrelation consistent (HAC)methods (op. cit.).The results in Pedersen (2020) justify applicability of t − statistic approaches in robust tests of un-correlatedness - property (i) - for stationary general GARCH (GGARCH) processes with symmetric innovations and tail indices ζ ≥ ζ ∈ (2 ,
4) that typical for financial returnsin developed markets). The arguments in Pedersen (2020) are based on the established asymptotic Many studies in finance and financial econometrics rely on the assumption of finite fourth or higher momentsfor the returns, including the studies that use heteroskedasticity and autocorrelation consistent inference approachesrequiring high-order moments of time series in consideration to be finite (see Remark 3.3 and the discussion inAppendix A). In addition, some of studies on stylized facts in financial and other markets have also dealt with verylarge powers of the returns in analogues of properties (i) and (ii), including, for instance, the analysis of efficiency inthe bitcoin market using the autocorrelations
Corr ( R pt , R pt − h ) of very large odd powers p of bitcoin returns equal toor even larger than p = 17 . The latter requires the assumption that the moments of the returns of order 34 are finite: E ( R t ) < ∞ (see Nadaraja and Chu (2017)). Such assumptions are problematic for real-world markets due to theirheavy-tailedness properties. ζ ≥ ζ ∈ (2 , . This paper provides new results that justify applicability of t − statistic robust inference approaches inthe analysis of properties (i’) and (ii”) and their implications for market (non-)efficiency, volatilityclustering and nonlinear dependence. The results do not impose symmetry of the processes inconsideration. Justification of applicability of t − statistic robust inference approaches in the contextrelies on asymptotic normality of group estimators of autocovariances of powers of absolute returnsand their signed versions in (i’) and (ii”) and new results on their asymptotic independence undervery general assumptions.One of the paper’s methodological contributions consists in the use of coupling methods to establishasymptotic independence, under general conditions, of group estimators of the proposed measuresof market (non-)efficiency which is the main difficulty in justification of applicability of robust t − statistic inference approaches in the context. Coupling methods may prove to be useful in furtherapplications of robust t − statistic approaches in other problems of econometrically justified infer-ence for GARCH-type processes and financial markets exhibiting nonlinear dependence, volatilityclustering and heavy-tailedness. The results presented in this paper hold for a general class of strictly stationary time series processes,that satisfy certain mixing and moment conditions stated below. In order to illustrate the main ideas,we consider an ongoing example given by the much celebrated GARCH(1, 1) process.Let Z = { ..., − , − , , , , ... } . Consider a GARCH(1, 1) process ( R t ) t ∈ Z , defined, given nonnegativeparameters ω, α, β, by R t = σ t Z t , t ∈ Z , (3.4)where ( Z t ) t ∈ Z is sequence of i.i.d . r.v.’s with mean zero and unit variance: E ( Z t ) = 0 , V ar ( Z t ) = 1 , σ t ) t ∈ Z is a volatility process σ t = ω + αR t − + βσ t − . (3.5)Throughout the paper, Z denotes a r.v. that has the same distribution as that of the r.v.’s ( Z t ) . As is well-known (see, among others, Nelson (1990); Davis and Mikosch (2000, 2009 a ); Mikoschand St˘aric˘a (2000), the volatility process ( σ t ) of a GARCH(1, 1) process (3.4)-(3.5) has a strictlystationary and ergodic version if and only if ω > E [log( αZ + β )] < . Also, in this case,stationarity of ( σ t ) implies stationarity of the GARCH(1, 1) process ( R t ) .
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In addition, under mildconditions on the distribution of Z t , e.g. if the distribution has a Lebesgue density, then the GARCHprocess is β -mixing with geometric rate (see e.g. Francq and Zakoan, 2006, Theorem 3). This inturn ensures that the process is also strongly, or α -, mixing with exponential decay, which (undersuitable conditions) enables us to apply a central limit theorem (CLT) to general transformationsof R t . We refer to Appendix C for additional details about α - and β -mixing processes.Under, essentially, the conditions listed above, the strictly stationary solution to (3.4)-(3.5) sat-isfies Kesten’s theorem (see Davis and Mikosch (1998); Mikosch and St˘aric˘a (2000); Davis andMikosch (2000, 2009 a ) and references therein). Specifically, the invariant distribution of stationaryGARCH(1, 1) process ( R t ) has power law tails (1.3) with tail index ζ > E [( αZ + β ) ζ/ ] = 1 . (3.6) Example 3.1
Let ( R t ) t ∈ Z be a striclty stationary GARCH(1, 1) process (3.4)-(3.5) with the tailindex ζ in (3.6), and let κ Z = E ( Z ) < ∞ denote the kurtosis of Z . From Kesten’s equation (3.6) itfollows that the tail index ζ of R t in is greater than 2, ζ > , and thus the (unconditional) varianceof R t is finite, V ar ( R t ) < ∞ , if and only if α + β < . Further, ζ < , and thus E ( R t ) < ∞ , ifand only if α κ Z + 2 αβ + β > , that is, if and only if ( α + β ) > − ( κ Z − α . Hence, the tailindex ζ ∈ (2 , , as in the case of developed markets (see the discussion in Section 1.1), if and onlyif − ( κ Z − α < ( α + β ) < . In the case of the commonly used standard normal innovations In the case of a GARCH( p , q ) model (3.4) with the volatility process σ t = ω + (cid:80) pi =1 α i R t − i + (cid:80) qj =1 β j σ t − j , where ω, α i , β j ≥ , sufficient conditions for existence of a stationary solution ( R t ) , σ t are given by ω > , (cid:80) pi =1 α i + (cid:80) qj =1 β j ≤ , certain restrictions on the distribution of Z, and some further technical conditions (see Bougerol andPicard (1992), the discussion in Davis and Mikosch (2000) and references therein). The condition (cid:80) pi =1 α i + (cid:80) qj =1 β j < V ar ( R t ) < ∞ , and thus for the second order stationarity ofGARCH( p , q ) process ( R t ) . In the case of ARCH(1) process with β = 0 and the standard normal Z, the condition E [log( α Z )] < < α < e γ ≈ . , where γ is Euler’s constant. These conditions illustrate the interplay between the degree of volatility persistence in GARCH(1, 1) model(3.4)-(3.5), as measured by the persistence parameter ρ = α + β, and that of heavy-tailedness in innovations ( Z t ) , asmeasured by their kurtosis k Z , in generating heavy tails of the process ( R t ) ∼ N (0 , , the conditions for ζ ∈ (2 , are − α < ( α + β ) < .
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Remark 3.1
We emphasize that the results in the paper hold for rather wide classes of time se-ries processes. In addition to the classic GARCH(1,1) process, the results hold for heavy-tailedGARCH( p, q ) time series (see, among others, the definition and discussion of properties of GARCH( p, q )processes in Davis and Mikosch (2009 a )), generalized GARCH (GGARCH) processes (e.g., Pedersen(2020)), and stochastic volatility processes (e.g., Davis and Mikosch (2001)), among others. Given a strictly stationary process ( R t ) t ∈ Z , we consider the following population autocovariance andautocorrelation functions of order h for measuring non-linear dependence in the process.For p > E [ | R t | p ] < ∞ , let γ | R | p ( h ) = Cov ( | R t | p , | R t − h | p ) , h = 0 , , . . . , (3.7) ρ | R | p ( h ) = Corr ( | R t | p , | R t − h | p ) = γ | R | p ( h ) γ | R | p (0) , h = 1 , . . . (3.8)The above measures are non-zero if the process ( R t ) t ∈ Z is conditionally heteroskedastic, i.e. if itexhibits volatility clustering. In order to measure the degree of efficiency, i.e. if R t is predictable withrespect to its lagged values, we define the following quantities. For s > E [ | R t | s ] < ∞ , γ (cid:48) R, | R | s sign ( R ) ( h ) = Cov ( R t , | R t − h | s sign ( R t − h )) , h = 0 , , . . . , (3.9)and for max { E [ | R t | s ] , E [ | R t | ] } < ∞ , ρ (cid:48) R, | R | s sign ( R ) ( h ) = Corr ( R t , | R t − h | s sign ( R t − h )) = γ (cid:48) R, | R | s sign ( R ) ( h ) (cid:112) γ R (0) γ | R | s sign ( R ) (0) , h = 0 , . . . . (3.10) Example 3.2
In the case where ( R t ) t ∈ Z is a GARCH(1,1) process with tail index ζ > , we havethat the quantities in (3.7) and (3.8) are defined if ζ > p . For instance, in order to define the ACFfor the squared returns ( p = 2 ), we need that ζ > . Likewise, the auto(cross)covariance in (3.9) isdefined if ζ > s , and the auto(cross)correlation in (3.10) is defined if ζ > { , s } . Remark 3.2
For s = 1 , we note that (3.9) and (3.10) are identical to the usual autocovariancesand autocorrelations, respectively. For s (cid:54) = 1 , (3.9) and (3.10) , are still able to detect market (non-)efficiency. In particular, if ( R t ) t ∈ Z is a martingale difference sequence, e.g., if it is a GARCHprocess, the quantities are equal to zero, just like the usual autocovariances and autocorrelations. Similarly, E ( R t ) = ∞ , as is often observed for financial returns in emerging and developing markets (see Section1.1), if and only if α + β ≥ . In the case of an ARCH(1) process ( R t ) with β = 0 in (3.4)-(3.5), the conditions for ζ ∈ (2 ,
4) become 1 / √ κ Z <α < . In the case of ARCH(1) with standard normal innovations Z ∼ N (0 ,
1) the condition ζ ∈ (2 ,
4) holds if andonly if 1 / √ < α < . Kesten’s equation (3.6) also motivates plug-in estimates of the tail index parameter ζ of a GARCH model fittedto a financial returns’ time series ( R t ) (see Chan et al. (2013); Zhang et al. (2019)).
11n the above notation, analogues (i’), (ii”) of stylized facts (i), (ii), (ii’) on absence of linear au-tocorrelations and presence of nonlinear dependence and volatility clustering in financial returnsbecome(i’) γ (cid:48) R, | R | s sign ( R ) ( h ) , ρ (cid:48) R, | R | s sign ( R ) ( h ) ≈ , even for small lags h = 1 , , ... (ii”) γ | R | p ( h ) , ρ | R | p ( h ) >> , even for large lags h > . In the next section, we consider estimation of the above quantities and discuss the asymptoticproperties of their estimators.
Let ( R t ) t =1 ,...,T be a sample of observations. We now define the sample versions of the auto(cross)covarianceand auto(cross)correlations in (3.7)-(3.10). Denote by ˆ µ R , ˆ µ | R | p , and ˆ µ | R | s sign ( R ) , respectively, thesample means of R t , | R t | p , and | R t | s sign ( R t ), for p, s >
0, i.e.ˆ µ R = 1 T T (cid:88) t =1 R t , ˆ µ | R | p = 1 T T (cid:88) t =1 | R t | p , ˆ µ | R | s sign ( R ) = 1 T T (cid:88) t =1 | R t | s sign ( R t ) . (3.11)The sample versions of (3.7) and (3.8) are given, respectively, byˆ γ | R | p ( h ) = 1 T T (cid:88) t = h +1 ( | R t | p − ˆ µ | R | p )( | R t − h | p − ˆ µ | R | p ) (3.12)ˆ ρ | R | p ( h ) = ˆ γ | R | p ( h )ˆ γ | R | p (0) . (3.13)Likewise, the sample versions of (3.9) and (3.10) areˆ γ (cid:48) R, | R | s sign ( R ) ( h ) = 1 T T (cid:88) t = h +1 ( R t − ˆ µ R )( | R t − h | s sign ( R t − h ) − ˆ µ | R | s sign ( R ) ) , (3.14)ˆ ρ (cid:48) R, | R | s sign ( R ) ( h ) = ˆ γ (cid:48) R, | R | s sign ( R ) ( h ) (cid:112) ˆ γ R (0)ˆ γ | R | s sign ( R ) (0) . (3.15)The following Theorems 3.1 and 3.2 follow directly from the general results presented in Appendix B(Theorem B.2; see also Davis and Mikosch (1998); Mikosch and St˘aric˘a (2000); Francq and Zako¨ıan(2006); Lindner (2009)), relying on a central limit theorem for α -mixing processes (see Theorem18.5.3 of Ibragimov and Linnik (1971)). Specifically, Theorem 3.1 provides a basis for asymptoticinference on property (ii”) - analogue of properties ((ii)), (ii’) - on the presence of nonlinear depen-dence and volatility clustering in financial returns. Likewise, Theorem 3.2 provides a basis for testing12nd asymptotic inference on analogue (i’) of stylized fact (i) on the absence of linear autocorrelationsin financial returns. Theorem 3.1
Let ( R t ) t ∈ Z be a strictly stationary α -mixing process. For p > , assume that thereexists a δ > such that E [ | R t | p + δ ] < ∞ and such that the mixing coefficients, α ( n ) , satisfy (cid:80) ∞ n =1 α ( n ) δ/ (2+ δ ) < ∞ . Then, with ˆ γ | R | p ( h ) and ˆ ρ | R | p ( h ) defined in (3.12) and (3.13) , respectively,for a fixed integer m , √ T (ˆ γ | R | p ( h ) − γ | R | p ( h )) h =0 , ,...,m → d ( G h,p ) h =0 ,...,m , (3.16) √ T ( ˆ ρ | R | p ( h ) − ρ | R | p ( h )) h =1 ,...,m → d ( H h,p ) h =1 ,...,m , (3.17) where the limits are multivariate Gaussian with mean zero. Theorem 3.2
Let ( R t ) t ∈ Z be a strictly stationary α -mixing process. For s > , assume that thereexists a δ > such that E [ | R t | s )+ δ ] < ∞ and such that the mixing coefficients, α ( n ) , satisfy (cid:80) ∞ n =1 α ( n ) δ/ (2+ δ ) < ∞ . Then with ˆ γ (cid:48) R, | R | s sign ( R ) ( h ) defined in (3.14) , √ T ( ˆ γ (cid:48) R, | R | s sign ( R ) ( h ) − γ (cid:48) R, | R | s sign ( R ) ( h )) h =0 , ,...,m → d ( G (cid:48) h,s ) h =0 , ,...,m , (3.18) where ( G (cid:48) h,s ) h =0 , ,...,m is multivariate Gaussian with mean zero.If, in addition, γ (cid:48) R, | R | s sign ( R ) ( h )) h =1 ,...,m = (0 , . . . , , with ˆ ρ (cid:48) R, | R | s sign ( R ) ( h ) defined in (3.15) , then √ T ( ˆ ρ (cid:48) R, | R | s sign ( R ) ( h )) h =1 ,...,m → d (( γ R (0) γ | R | s sign ( R ) (0)) − / G (cid:48) h,s ) h =1 ,...,m . (3.19) If, in addition, max { E [ | R | δ ] , | R | s + δ ] } < ∞ , then √ T ( ˆ ρ (cid:48) R, | R | s sign ( R ) ( h ) − ρ (cid:48) R, | R | s sign ( R ) ( h )) h =1 ,...,m → d ( H (cid:48) h,s ) h =1 ,...,m , (3.20) where ( H (cid:48) h,s ) h =1 ,...,m is multivariate Gaussian with mean zero. Remark 3.1
In Theorem 3.2, the moment conditions for asymptotic normality of sample autoco-variances in (3.18) are generally weaker than those in the case of autocorrelations in (3.20). Thisis due to the necessity of additional moment assumptions needed for asymptotic normality of sam-ple variances of R t and | R t | s sign ( R t ) required for asymptotic normality of sample autocorrelationsin the case where the true correlation are non-zero (see also Appendix B). In the case where onetests for nullity of autocorrelations, corresponding to (3.19) , the convergence of the sample autocor-relations only relies on asymptotic normality of the sample autocovariances and consistency of thesample variances, which in turn requires the same moment conditions as in the case of convergenceof sample autocovariances (see Appendix B for additional details). emark 3.2 The formulas for limiting variance-covariance matrices in Theorems 3.1 and 3.2 areprovided in the case of autocovariances and autocorelations,
Cov ( f ( R t ) , g ( R t − h )) and Corr ( f ( R t ) , g ( R t − h )) , for general functions f and g of R t in Appendix B. Remark 3.3
Similar to Appendix B and the analysis of sample linear autocovariances and auto-correlations of GARCH time series and those of their squares in, e.g., Davis and Mikosch (1998);Mikosch and St˘aric˘a (2000); Francq and Zako¨ıan (2006), the limiting distributions in Theorems3.1 and 3.2 are not particularly useful in practice since the structure of the asymptotic variance-covariance matrices are complicated. For instance, for the case of (3.16), the asymptotic variance-covariance matrix depends on autocovariances of (essentially) any order of ( | R t | p | R t − h | p ) h =0 ,...,m , asgiven in (B.38) in Appendix E. Under suitable conditions, including more restrictive moment con-ditions, the limiting variance-covariance matrices of sample autocovariances may be estimated byHAC-type estimators (see Newey and West (1987), Andrews (1991)). For instance, following theconditions of Newey and West (1987, Theorem 2), one has to assume that E [ | R t | p (4+ (cid:15) ) ] < ∞ forsome (cid:15) > . The conditions are rather restrictive for financial applications as, for instance, theirapplication with p = 1 requires that the tail index ζ > for the GARCH(1,1) process. It is fur-ther important that HAC inference methods often have poor finite sample properties, even in ratherstandard inference problems (see Ibragimov and M¨uller (2010, 2016) and references therein). Example 3.3
As discussed in Section 3.1, under suitable conditions, GARCH(1,1) processes are α -mixing with exponential decay, and hence satisfy the conditions on the mixing coefficients in The-orems 3.1 and 3.2. In particular, Theorem 3.1 applies in the case where the GARCH process hastail index ζ > p . Noting that the GARCH(1,1) process is a martingale difference sequence, we havethat (3.18) and (3.19) in Theorem 3.2 hold if the tail index ζ > s ) . Remark 3.4
In the case where ( R t ) t ∈ Z is a GARCH(1,1) process, and the moment conditions ofTheorems 3.1 and 3.2 are not satisfied, the rate of convergence of the sample autcovariances andautocorrelations are expected to be slower than √ T and depend on the tail index ζ as well as thepowers p and s , see e.g. Davis and Mikosch (2009 a ). For instance, for the case p = 1 and ζ ∈ (2 , it holds that T − /ζ (ˆ γ | R | p ( h ) − γ | R | p ( h )) has an infinite variance stable limiting distribution withindex of stability given by ζ/ . A similar result applies to T − /ζ (ˆ γ | R | p ( h ) − γ | R | p ( h )) if p = 2 and ζ ∈ (4 , , where the index of stability of the limiting variable is ζ/ . In contrast, for the cases { p = 1 and ζ ∈ (0 , } and { p = 2 and ζ ∈ (0 , } , ˆ γ | R | p ( h ) has a non-degenerate stable limit.We refer to Ibragimov et al. (2015) and the references therein for a review of properties of stabledistributions. The same reasoning applies to the sample autocorrelation ˆ ρ | R | p ( h ) where the randomlimits depend on non-Gaussian stable vector. We emphasize that the non-Gaussian stable limitingvariables are complex and hard to describe, and hence practically useless for testing a hypothesisabout autocovariances and autocorrelations. For the case of stochastic volatility processes with heavy-tailed noise, in can be shown (under suitable conditions)that the rate of convergence of the sample autocovariances and autocorrelations are faster than √ T for the cases { p = 1 and ζ ∈ (2 , } and { p = 2 and ζ ∈ (4 , } , which highlights a difference between GARCH and stochasticvolatility processes, see Davis and Mikosch (2009 b ) for additional details.
14n the next section we describe how the results in Theorems 3.1 and 3.2 can be used in relation torobust inference on autocovariances and autocorrelations of the form (3.7)-(3.10).
The previous section has emphasized several problems with asymptotic inference on key stylized factsof financial markets using normal convergence for sample autocovariances and autocorrelations andHAC inference on limiting variance-covariance matrices, including restrictive assumptions on theirapplicability and poor finite sample properties of HAC inference approaches. These problems becomeeven more severe under convergence of sample autocovariances and autocorrelations to functions ofnon-Gaussian stable r.v.’s (see Remark 3.4). This motivates the development and applicationsof robust approaches to inference on measures of market (non-)efficiency, volatility clustering andnonlinear dependence, including the autocovariances and autocorrelations of powers | R t | p of financialreturns R t . This section presents robust inference methods based on t − statistics in group estimatorsdiscussed in Ibragimov and M¨uller (2010, 2016) (see also Section 3.3 in Ibragimov et al. (2015) andAppendix A in this paper). We provide the results that demonstrate applicability of the methodsin the context of inference on stylized facts of financial markets. The results also show that themethods can be used for a unified treatment and inference on the general class of measures ofmarket (non-)efficiency, nonlinear dependence and volatility clustering, including those in the formof autocovariances and autocorrelations of powers of absolute returns dealt with in this paper.The t − statistic robust approach discussed in Appendix A is used for inference on the parameter β of a general strictly stationary process ( R t ) t ∈ Z given by the population autocovariances β = γ | R | p ( h ) ,γ R, | R | s sign ( R ) ( h ) , p, s > , discussed in Section 3.1 (see properties (i’), (ii”)). Let R , R , ..., R T be a sample of observations. Following the t − statistic approach, one partitionsthe sample into a fixed number q ≥ R k with ( j − T /q 05 if the absolute value | t β | exceeds the (1 − α/ 2) percentile of the Student- t distribution with q − t − statistic approachto robust inference on autocovariances γ | R | p ( h ) , γ (cid:48) R, | R | p sign ( R ) ( h ) and robust tests of properties ((i’)),((ii”)) is asymptotically valid under the asymptotic normality and asymptotic independence of thegroup sample autocovariances ˆ γ j, | R | p ( h ) and ˆ γ (cid:48) j,R | R | p sign ( R ) ( h ) (group estimators ˆ β j in this context).Asymptotic normality of the above group sample autocovariances hold as long it holds for the fullsample autocovariances ˆ γ | R | p ( h ) and ˆ γ (cid:48) R, | R | s sign ( R ) ( h ) (see Theorems 3.1 and 3.2).The following lemma establishes asymptotic independence of the group sample autocovariances andautocorrelations under the conditions in Theorems 3.1 and 3.2 and thus completes the justificationof the applicability of the robust t − statistic approaches in the settings considered. Lemma 3.1 Suppose that the strictly stationary process ( R t ) t ∈ Z is β -mixing. Under the assumptionsof Theorem 3.1, the centered and scaled group sample autocovariances, (cid:112) T /q (ˆ γ i, | R | p ( h ) − γ | R | p ( h )) and (cid:112) T /q (ˆ γ j, | R | p ( h ) − γ | R | p ( h )) , are asymptotically independent for i, j = 1 , , ..., q, with i (cid:54) = j . Likewise,under the conditions of Theorem 3.2, for the sample autocovariances, (cid:112) T /q ( ˆ γ (cid:48) i,R, | R | s sign ( R ) ( h ) − γ (cid:48) R, | R | s sign ( R ) ) and (cid:112) T /q ( ˆ γ (cid:48) j,R, | R | s sign ( R ) ( h ) − γ (cid:48) R, | R | s sign ( R ) ) are asymptotically independent for i, j =1 , , ..., q, with i (cid:54) = j . The asymptotic independence property does also apply to the group sampleautocorrelations ˆ ρ i, | R | p ( h ) and ˆ ρ (cid:48) i,R, | R | s sign ( R ) , i, = 1 , , ..., q , under the assumptions in Theorem 3.1and 3.2, respectively. emark 3.3 The proof of Lemma 3.1 is given in Appendix D. The asymptotic independence relieson an exact coupling argument that holds for β -mixing processes. In the proof it is only used that thesample autocovariances (or autocorrelations) have some limiting distribution when suitably centeredand scaled. Hence, the limiting distribution does not necessarily have to be Gaussian, but could,for instance, be non-Gaussian stable, as discussed in Remark 3.4. A similar argument was used byPedersen (2020) in order to show asymptotic independence of group-based least squares estimators forthe autoregressive coefficients in general heavy-tailed AR-GARCH-type processes under Gaussian andnon-Gaussian stable limits. Note that an alternative approach to show the asymptotic independenceunder Gaussian limits is to show that the joint Gaussian limiting distribution of the group-basedestimators has a block-diagonal covariance matrix. Example 3.4 As discussed in Section 3.1, under suitable conditions, the GARCH(1,1) is β -mixing(with geometric rate) and, hence, satisfies Lemma 3.1 under the conditions on the tail index ζ discussed in Example 3.3. For instance, from Example 3.3, the limits of the (centered and scaled)group estimators ˆ γ j, | R | p ( h ) and ˆ ρ j, | R | p ( h ) are normal if p < ζ/ . Likewise, from Example 3.3, thelimiting distributions of ˆ γ (cid:48) j,R, | R | s sign ( R ) ( h ) and ˆ ρ (cid:48) j,R, | R | s sign ( R ) ( h ) are normal if s < ζ/ − . Forthe latter case with h > , suppose instead that s ) > ζ > s . Then similar to Pedersen(2020) for the case s = 1 , the limiting distributions of group sample autocovariances ˆ γ (cid:48) j,R, | R | s sign ( R ) ( h ) are stable, see also Remark 3.4. Under symmetric innovations, Z , the stable distribution becomessymmetric, and hence given by a mixture of normal distributions. In such a situation, as pointedout by Ibragimov and M¨uller (2010), the t -statistic approach is also applicable, provided that theasymptotic independence still applies, which follows directly by the arguments given in Appendix D. Lemma 3.1 justifies the applicability of robust t − statistic based approaches in inference on theautocovariances ˆ γ | R | p ( h ) of powers | R t | p of absolute values of R t and on stylized fact (ii”) underappropriate moment requirements given in Theorem 3.1, i.e. E [ | R t | p + δ ] < ∞ for some δ > γ (cid:48) R, | R | s sign ( R ) ( h ) and stylized fact (i’) in the case E [ | R t | s )+ δ ] < ∞ . From Remark 3.4 and thediscussion in Appendix A and Example 3.4, it is further follows that the t − statistic based robustapproach is also applicable in inference on autocovariances ˆ γ (cid:48) R, | R | s sign ( R ) ( h ) and stylized fact (i’) inthe case where the distributional limits of (full-sample) sample autocovariances (and those of thecorresponding group sample autocovariances) are symmetric stable mixtures of normals.It is further important to note that, in contrast to HAC inference approaches with estimates oflimiting variance-covariance matrices that depend on the powers p and s used in measures of mar-ket (non-)efficiency and volatility clustering γ | R | p ( h ) and γ (cid:48) R, | R | s sign ( R ) ( h ) , the robust t − statistic ap-proaches described in this section can be used irrespective of the powers p and s in the measuresconsidered. 17 Robust inference methods: Finite sample properties In this section, we present the numerical results on finite sample properties of robust t − statisticinference approaches and their comparisons with HAC inference procedures in the analysis of stylizedfacts (i’) and (ii”) of market (non-)efficiency in comparison to standard approaches. To investigate finite sample performance of HAC and robust t − statistic inference approaches intesting for stylized fact (i’): ρ (cid:48) R, | R | s sign ( R ) ( h ) = 0, the data is generated from the following DGPs: R t = φR t − + ε t , (4.24) ε t = σ t Z t , t = 2 , . . . , T, (4.25)where the GARCH-type process ( ε t ) follows one of the following three models on the volatilitydynamics and the distribution of the innovations ( Z t ) . (Symmetric) ARCH(1) with normal innovations σ t = 0 . π / / ε t − , (4.26)and ( Z t ) is a sequence of i.i.d. standard normal N (0 , 1) r.v.’s.2. ARCH(1) with asymmetric innovations : σ t follows (4.26) and ( Z t ) is a sequence of i.i.d. r.v.’sthat have an asymmetric Student- t distribution with 50 degree of freedom and the skewnessparameter of 0.5 (and unit variance; see Section 6.7 in Christoffersen (2012)), denoted t (50 , . . ARCH(1) with asymmetric heavy-tailed innovations : σ t follows (4.26) and ( Z t ) is a sequenceof i.i.d. r.v.’s that have an asymmetric Student- t distribution with 3 degree of freedom andthe skewness parameter of 0.5, denoted t (3 , . GJR-GARCH(1,1,1) with normal innovations : σ t = 0 . . | ε t − | − ε t − ) + 0 . σ t − (4.27)where ( Z t ) is a sequence of i.i.d. standard normal N (0 , 1) r.v.’s.By Kesten’s equation (3.6) in Section 3, for the first ARCH process (with standard normal inno-vations) the value α = π / / ζ = 3 in (iii). The second and thirdARCH processes have the same α = π / / t − distributed innovations thatare more heavy-tailed than normal. By Kesten’s equation (3.6), the tail index ζ in (iii) for theseprocesses satisfies 2 < ζ < 3. For the choice of the parameters of the GJR-GARCH(1,1,1) model,18he tail ζ equals to 3 as in the case of ARCH with normal innovations. 27 28 In the numerical analysis of finite sample properties, we present the results for HAC tests based on theHAC t − statistics of (full-sample) sample autocorrelations ˆ ρ R (1) , ˆ ρ (cid:48) R, | R | . sign ( R ) (1), ˆ ρ (cid:48) R, | R | . sign ( R ) (1),ˆ ρ (cid:48) R, | R | . sign ( R ) (1) and for robust inference approaches based on t − statistics in their group counter-parts with q = 4 , , 12 and 16. For the above DGPs, in order to have asymptotic normality forˆ ρ (cid:48) R, | R | s sign ( R ) ( h ) under ρ (cid:48) R, | R | s sign ( R ) ( h ) = 0, s should be less than 0.5. In this section, we analyse finite sample size properties of the tests by setting φ = 0 in (4.24). Thesample size is 5,000 and the number of replications is 10,000. The results are presented in Table 1.For all cases, the standard HAC-based tests are oversized, and the largest oversizing is observed forthe asymmetric ARCH model with heavy-tailed innovations, t (3 , . 5) especially in tests based on thefirst-order sample autocorrelations ˆ ρ R (1). For the robust t − statistic approaches, the size control isvery good even for the tests based on group estimates of the first-order autocorrelations ρ R (1), againexcept for the case of the asymmetric ARCH model with t (3 , . t − statistic inference approaches inapplicable. The same is the case for bothHAC and robust t − statistic approaches based on (full-sample and group) sample autocorrelationsˆ ρ (cid:48) R, | R | . sign ( R ) (1) (as indicated above, one needs s to be smaller than 0.5 for asymptotic normalityof sample analogues of these autocorrelations). The robust t − statistic approach based on groupestimators of ρ (cid:48) R, | R | . sign ( R ) (1) and ρ (cid:48) R, | R | . sign ( R ) (1) are slightly oversized with better size controlfor q = 4 and for (the second best size control) q = 8. Overall, the best tests in term of finitesample size are the robust t − statistic tests based on group estimates ˆ ρ (cid:48) R, | R | . sign ( R ) (1) with q = 4and q = 8. [Table 1 about here.] For the numerical analysis of size-adjusted power properties of inference on market (non-)efficiency,we simulate the DGPs considered in the previous section with φ ranging from 0 to 0.5. Figures1, 2, 3 and 4 present the results on finite size-adjusted power for Model 1 (ARCH with normalinnovations), Model 2 (ARCH with asymmetric innovations), Model 3 (ARCH with asymmetricheavy-tailed innovations) and Model 4 (GJR-GARCH with normal innovations), respectively, for The parameters of the GJR-GARCH process that correspond to the tail index ζ = 3 were found using simulationsfrom the analogue of Kesten’s equation (3.6) for the stochastic recurrence (difference) equation defining the process(see, among others, Mikosch and St˘aric˘a (2000)). t − statistic approaches based on groupsample autocorrelations with q = 8 . In the case of the normal ARCH (Model 1), the power curvesfor HAC and t − statistic-based tests are very close to each other, except for the case of tests basedon ˆ ρ (cid:48) R (1) that have lower size-adjusted power. For the asymmetric case (Model 2), the resultsare virtually the same. For the asymmetric case with heavy-tailed innovations (Model 3), theconclusions are also similar, with the most powerful tests being those based on ˆ ρ (cid:48) R, | R | . sign ( R ) (1)and ˆ ρ (cid:48) R, | R | . sign ( R ) (1) (and latter is slightly more powerful than the former). Similar conclusions alsohold in the GJR-GARCH case (Model 4).[Figures 1-4 about here.]Additionally, we provides the numerical results on comparisons of size-adjusted power properties ofrobust t − statistic inference approaches with the different numbers q of groups. According to Figures5-20 presented in Appendix F.1, the power of robust t − statistic tests increases as the number ofgroups q increases, but the differences between the power curves become negligble as the power s inthe dependence measures ˆ ρ (cid:48) R, | R | s sign ( R ) (1) decreases (except for the test with q = 4 groups). For the analysis of stylized fact (ii”) on nonlinear dependence and volatility clustering based onautocrrelations of powers of absolute returns, the data is generated from the ARCH DGP’s similarto Section 4.1, so that R t = σ t Z t , t = 2 , . . . , T, (4.28)where the dynamics of volatility σ t follows (4.26) and the innovations Z t are i.i.d. r.v.’s with standardnormal or an asymmetric Student- t distribution with the skewness parameter 0.5 and the numberof degrees of freedom equal to 50 (ARCH with asymmetric innovations) or 3 (the case of ARCHwith asymmetric heavy-tailed innovations). As in Section 4.1, we note that the tail index ζ in (iii)equals to 3 in the case of the above ARCH process with normal innovations and is smaller than 3:2 < ζ < t distributedinnovations.We provide the numerical results for HAC tests based on estimators ˆ ρ R (1) ˆ ρ | R | (1), ˆ ρ | R | . (1),ˆ ρ | R | . (1), ˆ ρ | R | . (1) and for robust t − statistic approaches based on the group counterparts of theestimators with q = 4 , , 12 and 16. The comparisons are based on the coverage level of the corre-sponding confidence intervals for the unknown population autocorrelations estimators ρ R (1) , ρ | R | (1), ρ | R | . (1), ρ | R | . (1), ρ | R | . (1) for different values of the ARCH parameter α ranging from 0 to 1( α = 1 implies infinite second moment and ζ = 2 , so that the power p in (ii”) should be smallerthan 0.5). 20igures 21-25 present the numerical results on the coverage of confidence intervals in the normalARCH case for the tests based on ˆ ρ R (1) (Figure 21), ˆ ρ | R | (1) (Figure 22), ˆ ρ | R | . (1) (Figure 23),ˆ ρ | R | . (1) (Figure 24) and ˆ ρ | R | . (1) (Figure 25). One can see very unstable coverage for the testsbased on ˆ ρ R (1) (Figure 21) and ˆ ρ | R | (1) (Figure 22) due to the loss of asymptotic normality. Thecoverage improves as the power p becomes smaller. The best coverage for all range of α ’s is observedfor robust t − statistic tests based on group estimates ˆ ρ | R | . (1) with q = 4, and the second best testis with q = 8. [Figures 21-25 about here.]The qualitative conclusions are similar for the case of t (50 , . t (3 , . In this section we revisit a recent study by Baltussen et al. (2019) on linear dependence in returnson major stock market indexes. Specifically, Baltussen et al. (2019) present empirical results basedon HAC inference approaches applied to the first-order autocorrelations, that is, ρ R (1), that suggestthat serial dependence in daily returns on 20 major market indexes covering 15 countries in NorthAmerica, Europe, and Asia was significantly positive until the end of the 1990s, and switched tobeing significantly negative since the 2000s. In light of Theorem 3.2 (with s = 1), asymptoticnormality of the sample autocorrelation relies on finite fourth-order moments of the process, and inthe case where such moment conditions are not satisfied, the limiting distribution may be a functionof non-Gaussian stable variables (Remark 3.4). We consider daily percentage returns on the majorstock indexes from March 3, 1999 to January 14, 2020. The second and third columns of Table2 present, respectively, (bias-corrected) log-log rank-size estimates of the tail indices for the returntime series and their 95% confidence intervals (see Gabaix and Ibragimov (2011) and Ch. 3 inIbragimov et al. (2015)). Importantly, the tail index for all of the returns time series appear to besmaller than 4, and the left end-points of the confidence intervals vary from 2.53 to 3.37 across thereturn series, so that the confidence intervals either lie on the left of the value of 4 or contain it.This implies that inference on linear autocorrelations, ρ R ( h ), using asymptotic normality for theirsample analogues, ˆ ρ R ( h ), is invalid .Columns 4, 7, 9 and 11 in Table 2 provide the full-sample estimates of the autocorrelations ρ R (1)and ρ (cid:48) R, | R | s sign ( R ) (1) in properties (i) and (i’) and those of the multi-period autocorrelation MAC(5)(the weighted sum of the first five autocorrelation coefficients of order h = 1 , , , , 5) used in The data series were retrieved from Bloomberg. We choose the same starting date (March 3, 1999) as Baltussenet al. (2019), but extend the sample until January 14, 2020 (Baltussen et al. use data series ending on December 31,2016). ρ R (1) andMAC(5).Column 8 in Table 2 presents (95% confidence) intervals (A.30) that can be used in testing the hy-pothesis that the (non-)efficiency dependence measure ρ R, | R | s sign ( R ) (1) equals zero: ρ R, | R | s sign ( R ) (1) =0 using the robust t − statistic tests with q = 8 groups as discussed in Appendix A.We choose the power s based on the left end-points of the confidence intervals for the tail index.Specifically, for a given series, if an end-point exceeds 3, we set s = 0 . 5, and if the end-point liesbetween 2.5 and 3.0, we set s = 0 . 25, so that the moment conditions for asymptotic normalityof the group-based estimators are satisfied. We see that for most of the series (17 out of 20),the considered intervals in (A.30) based on robust t − statistic approaches applied to inference on ρ R, | R | s sign ( R ) (1) contain zero, and for two of the series the confidence intervals lie on the right of zero.Hence, following the robust t − statistic tests (see Appendix A), we find little evidence for significantlynegative (serial) dependence in the return series, in contrast to the conclusions in Baltussen et al.(2019). The same qualitative conclusions apply if we set s = 0 . Column 10 in Table 2), andalso hold for inference on the MAC(5) measure (Column 13 of the table).[Table 2 about here.]Tables 3-5 provide the results on testing for nonlinear dependence and volatility clustering in theindices’ returns using 95% confidence intervals constructed on the base of robust t − statistic ap-proaches applied to the first-order autocorrelations ˆ ρ R (1) , ˆ ρ | R | (1) and ˆ ρ | R | p (1) in the above period. Overall, the tables confirm the presence of nonlinear dependence and volatility clustering in thereturns on the financial indices. [Tables 3-5 about here.] The results in this paper justify applicability of general t − statistic robust inference methods inIbragimov and M¨uller (2010, 2016) in the context of inference on measures of market (non-)efficiency,volatility clustering and nonlinear dependence based on powers of absolute values of financial returns.The focus of this paper was on the case of testing of such stylized facts of financial markets in form (i)- Under the hypothesis ρ R, | R | s sign ( R ) (1) = 0 in consideration. The HAC-based approach used by Baltussen et al. (2019) corresponds to the 95% confidence intervals reportedin Column 5 in Table 2. Although theoretically unjustified in light of the tail index estimates, we note that theseconfidence intervals do not provide much evidence for significant negative serial dependence. The discrepancy betweenthe conclusion made from Table 2 and the ones obtained by Baltussen et al. (2019) is most likely due to differentchoices of sample periods (we apply a longer sample) and the fact that Baltussen et al. (2019) rely on 90% confidenceintervals. The tables also present the results on tail index estimation and HAC confidence intervals for the considereddependence measures. A Robust t − statistic inference approaches Suppose we want to conduct inference about some scalar parameter β of an autocorrelated, het-erogenous and possibly heavy-tailed time series ( X t ) (e.g., a parameter of a predictive regressionof returns R t on some explanatory factor, e.g., the dividend to price ratio or an autocovariancecoefficient of returns R t or powers of their absolute values) using a large data set of T observations X , X , ..., X T . For a wide range of time series models and estimators ˆ β of β, it is known that the dis-tribution of ˆ β is approximately normal in large samples, that is, √ T ( ˆ β − β ) → d N (0 , σ ) as T → ∞ . If the autocorrelations in ( Y t ) are pervasive and pronounced enough, then it will be challenging toconsistently estimate the limiting variance σ , e.g., by HAC approaches, and inference proceduresfor β that ignore the sampling variability of a candidate consistent estimator ˆ σ (e.g., based onHAC standard error of ˆ β ) is expected to have poor finite sample properties (see Remark 3.3 and thediscussion in Ibragimov and M¨uller (2010) and Section 3.3 in Ibragimov et al. (2015)).The results in Ibragimov and M¨uller (2010, 2016) provide the following general approach to robustinference about an arbitrary parameter β of a time series or an economic or financial model underheterogeneity, correlation and heavy-tailedness of a largely unknown form. Consider a partitionof the original data sample X , X , ..., X T into a fixed number q ≥ X s with ( j − T /q < s ≤ jT /q. Denote by ˆ β j the estimator of β using observationsin group j only (e.g., the OLS estimate of the predictive regression or the sample autocovariancecoefficient of returns R t or powers of their absolute values for observations in group j ). Suppose thatthe group estimators ˆ β j are asymptotically normal: √ T ( ˆ β j − β ) → d N (0 , σ j ) and, also, √ T ( ˆ β i − β )and √ T ( ˆ β j − β ) are asymptotically independent for i (cid:54) = j (the asymptotic normality of √ T ( ˆ β j − β )typically follows from the same reasoning as the asymptotic normality of the full sample estimatorˆ β ). The condition of asymptotic independence of ˆ β i and ˆ β j is a condition on the degree of (weak)dependence in time series ( X t ) . In addition to time series models, the approach can be used in panel, clustered, spatially correlated and othersettings (see Ibragimov and M¨uller (2010, 2016) and Section 3.3 in Ibragimov et al. (2015)). In general settings, the argument for asymptotic independence of group estimators of β depends on the choiceof groups and the details of the application (see Section 4 in Ibragimov and M¨uller (2010) for the discussion of sucharguments for different econometric models, including time series, panel, clustering and spatially correlated settings).Section 3.4 in the working paper version Ibragimov and M¨uller (2007) of Ibragimov and M¨uller (2010) presents asimple-to-implement test of whether additional assumptions required for consistent variance estimation hold in the t − statistic inference approach in Ibragimov and M¨uller (2010), one can performan asymptotically valid test of level α, α ≤ . 05 of H : β = β against β (cid:54) = β be rejecting H when | t β | exceeds the (1 − α/ 2) percentile cv α of the Student- t distribution with q − t β is the usual t − statistic in group estimators ˆ β j , j = 1 , , ..., q : t β = √ q ˆ β − β s ˆ β (A.29)with ˆ β = q − (cid:80) qj =1 ˆ β j and s β = ( q − − (cid:80) qj =1 ( ˆ β j − ˆ β ) , respectively the sample mean and samplevariance of ˆ β j , j = 1 , ..., q. As usual, the robust t − statistic tests of level α the null hypothesis H : β = β against β (cid:54) = β can be also conducted by checking whether the interval( ˆ β − cv α s β / √ q, ˆ β − cv α s β / √ q ) (A.30)contains β , where, as before, cv α denotes the (1 − α/ 2) percentile of the Student- t distribution with q − β with an asymptoticcoverage of at least 1 − α (see Ibragimov and M¨uller (2007)). As discussed in Ibragimov and M¨uller (2010), the t − statistic approach provides a number of im-portant advantages over the existing methods. In particular, it can be employed when data arepotentially heterogeneous and correlated in a largely unknown way. In addition, the approach issimple to implement and does not need new tables of critical values. The assumptions of asymptoticnormality for group estimators in the approach are explicit and easy to interpret, in contrast toconditions that imply validity of alternative procedures. The numerical results in Ibragimov and M¨uller (2010) demonstrate that, for many dependence andheterogeneity settings considered in the literature and typically encountered in practice for timeseries, panel, clustered and spatially correlated data, the choice q = 8 or q = 16 leads to robust testswith attractive finite sample performance. data at hand. This test can be used to decide between the use of HAC methods and robust t − statistics approaches. As discussed in Ibragimov and M¨uller (2010), the t − statistic approach to robust inference can also be used fortest levels α ≤ −√ ≈ . ..., where Φ( x ) is the standard normal cdf. Naturally, these conclusions on the tests and confidence intervals hold if the conditions on applicability of the t − statistic approach are satisfied in the context; in particular, the robust t − statistic tests of H may be conductedas described above if the group estimators ˆ β j , are asymptotically normal and asymptotically independent under thenull hypothesis H : β = β . Note that there is a typo in formulation of confidence intervals in Section 2.2 in Ibragimov and M¨uller (2007);the intervals should account for the division by √ q, as in (A.30). As is shown in Ibragimov and M¨uller (2010), the t − statistic based approach to robust inference efficiently exploitsthe information contained in these regularity assumptions, both in the small sample settings (uniformly most powerfulscale invariant test against a benchmark alternative with equal variances) and also in the asymptotic frameworks. The asymptotic efficiency results for t − statistic based robust inference further imply that it is not possible touse data dependent methods to determine the optimal number of groups q to be used in the approach when the only t − statistic approach provides a formal justification for the widespread FamaMacBeth methodfor inference in panel regressions with heteroskedasticity (see Fama and MacBeth (1973)). In theapproach, one estimates the regression separately for each year, and then tests hypotheses about thecoefficient of interest using the t − statistic of the resulting yearly coefficient estimates. The Fama-MacBeth approach is a special case of the t − statistic based approach to inference, with observationsof the same year collected in a group.Importantly for inference on heavy-tailed time series generated by GARCH-type models and typicalfor real-world financial markets (see the discussion in Section 1.1), the t − statistic robust inferenceapproach remains valid as long as the group estimators ˆ β j , j = 1 , , ..., q, are asymptotically inde-pendent and convergence (at an arbitrary rate) to heavy-tailed scale mixtures of normals. Namely,the approach is asymptotically valid if m T ( ˆ β j − β ) qj =1 → d Z j V j , j = 1 , , ..., q, for some real sequence m T , where Z j ∼ i.i.d. N (0 , , the random vector { V j } qj =1 is independent of the vector { Z j } qj =1 andmax j | V j | > Z, aslimits of sample linear autocovariances ˆ γ R ( h ) of stationary GARCH-type processes (e.g., GARCH(1,1) processes (3.4)-(3.5)) with tail indices ζ < γ (cid:48) R, | R | s sign ( R ) ( h ) , s > , in Theorem 3.2 in the case of GARCH-type processes with tail indices ζ < s (see Remark 3.4).The results in Ibragimov and M¨uller (2016) provide analogues of the above t − statistic approachesthat can be used in inference on equality of two parameters of economic or financial models, withnatural applications in tests for structural breaks (e.g., in parameters of GARCH-type models,including their tail indices or measures of market (non-)efficiency, volatility clustering and nonlineardependence considered in this paper). B Asymptotic normality of general sample autocrosscovari-ances and -correlations In order to show that the sample auto(cross)covariances and -correlations have asymptotic Gaussianlimits, we rely on the following central limit theorem (CLT) for strictly stationary α -mixing, i.e.strongly mixing, processes. We refer to Appendix C for additional details about mixing processes. assumption imposed on the data generating process is that of asymptotic normality for the group estimators ˆ β j . Theorem B.1 (Theorem 18.5.3 of Ibragimov and Linnik (1971)) Let ( R t ) t ∈ Z be an R -valuedmean zero stationary strongly mixing process with mixing coefficient α ( n ) and let E [ | R t | δ ] < ∞ for some δ > . If ∞ (cid:88) n =1 α ( n ) δ/ (2+ δ ) < ∞ , (B.31) then σ ≡ E [ R ] + 2 (cid:80) ∞ j =1 E [ R R j ] < ∞ , and T − / (cid:80) Tt =1 R t d → N (0 , σ ) . We have the following result. Theorem B.2 Let ( R t ) t ∈ Z be an R -valued strictly stationary and strongly mixing process. Let f : R → R and g : R → R be measurable functions. Consider the sample autocrosscovariance functionof f ( R t ) and g ( R t ) for some order h ≥ , ˆ γ T,f ( R ) ,g ( R ) ( h ) = 1 T T (cid:88) t =1 f ( R t ) g ( R t − h ) − ( 1 T T (cid:88) t =1 f ( R t ))( 1 T T (cid:88) t =1 g ( R t − h )) , (B.32) and their population versions, γ f ( R ) ,g ( R ) ( h ) = Cov( f ( R t ) , g ( R t − h )) = E [ f ( R t ) g ( R t − h )] − E [ f ( R t )] E [ g ( R t − h )] . (B.33) Likewise, consider the sample cross autocrosscorrelation functions, ˆ ρ T,f ( R ) ,g ( R ) ( h ) = ˆ γ T,f ( R ) ,g ( R ) ( h ) (cid:112) ˆ γ T,f ( R ) ,f ( R ) (0)ˆ γ T,g ( R ) ,g ( R ) (0) , h ≥ , (B.34) and their population versions ρ f ( R ) ,g ( R ) ( h ) = γ f ( R ) ,g ( R ) ( h ) (cid:112) γ f ( R ) ,f ( R ) (0) γ g ( R ) ,g ( R ) (0) . (B.35) Suppose that there exists a δ > such that max { E [ | f ( R t )) | δ ] , E [ | g ( R t )) | δ ] } < ∞ and max h =0 ,...,m { E [ | f ( R t ) g ( R t − h ) | δ ] } < ∞ , (B.36) and such that the mixing coefficients of ( R t ) t ∈ Z satisfy (B.31) . Then √ T (ˆ γ T,f ( R ) ,g ( R ) ( h ) − γ f ( R ) ,g ( R ) ( h )) h =0 ,...,m d → ( G h,f ( R ) ,g ( R ) ) h =0 ,...,m , (B.37) where ( G h,f ( R ) ,g ( R ) ) h =0 ,...,m is an ( m + 1) -dimensional Gaussian vector with zero mean and covariance atrix given by Γ = Var( Y ) + 2 ∞ (cid:88) k =1 Cov( Y , Y k ) , (B.38) where Y t = ( Y t,h ) h =0 ,...,m , Y t,h = ( f ( R t ) − E [ f ( R t )])( g ( R t − h ) − E [ g ( R t − h )]) − γ f ( R ) ,g ( R ) ( h ) .If, in addition, γ f ( R ) ,g ( R ) ( h )) h =1 ,...,m = (0 , . . . , , then √ T ( ˆ ρ T,f ( R ) ,g ( R ) ( h )) h =1 ,...,m d → (cid:0) ( γ f ( R ) ,f ( R ) (0) γ g ( R ) ,g ( R ) (0)) − / G h,f ( R ) ,g ( R ) (cid:1) h =1 ,...,m . (B.39) Suppose that, in addition, there exists a δ > such that max { E [ | f ( R t ) | δ ] , E [ | g ( R t ) | δ ] } < ∞ (B.40) and such that (B.31) holds. Then √ T ( ˆ ρ T,f ( R ) ,g ( R ) ( h ) − ρ f ( R ) ,g ( R ) ( h )) h =1 ,...,m d → ( ˜ G h,f ( R ) ,g ( R ) ) h =1 ,...,m , (B.41) where ( ˜ G h,f ( R ) ,g ( R ) ) h =0 ,...,m is a Gaussian vector with mean zero and covariance given by A Γ † A (cid:48) , where A is constant matrix defined in (B.44) and Γ † = Var( Y † ) + 2 ∞ (cid:88) k =1 Cov( Y † , Y † k ) , (B.42) with Y † t = ( Y (cid:48) t , V t, , V t, ) (cid:48) , V t, = ( f ( R t ) − E [ f ( R t )]) − γ f ( R ) ,f ( R ) (0) , V t, = ( g ( R t ) − E [ g ( R t )]) − γ g ( R ) ,g ( R ) (0) . Proof: Firstly, note that strong mixing is a property about the σ -field generated by ( R t ) t ∈ Z (see Appendix C). Since f and g are measurable, it follows that the process ( V t ) t ∈ Z , with V t =( f ( R t ) , g ( R t ) , g ( R t − ) , . . . , g ( R t − m )), is also strictly stationary and strongly mixing with mixing co-efficients satisfying (B.31) (see also Francq and Zako¨ıan (2006)). Likewise, the same property appliesto the process ( ˜ V t,h ) t ∈ Z with ˜ V t,h = f ( R t ) g ( R t − h ), h = 0 , . . . , m . This allows us to apply TheoremB.1 to the processes ( V t ) t ∈ Z and ( ˜ V t,h ) t ∈ Z , under the assumed integrability conditions. Next, notethatˆ γ T,f ( R ) ,g ( R ) ( h ) − γ f ( R ) ,g ( R ) ( h ) = 1 T T (cid:88) t =1 ( f ( R t ) − E [ f ( R t )])( g ( R t − h ) − E [ g ( R t − h )]) − γ f ( R ) ,g ( R ) ( h ) − ( 1 T T (cid:88) t =1 ( f ( R t ) − E [ f ( R t )]))( 1 T T (cid:88) t =1 ( g ( R t − h ) − E [ g ( R t − h )) . Since max { E [ | f ( R t )) | δ ] , E [ | g ( R t )) | δ ] } < ∞ for some δ > T (cid:80) Tt =1 f ( R t ) − E [ f ( R t )] and T (cid:80) Tt =1 g ( R t − h ) − E [ g ( R t − h )] are O p ( T − / ), so that √ T (ˆ γ T,f ( R ) ,g ( R ) ( h ) − γ f ( R ) ,g ( R ) ( h )) = 1 √ T T (cid:88) t =1 (( f ( R t ) − E [ f ( R t )])( g ( R t − h ) − E [ g ( R t − h )]) − γ f ( R ) ,g ( R ) ( h )) + o p (1) . Let Y t,h = ( f ( R t ) − E [ f ( R t )])( g ( R t − h ) − E [ g ( R t − h )]) − γ f ( R ) ,g ( R ) ( h ). Using (B.36), there exists a δ > E [ | Y t,h | δ ] < ∞ . Using that E [ Y t,h ] = 0 and that ( Y t,h : t ∈ Z ) isstrongly mixing satisfying (B.31), √ T (ˆ γ T,f ( R ) ,g ( R ) ( h ) − γ f ( R ) ,g ( R ) ( h )) d → G h,f ( R ) ,g ( R ) by Theorem B.1.The joint convergence in (B.37) is obtained by similar arguments applied to linear combinations of √ T (ˆ γ T,f ( R ) ,g ( R ) ( h ) − γ f ( R ) ,g ( R ) ( h )) h =0 ,...,m and an application of the Cram´er-Wold device.Next, we consider the limiting distribution of the sample correlations. The limiting distributionfor the case where γ f ( R ) ,g ( R ) ( h )) h =1 ,...,m = (0 , . . . , 0) is immediate, by noting that (ˆ γ T,f ( R ) ,f ( R ) (0) − γ f ( R ) ,f ( R ) (0)) and (ˆ γ T,g ( R ) ,g ( R ) (0) − γ g ( R ) ,g ( R ) (0)) are o p (1) and by an application of Slutsky’s theorem.Next, using (B.40), it holds by arguments similar to the ones given above that √ T (ˆ γ T,f ( R ) ,g ( R ) ( h ) − γ f ( R ) ,g ( R ) ( h )) h =0 ,...,m ˆ γ T,f ( R ) ,f ( R ) (0) − γ f ( R ) ,f ( R ) (0)ˆ γ T,g ( R ) ,g ( R ) (0) − γ g ( R ) ,g ( R ) (0) d → G † , (B.43)where G † is an ( m + 3)-dimensional Gaussian vector with mean zero and covariance given by Γ † defined in (B.42). Let x = ( x , . . . , x m +3 ) (cid:48) ∈ R m +3 and define the function ˜ g : R m +3 → R m +1 as˜ g ( x ) = ( x √ x m +2 x m +3 , . . . , x m +1 √ x m +2 x m +3 ) (cid:48) . Define the matrix A = ∂ ˜ g ( x ) ∂x (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) x = γ † , γ † = (( γ f ( R ) ,g ( R ) ( h )) (cid:48) h =0 ,...,m , γ f ( R ) ,f ( R ) (0) , γ g ( R ) ,g ( R ) (0)) (cid:48) . (B.44)The convergence in (B.41) is then obtained by an application of the delta method. (cid:3) C α - and β -mixing Let (Ω , F , P ) be a probability space.The β -mixing coefficient between two σ -fields A and B , A , B ⊂ F , is given by β ( A , B ) := 12 sup I (cid:88) i =1 J (cid:88) j =1 | P ( A i ∩ B j ) − P ( A i ) P ( B j ) | , A i : i = 1 , . . . , I ) and ( B j : j = 1 , . . . , J ) ofΩ with A i ∈ A and B j ∈ B .Let ( x t : t ∈ Z ) be a sequence of r.v.’s. Define the σ -fields F n := σ ( x t : t ∈ Z , t ≤ n ) , G n := σ ( x t : t ∈ Z , t ≥ n ) . The β -mixing coefficients of ( x t : t ∈ Z ) are given by β ( k ) := sup n ∈ Z β ( F n , G n + k ) . Note that if ( x t : t ∈ Z ) is strictly stationary, β ( k ) = β ( F , G k ). The sequence ( x t : t ∈ Z ) is saidto be β -mixing if β ( k ) → k → ∞ . Likewise, the strictly stationary sequence ( x t : t ∈ Z ) has α -mixing coefficients α ( k ) := sup A ∈F ,B ∈G k | P ( A ∩ B ) − P ( A ) P ( B ) | , and is said to be α -mixing if α ( k ) → k → ∞ . Notice that for a strictly stationary process, β ( k ) ≥ α ( k ), and hence a β -mixing process is also α -mixing. D On the validity of Lemma 3.1 Suppose that ( x t : t ∈ Z ) is strictly stationary and β -mixing. For the sake of clarity we focus on theasymptotic independence of group-based estimators for autocovariances. The sample autocorrela-tions are dealt with in a similar fashion. Specifically, let f : R → R and g : R → R be measurablefunctions, and define y t = ( y t, , y t, ) (cid:48) = ( f ( x t ) , g ( x t − )) (cid:48) . Since f and g are measurable, ( y t : t ∈ Z )is strictly stationary and β -mixing. With i, j ∈ Z , 0 ≤ i < j , letˆ φ i,j := ( j − i + 1) − j (cid:88) t = i y t, y t, − (cid:32) ( j − i + 1) − j (cid:88) t = i y t, (cid:33) (cid:32) ( j − i + 1) − j (cid:88) t = i y t, (cid:33) − φ , where φ = E [ y t, y t, ] − E [ y t, ] E [ y t, ] , and suppose that for some deterministic sequence ( a T ), satisfying a T → ∞ , a T ˆ φ ,T → d Z, , (D.45)for some random variable Z (potentially non-Gaussian).29uppose that we split the sample into two equi-sized groups , such that we have the group-basedestimators ˆ φ , (cid:98) T/ (cid:99) and ˆ φ (cid:98) T/ (cid:99) +1 , (cid:98) T/ (cid:99) . We seek to show that a (cid:98) T/ (cid:99) ˆ φ , (cid:98) T/ (cid:99) and a (cid:98) T/ (cid:99) ˆ φ (cid:98) T/ (cid:99) +1 , (cid:98) T/ (cid:99) areasymptotically independent. By the Cramr-Wold device, the asymptotic independence holds, if weshow that for any constants ( k , k ) ∈ R , k a (cid:98) T/ (cid:99) ˆ φ , (cid:98) T/ (cid:99) + k a (cid:98) T/ (cid:99) ˆ φ (cid:98) T/ (cid:99) +1 , (cid:98) T/ (cid:99) → d k Z (1) + k Z (2) where Z (1) and Z (2) are independent copies of Z . Let ˜ T := ˜ T ( T ) be an increasing sequence ofpositive integers satisfying ˜ T = o ( T ) as T → ∞ . It holds that a (cid:98) T/ (cid:99) ˆ φ (cid:98) T/ (cid:99) +1 , (cid:98) T/ (cid:99) = a (cid:98) T/ (cid:99) ˆ φ (cid:98) T/ (cid:99) +1 , (cid:98) T/ (cid:99) +1+ ˜ T + a (cid:98) T/ (cid:99) ˆ φ (cid:98) T/ (cid:99) +2+ ˜ T , (cid:98) T/ (cid:99) =: S (1) T + S (2) T , (D.46)where it holds, due to (D.45), that S (1) T = o p (1) . (D.47)Let ( y (cid:63)t : t ∈ Z ) denote a sequence with the same distribution as ( y t : t ∈ Z ) and independent of F (cid:98) T/ (cid:99) . By Theorem 5.1 of Rio (2017), P ( y (cid:63)t (cid:54) = y t for some t ≥ (cid:98) T / (cid:99) + k ) = β ( F (cid:98) T/ (cid:99) , G (cid:98) T/ (cid:99) + k ) , (D.48)where β ( F (cid:98) T/ (cid:99) , G (cid:98) T/ (cid:99) + k ) is defined in Appendix C. Letˆ φ (cid:63)i,j := ( j − i + 1) − j (cid:88) t = i y (cid:63)t, y (cid:63)t, − (cid:32) ( j − i + 1) − j (cid:88) t = i y (cid:63)t, (cid:33) (cid:32) ( j − i + 1) − j (cid:88) t = i y (cid:63)t, (cid:33) − φ , and note that S (2) T = a (cid:98) T/ (cid:99) ˆ φ (cid:63) (cid:98) T/ (cid:99) +2+ ˜ T , (cid:98) T/ (cid:99) + (cid:16) a (cid:98) T/ (cid:99) ˆ φ (cid:98) T/ (cid:99) +2+ ˜ T , (cid:98) T/ (cid:99) − a (cid:98) T/ (cid:99) ˆ φ (cid:63) (cid:98) T/ (cid:99) +2+ ˜ T , (cid:98) T/ (cid:99) (cid:17) . For any ε > 0, using (D.48) and that ( y t : t ∈ Z ) is strictly stationary and β -mixing, P (cid:104)(cid:12)(cid:12)(cid:12) a (cid:98) T/ (cid:99) ˆ φ (cid:98) T/ (cid:99) +2+ ˜ T , (cid:98) T/ (cid:99) − a (cid:98) T/ (cid:99) ˆ φ (cid:63) (cid:98) T/ (cid:99) +2+ ˜ T , (cid:98) T/ (cid:99) (cid:12)(cid:12)(cid:12) > ε (cid:105) ≤ P (cid:104) y (cid:63)t (cid:54) = y t for some t ≥ (cid:98) T / (cid:99) + 2 + ˜ T (cid:105) = β ( F (cid:98) T/ (cid:99) , G (cid:98) T/ (cid:99) +2+ ˜ T )= β (2 + ˜ T ) = o (1) , so that S (2) n = a (cid:98) T/ (cid:99) ˆ φ (cid:63) (cid:98) T/ (cid:99) +2+ ˜ T , (cid:98) T/ (cid:99) + o p (1) . (D.49) In the case of more groups, one has to repeat the coupling argument. a (cid:98) T/ (cid:99) ˆ φ (cid:98) T/ (cid:99) +1 , (cid:98) T/ (cid:99) = a (cid:98) T/ (cid:99) ˆ φ (cid:63) (cid:98) T/ (cid:99) +2+ ˜ T , (cid:98) T/ (cid:99) + o p (1) . Using (D.45), we then obtain that for any ( k , k ) ∈ R , k a (cid:98) T/ (cid:99) ˆ φ , (cid:98) T/ (cid:99) + k a (cid:98) T/ (cid:99) ˆ φ (cid:98) T/ (cid:99) +1 , (cid:98) T/ (cid:99) = k a (cid:98) T/ (cid:99) ˆ φ , (cid:98) T/ (cid:99) + k a (cid:98) T/ (cid:99) ˆ φ (cid:63) (cid:98) T/ (cid:99) +2+ ˜ T , (cid:98) T/ (cid:99) + o p (1) w → k Z (1) + k Z (2) , where Z (1) and Z (2) are copies of Z , and Z (1) and Z (2) are independent since a (cid:98) T/ (cid:99) ˆ φ (cid:63) (cid:98) T/ (cid:99) +2+ ˜ T , (cid:98) T/ (cid:99) is independent of F (cid:98) T/ (cid:99) . E HAC estimation of asymptotic covariance matrix Note that the limiting distributions in Theorem B.2 are not particularly useful in practice since theasymptotic covariance matrices are unknown. We note that under suitable conditions the matricesmay be estimated by HAC-type estimators, see e.g. the seminal paper by Newey and West (1987).In particular, assuming that E [ (cid:107) Y † t (cid:107) (cid:15) ] < ∞ for some (cid:15) > 0, an application of Newey and West(1987, Theorem 2) yields that the matrix Γ † defined in (B.42) may be estimated byˆΓ † = ˆΩ † + N T (cid:88) j =1 w j ( N T ) ˆΩ † j , ˆΩ † j = 1 T T (cid:88) t =1 ˆ Y † t ˆ Y †(cid:48) t − j , (E.50)where w j ( N T ) are suitable weights and N T is an increasing sequence (depending on T ), and ˆ Y † t =( ˆ Y (cid:48) t , ˆ V t, , ˆ V t, ) (cid:48) with ˆ Y t = ( ˆ Y t,h ) h =0 ,...,m ,ˆ Y t,h = ( f ( R t ) − T T (cid:88) t =1 f ( R t ))( g ( R t − h ) − T T (cid:88) t =1 g ( R t )) − ˆ γ T,f ( R ) ,g ( R ) ( h )ˆ V t, = ( f ( R t ) − T T (cid:88) t =1 f ( R t )) − ˆ γ T,f ( R ) ,f ( R ) (0) , ˆ V t, = ( g ( R t ) − T T (cid:88) t =1 g ( R t )) − ˆ γ T,g ( R ) ,g ( R ) (0) . Remark E.1 The covariance matrix A Γ † A (cid:48) of the sample correlation function can be estimated by ˆ A ˆΓ † ˆ A (cid:48) with ˆΓ † given in (E.50) and ˆ A given by (B.44) with γ † replaced by ˆ γ † = ((ˆ γ T,f ( R ) ,g ( R ) ( h )) (cid:48) h =0 ,...,m , ˆ γ T,f ( R ) ,f ( R ) (0) , ˆ γ T,g ( R ) ,g ( R ) (0)) (cid:48) Remark E.2 In case we want to estimate the asymptotic covariance matrix Γ in (B.38) we mayapply an estimator similar to the one stated in (E.50) with ˆ Y † t replaced by ˆ Y t . Additional simulations results F.1 Size-adjusted powers for ρ (cid:48) R, | R | s sign ( R ) ( h ) [Figures 5-20 about here.] F.2 Coverage levels for ρ | R | p ( h ) [Figures 26-35 about here.] References Andrews, D. W. K. (1991), ‘Heteroskedasticity and autocorrelation consistent covariance matrixestimation’, Econometrica , 817–858.Baltussen, G., van Bekkum, S. and Da, Z. 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(2019), ‘Inference for the tail index of a GARCH(1,1) model and anAR(1) model with ARCH(1) errors’, Econometric Reviews , 151–169.35 a b l e : S i ze o f t e s t s f o r a b s e n ce o f a u t o c o rr e l a t i o n s , h = ρ R ( h ) ρ R, | R | . sign ( R ) ( h ) ρ (cid:48) R, | R | . sign ( R ) ( h ) ρ (cid:48) R, | R | . sign ( R ) ( h ) ρ R ( h ) ρ (cid:48) R , | R | . s i g n ( R ) ( h ) ρ (cid:48) R , | R | . s i g n ( R ) ( h ) ρ (cid:48) R , | R | . s i g n ( R ) ( h ) q A R C H ( ) , N ( , ) . . . . . . . . . . . . . . . . . . . . A R C H ( ) , t ( , . ) . . . . . . . . . . . . . . . . . . . . A R C H ( ) , t ( , . ) . . . . . . . . . . . . . . . . . . . . G J R - G A R C H , N ( , ) . . . . . . . . . . . . . . . . . . . . a b l e : E m p i r i c a l r e s u l t s , t e s t i n g f o r e ffi c i e n t m a r k e t h y p o t h e s i s , h = S e r i e s Estimateoftailindex CIfortailindex ˆ ρ R ( h ) CIfor ρ R ( h )(HAC) Interval(A.30)for ρ R ( h )( q =8) ˆ ρ (cid:48) R, | R | s sign ( R ) ( h ) Interval(A.30)for ρ (cid:48) R, | R | s sign ( R ) ( h )( q =8) ˆ ρ (cid:48) R, | R | . sign ( R ) ( h ) Interval(A.30)for ρ (cid:48) R, | R | . sign ( R ) ( h )( q =8) EstimateofMAC(5) CIforMAC(5)(HAC) Interval(A.30)forMAC(5)( q =8) S & P . [ . , . ] - . [ - . , - . ] ∗ [ - . , - . ] ∗ - . [ - . , . ] - . [ - . , . ] - . [ - . , - . ] ∗ [ - . , - . ] ∗ F T S E . [ . , . ] - . [ - . , - . ] ∗ [ - . , . ] - . [ - . , . ] - . [ - . , . ] - . [ - . , - . ] ∗ [ - . , - . ] ∗ D J E S I . [ . , . ] - . [ - . , . ][ - . , . ] - . [ - . , . ] - . [ - . , . ] - . [ - . , - . ] ∗ [ - . , . ] T O P I X . [ . , . ] . [ - . , . ][ - . , . ] . [ - . , . ] . [ - . , . ] - . [ - . , . ][ . , . ] A S X . [ . , . ] - . [ - . , . ][ - . , . ] - . [ - . , . ] - . [ - . , . ] - . [ - . , - . ] ∗ [ - . , . ] T S E . [ . , . ] - . [ - . , . ][ - . , . ] - . [ - . , . ] . [ - . , . ] - . [ - . , . ][ - . , . ] C A C . [ . , . ] - . [ - . , . ][ - . , . ] - . [ - . , . ] - . [ - . , . ] - . [ - . , - . ] ∗ [ - . , . ] D AX . [ . , . ] - . [ - . , . ][ - . , . ] - . [ - . , . ] - . [ - . , . ] - . [ - . , . ][ - . , . ] I B E X . [ . , . ] . [ - . , . ][ - . , . ] . [ - . , . ] . [ - . , . ] - . [ - . , - . ] ∗ [ - . , . ] M I B . [ . , . ] - . [ - . , . ][ - . , . ] - . [ - . , . ] - . [ - . , . ] - . [ - . , - . ] ∗ [ - . , - . ] ∗ A E X I nd e x . [ . , . ] - . [ - . , . ][ - . , . ] . [ - . , . ] . [ - . , . ] - . [ - . , . ][ - . , . ] O M X S t o c k h o l m . [ . , . ] . [ - . , . ][ - . , . ] . [ - . , . ] . [ - . , . ] - . [ - . , . ][ - . , . ] S M I . [ . , . ] . [ - . , . ][ - . , . ] . [ - . , . ] . [ - . , . ] - . [ - . , . ][ - . , . ] N i kk e i . [ . , . ] - . [ - . , - . ] ∗ [ - . , - . ] ∗ - . [ - . , . ] - . [ - . , . ] - . [ - . , - . ] ∗ [ - . , - . ] ∗ H S I . [ . , . ] - . [ - . , . ][ - . , . ] . [ . , . ] ∗ . [ . , . ] ∗ - . [ - . , . ][ - . , . ] N a s d a q . [ . , . ] - . [ - . , - . ] ∗ [ - . , - . ] ∗ - . [ - . , - . ] ∗ - . [ - . , . ] - . [ - . , - . ] ∗ [ - . , - . ] ∗ NY S E . [ . , . ] - . [ - . , - . ] ∗ [ - . , . ] - . [ - . , . ] - . [ - . , . ] - . [ - . , . ][ - . , . ] R u ss e ll . [ . , . ] - . [ - . , - . ] ∗ [ - . , . ] - . [ - . , . ] - . [ - . , . ] - . [ - . , - . ] ∗ [ - . , . ] S & P . [ . , . ] - . [ - . , . ][ - . , . ] . [ - . , . ] . [ - . , . ] - . [ - . , - . ] ∗ [ - . , - . ] ∗ K O S P I . [ . , . ] . [ - . , . ][ . , . ] ∗ . [ . , . ] ∗ . [ - . , . ] - . [ - . , . ][ - . , . ] K O S P I . [ . , . ] . [ - . , . ][ . , . ] ∗ . [ . , . ] ∗ . [ . , . ] ∗ - . [ - . , . ][ - . , . ] a b l e : E m p i r i c a l r e s u l t s , t e s t i n g f o r n o n - li n e a r i t y , h = S e r i e s Estimateoftailindex CIfortailindex ˆ ρ R ( h ) CIfor ρ R ( h )(HAC) CIfor ρ R ( h )( q =8) ˆ ρ | R | ( h ) CIfor ρ | R | ( h )(HAC) CIfor ρ | R | ( h )( q =8) ˆ ρ | R | . ( h ) CIfor ρ | R | . ( h )( q =8) ˆ ρ | R | . ( h ) CIfor ρ | R | . ( h )( q =8) S & P . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ - . , . ] F T S E . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) D J E S I . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) T O P I X . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ - . , . ] . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ - . , . ] A S X . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ - . , . ] T S E . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) C A C . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) D AX . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) I B E X . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) M I B . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) A E X I nd e x . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) O M X S t o c k h o l m . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) S M I . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) N i kk e i . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ - . , . ] . [ . , . ] (cid:63) [ - . , . ] . [ - . , . ] . [ - . , . ] H S I . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ - . , . ] . [ . , . ] (cid:63) [ - . , . ] . [ - . , . ] . [ - . , . ] N a s d a q . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) NY S E . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) R u ss e ll . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ - . , . ] S & P . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) K O S P I . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ - . , . ] . [ . , . ] (cid:63) [ - . , . ] . [ - . , . ] . [ - . , . ] a b l e : E m p i r i c a l r e s u l t s , t e s t i n g f o r n o n - li n e a r i t y , h = S e r i e s Estimateoftailindex CIfortailindex ˆ ρ R ( h ) CIfor ρ R ( h )(HAC) CIfor ρ R ( h )( q =8) ˆ ρ | R | ( h ) CIfor ρ | R | ( h )(HAC) CIfor ρ | R | ( h )( q =8) ˆ ρ | R | . ( h ) CIfor ρ | R | . ( h )( q =8) ˆ ρ | R | . ( h ) CIfor ρ | R | . ( h )( q =8) S & P . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) F T S E . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) D J E S I . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) T O P I X . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) A S X . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ - . , . ] T S E . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) C A C . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) D AX . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) I B E X . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) M I B . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) A E X I nd e x . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) O M X S t o c k h o l m . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) S M I . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) N i kk e i . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) H S I . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) N a s d a q . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) NY S E . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) R u ss e ll . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ - . , . ] S & P . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) K O S P I . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) a b l e : E m p i r i c a l r e s u l t s , t e s t i n g f o r n o n - li n e a r i t y , h = S e r i e s Estimateoftailindex CIfortailindex ˆ ρ R ( h ) CIfor ρ R ( h )(HAC) CIfor ρ R ( h )( q =8) ˆ ρ | R | ( h ) CIfor ρ | R | ( h )(HAC) CIfor ρ | R | ( h )( q =8) ˆ ρ | R | . ( h ) CIfor ρ | R | . ( h )( q =8) ˆ ρ | R | . ( h ) CIfor ρ | R | . ( h )( q =8) S & P . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) F T S E . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) D J E S I . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) T O P I X . [ . , . ] (cid:63) . [ - . , . ][ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) A S X . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ - . , . ] T S E . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) C A C . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) D AX . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) I B E X . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) M I B . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) A E X I nd e x . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) O M X S t o c k h o l m . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) S M I . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) N i kk e i . [ . , . ] (cid:63) . [ - . , . ][ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) H S I . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ - . , . ] N a s d a q . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) NY S E . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) R u ss e ll . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) S & P . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ . , . ] (cid:63) K O S P I . [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) [ . , . ] (cid:63) . [ . , . ] (cid:63) . [ - . , . ] ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ϕ Figure 1: Size-adjusted power for ARCH(1) with N (0 , 1) noise. ρ R ( h ) , HAC : , ρ R, | R | . sign ( R ) ( h ) , HAC : · ρ R, | R | . sign ( R ) ( h ) , HAC : , ρ R, | R | . sign ( R ) ( h ) , HAC : , ρ R ( h ), q = 8 : (cid:3) , ρ R, | R | . sign ( R ) ( h ), q = 8 : (cid:52) , ρ R, | R | . sign ( R ) ( h ), q = 8 : (cid:53) , ρ R, | R | . sign ( R ) ( h ), q = 8 : (cid:13) ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ϕ Figure 2: Size-adjusted power for ARCH(1) with t (3 , . 5) noise. ρ R ( h ) , HAC : , ρ R, | R | . sign ( R ) ( h ) , HAC : · ρ R, ( h ) , HAC : , ρ R, | R | . sign ( R ) ( h ) , HAC : , ρ R ( h ), q = 8 : (cid:3) , ρ R, | R | . sign ( R ) ( h ), q = 8 : (cid:52) , ρ R, | R | . sign ( R ) ( h ), q = 8 : (cid:53) , ρ R, | R | . sign ( R ) ( h ), q = 8 : (cid:13) ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ϕ Figure 3: Size-adjusted power for ARCH(1) with t (50 , . 5) noise. ρ R ( h ) , HAC : , ρ R, | R | . sign ( R ) ( h ) , HAC : · ρ R, ( h ) , HAC : , ρ R, | R | . sign ( R ) ( h ) , HAC : , ρ R ( h ), q = 8 : (cid:3) , ρ R, | R | . sign ( R ) ( h ), q = 8 : (cid:52) , ρ R, | R | . sign ( R ) ( h ), q = 8 : (cid:53) , ρ R, | R | . sign ( R ) ( h ), q = 8 : (cid:13) ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ϕ Figure 4: Size-adjusted power for GJR-GARCH(1,1,1) with N (0 , 1) noise. ρ R ( h ) , HAC : , ρ R, | R | . sign ( R ) ( h ) , HAC : · ρ R, | R | . sign ( R ) ( h ) , HAC : , ρ R, | R | . sign ( R ) ( h ) , HAC : , ρ R ( h ), q = 8 : (cid:3) , ρ R, | R | . sign ( R ) ( h ), q = 8 : (cid:52) , ρ R, | R | . sign ( R ) ( h ), q = 8 : (cid:53) , ρ R, | R | . sign ( R ) ( h ), q = 8 : (cid:13) ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ϕ Figure 5: Size-adjusted power for ARCH(1) with N (0 , 1) noise, ρ R ( h ).HAC: , q = 4 : (cid:3) , q = 8 : (cid:52) , q = 12 : (cid:53) , q = 16 : (cid:13) ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ϕ Figure 6: Size-adjusted power for ARCH(1) with N (0 , 1) noise, ρ (cid:48) R, | R | . sign ( R ) ( h ).HAC: , q = 4 : (cid:3) , q = 8 : (cid:52) , q = 12 : (cid:53) , q = 16 : (cid:13) ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ α Figure 7: Size-adjusted power for ARCH(1) with N (0 , 1) noise, ρ (cid:48) R, | R | . sign ( R ) ( h ).HAC: , q = 4 : (cid:3) , q = 8 : (cid:52) , q = 12 : (cid:53) , q = 16 : (cid:13) ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ϕ Figure 8: Size-adjusted power for ARCH(1) with N (0 , 1) noise, ρ (cid:48) R, | R | . sign ( R ) ( h ).HAC: , q = 4 : (cid:3) , q = 8 : (cid:52) , q = 12 : (cid:53) , q = 16 : (cid:13) ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ϕ Figure 9: Size-adjusted power for ARCH(1) with t (3 , . 5) noise, ρ R ( h ).HAC: , q = 4 : (cid:3) , q = 8 : (cid:52) , q = 12 : (cid:53) , q = 16 : (cid:13) ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ϕ Figure 10: Size-adjusted power for ARCH(1) with t (3 , . 5) noise, ρ (cid:48) R, | R | . sign ( R ) ( h ).HAC: , q = 4 : (cid:3) , q = 8 : (cid:52) , q = 12 : (cid:53) , q = 16 : (cid:13) ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ϕ Figure 11: Size-adjusted power for ARCH(1) with t (3 , . 5) noise, ρ (cid:48) R, | R | . sign ( R ) ( h ).HAC: , q = 4 : (cid:3) , q = 8 : (cid:52) , q = 12 : (cid:53) , q = 16 : (cid:13) ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ϕ Figure 12: Size-adjusted power for ARCH(1) with t (3 , . 5) noise, ρ R, | R | . sign ( R ) ( h ).HAC: , q = 4 : (cid:3) , q = 8 : (cid:52) , q = 12 : (cid:53) , q = 16 : (cid:13) ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ϕ Figure 13: Size-adjusted power for ARCH(1) with t (50 , . 5) noise, ρ (cid:48) R ( h ).HAC: , q = 4 : (cid:3) , q = 8 : (cid:52) , q = 12 : (cid:53) , q = 16 : (cid:13) ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ϕ Figure 14: Size-adjusted power for ARCH(1) with t (50 , . 5) noise, ρ (cid:48) R, | R | . sign ( R ) ( h ).HAC: , q = 4 : (cid:3) , q = 8 : (cid:52) , q = 12 : (cid:53) , q = 16 : (cid:13) ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ϕ Figure 15: Size-adjusted power for ARCH(1) with t (50 , . 5) noise, ρ (cid:48) R, | R | . sign ( R ) ( h ).HAC: , q = 4 : (cid:3) , q = 8 : (cid:52) , q = 12 : (cid:53) , q = 16 : (cid:13) ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ϕ Figure 16: Size-adjusted power for ARCH(1) with t (50 , . 5) noise, ρ (cid:48) R, | R | . sign ( R ) ( h ).HAC: , q = 4 : (cid:3) , q = 8 : (cid:52) , q = 12 : (cid:53) , q = 16 : (cid:13) ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ϕ Figure 17: Size-adjusted power for GJR-GARCH(1,1,1) with N (0 , 1) noise ρ R ( h ).HAC: , q = 4 : (cid:3) , q = 8 : (cid:52) , q = 12 : (cid:53) , q = 16 : (cid:13) ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ϕ Figure 18: Size-adjusted power for GJR-GARCH(1,1,1)with N (0 , 1) noise, ρ (cid:48) R, | R | . sign ( R ) ( h ).HAC: , q = 4 : (cid:3) , q = 8 : (cid:52) , q = 12 : (cid:53) , q = 16 : (cid:13) ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ϕ Figure 19: Size-adjusted power for GJR-GARCH(1,1,1) with N (0 , 1) noise, ρ (cid:48) R, | R | . sign ( R ) ( h ).HAC: , q = 4 : (cid:3) , q = 8 : (cid:52) , q = 12 : (cid:53) , q = 16 : (cid:13) ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ϕ Figure 20: Size-adjusted power for GJR-GARCH(1,1,1) with N (0 , 1) noise, ρ (cid:48) R, | R | . sign ( R ) ( h ).HAC: , q = 4 : (cid:3) , q = 8 : (cid:52) , q = 12 : (cid:53) , q = 16 : (cid:13) .0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 α Figure 21: Coverage level for ARCH(1) with N (0 , 1) noise, ρ R ( h ).HAC: , q = 4: , q = 8: · q = 12: , qq 1) noise, ρ R ( h ).HAC: , q = 4: , q = 8: · q = 12: , qq = 16: α Figure 22: Coverage level for ARCH(1) with N (0 , 1) noise, ρ | R | ( h ).HAC: , q = 4: , q = 8: · q = 12: , q = 16:53 .0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 α Figure 23: Coverage level for ARCH(1) with N (0 , 1) noise, ρ | R | . ( h ).HAC: , q = 4: , q = 8: · q = 12: , qq 1) noise, ρ | R | . ( h ).HAC: , q = 4: , q = 8: · q = 12: , qq = 16: α Figure 24: Coverage level for ARCH(1) with N (0 , 1) noise, ρ | R | . ( h ).HAC: , q = 4: , q = 8: · q = 12: , q = 16:54 .0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 α Figure 25: Coverage level for ARCH(1) with N (0 , 1) noise, ρ | R | . ( h ).HAC: , q = 4: , q = 8: · q = 12: , qq 1) noise, ρ | R | . ( h ).HAC: , q = 4: , q = 8: · q = 12: , qq = 16: α Figure 26: Coverage level for ARCH(1) with t (50 , . 5) noise, ρ R ( h ).HAC: , q = 4: , q = 8: · q = 12: , q = 16:55 .0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 α Figure 27: Coverage level for ARCH(1) with t (50 , . 5) noise, ρ | R | ( h ).HAC: , q = 4: , q = 8: · q = 12: , qq 5) noise, ρ | R | ( h ).HAC: , q = 4: , q = 8: · q = 12: , qq = 16: α Figure 28: Coverage level for ARCH(1) with t (50 , . 5) noise, ρ | R | . ( h ).HAC: , q = 4: , q = 8: · q = 12: , q = 16:56 .0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 α Figure 29: Coverage level for ARCH(1) with t (50 , . 5) noise, ρ | R | . ( h ).HAC: , q = 4: , q = 8: · q = 12: , qq 5) noise, ρ | R | . ( h ).HAC: , q = 4: , q = 8: · q = 12: , qq = 16: α Figure 30: Coverage level for ARCH(1) with t (50 , . 5) noise ρ | R | . ( h ).HAC: , q = 4: , q = 8: · q = 12: , q = 16:57 .0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 α Figure 31: Coverage level for ARCH(1) with t (3 , . 5) noise, ρ R ( h ).HAC: , q = 4: , q = 8: · q = 12: , qq 5) noise, ρ R ( h ).HAC: , q = 4: , q = 8: · q = 12: , qq = 16: α Figure 32: Coverage level for ARCH(1) with t (3 , . 5) noise, ρ | R | ( h ).HAC: , q = 4: , q = 8: · q = 12: , q = 16:58 .0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 α Figure 33: Coverage level for ARCH(1) with t (3 , . 5) noise, ρ | R | . ( h ).HAC: , q = 4: , q = 8: · q = 12: , qq 5) noise, ρ | R | . ( h ).HAC: , q = 4: , q = 8: · q = 12: , qq = 16: α Figure 34: Coverage level for ARCH(1) with t (3 , . 5) noise, ρ | R | . ( h ).HAC: , q = 4: , q = 8: · q = 12: , q = 16:59 .0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 α Figure 35: Coverage level for ARCH(1) with t (3 , . 5) noise, ρ | R | . ( h ).HAC: , q = 4: , q = 8: · q = 12: , qq