New robust inference for predictive regressions
NNew robust inference for predictive regressions βRustam Ibragimov π,π , Jihyun Kim π , Anton Skrobotov π,π π Imperial College Business School, Imperial College London π Toulouse School of Economics, University of Toulouse Capitole π Russian Presidential Academy of National Economy and Public Administration π Saint Petersburg University
Abstract
We propose two robust methods for testing hypotheses on unknown parameters of pre-dictive regression models under heterogeneous and persistent volatility as well as endoge-nous, persistent and/or heavy-tailed regressors and errors. The proposed robust testingapproaches are applicable both in the case of discrete and continuous time models. Bothof the methods use the Cauchy estimator to effectively handle the problems of endogene-ity, persistence and/or heavy-tailedness in regressors and errors. The difference betweenthe two methods is how the heterogeneous volatility is controlled. The first method relieson robust π‘ -statistic inference using group estimators of a regression parameter of interestproposed in Ibragimov and MΒ¨uller (2010). It is simple to implement, but requires the ex-ogenous volatility assumption. To relax the exogenous volatility assumption, we proposeanother method which relies on the nonparametric correction of volatility. The proposedapproaches perform well compared with widely used alternative inference procedures interms of their finite sample properties. Keywords : predictive regressions, robust inference, near nonstationarity, heavy tails,nonstationary volatility, endogeneity.
JEL Codes : C12, C22 β The authors thank Walter Distaso, Jean-Marie Dufour, Siyun He, Nour Meddahi, Mikkel Plagborg-M ΓΈ ller,Aleksey Min, Ulrich K. MΒ¨uller, Rasmus S. Pedersen, Artem Prokhorov, Rogier Quaedvlieg, Robert Taylor andthe participants at the Center for Econometrics and Business Analytics (CEBA, St. Petersburg State University)seminar series and the session on Econometrics of Time Series at the 12th World Congress of the EconometricSociety for helpful comments and suggestions. Rustam Ibragimov and Anton Skrobotovβ research for this paperwas supported in part by a grant from the Russian Science Foundation (Project No. 20-18-00365). Jihyun Kimis grateful to the French Government and the ANR for support under the Investissements dβAvenir program;Grant ANR-17-EURE-0010. Address correspondence to Rustam Ibragimov, Imperial College Business School,South Kensignton Campus, London SW7 2AZ, the United Kingdom; e-mail: [email protected]. a r X i v : . [ ec on . E M ] A ug Introduction
Many papers in the literature have focused on econometric analysis of predictive regressionsin economics and finance under the problems of endogeneity and potential nonstationarity inregressors and errors (see Phillips, 2015, for up-to-date review). One of the main and popularapproaches was proposed by Phillips and Magdalinos (2009). The authors suggest using self-generated instrumental variables (IVs) to eliminate the influence of endogeneity in predictiveregressions and conduct standard asymptotic inference based on normal distributions. Phillipsand Lee (2013), Kostakis et al. (2015) and Xu (2017) provide extensions of the approach inPhillips and Magdalinos (2009) to wider classes of models, including long-horizon predictabilitymodels and cointegration of an unknown form among predictors. Phillips and Lee (2016) analyseIV-based methods when some regressors may be integrated and some of the predictors aremildly explosive. Hosseinkouchack and Demetrescu (2016) investigate finite sample propertiesof the IV-based tests. The authors propose a bias corrected version of the test statistic, obtainasymptotic expansions to correct a limiting distribution, and provide approximations for itsquantiles. Demetrescu and Rodrigues (2017) propose another type of bias correction of theIV-based test statistic. Li et al. (2017) adopt a conceptually different approach using empiricallikelihood methods for uniform inference.Choi et al. (2016) propose an inference approach based on the Cauchy test with random timethat allows for the presence of (nearly) nonstationary dynamics of the (stochastic) volatilityprocess displayed by the innovations of the model. The IV-based inference proposed in Choiet al. (2016) uses the Cauchy estimator to eliminate the effects of endogeneity, and appliestime change to handle the problem of potential nonstationary in the volatility process (so that,volatility time is used instead of calendar time). The approach is based on the results that, incontrast to OLS estimators of the model parameters that have nonstandard asymptotics, theCauchy estimator remains asymptotically normal under the problems of near nonstationarity inthe regressors (see also So and Shin, 1999, Chang, 2002, 2012, for the use of Cauchy and otherIV estimators in testing for unit roots). The estimatorsβ consistent standard errors are employedto obtain standard normal convergence for the π‘ -statistics of the predictive regression modelparameters. The inference approaches based on Cauchy estimators are shown to have betterfinite sample properties than widely used alternative testing methods, including the Bonferroni π -test proposed by Campbell and Yogo (2006) and the restricted likelihood ratio test of Chenand Deo (2009). One of the limitations of the time change approach is that it is applicable only for the data A number of works in econometrics have also focused on robust inference using conceptually related signtests applied to different time series regression models under general assumptions (see, among others, Dufourand Hallin (1993), Campbell and Dufour (1995), So and Shin (2001), Brown and Ibragimov (2019), Kim andMeddahi (2020), and references therein). π‘ -test with the parameterβsgroup estimates and critical values of a Student- t distribution. This results in valid and in somesense efficient inference when the groups are chosen in a way that ensures that the parame-ter estimators are asymptotically independent, unbiased and Gaussian with possibly differentvariances. The π‘ -statistic robust inference approaches have been shown to have appealing finitesample performance for time series, panel, clustered and spatially correlated data exhibitingheterogeneity, dependence and heavy-tailedness of largely unknown form that are typical forreal-world financial and economic markets (see, in addition to the above references, the recentstudy by Esarey and Menger, 2019, the review and discussion of the approaches in Section 3.3 inIbragimov et al., 2015; and also Ibragimov and MΒ¨uller, 2016, for their extensions to robust testson equality of two parameters and the discussion of applications in the analysis of treatmenteffects and structural breaks, among other areas). In this paper, we propose two robust methods for testing hypotheses on parameters ofpredictive regression models under heterogeneous and persistent volatility as well as endogenous,persistent and/or heavy-tailed regressors and errors. The inference approaches proposed in the The π‘ -statistic robust inference approach proposed in Ibragimov and MΒ¨uller (2010) provides a formal jus-tification for the widespread FamaβMacBeth method for inference in panel regressions (see Fama and Mac-Beth, 1973). Following the method, one estimates the regression separately for each year, and then tests hy-potheses about the coefficient of interest using the π‘ -statistic of the resulting yearly coefficient estimates. TheFamaβMacBeth approach is a special case of the π‘ -statistic based approach to inference, with observations ofthe same year collected in a group. See, among others, Bloom et al. (2013), Krueger et al. (2017), Blinder and Watson (2016), Verner andGyongyosi (2018), Chen and Ibragimov (2019) and Gargano et al. (2019) for empirical applications of therobust inference approaches proposed in Ibragimov and MΒ¨uller (2010, 2016). The recent works by Pedersen(2019), Anatolyev (2019) and Ibragimov, Pedersen and Skrobotov (2019) provide applications of the approachesin robust inference on general classes of GARCH and AR-GARCH-type models exhibiting heavy-tailedness andvolatility clustering properties typical for real-world financial and economic markets. π‘ -statistic robust inference approach in Ibragimov and MΒ¨uller(2010) that is applied to asymptotically normal group Cauchy estimates of a predictive regres-sion parameter to handle the problems of heterogeneous volatility and persistence in regressorsand regression errors. In the context of time series predictive regressions (e.g., predictive re-gressions for stock returns), robust large sample inference is implemented as follows: The timeseries is partitioned into some number π β₯ groups of consecutive observations, the Cauchy es-timates of a predictive regression parameter are estimated for each group, and then a standard π‘ -test of level 5% with the resulting π Cauchy estimates of a regression parameter of interest isconducted.The above approach is simple to implement, but it requires the exogenous volatility assump-tion. To relax the exogenous volatility assumption, we propose another method which relies onthe nonparametric correction of volatility. The proposed methods perform well compared withwidely used alternative inference procedures in terms of their finite sample properties.The rest of the paper is organized as follows. Section 2 reviews the predictive regressionmodels and their assumptions. In Section 3, we propose robust inference methods and presentthe results on their asymptotic properties. Section 4 provides numerical results on finite sam-ple properties of the proposed robust inference approaches. Section 5 makes some concludingremarks. For simplicity, we first consider the case of predictive models without the intercept;and the analogues of inference methods proposed in the paper for predictive models with anintercept are provided in Appendix A. The proofs are provided in Appendix B.
Throughout the paper, we consider ( β± π‘ ) -adapted processes defined on a probability space (β¦ , β± π‘ , π ) equipped with an increasing filtration ( β± π‘ ) of sub- π -fields of β± π‘ . Our purpose is totest (un)predictability of the process ( π¦ π‘ ) (e.g., the time series of excess stock returns) basedon some covariate process ( π₯ π‘ ) (e.g., the time series of price-to-dividend ratios). As usual, weconsider the linear predictive regression model π¦ π‘ = πΌ + π½π₯ π‘ β + π’ π‘ , π‘ = 1 , ..., π. (1)The regression errors π’ π‘ are assumed to satisfy π’ π‘ = π£ π‘ π π‘ . with a volatility process π£ π‘ . We suppose that the following assumptions hold.4 ssumption 2.1. (a) ( π π‘ ) is a martingale-difference sequence (MDS) with respect to the filtra-tion ( β± π‘ ) with conditional variances πΈ ( π π‘ |β± π‘ β ) = 1 , and (b) ( π£ π‘ ) is an ( β± π‘ β ) -adapted processsuch that πΈ ( π’ π‘ |β± π‘ β ) = π£ π‘ > for all π‘ β₯ . The hypothesis of unpredictability of ( π¦ π‘ ) (e.g., that of stock returnsβ unpredictability)corresponds to the hypothesis π½ = 0 in predictive model (1). It is well-known that the standardOLS estimator of π½ is not Gaussian asymptotically under π» : π½ = 0 if either ( π₯ π‘ ) is (nearly)nonstationary (see Elliott and Stock, 1994, Phillips, 1987 b , Giraitis and Phillips, 2006, Phillipsand Magdalinos, 2007) or is stationary with infinite second moment (e.g., Granger and Orr,1972, Embrechts et al., 1997, Ibragimov et al., 2015, and references therein), even when π£ π‘ = π is constant.In this paper, we propose robust inference approaches based on the Cauchy estimator. Forsimplicity, we let πΌ = 0 , and, as usual, define the least square estimator Λ π½ and the Cauchyestimator Λ π½ , respectively, as Λ π½ = (οΈ π βοΈ π‘ =1 π₯ π‘ β )οΈ β π βοΈ π‘ =1 π₯ π‘ β π¦ π‘ and Λ π½ = (οΈ π βοΈ π‘ =1 | π₯ π‘ β | )οΈ β π βοΈ π‘ =1 π ππ ( π₯ π‘ β ) π¦ π‘ , where π ππ ( Β· ) is the sign function defined as π ππ ( π₯ ) = 1 for π₯ β₯ , and π ππ ( π₯ ) = β for π₯ < . Under π½ = 0 , the numerator of Λ π½ becomes βοΈ ππ‘ =1 π£ π‘ π π‘ , where π π‘ = π ππ ( π₯ π‘ β ) π π‘ is an MDS withrespect to β± π‘ with unit variance. Therefore, if π£ π‘ is observable and is bounded almost surely, onecan construct a robust test for the null hypothesis π» : π½ = 0 against the alternative π» : π½ ΜΈ = 0 using the statistic π‘ ( π£ ) = 1 π / π β βοΈ π‘ =1 π ππ ( π₯ π‘ ) π¦ π‘ +1 π£ π‘ +1 since π‘ ( π£ ) = {οΈ π β / βοΈ ππ‘ =1 π π‘ β π N (0 , under π½ = 0 , π½π β / βοΈ π β π‘ =1 | π₯ π‘ /π£ π‘ | + π β / βοΈ ππ‘ =1 π π‘ β π π ππ ( π½ ) Γ β under π½ ΜΈ = 0 by a martingale CLT, as long as π β / βοΈ π β π‘ =1 | π₯ π‘ /π£ π‘ | β π β under π½ ΜΈ = 0 .Let us define two continuous time partial sum processes ( π ππ , π ππ : 0 β€ π β€ by π ππ = 1 π / π π ] βοΈ π‘ =1 π π‘ and π ππ = 1 π / π π ] βοΈ π‘ =1 π π‘ , (2)with π π‘ = π ππ ( π₯ π‘ β ) π π‘ . The stocastic process ( π π , π π ) takes values in D R Γ R [0 , , where D πΈ [0 , denotes the space of c`adl`ag functions from [0 , to πΈ β R π for some positive integer π . In what For simplicity, we consider a model without intercept, and the inference procedures under πΌ ΜΈ = 0 arepresented in the Appendix. β π β denotes weak convergence with respect to the Skorokhod metric (e.g., Billingsley,2013).Under Assumption 2.1, both π π‘ and π π‘ are MDSs with respect to the filtration ( β± π‘ ) with πΈ ( π π‘ |β± π‘ β ) = πΈ ( π π‘ |β± π‘ β ) = 1 . Therefore, (οΈ π π , π π )οΈ β π ( π, π ) , in D R Γ R [0 , , where π and π are standard Brownian motions, by the usual functional centrallimit theorem. In general, E π π π π ΜΈ = 0 and E π π = π lim π ββ π β βοΈ ππ‘ =1 π ππ ( π₯ π‘ β ) .We define a stochastic process π π on D R + [0 , as π π ( π ) = π£ [ π π ] . We assume that π π has alimiting process π defined over β€ π β€ such that ( π π , π π , π π ) converges to ( π, π, π ) jointly.Specifically, we assume that the following assumption holds. Assumption 2.2.
There exists a positive process π on D R + [0 , such that ( π π ) β π.π . ( π ) in D R + [0 , and ( π π , π π , π π ) β π ( π, π, π ) in D R Γ R Γ R + [0 , , where π and π are Brownian motions with respect to the filtration to which π , π and π are adapted. The above assumptions hold for wide classes of models such as models with nonstationaryvolatility, regime switching and structural breaks in volatility. It also holds for the processeswith π£ π‘ = π ( π‘/π ) , where π ( π ) is a deterministic function on [0 , , considered by Cavaliereand Taylor (2007), ? , ? and ? , among others. The assumptions also hold for processes withnonstationary volatilities considered by Hansen (1995) and Chung and Park (2007), who assumethat π£ π‘ is a smooth positive transformation of a (near-)integrated process. One should notethat Assumption 2.1 on volatility is more general and allows the volatility to be stochastic anddiscontinuous, which are desirable properties for modelling financial returnsβ time series. Assumptions 2.1 and 2.2 rule out some cases of globally homoskedastic processes, such as stationary GARCHprocesses. However, it can be shown that the results stated in this paper continue to hold under the weakercondition that ( π π‘ ) is an MDS with respect to ( β± π‘ ) with conditional heteroskedasticity, say πΈ ( π π‘ |β± π‘ β ) = β π‘ ,satisfying some regularity conditions (see, e.g, Chang and Park (2002), Assumption 1; Boswijk et al. (2016),Assumption 2 (b)). Under these more general conditions, one may assume that ( π π‘ ) is a stationary GARCHprocess. The resulting conditional heteroskedasticity properties of the regression error ( π’ π‘ ) become πΈ ( π’ π‘ |β± π‘ β ) = π£ π‘ β π‘ , where π£ π‘ and β π‘ correspond to, respectively, persistent and transient components in the conditionalvolatility of π’ π‘ . Assumption 2.2 is a simplified version of the condition π£ [ π π ] /π π β π π π considered by ? . We rule out theexplosive volatility settings with π π β β , and consider the stable volatility processes with π π = 1 for simplicity.The results in the paper can be obtained under the explosive volatility assumption with π π β β at the cost ofa more involved analysis. ( π π , π π , π π ) and associated stochastic integrals as well as the OLS ( Λ π½ ) and Cauchy ( Λ π½ ) estima-tors. In the sequel, for two sequences of random variables π΄ π and π΅ π , the notation β π΄ π β π π΅ π "means that both π΄ π /π΅ π and π΅ π /π΄ π are stochastically bounded. Lemma 2.1.
Under Assumptions 2.1 and 2.2, one has (οΈ π π , π π , π π , β«οΈ π π ( π ) ππ ππ , β«οΈ π π ( π ) ππ ππ )οΈ β π (οΈ π, π, π, β«οΈ π ( π ) ππ π , β«οΈ π ( π ) ππ π )οΈ in D R Γ R Γ R + Γ R Γ R [0 , . Lemma 2.2.
Let Assumptions 2.1 and 2.2 hold. Suppose that there exist two diverging sequences ( π π ) and ( π π ) of positive real numbers such that βοΈ ππ‘ =1 | π₯ π‘ β | β π π π and βοΈ ππ‘ =1 π₯ π‘ β β π π π . Thenone has Λ π½ β π½ = π π ( π β / π ) and Λ π½ β π½ = π π ( π / /π π ) . Example 2.1.
For some processes, one may explicitly obtain the sequences ( π π , π π ) for whichLemma 2.2 holds.(a) For weakly stationary processes ( π₯ π‘ ) with πΈ | π₯ π‘ | < β , π π = π π = π ;(b) For the case of unit root and near unit root time series ( π₯ π‘ ) , π π = π / and π π = π (seePhillips, 1987 a , 1987 b , Giraitis and Phillips, 2006, Phillips and Magdalinos, 2007, Ibragimovand Phillips, 2008, and references therein);(c) For fractionally integrated πΌ ( π ) processes ( π₯ π‘ ) with < π < / , π π = π π +3 / β π ( π ) and π π = π π +2 β π ( π ) , for some slowly varing functions β π , β π (see Baillie, 1996; Lemma 3.4 inPhillips, 1999; Kim and Phillips, 2006, and references therein);(d) For πΌ -stable processes ( π₯ π‘ ) with < πΌ < : π π = π and π π = π /πΌ β ( π ) for some slowlyvarying function β (see Embrechts et al., 1997, and Phillips and Solo, 1992);(e) For πΌ -stable ( π₯ π‘ ) with < πΌ < : π π = π /πΌ β ( π ) for some slowly varying function β , and π π = π π (see Embrechts et al., 1997, Phillips and Solo, 1992, and references therein). We propose two new methods for robust inference on the parameter π½ in predictive regression(1). The first method relies on the π‘ -statistic based robust inference using group estimates ofthe parameter π½ proposed in Ibragimov and MΒ¨uller (2010) (see also Ibragimov and MΒ¨uller,2016, and Section 3.3 in Ibragimov et al., 2015). The second method relies on nonparametric7stimation of the volatility process π . The first approach is simple to implement, and doesnot require the estimation of π . For the approach, however, it is required that independencebetween π and π holds as in the following assumption. Assumption 3.1. (Volatility exogeneity) The processes π and π in Assumption 2.2 are inde-pendent each other. Under Assumption 3.1, β«οΈ π ( π ) ππ π has a scale mixture of normals (mixed normal) distribu-tion: β«οΈ π π ( π ) ππ π = π MN (0 , β«οΈ π π ( π ) ππ ) (here and in what follows, MN (0 , Ξ£) denotes a normalrandom vector with the random variance-covariance matrix Ξ£ ). If the independence condition,Assumption 3.1, is violated, then β«οΈ π ( π ) ππ π becomes a non-Gaussian martingale in general. The robust test is based on the statistic π := 1 β π π βοΈ π‘ =1 π ππ ( π₯ π‘ β ) π¦ π‘ , (3)which is the (scaled) numerator of the Cauchy estimator. Consider a partition of the data intosome fixed number π β₯ of (approximately) equal groups of consecutive observations: Theobservations with the index π‘, π‘ = 1 , ..., π, is element of group π if π‘ β π’ π = { π : ( π β π /π <π β€ ππ /π } , π = 1 , . . . , π. Following the π‘ -statistic approaches, one calculates π statistics givenby numerators of the Cauchy estimator in (3) using the observations in each of the groups. For π = 1 , . . . , π, denote by π π the statistics π calculated using the observations in the π -th group.The following proposition justifies applicability of the π‘ -statistic based robust inferenceapproach in Ibragimov and MΒ¨uller (2010) in the settings considered. Proposition 3.1.
Let Assumptions 2.1, 2.2 and 3.1 hold. For any fixed π β₯ , one has ( π , π , . . . , π π ) β π MN (0 , Ξ£ π ) , where Ξ£ π is a π -dimensional diagonal matrix with π th element β«οΈ π/π ( π β /π π ( π ) ππ . Denote by π‘ π ( π ) the π‘ -statistic calculated using the π group numerators π π , π = 1 , ..., π, ofthe Cauchy estimators: π‘ π ( π ) = β π Γ π /π π , (4)with π = π β βοΈ ππ =1 π π and π π = ( π β β βοΈ ππ =1 ( π π β π ) . Following Ibragimov and MΒ¨uller (2010), convergence of the group statistics π π to scale mix-ture of normals in Proposition 3.1 allows one to construct asymptotically valid test of level πΌ β€ . of the unpredictability null hypothesis π» : π½ = 0 against π» : π½ ΜΈ = 0 by rejecting8 when the absolute value | π‘ π ( π ) | of the π‘ -statistic π‘ π ( π ) exceeds the (1 β πΌ/ percentile of aStudent- π‘ distribution with π β degrees of freedom. The proposed approach is simple to implement. Moreover, it is robust to volatility persis-tence as well as to distributional properties (e.g., moment assumptions and heavy-tailedness)in the covariate process ( π₯ π‘ ) as the employed numerators π π of group Cauchy estimators of π½ are based on signs of π₯ π‘ . However, the volatility exogeneity assumption may not hold when,in particular, there exist leverage effects. As an alternative method, we propose another in-ference approach in the next section. It will be shown that the alternative test is robust notonly to the distributional properties of ( π₯ π‘ ) and to persistence in volatility but also to volatilityendogeneity. This section proposes tests for (un)predictability in regressions (1) which are robust tovolatility endogeneity. The testing approaches rely on a nonparametric estimator Β― π for π , andthe test statistics have the following form: π‘ (Β― π ) = 1 β π π β βοΈ π‘ =1 π ππ ( π₯ π‘ ) π¦ π‘ +1 Β― π ( π‘/π ) . (5) One-sided tests are conducted in a similar fashion. As discussed in Ibragimov and MΒ¨uller (2016), ? , the π‘ -statistic approaches to asymptotic robust inferenceabout a parameter of interest are valid under the assumptions that the group estimates of the parameter areasymptotically normal or scale mixtures of normals and are asymptotically independent. These assumptions aresatisfied under many dependence and heterogeneity settings of largely unknown form typically encountered inpractice. From the results in Section 4.2 in ? it further follows that. if only the above regularity assumptions areimposed on the structure of dependence and heterogeneity in the model considered, then it is not possible to usedata-driven methods to determine an optimal number of groups π. At the same time, according to the numericalresults in Ibragimov and MΒ¨uller (2016), ? and Section 3.3 in Ibragimov et al. (2015), for many dependence andheterogeneity settings considered in the literature and typically encountered in practice for time series, panel,clustered and spatially correlated data, the choice π = 4 , or π = 16 leads to robust tests with attractive finitesample performance. One can show that the π‘ -statistic approach to robust inference in predictive regressions is also valid if,instead of Assumptions 2.2 and 3.1, the errors ( π’ π‘ ) in regression (1) are assumed to follow a stationary GARCHprocess (so that, e.g., in the case of GARCH(1, 1) process, the dynamics of the volatility ( π£ π‘ ) is given by π£ π‘ = π + πΎ π’ π‘ β + πΎ π£ π‘ β , π > , πΎ , πΎ β₯ ). Namely, using the martingale CLT and coupling arguments similarto Pedersen (2019) and Ibragimov, Pedersen and Skrobotov (2019), one can show that the results in Proposition3.1 on weak convergence and asymptotic independence of group statistics π π , π = 1 , ..., π, hold under appropriateconditions on the GARCH parameters in the dynamics of the volatility ( π£ π‘ ) and moment assumptions on theinnovations ( π π‘ ) of the GARCH process ( π’ π‘ ) .
9n our framework, the adaptedness of a volatility estimator, Β― π , is important and simplifies theproof of the asymptotic normality of π‘ (Β― π ) under π½ = 0 . Suppose that Β― π ( π ) is adapted to ( β± [ π π ] ) for β€ π β€ , and Β― π ( π ) β π π ( π ) uniformly in [0 , . Then, under the condition ( π π ) β π.π . ( π ) in Assumption 2.2, it follows from Lemma 2.1 and the continuous mapping theorem that π‘ (Β― π ) = 1 β π π β βοΈ π‘ =1 π ππ ( π₯ π‘ ) π’ π‘ +1 Β― π ( π‘/π ) = 1 β π π β βοΈ π‘ =1 π ππ ( π₯ π‘ ) π π‘ +1 π π (( π‘ + 1) /π )Β― π ( π‘/π ) β π β«οΈ π ( π ) π ( π ) ππ π = N (0 , . The weak convergence requires the adaptedness of Β― π ( π‘/π ) to the filtration β± π‘ (see, e.g., Theorem4.6 in Kurtz and Protter, 1991, and also the analysis of convergence to stochastic integrals usingmartingale convergence methods in Ibragimov and Phillips, 2008).To construct an adapted volatility estimator, we consider a recursive least square estimator Λ π½ ( π ) for π β [0 , defined as Λ π½ ( π ) = βοΈ [ π π ] π‘ =1 π₯ π‘ β π¦ π‘ βοΈ [ π π ] π‘ =1 π₯ π‘ β for /π β€ π β€ , and Λ π½ ( π ) = 0 for β€ π < /π . It is important to note that ( Λ π½ ( π ) , β€ π β€ is adapted to the filteration ( β± [ π π ] , β€ π β€ for all π > . Consequently, the correspondingresiduals Λ π’ π‘ ( π ) = π¦ π‘ β Λ π½ ( π ) π₯ π‘ β are also adapted to ( β± [ π π ] ) as long as π‘ β€ [ π π ] . Define a (partial sum) stochastic process π π ( π ) for π β [0 , by π π ( π ) = [ π π ] βοΈ π‘ =1 π₯ π‘ β . We suppose that the following assumption holds.
Assumption 3.2.
There exists a diverging sequence ( π π ) of positive real numbers such that π π π π β π π, for some positive stochastic process π ( π ) for π β [0 , . One has the following proposition.
Proposition 3.2.
Let Assumptions 2.1, 2.2 and 3.2 hold. If β β and βπ β β as π β β ,then sup β β€ π β€ | Λ π½ ( π ) β π½ | = π π (οΈ (log( π ) /π π ) / )οΈ . For the robust tests, we consider two nonparametric estimators Λ π and Λ π defined, respec-tively, by Λ π ( π ) = βοΈ ππ‘ =1 Λ π’ π‘ πΎ β ( π β π‘/π ) βοΈ ππ‘ =1 πΎ β ( π β π‘/π ) , β β€ π β€
1; Λ π ( π ) = Λ π ( β ) , β€ π < β, Λ π’ π‘ = π¦ π‘ β Λ π½π₯ π‘ β , and Λ π ( π ) = βοΈ ππ‘ =1 Λ π’ π‘ ( π ) πΎ β ( π β π‘/π ) βοΈ ππ‘ =1 πΎ β ( π β π‘/π ) , β β€ π β€
1; Λ π ( π ) = Λ π ( β ) , β€ π < β, with Λ π’ π‘ ( π ) = π¦ π‘ β Λ π½ ( π ) π₯ π‘ β . Here πΎ β ( π‘ ) = πΎ ( π‘/β ) with a kernel function πΎ and bandwidth β .We suppose that the kernel function πΎ satisfies the following assumption. Assumption 3.3. (a) πΎ has a compact support [0 , , (b) sup π | πΎ ( π ) | < β , and (c) | πΎ ( π ) β πΎ ( π ) | β€ πΆ | π β π | for all π, π β [0 , and some constant πΆ > . Part (a) of Assumption 3.3 implies that πΎ is a one-side kernel. Under this condition, onlythe residuals (Λ π’ π‘ ( π ) , π‘ β€ [ π π ]) are used in the volatility estimator Λ π ( π ) . Consequently, (Λ π ( π )) is adapted to ( β± [ π π ] ) since (Λ π’ π‘ ( π ) , π‘ β€ [ π π ]) are ( β± [ π π ] ) -adapted. Similarly, only the residuals (Λ π’ π‘ , π‘ β€ [ π π ]) are used in the volatility estimator Λ π ( π ) . However, (Λ π’ π‘ , π‘ β€ [ π π ]) are not adaptedto ( β± [ π π ] ) for any π < since all of the observations ( π¦ π‘ , π₯ π‘ β ) , π‘ = 1 , , Β· Β· Β· , π in the sample areused in the least square estimator Λ π½. In the rest of this section, we analyze the convergences of the volatility estimators Λ π and Λ π , and use these results to show the validity of the proposed test with the test statistic π‘ (Λ π ) in(5) based on the adapted volatility estimator Λ π . Moreover, we show the asymptotic normality,under π½ = 0 , of π‘ (Λ π ) with the non-adapted volatility estimator Λ π by approximating it by π‘ (Λ π ) .In principle, one can directly prove the validity of the test with π‘ (Λ π ) without introducing theintermediate step for the asymptotic equivalence of π‘ (Λ π ) and π‘ (Λ π ) . It seems that the presentedarguments for asymptotic results are more constructive than the direct analysis of convergenceof π‘ (Λ π ) . In the arguments, the test statistic π‘ (Λ π ) is decomposed into the martingale part π‘ (Λ π ) andthe non-martingale part π‘ (Λ π ) β π‘ (Λ π ) , and then each term is analyzed separately thus emphasizing,in particular, the martingale convergence essence of the asymptotic results obtained. We first provide a set of sufficient conditions under which sup π β I β | Λ π ( π ) β π ( π ) | β π and sup π β I β | Λ π ( π ) β π ( π ) | β π , for a measurable set I β β [0 , such that the Lebesgue measure of { [0 , β I β } converges to zeroas β β . We then use the results to construct the robust inference based on the test statistic π‘ (Β― π ) in (5) for Β― π = Λ π, Λ π .We suppose that the following conditions hold. See also Ibragimov and Phillips (2008) for application of general martingale convergence results in theunification of the asymptotic analysis of stationary, explosive, unit root, and local to unity autoregressions, aswell as some general nonlinear time series regressions. ssumption 3.4. (a) sup π‘ β₯ E ( | π’ π‘ | π |β± π‘ β ) < β a.s., for some π > such that π log π β π / ( βπ ) π = π (1) for some β β and β π β β ; (b) π has finitely many jumps in [0 , with probability one;(c) There exists a sequence ( π β ) of positive real numbers satisfying sup π β [0 , sup π β πβ β€ π <π | π ( π β ) β π ( π ) | = π π ( π β ) for some π > (d) Let π½ be the set of volatility discontinuity points: π½ = { π β [0 , | π ( π ) ΜΈ = π ( π β ) } . Further, let π½ β be the collection of β -neighborhoods of discontinuity points, i.e., π½ β = { ( π β β, π + β ) | π β π½ } β© [0 , . There exists a sequence ( π β,π ) of positive real numbers satisfying sup π β [0 , β π½ β | ( π π ) ( π ) β π ( π ) | = π π ( π β,π ); The conditions in part (a) of Assumption 3.4 are commonly used in the literature on uniformconvergence of nonparametric estimators. Under log
π / ( β π ) = π (1) , part (a) of Assumption3.4 holds if sup π‘ β₯ E ( | π’ π‘ | |β± π‘ β ) < β with probability one.In the proposed robust tests, we allow for jumps in the limiting volatility process π . Thecondition in part (b) of Assumption 3.4 implies that the jump component of π is of finiteactivity thus allowing one to effectively handle the discontinuities in the volatility process.Part (c) of Assumption 3.4 provides a modulus of continuity of π over its continuous points.Part (d) of the assumption provides a convergence rate of | ( π π ) β π | over continuity pointsof π .The following convergence results for Λ π and Λ π hold. Theorem 3.3.
Let Assumptions 2.1, 2.2 and 3.2-3.4 hold. If β π β β , then sup π β [ β, β π½ β | Λ π ( π ) β π ( π ) | = π π ( π β ) + π π ( π β,π ) + π π (οΈ log / ( π ) / ( βπ ) )οΈ , sup π β [ β, β π½ β | Λ π ( π ) β π ( π ) | = π π ( π β ) + π π ( π β,π ) + π π (log( π ) / ( βπ )) . In Theorem 3.3, the rate of convergence for π β π½ is not clear. Under an additional condition,the following corollary provides uniform convergence rates over the unit interval including π β π½ . Corollary 3.4.
Let the assumptions in Theorem 3.3 hold. If, in addition, the following conditionholds:(e) There exists a sequence ( π π ) of positive real number satisfying sup β€ π β€ | ( π π ) ( π ) β π ( π ) | = π π ( π π ) . hen one has sup β β€ π β€ | Λ π ( π ) β π ( π ) | = π π ( π β ) + π π ( π π ) + π π (οΈ log / ( π ) / ( βπ ) )οΈ , sup β β€ π β€ | Λ π ( π ) β π ( π ) | = π π ( π β ) + π π ( π π ) + π π (log( π ) / ( βπ )) . Condition (e) in Corollary 3.4 relies on the uniform metric, and is stronger than condition(d) in Assumption 3.4.
We consider the asymptotic behavior of the test statistic π‘ (Β― π ) in (5) with Β― π = Λ π * , where Λ π * = max { Λ π, π β,π } for a sequence ( π β,π ) of positive real numbers. We note again that Λ π ( π‘/π ) isadapted to β± π‘ , and so is Λ π * ( π‘/π ) . Therefore, π’ π‘ +1 / Λ π * ( π‘/π ) is an MDS with respect to β± π‘ with E [ π’ π‘ +1 / Λ π * ( π‘/π ) | β± π‘ ] = 0 .We suppose that the following assumption holds. Assumption 3.5. (a) For some π > , one has π ( π ) β₯ π for all β€ π β€ (b) π β,π β π β (0 , π ) , β/π β,π β and log ( π ) / ( π β,π β π ) β . Part (a) in Assumption 3.5 is standard. Part (b) of the assumption provides sufficientconditions to ensure the estimated volatility Λ π * is large enough so that the test statistic π‘ (Λ π * ) is well behaved. Theorem 3.5.
Let the conditions in Theorem 3.3, and Assumption 3.5 hold. Further, let π β , π β,π β as β β and π β β . One has(a) If π½ = 0 , then π‘ (Λ π * ) β π N (0 , . (b) If π½ ΜΈ = 0 and βοΈ π β π‘ =1 | π₯ π‘ | / β π β π β , then | π‘ (Λ π * ) | β π β . Hansen (1995) provides a nonparametric GLS method for regression models with nonsta-tionary volatility using the estimator Λ π to correct the heteroskedasticity. In this case, as wasmentioned above, Λ π ( π‘/π ) is not adapted to β± π‘ , and hence, π’ π‘ +1 / Λ π ( π‘/π ) is not an MDS withrespect to β± π‘ . If one constructs a statistic π‘ (Λ π * ) = π β βοΈ π‘ =1 π ππ ( π₯ π‘ ) π¦ π‘ +1 Λ π * ( π‘/π ) , Λ π * = max { Λ π, π β,π } , then the arguments for asymptotic normality of π‘ (Λ π ) β π N (0 , cannot rely on direct application of martingale convergence results. The following theorem shows the validity of the test with π‘ (Λ π * ) by approximating it by π‘ (Λ π * ) instead of applying a martingale CLT directly to π’ π‘ +1 / Λ π ( π‘/π ) . Theorem 3.6.
Under the conditions in Theorem 3.5, π‘ (Λ π * ) is asymptotically equivalent to π‘ (Λ π * ) , that is π‘ (Λ π * ) = π‘ (Λ π * )(1 + π π (1)) for π½ β R . One should note that that the assumptions on the limiting volatility π are more generalthan those in Hansen (1995) and other works in the literature on the topic. In particular, theassumptions in Hansen (1995) do not allow for structural changes or regime switching in thevolatility process as the limiting volatility is assumed to have continuous sample paths almostsurely. In contrast, the limiting volatility is allowed to have an arbitrary number of jumps inthis paper, and hence, structural changes or regime switching are allowed. To our knowledge,the current paper is the first one in the literature that considers nonparametric estimation ofpersistent volatility allowing an arbitrary number of jumps. This section provides the numerical results on finite sample performance of the proposedrobust π‘ -statistic based tests π‘ π ( π ) , π β { , , , } , as well as the robust tests based on con-sistent volatility estimation π‘ (Λ π ) (with adapted volatility estimator) and π‘ (Λ π ) (with nonadaptedvolatility estimator). We present the comparisons of the finite sample properties of the testswith the βidealβ π‘ -test π‘ ( π ) that uses the true π½ in estimation of π , the test proposed by Choiet al. (2016) (denoted as Cauchy RT; RT for random time) and two other tests also consideredin Choi et al. (2016): the Bonferroni π -test of Campbell and Yogo (2006) (denoted as BQ) andthe restricted likelihood ratio test of Chen and Deo (2009) (denoted as RLRT). In the analysisof the testsβ performance, we follow Choi et al. (2016) and use recursive de-meaning to allowfor the non-constant mean. In particular, the arguments for Theorem 3 in Hansen (1995) on asymptotic normality of the π‘ -statistic ona predictive regression parameter based on the adjusted nonparametric volatility estimator Λ π ππ therein appearto be incomplete as Λ π ππ is not adapted to the filtration ( β± ππ ) dealt with in the paper not only for π < π but alsofor π β₯ π . However, the conclusion of the theorem in the paper remains valid and can be shown in a similarmanner to our approaches in Theorems 3.5 and 3.6. We note again that stationary GARCH-type models do not satisfy Assumptions 2.1 and 2.2, and hence, theasymptotic validity of the tests based on (full-sample) π‘ -statistics π‘ (Λ π * ) and π‘ (^ π * ) with nonparametric estimatorsof volatility is not formally established in the case of a stationary GARCH-type ( π’ π‘ ) . As noted in footnote 4,however, one may allow those models in the tests with nonparametric volatility estimators and show theirasymptotic validity. These extensions are left for future research, and will be present elsewhere.
14s in Choi et al. (2016), we first simulate our DGP in continuous time. The data is generatedusing the following DGP: ππ π‘ = Β― π½π π π‘ ππ‘ + ππ π‘ , ππ π‘ = π π‘ (οΈ ππ π‘ + β«οΈ R π₯ Ξ( ππ‘, ππ₯ ) )οΈ , (6) ππ π‘ = β Β― π π π π‘ ππ‘ + π π‘ ππ π‘ , (7)where π π‘ and π π‘ are Brownian motions with πΈ ( π π‘ π π‘ ) = β . π‘ . We set the constant term inthe predictive regression to be zero without loss of generality and use recursive de-meaning. Weassume that the continuous time models are observed at πΏ -intervals over π years with πΏ = 1 / ,which corresponds to daily observations of size π . The volatility process considered in the numerical results is assumed to follow one of thefollowing models:β’ Model CNST.
Constant volatility : π π‘ = π , π = 1 .β’ Model SB. Structural break in volatility : π + ( π β π )1 { π‘/π β₯ / } with π = 1 and π = 4 .β’ Model GBM. Geometric Brownian motion : ππ π‘ =
12 Β― π π π π‘ ππ‘ + Β― π β π π π‘ ππ π‘ , where π π‘ is aBrownian motion with πΈ ( π π‘ π π‘ ) = β . π‘ , and Β― π = 9 .β’ Model RS. Regime switching : π π‘ = π (1 β π π‘ ) + π π π‘ , where π π‘ be a homogeneous Markovprocess indicating the current state of the world which is independent of both π π‘ and π π‘ with the state space { , } and the transition matrix π π‘ = (οΈ . . . . )οΈ + (οΈ . β . β . . )οΈ exp (οΈ β Β― ππ π‘ )οΈ , where Β― π = 60 , π = 1 and π = 4 . The process π π‘ is initialized by its invariant distribution.We set the number of years π β { , , } (which correspond to 60, 240 and 600 monthly data)and consider the values Β― π β { , , } for the persistence parameter Β― π of π π‘ in (7).Importantly, as indicated before, in contrast to the Cauchy RT test in Choi et al. (2016),the robust tests proposed in this paper are applicable, not only in the continuous time models,but also in the discrete time framework. We consider the following discrete time models in theanalysis of finite sample performance of the tests: π¦ π‘ = Β― π½π π₯ π‘ β + π π,π‘ π π‘ , π₯ π‘ = (οΈ β Β― π π )οΈ π₯ π‘ β + π π,π‘ π π‘ , (8) The set-up is similar to Choi et al. (2016); see that paper for further details on generating the continuoustime models. π‘ = 2 , . . . , π , where π β { , , } (the same number of monthly observations as incontinuous time simulations) and the same values of Β― π½ and Β― π . Here the innovations ( π π‘ , π π‘ ) areassumed to be multivariate normal with the correlation coefficient β π π,π‘ and π π,π‘ , we consider three specifications: Model CNSTand Model SB as in the continuous time setup, and GARCH volatility dynamics with π π,π‘ = 1 + πΌπ π‘ β + π½π π,π‘ β , π π,π‘ = 1 + πΌπ π‘ β + π½π π,π‘ β . In the numerical analysis, we consider the ARCH(1) processes with π½ = 0 , πΌ = 0 . (whichhas a stationary distribution with an infinite fourth moment); π½ = 0 , πΌ = 0 . (which impliesan infinite third moment); IGARCH(1,1) models with πΌ = 0 . , π½ = 0 . and πΌ = 0 . , π½ = 0 . (in the models, the second moments are infinite). In this section, we analyse finite sample size properties of the (un)predictability tests bysetting Β― π½ = 0 in regression models (6) and (8). The numerical results on finite sample sizeproperties are presented in Tables 1-3.Table 1 provides the finite sample size results for models CNST, SB, GBM and RS. Thefinite sample size values for the OLS, BQ, RLRT and Cauchy RT tests are exactly the same asthose reported in Choi et al. (2016). These numerical results show that the size of the OLS, BQand RLRT tests is highly distorted for most of the time-varying volatility models considered.The π‘ -statistic based tests π‘ π ( π ) have good size control for moderate π such as π = 12 and π = 16 (the tests appear to be somewhat undersized for smaller π ). The size of π‘ ( π ) and π‘ ( π ) is close to the nominal level and is competitive with the Cauchy RT test for all DGPs. We notethat in the Geometric Brownian Motion case, when the assumption of volatility exogeneity isviolated, one observes slight conservative size distortions in the robust π‘ -statistic based tests,but, in general, the π‘ ( π ) and π‘ ( π ) tests are also competitive with the Cauchy RT test in thiscase as well.The size of the tests with the nonparametric estimator of the volatility process is close tothe nominal level under the βidealβ volatility estimator with the true π½. If one uses the adaptedor non-adapted volatility estimators (the tests based on π‘ (Λ π ) and π‘ (Λ π ) , respectively), the sizedistortions are higher, but they are not severe and decrease as the sample size increases. The size See, among others, Mikosch and StΛaricΛa (2000), Davis and Mikosch (1998), Pedersen (2019), Ibragimov,Pedersen and Skrobotov (2019), and references therein for the results on moment properties of GARCH processesand their importance in robust econometric inference, including that on autocorrelation functions. Note that, as discussed at footnotes 4, 5 and 8, one may establish the validity of the π‘ -statistic robustinference approach and the inference approach based on nonparametric volatility estimators under the consideredGARCH dynamics in the regression errors π’ π‘ = π π,π‘ π π‘ , π‘ (Λ π ) based on the non-adapted volatility estimator for which validity is establishedin Theorem 3.6, is uniformly better than that for the test based on π‘ (Λ π ) .As mentioned before, the Cauchy RT test is inapplicable in the discrete time settings.Table 2 provides the numerical results on finite sample size properties of all the tests exceptCauchy RT in the discrete time settings with Model CNST and Model SB. The quantitativeand qualitative comparisons of the size properties of the tests in the table are similar to thecontinuous time case. Table 3 provides the numerical results on finite sample properties of alltests except Cauchy RT in the case of the discrete time settings with GARCH processes. Thecomparisons and conclusions on the finite sample size performance of the tests in the table issimilar to those for Tables 1 and 2. Figures 1-10 present the finite sample power properties of the tests considered. In our simu-lations, we consider the DGPs in (6) in continuous time and (8) in discrete time with Β― π½ rangingfrom 0 to 20. All the power curves presented in the figures are size-adjusted. Taking into accountthe results on finite sample size performance of the tests considered and their comparisons, wemainly focus on the three tests: the Cauchy RT, the robust π‘ -statistic based test with π‘ ( π ) ,and the test based on π‘ (Λ π ) with non-adaptive volatility estimation, in the analysis of finitesample power properties. For comparison, the analysis also provides the numerical results onfinite sample power of the OLS, BQ and RLRT tests.Figure 1 presents the results on finite sample power properties of the tests for the constantvolatility case. One can see that the power curves of the Cauchy RT test, the robust π‘ -statistictest with π‘ ( π ) and those of the test based on π‘ (Λ π ) are very close to each other.In Figure 2 for the case of the structural break in volatility, one observes that the CauchyRT test appears to be superior to other testing approaches (except the cases with Β― π = 0 andlarge Β― π½ ). At the same time, the second-best test π‘ (Λ π ) and the robust π‘ -statistic test with π‘ ( π ) also appear to have good finite sample power properties.Figure 3 provides the numerical results on finite sample properties of the tests in the geo-metric Brownian motion case. For the case of a unit root regressor, the power properties of thetest based on π‘ (Λ π ) appear to outperform those of both the test based on π‘ ( π ) and the CauchyRT test. However, the finite sample power performance of the Cauchy RT test improves in thecase of near unit root regressor with Β― π = 5 and Β― π = 20 .As this is a case with endogenous volatility, asymptotic mixed normality does not hold forthe group statistics (the numerators of group Cauchy estimators of the parameter π½ ) used incalculation of the π‘ -statistic π‘ ( π ) . Interestingly, however, the robust π‘ -statistic test π‘ ( π ) has The finite sample properties of the robust π‘ -statistic based test with π‘ ( π ) are very similar to those with π‘ ( π ) and the numerical results for it are omitted for brevity. π₯ π‘ with Β― π = 0 .The power curves for the regime switching case presented in Figure 4 demonstrate that thetest based on π‘ (Λ π ) has better power properties than other tests in the case Β― π = 0 . For thecase of the near unit root persistence in the regressor, the power properties of the Cauchy RTtest appear to be better than those of the test based on π‘ (Λ π ) for relatively small sample sizes(small values of π ). However, as the sample size increases, the test based on π‘ (Λ π ) becomes morepowerful than the Cauchy RT test (see, e.g., Figure 4 for the case Β― π = 5 and π = 50 ). For largedeviations from a unit root regressors ( Β― π = 20 ), the Cauchy RT is more powerful than othertests, but the power curves appear to be very similar to each other.Figures 5-10 present the numerical results on power properties under discrete time settingsfor all the tests considered except Cauchy RT that is inapplicable in the discrete time settings.The results in the figures are provided for the cases of constant volatility; the structural break involatility; the GARCH cases, respectively. For all the cases the conclusions on power propertiesof the tests and their comparisons are virtually the same as in continuous time framework.Overall, the numerical results on finite sample properties of the tests indicate good perfor-mance of the test based on π‘ (Λ π ) and the robust π‘ -statistic test based on π‘ ( π ) in comparisonto the Cauchy RT. Again, the latter test is inapplicable in the discrete time settings. Theirrelative finite sample performances vary across different models. Which test should be used inpractice depends on the availability of high frequency data as well as the size-power trade-offfor a specific model. On the one hand, the test based on π‘ (Λ π ) often outperforms the robust π‘ -statistic test based on π‘ ( π ) in terms of power properties. On the other hand, the latter testhas a better finite sample size while the test based on π‘ (Λ π ) is slightly oversized. Importantly, therobust π‘ -statistic-based tests like π‘ ( π ) and π‘ ( π ) are simple to construct and do not requireany estimation of the volatility process. Therefore, it seems fair to say that the proposed twotests in this paper and the Cauch RT are complementing, rather than substituting, each other. In this paper, we propose two robust methods for testing hypotheses on parameters ofpredictive regression models under heterogeneous and persistent volatility as well as endogenous,persistent and/or heavy-tailed regressors and errors. The inference approaches proposed in thepaper are applicable both in the case of continuous and discrete time models. Both of themethods use the Cauchy estimator to effectively handle the problems of endogeneity, persistenceand/or heavy-tailedness in regressors and regression errors.The first method relies on robust π‘ -statistic inference using group estimators of a regressionparameter of interest proposed in Ibragimov and MΒ¨uller (2010). It is simple to implement, but18equires the exogenous volatility assumption. To relax the exogenous volatility assumption,we propose another method which relies on the nonparametric correction of volatility. Theproposed methods perform well compared with widely used alternative inference procedures interms of their finite sample properties for various dependence and persistence settings observedin real-world financial and economic markets. 19 ppendix A: Model with Intercept In this section, we extend the robust inference methods in Section 3 to model (1) withan intercept. For our methods, we define Λ π½ and Λ π½ ( π ) , respectively, by the least square andrecursive least square estimators for π½ in the regression model (1) with intercept. We alsodefine a modified Cauchy estimator Λ π½ as Λ π½ = βοΈ ππ‘ =1 ( π ππ ( π₯ π‘ ) β Β― π π ) π¦ π‘ +1 βοΈ ππ‘ =1 | π₯ π‘ | β π Β― π π Β― π₯ π , where Β― π₯ π = π β βοΈ ππ‘ =1 π₯ π‘ and Β― π π = π β βοΈ ππ‘ =1 π ππ ( π₯ π‘ ) .The robust inference procedures for the model with intercept will be similar to the modelwithout intercept in Section 3, but they will be constructed from Λ π½ , Λ π½ and Λ π½ for the modelwith intercept, defined as above, instead of the estimators for the model without intercept inSections 2 and 3.To show the validity of our methods, we let π ππ = π β βοΈ [ π π ] π‘ =1 π ππ ( π₯ π‘ β ) and assume that Assumption A.1.
There exists a positive process π on D R + [0 , such that ( π π ) β π.π . ( π ) in D R + [0 , and ( π π , π π , π π , π π ) β π ( π, π, π, π ) in D R Γ R Γ R Γ R + [0 , , where π is a c`adl`ag process, π and π are Brownian motions with respectto the filtration to which π , π , π and π are adapted. Under Assumption A.1, we have π π = π π π π + (1 β π π ) / π β₯ π , (A.1)where E [ ππ π‘ ππ β₯ π‘ ] = 0 . In particular, if π½ = 0 , then the numerator of Λ π½ satisfies, regardless ofwhether πΌ = 0 , the following asymptotics β π π βοΈ π‘ =1 ( π ππ ( π₯ π‘ ) β Β― π π ) π¦ π‘ +1 = 1 β π π βοΈ π‘ =1 ( π ππ ( π₯ π‘ ) β Β― π π ) π£ π‘ +1 π π‘ +1 = π΄ π β π΅ π , where π΄ π = 1 β π π βοΈ π‘ =1 π ππ ( π₯ π‘ ) π£ π‘ +1 π π‘ +1 β π β«οΈ π π ππ π β‘ π΄,π΅ π = 1 β π Β― π π π βοΈ π‘ =1 π£ π‘ +1 π π‘ +1 β π π β«οΈ π π ππ π β‘ π΅. π½ = 01 β π π βοΈ π‘ =1 ( π ππ ( π₯ π‘ ) β Β― π π ) π¦ π‘ +1 β π π΄ β π΅. As in Section 3, we propose two robust inference methods depending upon whether thefollowing assumption holds or not.
Assumption A.2. (Volatility exogeneity) π is independent of π and π , and π is independentof π and π . A.1 Robust Inference Under Exogenous Volatility
Under Assumption A.2, π΄ β π΅ becomes a mixed normal random variable since π΄ β π΅ = β«οΈ π π ππ π β π β«οΈ π π ππ π = β«οΈ π π (1 β π π ) / ππ β₯ π β β«οΈ π π ( π β π π ) ππ π = π MN (οΈ , β«οΈ π π (1 + π β π π π ) ππ )οΈ due, in particular, to (A.1). We then can define a modified version of the statistic (3) as π := 1 β π π βοΈ π‘ =1 ( π ππ ( π₯ π‘ ) β Β― π π ) π¦ π‘ +1 , and construct the π group statistic ( π π , π = 1 , Β· Β· Β· , π ) correspondingly. Corollary A.1.
Let Assumptions 2.1, A.1 and A.2 hold. Then the conclusion in Proposition 3.1remains valid with Ξ£ π being a π -dimensioanl diagonal matrix with π th element β«οΈ π/π ( π β /π π ( π )(1 + π β π π π ) ππ regardless of whether πΌ = 0 . Due to Corollary A.1, we can follow Ibragimov and MΒ¨uller (2010) to construct a robustinference as we did in Section 3.1.
A.2 Robust Inference Under Endogenous Volatility
To generalize the method in Section 3.2, we construct Λ π and Λ π , respectively, from Λ π’ π‘ = π¦ π‘ β Λ πΌ β Λ π½π₯ π‘ β and Λ π’ π‘ ( π ) = π¦ π‘ β Λ πΌ ( π ) β Λ π½ ( π ) π₯ π‘ β , as in Section 3.2. The uniform convergence results,such as Theorem 3.3 and Corollary 3.4, for Λ π and Λ π can be obtained. However, we will notformally develop the uniform convergence results here since they are relatively straightforward,given the uniform convergences in Theorem 3.3 and Corollary 3.4.21e then consider a modified version of the test statistic (5) as π‘ (Β― π ) β‘ βοΈ π (1 β (Β― π π ) ) π βοΈ π‘ =1 ( π ππ ( π₯ π‘ ) β Β― π π ) π¦ π‘ +1 Β― π ( π‘/π ) for Β― π = Λ π * , Λ π * . Under π½ = 0 , it is quite clear that the modified statistic π‘ (Β― π ) satisfies that π‘ (Β― π ) = β π βοΈ π (1 β (Β― π π ) ) ( πΆ π β π· π ) , where πΆ π = 1 β π π βοΈ π‘ =1 π ππ ( π₯ π‘ ) π£ π‘ +1 Β― π ( π‘/π ) π π‘ +1 β π π ,π· π = 1 β π Β― π π π βοΈ π‘ =1 π£ π‘ +1 Β― π ( π‘/π ) π π‘ +1 β π π π , as long as Λ π * and Λ π * converge to π sufficiently quickly as in Theorem 3.3. Moreover, we have π π = π π π π + (1 β π π ) / π β₯ π by (A.1), and therefore, π‘ (Β― π ) β π βοΈ β π (1 β π ) / π β₯ = π N (0 , , which implies that the test using π‘ (Β― π ) with Β― π = Λ π * , Λ π * becomes a robust inference for π½ = 0 regardless of whether πΌ = 0 . Appendix B: Proofs
B.1 Useful Lemmas
Lemma B.1.
Let Assumption 3.3 hold. If β π β β , then for π = πΎ, πΎ and π β = π ( π‘/β ) , wehave sup β€ π β€ βββββ βπ π βοΈ π‘ =1 π β ( π β π‘/π ) β β«οΈ π/β π ( π ) ππ βββββ = π (1 / ( β π )) . Proof.
We only prove the result for the case π = πΎ since the arguments in the case π = πΎ aresimilar. Define the function πΌ π,β : [0 , β { , } for π β [0 , and β > by πΌ π,β ( π ) = 1 { π β β β€ π β€ π } . We write π βοΈ π‘ =1 β«οΈ π‘π‘ β | πΎ β ( π β π‘/π ) β πΎ β ( π β π /π ) | ππ = π΄ ( π ) + π΅ ( π ) + πΆ ( π ) , π΄ ( π ) = π βοΈ π‘ =1 (οΈβ«οΈ π‘π‘ β | πΎ β ( π β π‘/π ) β πΎ β ( π β π /π ) | ππ )οΈ πΌ π,β β /π ( π‘/π ) ,π΅ ( π ) = π βοΈ π‘ =1 (οΈβ«οΈ π‘π‘ β | πΎ β ( π β π‘/π ) β πΎ β ( π β π /π ) | ππ )οΈ (οΈ β πΌ π,β +1 /π ( π‘/π ) )οΈ ,πΆ ( π ) = π βοΈ π‘ =1 (οΈβ«οΈ π‘π‘ β | πΎ β ( π β π‘/π ) β πΎ β ( π β π /π ) | ππ )οΈ (οΈ πΌ π,β +1 /π ( π‘/π ) β πΌ π,β β /π ( π‘/π ) )οΈ . By parts (a) and (c) of Assumption 3.3, we have sup β€ π β€ π΄ ( π ) β€ πΆ π βοΈ π‘ =1 β«οΈ π‘π‘ β | π‘ β π | βπ ππ β€ πΆ β . (B.1)Moreover, β«οΈ π‘π‘ β | πΎ β ( π β π‘/π ) β πΎ β ( π β π /π ) | ππ = 0 for all π‘ satisfying πΌ π,β +1 /π ( π‘/π ) = 0 by part(a) of Assumption 3.3. Hence, sup β€ π β€ π΅ ( π ) = 0 . (B.2)As for πΆ ( π ) , we have sup β€ π β€ βοΈ ππ‘ =1 (οΈ πΌ π,β +1 /π ( π‘/π ) β πΌ π,β β /π ( π‘/π ) )οΈ β€ , from which, to-gether with part (b) of Assumption 3.3, it follows that sup β€ π β€ πΆ ( π ) β€ β€ π β€ | πΎ ( π ) | sup β€ π β€ βββββ π βοΈ π‘ =1 (οΈ πΌ π,β +1 /π ( π‘/π ) β πΌ π,β β /π ( π‘/π ) )οΈβββββ = π (1) . (B.3)The stated result for π = πΎ follows immediately from (B.1)-(B.3) since βπ β«οΈ π πΎ β ( π β π /π ) ππ = 1 β β«οΈ πΎ β ( π β π ) ππ = β«οΈ π/β πΎ ( π ) ππ (B.4)by part (a) of Assumption 3.3 and the change of variable in integrals. This completes theproof. Lemma B.2.
Let Assumptions 3.3 and 3.4(c) hold. If β π β β , then sup β€ π β€ βββββ βπ π βοΈ π‘ =1 π π‘/π πΎ β ( π β π‘/π ) β π π β β«οΈ π/β πΎ ( π ) ππ βββββ = π π ( π β ) Proof.
By Assumption 3.3 (a), we can write βπ π βοΈ π‘ =1 π π‘/π πΎ β ( π β π‘/π ) β π π β β«οΈ π/β πΎ ( π ) ππ = π΄ ( π ) + π΅ ( π ) + πΆ ( π ) + π· ( π ) β β€ π β€ , where π΄ ( π ) = 1 βπ π βοΈ π‘ =1 β«οΈ π‘π‘ β π π‘/π { π‘ < [ π π ] } ( πΎ β ( π β π‘/π ) β πΎ β ( π β π /π )) ππ ,π΅ ( π ) = 1 βπ π βοΈ π‘ =1 β«οΈ π‘π‘ β (οΈ π π‘/π { π‘ < [ π π ] } β π π β )οΈ πΎ β ( π β π /π ) ππ ,πΆ ( π ) = π π β (οΈ βπ β«οΈ π πΎ β ( π β π /π ) ππ β β«οΈ π/β πΎ ( π ) ππ )οΈ ,π· ( π ) = 1 βπ π βοΈ π‘ =1 π π‘/π { π‘ = [ π π ] } πΎ β ( π β π‘/π ) . Note that π is bounded with probability one since π β D R + [0 , . It then follows from theproof of Lemma B.1 that sup β€ π β€ | π΄ ( π ) | = π π (1 / ( β π )) , and sup β€ π β€ | πΆ ( π ) | = 0 with probability one due to (B.4). We also have sup β€ π β€ | π· ( π ) | = sup β€ π β€ ββββ βπ π π π ] /π πΎ β ( π β [ π π ] /π ) ββββ = π π (1 / ( βπ )) . As for π΅ ( π ) , we can deduce from Assumptions 3.3(a) and 3.4(c) together with Lemma B.1that sup β€ π β€ | π΅ ( π ) | β€ sup β€ π β€ sup π β β β /π β€ π <π | π π β π π β | Γ π (1) = π π ( π β ) since β + 1 /π < πβ for any π > as long as βπ is sufficiently large. Lemma B.3.
Let Assumptions 3.3 and 3.4(d) hold. If β π β β , then sup π β [0 , β π½ β βββββ βπ π βοΈ π‘ =1 (οΈ π£ π‘ β π π‘/π )οΈ πΎ β ( π β π‘/π ) βββββ = π π ( π β,π ) .Proof. By construction, we have π£ π‘ = π ππ‘/π , and hence, sup π β [0 , β π½ β βββββ π βοΈ π‘ =1 (οΈ π£ π‘ β π π‘/π )οΈ πΎ β ( π β π‘/π ) βββββ = sup π β [0 , β π½ β βββββ π βοΈ π‘ =1 (οΈ ( π ππ‘/π ) β π π‘/π )οΈ πΎ β ( π β π‘/π ) βββββ β€ sup π β [0 , β π½ β sup π β β β€ π β€ π + β ββ ( π ππ ) β π π ββ π βοΈ π‘ =1 | πΎ β ( π β π‘/π ) | = π π ( π β,π βπ ) , Lemma B.4.
Let Assumptions 3.3 and Condition (e) in Corollary 3.4 hold. If β π β β , then sup β€ π β€ βββββ βπ π βοΈ π‘ =1 (οΈ π£ π‘ β π π‘/π )οΈ πΎ β ( π β π‘/π ) βββββ = π π ( π π ) . Proof.
This can be shown similar to the proof of Lemma B.3 using Condition (e) in Corollary3.4 in place of part (d) in Assumption 3.4.In what follows, we write Λ π ( π ) = Λ π π + Λ π π + Λ π π ( π ) + Λ π π ( π ) and Λ π ( π ) = Λ π π ( π ) + Λ π π ( π ) + Λ π π ( π ) +Λ π π ( π ) , where Λ π π ( π ) = Λ π π ( π ) = βοΈ ππ‘ =1 π£ π‘ πΎ β ( π β π‘/π ) βοΈ ππ‘ =1 πΎ β ( π β π‘/π ) , Λ π π ( π ) = Λ π π ( π ) = βοΈ ππ‘ =1 π£ π‘ ( π π‘ β πΎ β ( π β π‘/π ) βοΈ ππ‘ =1 πΎ β ( π β π‘/π ) , Λ π π ( π ) = ( Λ π½ ( π ) β π½ ) βοΈ ππ‘ =1 π₯ π‘ β πΎ β ( π β π‘/π ) βοΈ ππ‘ =1 πΎ β ( π β π‘/π ) , Λ π π ( π ) = 2( Λ π½ ( π ) β π½ ) βοΈ ππ‘ =1 π₯ π‘ β π£ π‘ π π‘ πΎ β ( π β π‘/π ) βοΈ ππ‘ =1 πΎ β ( π β π‘/π ) , Λ π π ( π ) = ( Λ π½ β π½ ) βοΈ ππ‘ =1 π₯ π‘ β πΎ β ( π β π‘/π ) βοΈ ππ‘ =1 πΎ β ( π β π‘/π ) , Λ π π ( π ) = 2( Λ π½ β π½ ) βοΈ ππ‘ =1 π₯ π‘ β π£ π‘ π π‘ πΎ β ( π β π‘/π ) βοΈ ππ‘ =1 πΎ β ( π β π‘/π ) . Lemma B.5.
Let Assumptions 2.1-2.2, 3.3 and 3.4(b-c) hold, and let β π β β .(a) Under Assumption 3.4(d), sup π β [ β, β π½ β | Λ π π ( π ) β π ( π ) | = π π ( π β ) + π π ( π β,π ) . (b) Under Condition (e) in Corollary 3.4, sup π β [ β, | Λ π π ( π ) β π ( π ) | = π π ( π β ) + π π ( π π ) . Proof of Lemma B.5.
Part (a) follows from Lemmas B.1-B.3, and part (b) follows from LemmasB.1-B.2 and B.4.
Lemma B.6.
Let Assumptions 2.1-2.2 and 3.3-3.4(a) hold. If β π β β , then sup β β€ π β€ | Λ π π ( π ) | = π π (οΈ(οΈ ( βπ ) β log π )οΈ / )οΈ = π π (1) Proof of Lemma B.6.
By Lemma B.1, we have sup β€ π β€ π βοΈ π‘ =1 πΎ β ( π β π‘/π ) = π ( βπ )
25s long as β π β β . It then follows from Theorem 2.1 of Wang and Chen (2014) and Assump-tions 2.1, 3.3 and 3.4(a) that sup β€ π β€ βββββ π βοΈ π‘ =1 π£ π‘ ( π π‘ β πΎ β ( π β π‘/π ) βββββ = π π (( βπ log π ) / ) . This, together with Lemma B.1, implies the stated results from Assumption 3.4(a) it followsthat ( βπ ) β log π = π (1) . Lemma B.7.
Let Assumptions 2.1-3.2 and 3.3-3.4(a) hold. If β π β β , then sup β β€ π β€ | Λ π π ( π ) | = π π (log π / ( βπ )) , sup β β€ π β€ | Λ π π ( π ) | = π π (log π / ( βπ )) , sup β β€ π β€ | Λ π π ( π ) | = π π (1 / ( βπ )) , sup β β€ π β€ | Λ π π ( π ) | = π π (οΈ (log( π )) / / ( βπ ) )οΈ . Proof of Lemma B.7.
We have sup β β€ π β€ | Λ π½ ( π ) β π½ | = π π ((log( π ) /π π ) / ) by Proposition 3.2. Itcan then be deduced from Lemma B.1, Assumptions 3.2 and 3.3(b) that sup β β€ π β€ βββββ βοΈ ππ‘ =1 π₯ π‘ β πΎ β ( π β π‘/π ) βοΈ ππ‘ =1 πΎ β ( π β π‘/π ) βββββ β€ sup β€ π β€ | πΎ ( π ) | sup β β€ π β€ βββββ βοΈ ππ‘ =1 π₯ π‘ β βοΈ ππ‘ =1 πΎ β ( π β π‘/π ) βββββ = π π ( π π / ( βπ )) . Moreover, using Assumption 3.2, one can show that sup β β€ π β€ βββββ βοΈ ππ‘ =1 π₯ π‘ β π£ π‘ π π‘ πΎ β ( π β π‘/π ) βοΈ ππ‘ =1 πΎ β ( π β π‘/π ) βββββ = π π (( π π log( π )) / / ( βπ )) , in a similar manner to the proof of Lemma B.6. The stated results for Λ π π and Λ π π follow im-mediately. Similarly, we have the results for Λ π π and Λ π π since | Λ π½ β π½ | = π π ( π β / π ) by Lemma2.2. Lemma B.8.
Let the conditions in Lemma B.7 hold. If Assumption 3.5 holds, then π β βοΈ π‘ =1 (οΈ π * ( π‘/π ) β π * ( π‘/π ) )οΈ = π π (1) Proof of Lemma B.8.
We have π β βοΈ π‘ =1 (οΈ π * ( π‘/π ) β π * ( π‘/π ) )οΈ = π β βοΈ π‘ =1 (οΈ Λ π * ( π‘/π ) β Λ π * ( π‘/π )Λ π * ( π‘/π )Λ π * ( π‘/π )(Λ π * ( π‘/π ) + Λ π * ( π‘/π )) )οΈ β€ π β,π π βοΈ π‘ =1 (Λ π * ( π‘/π ) β Λ π * ( π‘/π )) β€ ππ β,π (οΈ sup β€ π β€ | Λ π π ( π ) | + sup β€ π β€ | Λ π π ( π ) | + sup β€ π β€ | Λ π π ( π ) | + sup β€ π β€ | Λ π π ( π ) | )οΈ = π π (οΈ log ( π ) / ( π β,π β π ) )οΈ , Λ π * and Λ π * , and the last line holds dueto Lemma B.7. The stated result then follows from Assumption 3.5(b). This completes theproof. B.2 Proofs of the Main Resutls
Proof of Lemma 2.1.
The lemma follows from Kurtz and Protter (1991) (see also Theorem 2.1of Hansen, 1992, and Ibragimov and Phillips, 2008).
Proof of Lemma 2.2.
The stated results follow immediately from Lemma 2.1 under the condi-tions in the lemma.
Proof of Proposition 3.1.
The stated result follows immediately form Lemma 2.2 and Assump-tion 3.1.
Proof of Proposition 3.2.
The proposition follows from the well-known martingale exponentialinequalities (see, e.g., de la PeΛna, 1999).
Proofs of Theorem 3.3 and Corollary 3.4.
The stated results follow from Lemmas B.5-B.7.
Proof of Theorem 3.5.
Let π½ β = { [ π β β, π + β ) | π β π½ } β© [0 , . Define Λ π β ( π ) = β§β¨β© Λ π * ( π ) for π β [ β, β π½ β ,π π ( π ) for π β [0 , β ) βͺ π½ β . Clearly, (Λ π β ( π‘/π )) is a c`adl`ag function on [0 , , and is adapted to ( β± π‘ ) for all β and π. Therefore, Λ π β β π π in D R + [0 , , and hence, π‘ (Λ π β ) := 1 β π π β βοΈ π‘ =1 π ππ ( π₯ π‘ ) π£ π‘ +1 π π‘ +1 Λ π β ( π‘/π ) β π N (0 , under π½ = 0 , (B.5)and | π‘ (Λ π β ) | β π β under π½ ΜΈ = 0 . To complete the proof, it suffice to show that π‘ ( Λ π½ * ) = π‘ (Λ π β )(1 + π π (1)) for any π½ β R .Let π½ = 0 . Then π‘ (Λ π * ) β π‘ (Λ π β ) becomes a martingale, and hence, we may apply a martingaleCLT so that π‘ (Λ π * ) β π‘ (Λ π β ) = 1 β π π β βοΈ π‘ =1 π ππ ( π₯ π‘ ) π£ π‘ +1 π π‘ +1 (οΈ π * ( π‘/π ) β π β ( π‘/π ) )οΈ = 1 β π π β βοΈ π‘ =1 π ππ ( π₯ π‘ ) π£ π‘ +1 π π‘ +1 (οΈ π * ( π‘/π ) β π π ( π‘/π ) )οΈ { π‘/π β [0 , β ) βͺ π½ β } = π π ( β / /π β,π ) = π π (1) , (B.6)27ince π π β βοΈ π‘ =1 π£ π‘ +1 (οΈ π * ( π‘/π ) β π π ( π‘/π ) )οΈ { π‘/π β [0 , β ) βͺ π½ β } = π π ( β/π β,π ) = π π (1) by Assumption 3.5 and the fact that the set { π‘/π β [0 , β ) βͺ π½ β } contains at most πβπ numberof elements for some finite π with probability one due to Assumption 3.4(b).Now we let π½ ΜΈ = 0 . It follows from (B.5) and (B.6) that π‘ (Λ π * ) = π½ β π π β βοΈ π‘ =1 | π₯ π‘ | Λ π * ( π‘/π ) + π π (1) . Moreover, we may deduce from Theorem 3.3 and Assumption 3.5 that P { inf β€ π β€ Λ π * ( π ) β€ π β π } β for any π > . Therefore, π β βοΈ π‘ =1 | π₯ π‘ | Λ π * ( π‘/π ) β₯ (οΈ π π β βοΈ π‘ =1 | π₯ π‘ | )οΈ (1 + π π (1)) , from which, together with the condition βοΈ π β π‘ =1 | π₯ π‘ | / β π β β , we have the stated result. Proof of Theorem 3.6.
It suffices to show that π‘ (Λ π * ) β π‘ (Λ π * ) = π π (1) under π½ = 0 , since the proof under π½ ΜΈ = 0 is entirely analogous to that of π‘ (Λ π * ) . We have π‘ (Λ π * ) β π‘ (Λ π * ) = 1 β π π β βοΈ π‘ =1 π ππ ( π₯ π‘ ) π£ π‘ +1 π π‘ +1 (οΈ π ( π‘/π ) β π ( π‘/π ) )οΈ β€ (οΈ π π β βοΈ π‘ =1 π£ π‘ +1 π π‘ +1 )οΈ / (οΈ π β βοΈ π‘ =1 (οΈ π ( π‘/π ) β π ( π‘/π ) )οΈ )οΈ / = π π (1) , where the inequality follows from the Cauchy-Schwarz inequality, and the last equality holdsby Lemma B.8. This completes the proof. References
Anatolyev, S. (2019), βVolatility filtering in estimation of kurtosis (and variance)β,
DependenceModeling , 1β23.Baillie, R. T. (1996), βLong memory processes and fractional integration in econometricsβ, Jour-nal of Econometrics , 5β59. 28illingsley, P. (2013), Convergence of Probability Measures , Wiley Series in Probability andStatistics, Wiley.Blinder, A. S. and Watson, M. W. (2016), βPresidents and the US economy: An econometricexplorationβ,
American Economic Review , 1015β1045.Bloom, N., Eifert, B., Mahajan, A., McKenzie, D. and Roberts, J. (2013), βDoes managementmatter? Evidence from Indiaβ,
Quarterly Journal of Economics , 1β51.Boswijk, H. P., Cavaliere, G., Rahbek, A. and Taylor, A. R. (2016), βInference on co-integrationparameters in heteroskedastic vector autoregressionsβ,
Journal of Econometrics (1), 64 β85.Brown, D. and Ibragimov, R. (2019), βSign tests for dependent observationsβ,
Econometrics andStatistics , 1β8.Campbell, B. and Dufour, J.-M. (1995), βExact nonparametric orthogonality and random walktestsβ, Review of Economics and Statistics , 1β16.Campbell, J. and Yogo, M. (2006), βEfficient tests of stock return predictabilityβ, Journal ofFinancial Econometrics , 27β60.Cavaliere, G. and Taylor, A. R. (2007), βTesting for unit roots in time series models withnon-stationary volatilityβ, Journal of Econometrics , 919β947.Chang, Y. (2002), βNonlinear IV unit root tests in panels with cross-sectional dependencyβ,
Journal of Econometrics , 261β292.Chang, Y. (2012), βTaking a new contour: A novel approach to panel unit root testsβ,
Journalof Econometrics , 15β28.Chang, Y. and Park, J. Y. (2002), βOn the asymptotics of ADF tests for unit rootsβ,
EconometricReviews , 431β447.Chen, W. and Deo, R. (2009), βBias reduction and likelihood-based almost exactly sized hy-pothesis testing in predictive regressions using the restricted likelihoodβ, Econometric Theory , 1143β1179.Chen, Z. and Ibragimov, R. (2019), βOne country, two systems? The heavy-tailedness of ChineseA- and H- share marketsβ, Emerging Markets Review , 115β141.Choi, Y., Jacewitz, S. and Park, J. Y. (2016), βA reexamination of stock return predictabilityβ, Journal of Econometrics , 168β189. 29hung, H. and Park, J. Y. (2007), βNonstationary nonlinear heteroskedasticity in regressionβ,
Journal of Econometrics , 230β259.Davis, R. A. and Mikosch, T. (1998), βThe sample autocorrelations of heavy-tailed processeswith applications to ARCHβ,
Annals of Statistics .de la PeΛna, V. H. (1999), βA general class of exponential inequalities for martingales and ratiosβ, The Annals of Probability (1), 537β564.Demetrescu, M. and Rodrigues, P. (2017), Residual-augmented IVX predictive regression. Working Paper, Banco de Portugal .Dufour, J.-M. and Hallin, M. (1993), βImproved eaton bounds for linear combinations ofbounded random variables, with statistical applicationsβ,
Journal of the American Statis-tical Association , 1026β-1033.Elliott, G. and Stock, J. H. (1994), βInference in time series regression when the order ofintegration of a regressor is unknownβ, Econometric theory , 672β700.Embrechts, P., KlΒ¨uppelberg, C. and Mikosch, T. (1997), Modelling Extremal Events for Insur-ance and Finance , Springer.Engle, R. and Rangel, J. G. (2008), βThe spline-GARCH model for low-frequency volatility andits global macroeconomic causesβ,
Review of Financial Studies , 1187β1222.Esarey, J. and Menger, A. (2019), βPractical and effective approaches to dealing with clustereddataβ, Political Science Research and Methods , 541β559.Fama, E. F. and MacBeth, J. (1973), βRisk, return and equilibrium: Empirical testsβ, Journalof Political Economy , 607β636.Gargano, A., Pettenuzzo, D. and Timmermann, A. (2019), βBond return predictabilityβ, Man-agement Science , 508β540.Giraitis, L. and Phillips, P. C. B. (2006), βUniform limit theory for stationary autoregressionβ, Journal of Time Series Analysis , 51β60.Granger, C. W. J. and Orr, D. (1972), βInfinite variance and research strategy in time seriesanalysisβ, Journal of the American Statistical Association , 275ββ285.Hansen, B. E. (1992), βConvergence to stochastic integrals for dependent heterogeneous pro-cessesβ, Econometric Theory , 489β500.Hansen, B. E. (1995), βRegression with nonstationary volatilityβ, Econometrica , 1113β1132.30osseinkouchack, M. and Demetrescu, M. (2016), βFinite-sample size control of IVX-basedtests in predictive regressionsβ. Working paper, Goethe University Frankfurt and Christian-Albrechts-University of Kiel .Ibragimov, M., Ibragimov, R. and Walden, J. (2015),
Heavy-tailed Distributions and Robustnessin Economics and Finance , Vol. 214 of
Lecture Notes in Statistics , Springer.Ibragimov, R. and MΒ¨uller, U. (2010), β π‘ -statistic Based Correlation and Heterogeneity RobustInferenceβ, Journal of Business & Economic Statistics , 453β468.Ibragimov, R. and MΒ¨uller, U. K. (2016), βInference with few heterogeneous clustersβ, Review ofEconomics and Statistics , 83β96.Ibragimov, R., Pedersen, R. S. and Skrobotov, A. (2019), βNew approaches to robust inferenceon market (non-)efficiency, volatility clustering and nonlinear dependenceβ. Working paper,Imperial College Business School, the University of Copenhagen and the Russian PresidentialAcademy of National Economy and Public Administration .Ibragimov, R. and Phillips, P. C. B. (2008), βRegression asymptotics using martingale conver-gence methodsβ,
Econometric Theory , 888β947.Kim, C. S. and Phillips, P. C. B. (2006), βLog periodogram regression: The nonstaionary caseβ. Cowles Foundation Discussion Paper No. 1587 .Kim, J. and Meddahi, N. (2020), βVolatility regressions with fat tailsβ,
Journal of Econometrics ,Forthcoming.Kostakis, A., Magdalinos, T. and Stamatogiannis, M. (2015), βRobust econometric inference forstock return predictabilityβ,
Review of Financial Studies , 1506β1553.Krueger, A. B., Mas, A. and Niu, X. (2017), βThe evolution of rotation group bias: Will thereal unemployment rate please stand up?β, Review of Economics and Statistics , 258β264.Kurtz, T. G. and Protter, P. (1991), βWeak limit theorems for stochastic integrals and stochasticdifferential equationsβ, Annals of Probability , 1035β1070.Li, C., Li, D. and Peng, L. (2017), βUniform test for predictive regression with AR errorsβ, Journal of Business & Economic Statistics , 29β39.Mikosch, T. and StΛaricΛa, C. (2000), βLimit theory for the sample autocorrelations and extremesof a GARCH (1 , processβ, Annals of Statistics , 1427β1451.Pedersen, R. S. (2019), βRobust inference in conditionally heteroskedastic autoregressionsβ, Econometric Reviews . https://doi.org/10.1080/07474938.2019.1580950.31hillips, P. (2015), βHalbert White Jr. Memorial JFEC Lecture: Pitfalls and possibilities inpredictive regressionβ,
Journal of Financial Econometrics , 521β555.Phillips, P. C. B. (1987 a ), βTime-series regression with a unit rootβ, Econometrica , 277β301.Phillips, P. C. B. (1987 b ), βTowards a unified asymptotic theory for autoregressionβ, Biometrika , 535β547.Phillips, P. C. B. (1999), βDiscrete Fourier transforms of fractional processesβ. Cowles Founda-tion Discussion Paper No. 1243 .Phillips, P. C. B. and Magdalinos, T. (2009), Econometric inference in the vicinity of unity.
Working paper, Yale University and University of Nottingham .Phillips, P. C. B. and Solo, V. (1992), βAsymptotics for linear processesβ,
Annals of Statistics , 971β1001.Phillips, P. and Lee, J. (2013), βPredictive regression under various degrees of persistence androbust long-horizon regressionβ, Journal of Econometrics , 250β264.Phillips, P. and Lee, J. (2016), βRobust econometric inference with mixed integrated and mildlyexplosive regressorsβ,
Journal of Econometrics , 433β450.Phillips, P. and Magdalinos, T. (2007), βLimit theory for moderate deviations from a unit rootβ,
Journal of Econometrics , 115β130.So, B. and Shin, D. (1999), βCauchy estimators for autoregressive processes with applicationsto unit root tests and confidence intervalsβ,
Econometric Theory , 165β176.So, B. and Shin, D. (2001), βAn invariant sign tests for random walks based on recursive medianadjustmentβ, Journal of Econometrics , 197β229.Verner, E. and Gyongyosi, G. (2018), βHousehold debt revaluation and the real economy: Evi-dence from a foreign currency debt crisisβ,
Working Paper, MIT Sloan School of Management .Xu, K.-L. (2017), Testing for Return Predictability with Co-moving Predictors of UnknownForm.
Working paper, Indiana University Bloomington .32 able 1: Size (Continuous Time Models) Β― π = 0 Β― π = 5 Β― π = 20 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) able 2: Sizes (Discrete Time Models with CNST and SB) Β― π = 0 Β― π = 5 Β― π = 20 π‘ ( π ) π‘ (Λ π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) able 3: Sizes (Discrete Time Models with GARCH) Β― π = 0 Β― π = 5 Β― π = 20 πΌ = 0 . BQ 9.7 5.4 4.8 9.5 5.8 4.6 13.1 6.3 4.6 π = 4 RLRT 9.6 8.8 8.7 9.0 7.9 6.7 13.1 11.3 9.1q=4 4.5 4.9 4.8 4.7 4.7 4.3 4.8 4.6 4.8q=8 4.7 4.9 5.2 4.7 4.6 4.8 4.7 5.0 5.1q=12 4.5 4.9 5.2 4.6 4.6 4.5 4.7 4.6 5.4q=16 4.8 4.9 5.3 4.8 4.5 4.6 4.7 4.9 5.3 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) πΌ = 0 . BQ 10.2 6.2 5.2 10.4 7.1 5.8 14.7 8.3 6.8 π = 3 RLRT 10.7 10.0 9.2 10.6 10.0 8.1 15.9 14.9 12.9q=4 4.1 4.6 4.7 4.7 4.4 4.4 4.5 4.4 4.6q=8 4.6 4.7 4.8 4.5 4.3 4.6 4.7 4.7 4.6q=12 4.4 4.6 4.8 4.6 4.3 4.5 4.4 4.8 5.1q=16 4.7 4.6 5.1 4.7 4.5 4.6 4.6 4.8 5.0 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) πΌ = 0 . BQ 8.9 5.8 6.0 7.8 6.0 6.9 9.3 5.3 6.4 π½ = 0 . RLRT 9.1 10.1 11.5 5.7 8.3 9.8 6.2 9.0 11.5 π = 2 q=4 5.0 4.7 4.4 4.7 4.5 4.4 4.9 4.8 4.4q=8 4.9 4.9 4.5 5.0 4.9 4.9 5.1 5.2 4.4q=12 5.0 5.0 4.6 5.0 5.2 5.1 5.1 5.3 4.9q=16 5.0 5.0 4.9 5.1 5.1 5.1 5.1 5.3 4.8 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) πΌ = 0 . BQ 11.7 8.3 8.1 12.7 10.4 10.9 16.4 12.2 13.4 π½ = 0 . RLRT 13.3 13.0 12.7 13.7 14.7 15.1 20.2 23.7 23.2 π = 2 q=4 4.0 4.3 4.3 4.2 4.1 4.4 4.2 4.5 4.3q=8 4.2 4.2 4.3 4.5 4.0 4.6 4.6 4.7 4.3q=12 4.4 4.2 4.5 4.7 4.1 4.5 4.5 4.7 4.6q=16 4.5 4.4 4.4 4.9 4.2 4.8 4.6 4.8 4.5 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r C N S T O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r S B O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r G B M O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r R S O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r C N S T , d i s c r e t e t i m e O L S :, B o n f . Q : Β· R L R T :, π = : β , N P ::
Working paper, Indiana University Bloomington .32 able 1: Size (Continuous Time Models) Β― π = 0 Β― π = 5 Β― π = 20 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) able 2: Sizes (Discrete Time Models with CNST and SB) Β― π = 0 Β― π = 5 Β― π = 20 π‘ ( π ) π‘ (Λ π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) able 3: Sizes (Discrete Time Models with GARCH) Β― π = 0 Β― π = 5 Β― π = 20 πΌ = 0 . BQ 9.7 5.4 4.8 9.5 5.8 4.6 13.1 6.3 4.6 π = 4 RLRT 9.6 8.8 8.7 9.0 7.9 6.7 13.1 11.3 9.1q=4 4.5 4.9 4.8 4.7 4.7 4.3 4.8 4.6 4.8q=8 4.7 4.9 5.2 4.7 4.6 4.8 4.7 5.0 5.1q=12 4.5 4.9 5.2 4.6 4.6 4.5 4.7 4.6 5.4q=16 4.8 4.9 5.3 4.8 4.5 4.6 4.7 4.9 5.3 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) πΌ = 0 . BQ 10.2 6.2 5.2 10.4 7.1 5.8 14.7 8.3 6.8 π = 3 RLRT 10.7 10.0 9.2 10.6 10.0 8.1 15.9 14.9 12.9q=4 4.1 4.6 4.7 4.7 4.4 4.4 4.5 4.4 4.6q=8 4.6 4.7 4.8 4.5 4.3 4.6 4.7 4.7 4.6q=12 4.4 4.6 4.8 4.6 4.3 4.5 4.4 4.8 5.1q=16 4.7 4.6 5.1 4.7 4.5 4.6 4.6 4.8 5.0 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) πΌ = 0 . BQ 8.9 5.8 6.0 7.8 6.0 6.9 9.3 5.3 6.4 π½ = 0 . RLRT 9.1 10.1 11.5 5.7 8.3 9.8 6.2 9.0 11.5 π = 2 q=4 5.0 4.7 4.4 4.7 4.5 4.4 4.9 4.8 4.4q=8 4.9 4.9 4.5 5.0 4.9 4.9 5.1 5.2 4.4q=12 5.0 5.0 4.6 5.0 5.2 5.1 5.1 5.3 4.9q=16 5.0 5.0 4.9 5.1 5.1 5.1 5.1 5.3 4.8 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) πΌ = 0 . BQ 11.7 8.3 8.1 12.7 10.4 10.9 16.4 12.2 13.4 π½ = 0 . RLRT 13.3 13.0 12.7 13.7 14.7 15.1 20.2 23.7 23.2 π = 2 q=4 4.0 4.3 4.3 4.2 4.1 4.4 4.2 4.5 4.3q=8 4.2 4.2 4.3 4.5 4.0 4.6 4.6 4.7 4.3q=12 4.4 4.2 4.5 4.7 4.1 4.5 4.5 4.7 4.6q=16 4.5 4.4 4.4 4.9 4.2 4.8 4.6 4.8 4.5 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r C N S T O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r S B O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r G B M O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r R S O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r C N S T , d i s c r e t e t i m e O L S :, B o n f . Q : Β· R L R T :, π = : β , N P :: ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r S B , d i s c r e t e t i m e O L S :, B o n f . Q : Β· R L R T :, π = : β , N P ::
Working paper, Indiana University Bloomington .32 able 1: Size (Continuous Time Models) Β― π = 0 Β― π = 5 Β― π = 20 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) able 2: Sizes (Discrete Time Models with CNST and SB) Β― π = 0 Β― π = 5 Β― π = 20 π‘ ( π ) π‘ (Λ π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) able 3: Sizes (Discrete Time Models with GARCH) Β― π = 0 Β― π = 5 Β― π = 20 πΌ = 0 . BQ 9.7 5.4 4.8 9.5 5.8 4.6 13.1 6.3 4.6 π = 4 RLRT 9.6 8.8 8.7 9.0 7.9 6.7 13.1 11.3 9.1q=4 4.5 4.9 4.8 4.7 4.7 4.3 4.8 4.6 4.8q=8 4.7 4.9 5.2 4.7 4.6 4.8 4.7 5.0 5.1q=12 4.5 4.9 5.2 4.6 4.6 4.5 4.7 4.6 5.4q=16 4.8 4.9 5.3 4.8 4.5 4.6 4.7 4.9 5.3 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) πΌ = 0 . BQ 10.2 6.2 5.2 10.4 7.1 5.8 14.7 8.3 6.8 π = 3 RLRT 10.7 10.0 9.2 10.6 10.0 8.1 15.9 14.9 12.9q=4 4.1 4.6 4.7 4.7 4.4 4.4 4.5 4.4 4.6q=8 4.6 4.7 4.8 4.5 4.3 4.6 4.7 4.7 4.6q=12 4.4 4.6 4.8 4.6 4.3 4.5 4.4 4.8 5.1q=16 4.7 4.6 5.1 4.7 4.5 4.6 4.6 4.8 5.0 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) πΌ = 0 . BQ 8.9 5.8 6.0 7.8 6.0 6.9 9.3 5.3 6.4 π½ = 0 . RLRT 9.1 10.1 11.5 5.7 8.3 9.8 6.2 9.0 11.5 π = 2 q=4 5.0 4.7 4.4 4.7 4.5 4.4 4.9 4.8 4.4q=8 4.9 4.9 4.5 5.0 4.9 4.9 5.1 5.2 4.4q=12 5.0 5.0 4.6 5.0 5.2 5.1 5.1 5.3 4.9q=16 5.0 5.0 4.9 5.1 5.1 5.1 5.1 5.3 4.8 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) πΌ = 0 . BQ 11.7 8.3 8.1 12.7 10.4 10.9 16.4 12.2 13.4 π½ = 0 . RLRT 13.3 13.0 12.7 13.7 14.7 15.1 20.2 23.7 23.2 π = 2 q=4 4.0 4.3 4.3 4.2 4.1 4.4 4.2 4.5 4.3q=8 4.2 4.2 4.3 4.5 4.0 4.6 4.6 4.7 4.3q=12 4.4 4.2 4.5 4.7 4.1 4.5 4.5 4.7 4.6q=16 4.5 4.4 4.4 4.9 4.2 4.8 4.6 4.8 4.5 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r C N S T O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r S B O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r G B M O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r R S O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r C N S T , d i s c r e t e t i m e O L S :, B o n f . Q : Β· R L R T :, π = : β , N P :: ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r S B , d i s c r e t e t i m e O L S :, B o n f . Q : Β· R L R T :, π = : β , N P :: ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r A R C H ( πΌ = . ) , d i s c r e t e t i m e O L S :, B o n f . Q : Β· R L R T :, π = : β , N P ::
Working paper, Indiana University Bloomington .32 able 1: Size (Continuous Time Models) Β― π = 0 Β― π = 5 Β― π = 20 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) able 2: Sizes (Discrete Time Models with CNST and SB) Β― π = 0 Β― π = 5 Β― π = 20 π‘ ( π ) π‘ (Λ π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) able 3: Sizes (Discrete Time Models with GARCH) Β― π = 0 Β― π = 5 Β― π = 20 πΌ = 0 . BQ 9.7 5.4 4.8 9.5 5.8 4.6 13.1 6.3 4.6 π = 4 RLRT 9.6 8.8 8.7 9.0 7.9 6.7 13.1 11.3 9.1q=4 4.5 4.9 4.8 4.7 4.7 4.3 4.8 4.6 4.8q=8 4.7 4.9 5.2 4.7 4.6 4.8 4.7 5.0 5.1q=12 4.5 4.9 5.2 4.6 4.6 4.5 4.7 4.6 5.4q=16 4.8 4.9 5.3 4.8 4.5 4.6 4.7 4.9 5.3 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) πΌ = 0 . BQ 10.2 6.2 5.2 10.4 7.1 5.8 14.7 8.3 6.8 π = 3 RLRT 10.7 10.0 9.2 10.6 10.0 8.1 15.9 14.9 12.9q=4 4.1 4.6 4.7 4.7 4.4 4.4 4.5 4.4 4.6q=8 4.6 4.7 4.8 4.5 4.3 4.6 4.7 4.7 4.6q=12 4.4 4.6 4.8 4.6 4.3 4.5 4.4 4.8 5.1q=16 4.7 4.6 5.1 4.7 4.5 4.6 4.6 4.8 5.0 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) πΌ = 0 . BQ 8.9 5.8 6.0 7.8 6.0 6.9 9.3 5.3 6.4 π½ = 0 . RLRT 9.1 10.1 11.5 5.7 8.3 9.8 6.2 9.0 11.5 π = 2 q=4 5.0 4.7 4.4 4.7 4.5 4.4 4.9 4.8 4.4q=8 4.9 4.9 4.5 5.0 4.9 4.9 5.1 5.2 4.4q=12 5.0 5.0 4.6 5.0 5.2 5.1 5.1 5.3 4.9q=16 5.0 5.0 4.9 5.1 5.1 5.1 5.1 5.3 4.8 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) πΌ = 0 . BQ 11.7 8.3 8.1 12.7 10.4 10.9 16.4 12.2 13.4 π½ = 0 . RLRT 13.3 13.0 12.7 13.7 14.7 15.1 20.2 23.7 23.2 π = 2 q=4 4.0 4.3 4.3 4.2 4.1 4.4 4.2 4.5 4.3q=8 4.2 4.2 4.3 4.5 4.0 4.6 4.6 4.7 4.3q=12 4.4 4.2 4.5 4.7 4.1 4.5 4.5 4.7 4.6q=16 4.5 4.4 4.4 4.9 4.2 4.8 4.6 4.8 4.5 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r C N S T O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r S B O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r G B M O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r R S O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r C N S T , d i s c r e t e t i m e O L S :, B o n f . Q : Β· R L R T :, π = : β , N P :: ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r S B , d i s c r e t e t i m e O L S :, B o n f . Q : Β· R L R T :, π = : β , N P :: ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r A R C H ( πΌ = . ) , d i s c r e t e t i m e O L S :, B o n f . Q : Β· R L R T :, π = : β , N P :: ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r A R C H ( πΌ = . ) , d i s c r e t e t i m e O L S :, B o n f . Q : Β· R L R T :, π = : β , N P ::
Working paper, Indiana University Bloomington .32 able 1: Size (Continuous Time Models) Β― π = 0 Β― π = 5 Β― π = 20 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) able 2: Sizes (Discrete Time Models with CNST and SB) Β― π = 0 Β― π = 5 Β― π = 20 π‘ ( π ) π‘ (Λ π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) able 3: Sizes (Discrete Time Models with GARCH) Β― π = 0 Β― π = 5 Β― π = 20 πΌ = 0 . BQ 9.7 5.4 4.8 9.5 5.8 4.6 13.1 6.3 4.6 π = 4 RLRT 9.6 8.8 8.7 9.0 7.9 6.7 13.1 11.3 9.1q=4 4.5 4.9 4.8 4.7 4.7 4.3 4.8 4.6 4.8q=8 4.7 4.9 5.2 4.7 4.6 4.8 4.7 5.0 5.1q=12 4.5 4.9 5.2 4.6 4.6 4.5 4.7 4.6 5.4q=16 4.8 4.9 5.3 4.8 4.5 4.6 4.7 4.9 5.3 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) πΌ = 0 . BQ 10.2 6.2 5.2 10.4 7.1 5.8 14.7 8.3 6.8 π = 3 RLRT 10.7 10.0 9.2 10.6 10.0 8.1 15.9 14.9 12.9q=4 4.1 4.6 4.7 4.7 4.4 4.4 4.5 4.4 4.6q=8 4.6 4.7 4.8 4.5 4.3 4.6 4.7 4.7 4.6q=12 4.4 4.6 4.8 4.6 4.3 4.5 4.4 4.8 5.1q=16 4.7 4.6 5.1 4.7 4.5 4.6 4.6 4.8 5.0 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) πΌ = 0 . BQ 8.9 5.8 6.0 7.8 6.0 6.9 9.3 5.3 6.4 π½ = 0 . RLRT 9.1 10.1 11.5 5.7 8.3 9.8 6.2 9.0 11.5 π = 2 q=4 5.0 4.7 4.4 4.7 4.5 4.4 4.9 4.8 4.4q=8 4.9 4.9 4.5 5.0 4.9 4.9 5.1 5.2 4.4q=12 5.0 5.0 4.6 5.0 5.2 5.1 5.1 5.3 4.9q=16 5.0 5.0 4.9 5.1 5.1 5.1 5.1 5.3 4.8 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) πΌ = 0 . BQ 11.7 8.3 8.1 12.7 10.4 10.9 16.4 12.2 13.4 π½ = 0 . RLRT 13.3 13.0 12.7 13.7 14.7 15.1 20.2 23.7 23.2 π = 2 q=4 4.0 4.3 4.3 4.2 4.1 4.4 4.2 4.5 4.3q=8 4.2 4.2 4.3 4.5 4.0 4.6 4.6 4.7 4.3q=12 4.4 4.2 4.5 4.7 4.1 4.5 4.5 4.7 4.6q=16 4.5 4.4 4.4 4.9 4.2 4.8 4.6 4.8 4.5 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r C N S T O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r S B O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r G B M O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r R S O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r C N S T , d i s c r e t e t i m e O L S :, B o n f . Q : Β· R L R T :, π = : β , N P :: ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r S B , d i s c r e t e t i m e O L S :, B o n f . Q : Β· R L R T :, π = : β , N P :: ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r A R C H ( πΌ = . ) , d i s c r e t e t i m e O L S :, B o n f . Q : Β· R L R T :, π = : β , N P :: ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r A R C H ( πΌ = . ) , d i s c r e t e t i m e O L S :, B o n f . Q : Β· R L R T :, π = : β , N P :: ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r G A R C H ( πΌ = . , π½ = . ) , d i s c r e t e t i m e O L S :, B o n f . Q : Β· R L R T :, π = : β , N P ::
Working paper, Indiana University Bloomington .32 able 1: Size (Continuous Time Models) Β― π = 0 Β― π = 5 Β― π = 20 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) able 2: Sizes (Discrete Time Models with CNST and SB) Β― π = 0 Β― π = 5 Β― π = 20 π‘ ( π ) π‘ (Λ π ) π‘ (Λ π ) π‘ (^ π ) π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) able 3: Sizes (Discrete Time Models with GARCH) Β― π = 0 Β― π = 5 Β― π = 20 πΌ = 0 . BQ 9.7 5.4 4.8 9.5 5.8 4.6 13.1 6.3 4.6 π = 4 RLRT 9.6 8.8 8.7 9.0 7.9 6.7 13.1 11.3 9.1q=4 4.5 4.9 4.8 4.7 4.7 4.3 4.8 4.6 4.8q=8 4.7 4.9 5.2 4.7 4.6 4.8 4.7 5.0 5.1q=12 4.5 4.9 5.2 4.6 4.6 4.5 4.7 4.6 5.4q=16 4.8 4.9 5.3 4.8 4.5 4.6 4.7 4.9 5.3 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) πΌ = 0 . BQ 10.2 6.2 5.2 10.4 7.1 5.8 14.7 8.3 6.8 π = 3 RLRT 10.7 10.0 9.2 10.6 10.0 8.1 15.9 14.9 12.9q=4 4.1 4.6 4.7 4.7 4.4 4.4 4.5 4.4 4.6q=8 4.6 4.7 4.8 4.5 4.3 4.6 4.7 4.7 4.6q=12 4.4 4.6 4.8 4.6 4.3 4.5 4.4 4.8 5.1q=16 4.7 4.6 5.1 4.7 4.5 4.6 4.6 4.8 5.0 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) πΌ = 0 . BQ 8.9 5.8 6.0 7.8 6.0 6.9 9.3 5.3 6.4 π½ = 0 . RLRT 9.1 10.1 11.5 5.7 8.3 9.8 6.2 9.0 11.5 π = 2 q=4 5.0 4.7 4.4 4.7 4.5 4.4 4.9 4.8 4.4q=8 4.9 4.9 4.5 5.0 4.9 4.9 5.1 5.2 4.4q=12 5.0 5.0 4.6 5.0 5.2 5.1 5.1 5.3 4.9q=16 5.0 5.0 4.9 5.1 5.1 5.1 5.1 5.3 4.8 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) πΌ = 0 . BQ 11.7 8.3 8.1 12.7 10.4 10.9 16.4 12.2 13.4 π½ = 0 . RLRT 13.3 13.0 12.7 13.7 14.7 15.1 20.2 23.7 23.2 π = 2 q=4 4.0 4.3 4.3 4.2 4.1 4.4 4.2 4.5 4.3q=8 4.2 4.2 4.3 4.5 4.0 4.6 4.6 4.7 4.3q=12 4.4 4.2 4.5 4.7 4.1 4.5 4.5 4.7 4.6q=16 4.5 4.4 4.4 4.9 4.2 4.8 4.6 4.8 4.5 π‘ ( π ) π‘ (Λ π ) π‘ (^ π ) ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r C N S T O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r S B O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r G B M O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r R S O L S :, B o n f . Q : Β· R L R T :, C a u c h y R T :, π = : β , N P : ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r C N S T , d i s c r e t e t i m e O L S :, B o n f . Q : Β· R L R T :, π = : β , N P :: ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r S B , d i s c r e t e t i m e O L S :, B o n f . Q : Β· R L R T :, π = : β , N P :: ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r A R C H ( πΌ = . ) , d i s c r e t e t i m e O L S :, B o n f . Q : Β· R L R T :, π = : β , N P :: ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r A R C H ( πΌ = . ) , d i s c r e t e t i m e O L S :, B o n f . Q : Β· R L R T :, π = : β , N P :: ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r G A R C H ( πΌ = . , π½ = . ) , d i s c r e t e t i m e O L S :, B o n f . Q : Β· R L R T :, π = : β , N P :: ββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( a ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( b ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( c ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( d ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( e ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( f ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( g ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( h ) Β― π = , π = βββββββββββββββββββββββββββββββββββββββββ . . . . . . Ξ² ( i ) Β― π = , π = F i g u r e : P o w e r f o r G A R C H ( πΌ = . , π½ = . ) , d i s c r e t e t i m e O L S :, B o n f . Q : Β· R L R T :, π = : β , N P ::