Pub Date : 2022-08-04DOI: 10.1017/s0266466622000639
S. Johansen, M. Nielsen
It is well known that, under suitable regularity conditions, the normalized fractional process with fractional parameter d converges weakly to fractional Brownian motion (fBm) for $d>frac {1}{2}$ . We show that, for any nonnegative integer M, derivatives of order $m=0,1,dots ,M$ of the normalized fractional process with respect to the fractional parameter d jointly converge weakly to the corresponding derivatives of fBm. As an illustration, we apply the results to the asymptotic distribution of the score vectors in the multifractional vector autoregressive model.
{"title":"WEAK CONVERGENCE TO DERIVATIVES OF FRACTIONAL BROWNIAN MOTION","authors":"S. Johansen, M. Nielsen","doi":"10.1017/s0266466622000639","DOIUrl":"https://doi.org/10.1017/s0266466622000639","url":null,"abstract":"It is well known that, under suitable regularity conditions, the normalized fractional process with fractional parameter d converges weakly to fractional Brownian motion (fBm) for \u0000\u0000 \u0000 \u0000 \u0000$d>frac {1}{2}$\u0000\u0000 \u0000 . We show that, for any nonnegative integer M, derivatives of order \u0000\u0000 \u0000 \u0000 \u0000$m=0,1,dots ,M$\u0000\u0000 \u0000 of the normalized fractional process with respect to the fractional parameter d jointly converge weakly to the corresponding derivatives of fBm. As an illustration, we apply the results to the asymptotic distribution of the score vectors in the multifractional vector autoregressive model.","PeriodicalId":49275,"journal":{"name":"Econometric Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41669590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-03DOI: 10.1017/S0266466622000275
Shuo Li, Liuhua Peng, Xiaojun Song
Conditional value-at-risk (CVaR) and conditional expected shortfall (CES) are widely adopted risk measures which help monitor potential tail risk while adapting to evolving market information. In this paper, we propose an approach to constructing simultaneous confidence bands (SCBs) for tail risk as measured by CVaR and CES, with the confidence bands uniformly valid for a set of tail levels. We consider one-sided tail risk (downside or upside tail risk) as well as relative tail risk (the ratio of upside to downside tail risk). A general class of location-scale models with heavy-tailed innovations is employed to filter out the return dynamics. Then, CVaR and CES are estimated with the aid of extreme value theory. In the asymptotic theory, we consider two scenarios: (i) the extreme scenario that allows for extrapolation beyond the range of the available data and (ii) the intermediate scenario that works exclusively in the case where the available data are adequate relative to the tail level. For finite-sample implementation, we propose a novel bootstrap procedure to circumvent the slow convergence rates of the SCBs as well as infeasibility of approximating the limiting distributions. A series of Monte Carlo simulations confirm that our approach works well in finite samples.
{"title":"SIMULTANEOUS CONFIDENCE BANDS FOR CONDITIONAL VALUE-AT-RISK AND EXPECTED SHORTFALL","authors":"Shuo Li, Liuhua Peng, Xiaojun Song","doi":"10.1017/S0266466622000275","DOIUrl":"https://doi.org/10.1017/S0266466622000275","url":null,"abstract":"Conditional value-at-risk (CVaR) and conditional expected shortfall (CES) are widely adopted risk measures which help monitor potential tail risk while adapting to evolving market information. In this paper, we propose an approach to constructing simultaneous confidence bands (SCBs) for tail risk as measured by CVaR and CES, with the confidence bands uniformly valid for a set of tail levels. We consider one-sided tail risk (downside or upside tail risk) as well as relative tail risk (the ratio of upside to downside tail risk). A general class of location-scale models with heavy-tailed innovations is employed to filter out the return dynamics. Then, CVaR and CES are estimated with the aid of extreme value theory. In the asymptotic theory, we consider two scenarios: (i) the extreme scenario that allows for extrapolation beyond the range of the available data and (ii) the intermediate scenario that works exclusively in the case where the available data are adequate relative to the tail level. For finite-sample implementation, we propose a novel bootstrap procedure to circumvent the slow convergence rates of the SCBs as well as infeasibility of approximating the limiting distributions. A series of Monte Carlo simulations confirm that our approach works well in finite samples.","PeriodicalId":49275,"journal":{"name":"Econometric Theory","volume":"39 1","pages":"1009 - 1043"},"PeriodicalIF":0.8,"publicationDate":"2022-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43045091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-27DOI: 10.1017/S0266466622000342
P. Phillips
New methods are developed for identifying, estimating, and performing inference with nonstationary time series that have autoregressive roots near unity. The approach subsumes unit-root (UR), local unit-root (LUR), mildly integrated (MI), and mildly explosive (ME) specifications in the new model formulation. It is shown how a new parameterization involving a localizing rate sequence that characterizes departures from unity can be consistently estimated in all cases. Simple pivotal limit distributions that enable valid inference about the form and degree of nonstationarity apply for MI and ME specifications and new limit theory holds in UR and LUR cases. Normalizing and variance stabilizing properties of the new parameterization are explored. Simulations are reported that reveal some of the advantages of this alternative formulation of nonstationary time series. A housing market application of the methods is conducted that distinguishes the differing forms of house price behavior in Australian state capital cities over the past decade.
{"title":"ESTIMATION AND INFERENCE WITH NEAR UNIT ROOTS","authors":"P. Phillips","doi":"10.1017/S0266466622000342","DOIUrl":"https://doi.org/10.1017/S0266466622000342","url":null,"abstract":"New methods are developed for identifying, estimating, and performing inference with nonstationary time series that have autoregressive roots near unity. The approach subsumes unit-root (UR), local unit-root (LUR), mildly integrated (MI), and mildly explosive (ME) specifications in the new model formulation. It is shown how a new parameterization involving a localizing rate sequence that characterizes departures from unity can be consistently estimated in all cases. Simple pivotal limit distributions that enable valid inference about the form and degree of nonstationarity apply for MI and ME specifications and new limit theory holds in UR and LUR cases. Normalizing and variance stabilizing properties of the new parameterization are explored. Simulations are reported that reveal some of the advantages of this alternative formulation of nonstationary time series. A housing market application of the methods is conducted that distinguishes the differing forms of house price behavior in Australian state capital cities over the past decade.","PeriodicalId":49275,"journal":{"name":"Econometric Theory","volume":"39 1","pages":"221 - 263"},"PeriodicalIF":0.8,"publicationDate":"2022-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49241332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-22DOI: 10.1017/S0266466622000305
J. Hahn, Z. Liao, G. Ridder
In this paper, we reconsider the assumptions that ensure the identification of the production function in Olley and Pakes (1996, Econometrica 64, 1263–1297). We show that an index restriction plays a crucial role in the identification, especially if the capital stock is measured by the perpetual inventory method. The index restriction is not sufficient for identification under sample selectivity. The index restriction makes it possible to derive the influence function and the asymptotic variance of the Olley–Pakes estimator.
{"title":"IDENTIFICATION AND THE INFLUENCE FUNCTION OF OLLEY AND PAKES’ (1996) PRODUCTION FUNCTION ESTIMATOR","authors":"J. Hahn, Z. Liao, G. Ridder","doi":"10.1017/S0266466622000305","DOIUrl":"https://doi.org/10.1017/S0266466622000305","url":null,"abstract":"In this paper, we reconsider the assumptions that ensure the identification of the production function in Olley and Pakes (1996, Econometrica 64, 1263–1297). We show that an index restriction plays a crucial role in the identification, especially if the capital stock is measured by the perpetual inventory method. The index restriction is not sufficient for identification under sample selectivity. The index restriction makes it possible to derive the influence function and the asymptotic variance of the Olley–Pakes estimator.","PeriodicalId":49275,"journal":{"name":"Econometric Theory","volume":"39 1","pages":"1044 - 1066"},"PeriodicalIF":0.8,"publicationDate":"2022-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47227002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-22DOI: 10.1017/s0266466622000317
Ruixuan Liu, Zhengfei Yu
We propose two simple semiparametric estimation methods for ordered response models with an unknown error distribution. The proposed methods do not require users to choose any tuning parameters, and they automatically incorporate the monotonicity restriction of the unknown distribution function. Fixing finite-dimensional parameters in the model, we construct nonparametric maximum likelihood estimates for the error distribution based on the related binary choice data or the entire ordered response data. We then obtain estimates for finite-dimensional parameters based on moment conditions given the estimated distribution function. Our semiparametric approaches deliver root-n consistent and asymptotically normal estimators of the regression coefficient and threshold parameter. We also develop valid bootstrap procedures for inference. The advantages of our methods are borne out in simulation studies and a real data application.
{"title":"SIMPLE SEMIPARAMETRIC ESTIMATION OF ORDERED RESPONSE MODELS","authors":"Ruixuan Liu, Zhengfei Yu","doi":"10.1017/s0266466622000317","DOIUrl":"https://doi.org/10.1017/s0266466622000317","url":null,"abstract":"\u0000 We propose two simple semiparametric estimation methods for ordered response models with an unknown error distribution. The proposed methods do not require users to choose any tuning parameters, and they automatically incorporate the monotonicity restriction of the unknown distribution function. Fixing finite-dimensional parameters in the model, we construct nonparametric maximum likelihood estimates for the error distribution based on the related binary choice data or the entire ordered response data. We then obtain estimates for finite-dimensional parameters based on moment conditions given the estimated distribution function. Our semiparametric approaches deliver root-n consistent and asymptotically normal estimators of the regression coefficient and threshold parameter. We also develop valid bootstrap procedures for inference. The advantages of our methods are borne out in simulation studies and a real data application.","PeriodicalId":49275,"journal":{"name":"Econometric Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43135634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-14DOI: 10.1017/S0266466622000287
P. Phillips, Ying Wang
Functional coefficient (FC) regressions allow for systematic flexibility in the responsiveness of a dependent variable to movements in the regressors, making them attractive in applications where marginal effects may depend on covariates. Such models are commonly estimated by local kernel regression methods. This paper explores situations where responsiveness to covariates is locally flat or fixed. The paper develops new asymptotics that take account of shape characteristics of the function in the locality of the point of estimation. Both stationary and integrated regressor cases are examined. The limit theory of FC kernel regression is shown to depend intimately on functional shape in ways that affect rates of convergence, optimal bandwidth selection, estimation, and inference. In FC cointegrating regression, flat behavior materially changes the limit distribution by introducing the shape characteristics of the function into the limiting distribution through variance as well as centering. In the boundary case where the number of zero derivatives tends to infinity, near parametric rates of convergence apply in stationary and nonstationary cases. Implications for inference are discussed and a feasible pre-test inference procedure is proposed that takes unknown potential flatness into consideration and provides a practical approach to inference.
{"title":"LIMIT THEORY FOR LOCALLY FLAT FUNCTIONAL COEFFICIENT REGRESSION","authors":"P. Phillips, Ying Wang","doi":"10.1017/S0266466622000287","DOIUrl":"https://doi.org/10.1017/S0266466622000287","url":null,"abstract":"Functional coefficient (FC) regressions allow for systematic flexibility in the responsiveness of a dependent variable to movements in the regressors, making them attractive in applications where marginal effects may depend on covariates. Such models are commonly estimated by local kernel regression methods. This paper explores situations where responsiveness to covariates is locally flat or fixed. The paper develops new asymptotics that take account of shape characteristics of the function in the locality of the point of estimation. Both stationary and integrated regressor cases are examined. The limit theory of FC kernel regression is shown to depend intimately on functional shape in ways that affect rates of convergence, optimal bandwidth selection, estimation, and inference. In FC cointegrating regression, flat behavior materially changes the limit distribution by introducing the shape characteristics of the function into the limiting distribution through variance as well as centering. In the boundary case where the number of zero derivatives tends to infinity, near parametric rates of convergence apply in stationary and nonstationary cases. Implications for inference are discussed and a feasible pre-test inference procedure is proposed that takes unknown potential flatness into consideration and provides a practical approach to inference.","PeriodicalId":49275,"journal":{"name":"Econometric Theory","volume":"39 1","pages":"900 - 949"},"PeriodicalIF":0.8,"publicationDate":"2022-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49339204","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-14DOI: 10.1017/S0266466622000238
Luca Mucciante, Alessio Sancetta
This paper introduces a counting process for event arrivals in high-frequency trading, based on high-dimensional covariates. The novelty is that, under sparsity conditions on the true model, we do not need to impose any model penalty or parameters shrinkage, unlike Lasso. The procedure allows us to derive a central limit theorem to test restrictions in a two-stage estimator. We achieve this by the use of a sign constraint on the intensity which necessarily needs to be positive. In particular, we introduce an additive model to extract the nonlinear impact of order book variables on buy and sell trade arrivals. In the empirical application, we show that the shape and dynamics of the order book are fundamental in determining the arrival of buy and sell trades in the crude oil futures market. We establish our empirical results mapping the covariates into a higher-dimensional space. Consistently with the theoretical results, the estimated models are sparse in the number of parameters. Using this approach, we are also able to compare competing model hypotheses on the basis of an out-of-sample likelihood ratio type of test.
{"title":"ESTIMATION OF A HIGH-DIMENSIONAL COUNTING PROCESS WITHOUT PENALTY FOR HIGH-FREQUENCY EVENTS","authors":"Luca Mucciante, Alessio Sancetta","doi":"10.1017/S0266466622000238","DOIUrl":"https://doi.org/10.1017/S0266466622000238","url":null,"abstract":"This paper introduces a counting process for event arrivals in high-frequency trading, based on high-dimensional covariates. The novelty is that, under sparsity conditions on the true model, we do not need to impose any model penalty or parameters shrinkage, unlike Lasso. The procedure allows us to derive a central limit theorem to test restrictions in a two-stage estimator. We achieve this by the use of a sign constraint on the intensity which necessarily needs to be positive. In particular, we introduce an additive model to extract the nonlinear impact of order book variables on buy and sell trade arrivals. In the empirical application, we show that the shape and dynamics of the order book are fundamental in determining the arrival of buy and sell trades in the crude oil futures market. We establish our empirical results mapping the covariates into a higher-dimensional space. Consistently with the theoretical results, the estimated models are sparse in the number of parameters. Using this approach, we are also able to compare competing model hypotheses on the basis of an out-of-sample likelihood ratio type of test.","PeriodicalId":49275,"journal":{"name":"Econometric Theory","volume":"39 1","pages":"989 - 1008"},"PeriodicalIF":0.8,"publicationDate":"2022-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45007456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-03DOI: 10.1017/S0266466622000214
C. Robert
We consider high-frequency observations from a one-dimensional time-homogeneous diffusion process Y. We assume that the diffusion coefficient $sigma $ is continuously differentiable in y, but with a jump discontinuity at some level y, say $y=0$ . We first study sign-constrained kernel estimators of functions of the left and right limits of $sigma $ at $0$ . These functions intricately depend on both limits. We propose a method to extricate these functions by searching for bandwidths where the kernel estimators are stable by iteration. We finally provide an estimator of the discontinuity jump size. We prove its convergence in probability and discuss its rate of convergence. A Monte Carlo study shows the finite sample properties of this estimator.
{"title":"HOW LARGE IS THE JUMP DISCONTINUITY IN THE DIFFUSION COEFFICIENT OF A TIME-HOMOGENEOUS DIFFUSION?","authors":"C. Robert","doi":"10.1017/S0266466622000214","DOIUrl":"https://doi.org/10.1017/S0266466622000214","url":null,"abstract":"We consider high-frequency observations from a one-dimensional time-homogeneous diffusion process Y. We assume that the diffusion coefficient \u0000$sigma $\u0000 is continuously differentiable in y, but with a jump discontinuity at some level y, say \u0000$y=0$\u0000 . We first study sign-constrained kernel estimators of functions of the left and right limits of \u0000$sigma $\u0000 at \u0000$0$\u0000 . These functions intricately depend on both limits. We propose a method to extricate these functions by searching for bandwidths where the kernel estimators are stable by iteration. We finally provide an estimator of the discontinuity jump size. We prove its convergence in probability and discuss its rate of convergence. A Monte Carlo study shows the finite sample properties of this estimator.","PeriodicalId":49275,"journal":{"name":"Econometric Theory","volume":"39 1","pages":"848 - 880"},"PeriodicalIF":0.8,"publicationDate":"2022-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41629293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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