Peter M. Robinson's research while affiliated with The London School of Economics and Political Science and other places
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Publications (106)
We study a functional version of nonstationary fractionally integrated time series, covering the functional unit root as a special case. The time series taking values in an infinite-dimensional separable Hilbert space are projected onto a finite number of sub-spaces, the level of nonstationarity allowed to vary over them. Under regularity condition...
We develop refined inference for spatial regression models with predetermined regressors. The ordinary least squares estimate of the spatial parameter is neither consistent nor asymptotically normal, unless the elements of the spatial weight matrix uniformly vanish as sample size diverges. We develop refined testing of the hypothesis of no spatial...
In this paper, we consider efficiency improvement in a nonparametric panel data model with cross-sectional dependence. A Generalized Least Squares (GLS)-type estimator is proposed by taking into account this dependence structure. Parameterizing the cross-sectional dependence, a local linear estimator is shown to be dominated by this type of GLS est...
This paper studies stationary functional time series with long‐range dependence, and estimates the memory parameter involved. Semiparametric local Whittle estimation is used, where periodogram is constructed from the approximate first score, which is an inner product of the functional observation and estimated leading eigenfunction. The latter is o...
We study a functional version of fractionally integrated time series, covering the functional unit root as a special case. The functional time series are projected onto a finite number of sub-spaces, the level of nonstationarity allowed to vary over them. Under regularity conditions, we derive a weak convergence result for the projection of the fra...
The paper develops point estimation and asymptotic theory with respect to a semiparametric model for time series with moving mean and unconditional heteroscedasticity. These two features are modelled nonparametrically, whereas autocorrelations are described by a short memory stationary parametric time series model. We first study the usual least sq...
This paper studies stationary functional time series with long range dependence, and estimates the self-similarity parameter involved. Semiparametric local Whittle estimation is used, where the discrete Fourier transform and periodogram are constructed for the approximate first score which is an inner product of the functional observation and estim...
We discuss developments and future prospects for statistical modeling and inference for spatial data that have long memory. While a number of contributons have been made, the literature is relatively small and scattered, compared to the literatures on long memory time series on the one hand, and spatial data with short memory on the other. Thus, ov...
In a general class of semiparametric pure spatial models (having no explanatory variables) allowing nonlinearity in the parameter and the weight matrix, we propose adaptive tests and estimates which are asymptotically efficient in the presence of unknown, nonparametric distributional form. Feasibility of adaptive estimation is verified and its effi...
The article discusses statistical inference in parametric models for panel data. The models feature dynamics of a general nature, individual effects, and possible explanatory variables. The focus is on large-cross-section inference on Gaussian pseudo maximum likelihood estimates with temporal dimension kept fixed, partially complementing and extend...
We introduce methods and theory for functional or curve time series with long-range dependence. The temporal sum of the curve process is shown to be asymp-totically normally distributed, the conditions for this covering a functional version of fractionally integrated autoregressive moving averages. We also construct an estimate of the long-run cova...
Semiparametric panel data modelling and statistical inference with fractional stochastic trends, nonparametrically time-trending individual effects, and general cross-sectional correlation and heteroscedasticity in innovations are developed. The fractional stochastic trends allow for a wide range of nonstationarity, indexed by a memory parameter, n...
Pseudo maximum likelihood estimates are developed for higher-order spatial autoregressive models with increasingly many parameters, including models with spatial lags in the dependent variables both with and without a linear or nonlinear regression component, and regression models with spatial autoregressive disturbances. Consistency and asymptotic...
Central limit theorems are established for the sum, over a spatial region, of
observations from a linear process on a $d$-dimensional lattice. This region
need not be rectangular, but can be irregularly-shaped. Separate results are
established for the cases of positive strong dependence, short range
dependence, and negative dependence. We provide a...
We consider testing the null hypothesis of no spatial correlation against the
alternative of pure first order spatial autoregression. A test statistic based on the
least squares estimate has good first-order asymptotic properties, but these may not
be relevant in small- or moderate-sized samples, especially as (depending on properties
of the spatia...
An asymptotic theory is developed for series estimation of nonparametric and semiparametric regression models for cross-sectional data under conditions on disturbances that allow for forms of cross-sectional dependence and hetero-
geneity, including conditional and unconditional heteroskedascity, along with conditions on regressors that allow depen...
We propose a new and easy-to-use method for identifying cointegrated
components of nonstationary time series, consisting of an eigenalysis for a
certain non-negative definite matrix. Our setting is model-free, and we allow
the integer-valued integration orders of the observable series to be unknown,
and to possibly differ. Consistency of estimates...
We develop non-nested tests in a general spatial, spatio-temporal or panel data context. The spatial aspect can be interpreted quite generally, in either a geographical sense, or employing notions of economic distance, or when parametric modelling arises in part from a common factor or other structure. In the former case, observations may be regula...
In a panel data model with fixed effects, possible cross-sectional dependence is investigated in a spatial autoregressive setting. An Edgeworth expansion is developed for the maximum likelihood estimate of the spatial correlation coefficient. The expansion is used to develop more accurate interval estimates for the coefficient, and tests for cross-...
Nonparametric regression is developed for data with both a temporal and a cross-sectional dimension. The model includes additive, unknown, individual-specific components and allows also for cross-sectional and temporal dependence and conditional heteroscedasticity. A simple nonparametric estimate is shown to be dominated by a GLS-type one. Asymptot...
This paper develops consistency and asymptotic normality of parameter estimates for a higher-order spatial autoregressive model whose order, and number of regressors, are allowed to approach infinity slowly with sample size. Both least squares and instrumental variables estimates are examined, and the permissible rate of growth of the dimension of...
A dynamic panel data model is considered that contains possibly stochastic individual components and a common stochastic time trend that allows for stationary and nonstationary long memory and general parametric short memory. We propose four different ways of coping with the individual effects so as to estimate the parameters. Like models with auto...
Developments in research on stationary and nonstationary time series with long range dependence are reviewed.
Keywords:
long range dependence;
long memory;
parametric estimation;
semiparametric estimation;
statistical properties
We consider testing the null hypothesis of no spatial correlation against the alternative of pure first order spatial autoregression. A test statistic based on the least squares estimate has good first-order asymptotic properties, but these may not be relevant in small- or moderate-sized samples, especially as (depending on properties of the spatia...
We consider time series that, possibly after integer differencing or integrating or other detrending, are covariance stationary with spectral density that is regularly varying near zero frequency, and unspecified elsewhere. This semiparametric framework includes series with short, long and negative memory. We consider the consistency of the popular...
For testing lack of correlation against spatial autoregressive alternatives, Lagrange multiplier tests enjoy their usual computational advantages, but the (χ2) rst-order asymptotic approximation to critical values can be poor in small samples. We develop refined tests for lack of spatial error correlation in regressions, based on Edgeworth expansio...
Developments in research on stationary and nonstationary time series with long range dependence are reviewed.
A semiparametric model is proposed in which a parametric filtering of a nonstationary
time series, incorporating fractionally differencing with short memory correction,
removes correlation but leaves a nonparametric deterministic trend. Estimates
of the memory parameter and other dependence parameters are proposed, and shown
to be consistent and as...
Power law or generalized polynomial regressions with unknown real-valued
exponents and coefficients, and weakly dependent errors, are considered for
observations over time, space or space--time. Consistency and asymptotic
normality of nonlinear least-squares estimates of the parameters are
established. The joint limit distribution is singular, but...
We consider the estimation of parametric fractional time series models in
which not only is the memory parameter unknown, but one may not know whether it
lies in the stationary/invertible region or the nonstationary or noninvertible
regions. In these circumstances, a proof of consistency (which is a
prerequisite for proving asymptotic normality) ca...
Power law or generalized polynomial regressions with unknown real-valued exponents and coefficients, and weakly dependent errors, are considered for observations over time, space or space-time. Consistency and asymptotic normality of nonlinear least squares estimates of the parameters are established. The joint limit distribution is singular, but c...
Continuous Time Econometric ModellingBergstromA.R.Oxford University Press, 1991 - Volume 8 Issue 4 - Peter M. Robinson
Panel data, whose series length T is large but whose cross-section size N need not be, are assumed to have a common time trend. The time trend is of unknown form, the model includes additive, unknown, individual-speci…c com-ponents, and we allow for spatial or other cross-sectional dependence and/or heteroscedasticity. A simple smoothed nonparametr...
Central limit theorems are developed for instrumental variables estimates of linear and semi-parametric partly linear regression models for spatial data. General forms of spatial dependence and heterogeneity in explanatory variables and unobservable disturbances are permitted. We discuss estimation of the variance matrix, including estimates that a...
We consider a multivariate continuous-time process, generated by a system of linear stochastic differential equations, driven by white noise, and involving coefficients that possibly vary over time. The process is observable only at discrete, but not necessarily equally-spaced, time points (though equal spacing significantly simplifies matters). Su...
Disregarding spatial dependence can invalidate methods for analyzing cross-sectional and panel data. We discuss ongoing work on developing methods that allow for, test for, or estimate, spatial dependence. Much of the stress is on nonparametric and semiparametric methods.
Asset returns are frequently assumed to be determined by one or more common factors. We consider a bivariate factor model where the unobservable common factor and idiosyncratic errors are stationary and serially uncorrelated but have strong dependence in higher moments. Stochastic volatility models for the latent variables are employed, in view of...
Nonlinear functions of multivariate financial time series can exhibit long memory and fractional cointegration. However, tools for analysing these phenomena have principally been justified under assumptions that are invalid in this setting. Determination of asymptotic theory under more plausible assumptions can be complicated and lengthy. We discus...
We develop a sequence of tests for specifying the cointegrating rank of, possiblyfractional, multiple time series. Memory parameters of observables are treated asunknown, as are those of possible cointegrating errors. The individual test statisticshave standard null asymptotics, and are related to Hausman specification teststatistics: when the memo...
Efficient semiparametric and parametric estimates are developed for a
spatial autoregressive model, containing nonstochastic explanatory
variables and innovations suspected to be non-normal. The main stress is
on the case of distribution of unknown, nonparametric, form, where series
nonparametric estimates of the score function are employed in adap...
We provide a general class of tests for correlation in time series, spatial, spatio-temporal and cross-sectional data. We motivate our focus by reviewing how computational and theoretical difficulties of point estimation mount as one moves from regularly-spaced time series data, through forms of irregular spacing, and to spatial data of various kin...
Efficient semiparametric and parametric estimates are developed for aspatial autoregressive model, containing nonstochastic explanatoryvariables and innovations suspected to be non-normal. The main stress ison the case of distribution of unknown, nonparametric, form, where seriesnonparametric estimates of the score function are employed in adaptive...
Employing recent results of Robinson (2005) we consider the asymptotic properties ofconditional-sum-of-squares (CSS) estimates of parametric models for stationary timeseries with long memory. CSS estimation has been considered as a rival to Gaussianmaximum likelihood and Whittle estimation of time series models. The latter kinds ofestimate have bee...
A semiparametric bivariate fractionally cointegrated system is considered, integrationorders possibly being unknown and I (0) unobservable inputs having nonparametricspectral density. Two kinds of estimate of the cointegrating parameter ? are considered,one involving inverse spectral weighting and the other, unweighted statistics with a spectralest...
Efficient semiparametric and parametric estimates are developed for a spatial autoregressive model, containing non-stochastic explanatory variables and innovations suspected to be non-normal. The main stress is on the case of distribution of unknown, nonparametric, form, where series nonparametric estimates of the score function are employed in ada...
Empirical evidence has emerged of the possibility of fractional cointegration such that thegap, ß, between the integration order d of observable time series, and the integrationorder ? of cointegrating errors, is less than 0.5. This includes circumstances whenobservables are stationary or asymptotically stationary with long memory (so d < 1/2),and...
Smoothed nonparametric kernel spectral density estimates areconsidered for stationary data observed on a d-dimensional lattice.The implications for edge effect bias of the choice of kernel andbandwidth are considered. Under some circumstances the bias canbe dominated by the edge effect. We show that this problem can bemitigated by tapering. Some ex...
Strong consistency and asymptotic normality of the Gaussian pseudo-maximumlikelihood estimate of the parameters in a wide class of ARCH(8) processesare established. We require the ARCH weights to decay at least hyperbolically,with a faster rate needed for the central limit theorem than for the law of largenumbers. Various rates are illustrated in e...
Much time series data are recorded on economic and financial variables. Statistical modeling of such data is now very well developed, and has applications in forecasting. We review a variety of statistical models from the viewpoint of "memory", or strength of dependence across time, which is a helpful discriminator between different phenomena of in...
In a number of semiparametric models, smoothing seems necessary in order to obtain estimates of the parametric component which are asymptotically normal and converge at parametric rate. However, smoothing can inflate the error in the normal approximation, so that refined approximations are of interest, especially in sample sizes that are not enormo...
We consider a time series model involving a fractional stochastic component, whose integration order can lie in the stationary/invertible or nonstationary regions and be unknown, and additive deterministic component consisting of a generalised polynomial. The model can thus incorporate competing descriptions of trending behaviour. The stationary in...
The asymptotic distributions of cointegration tests are approximated using the Gamma distribution. The tests considered are for the I(1), the conditional I(1), as well as the I(2) model. Formulae for the parameters of the Gamma distributions are derived from response surfaces. The resulting approximation is flexible, easy to implement and more accu...
Cointegrated bivariate nonstationary time series are considered in a fractional context, without allowance for deterministic trends. Both the observable series and the cointegrating error can be fractional processes. The familiar situation in which the respective integration orders are 1 and 0 is nested, but these values have typically been assumed...
We consider the long memory and leverage properties of a model for the conditional variance of an observable stationary sequence, where the conditional variance is the square of an inhomogeneous linear combination of past values of the observable sequence, with square summable weights. This model, which we call linear ARCH (LARCH), specializes to t...
The semiparametric local Whittle or Gaussian estimate of the long memory parameter is known to have especially nice limiting distributional properties, being asymptotically normal with a limiting variance that is completely known. However in moderate samples the normal approximation may not be very good, so we consider a refined, Edgeworth, approxi...
Chinese policy-makers fear that an RMB appreciation will reduce low technology exports. We investigate this issue using data on China's exports to 30 countries. We find that an appreciation of the RMB would substantially reduce China's exports of clothing, furniture and footwear. We also find that an increase in foreign income, an increase in the C...
Econometric interest in the possibility of long memory has developed, as a flexible alternative to or compromise between the usual short memory or unit root prescriptions, for example in the context of modelling cointegrating or other relationships and in describing the dependence structure of nonlinear functions of financial returns. Semiparametri...
We show that it is possible to adapt to nonparametric disturbance autocorrelation in time series regression in the presence of long memory in both regressors and disturbances by using a smoothed nonparametric spectrum estimate in frequency–domain generalized least squares. When the collective memory in regressors and disturbances is sufficiently st...
This paper develops methods of investigating the existence and extent of cointegration in fractionally integrated systems. We focus on stationary series, with some discussion of extension to nonstationarity. The setting is semiparametric, so that modelling is effectively confined to a neighbourhood of frequency zero. We first discuss the definition...
Several semiparametric estimates of the memory parameter in standard long memory time series are now available. They consider only local behaviour of the spectrum near zero frequency, about which the spectrum is symmetric. However long-range dependence can appear as a spectral pole at any Nyqvist frequency (reflecting seasonal or cyclical long-memo...
We show that it is possible to adapt to nonparametric disturbance autocorrelation in time series regression in the presence of long memory in both regressors and disturbances by using a smoothed nonparametric spectrum estimate in frequency–domain generalized least squares. When the collective memory in regressors and disturbances is sufficiently st...
We consider a parametric spectral density with power-law behaviour about a fractional
pole at the unknown frequency !. The case of known !, especially ! = 0, is
standard in the long memory literature. When ! is unknown, asymptotic distribution
theory for estimates of parameters, including the (long) memory parameter, is significantly
harder. We stu...
The behaviour of averaged periodograms and cross-periodograms of a broad class of nonstationary processes is studied. The processes include nonstationary ones that are fractional of any order, as well as asymptotically stationary fractional ones. The cross-periodogram can involve two nonstationary processes of possibly different orders, or a nonsta...
This paper contributes empirically to our understanding of informed traders. It analyzes traders' characteristics in a foreign exchange electronic limit order market via anonymous trader identities. We use six indicators of informed trading in a cross-sectional multivariate approach to identify traders with high price impact. More information is co...
The seasonal structure of quarterly UK and Japanese consumption and income is examined by means of fractionally based tests proposed by Robinson (1994). These series were analysed from an autoregressive unit root viewpoint by Hylleberg, Engle, Granger and Yoo (HEGY, 1990) and Hylleberg, Engle, Granger and Lee (HEGL, 1993). We find that seasonal fra...
For a class of parametric ARCH models, Whittle estimation based on squared observations is shown to be [square root of n]-consistent and asymptotically normal. Our conditions require the squares to have short memory autocorrelation, by comparison with the work of Zaffaroni (1999, Gaussian Inference on Certain Long-Range Dependent Volatility Models,...
We establish valid Edgeworth expansions for the distribution of smoothed nonparametric spectral estimates, and of studentized versions of linear statistics such as the sample mean, where the studentization employs such a nonparametric spectral estimate. Particular attention is paid to the spectral estimate at zero frequency and, correspondingly, th...
or a particular conditionally heteroscedastic nonlinear (ARCH)
process for which the conditional variance of the observable sequence $r_t$ is
the square of an inhomogeneous linear combination of $r_s, s < t$, we give
conditions under which, for integers $l \geq 2, r_t^l$ has long memory
autocorrelation and normalized partial sums of $r_t^l$ converg...
Whittle pseudo-maximum likelihood estimates of parameters for stationary time series have been found to be consistent and asumptotically normal in the presence of long-range dependence. Generalizing the definition of the memory parameter d, we extend these results to include possibly nonstationary (0.5 = d < 1) or antipersistent (-0.5 < d < 0) obse...
We establish valid Edgeworth expansions for the distribution of smoothed nonparametric spectral estimates, and of studentized versions of linear statistics such as the same mean, where the studentization employs such a nonparametric spectral estimate. Particular attention is paid to the spectral estimate at zero frequency and, correspondingly, the...
This paper contributes empirically to our understanding of informed traders. It analyzes traders' characteristics in a foreign exchange electronic limit order market via anonymous trader identities. We use six indicators of informed trading in a cross-sectional multivariate approach to identify traders with high price impact. More information is co...
In Giraitis, Robinson, and Samarov (1997), we have shown that the optimal rate for memory parameter estimators in semiparametric long memory models with degree of "local smoothness" [beta] is n-r([beta]), r([beta])=[beta]/(2[beta]+1), and that a log-periodogram regression estimator (a modified Geweke and Porter-Hudak (1983) estimator) with maximum...
We establish valid theoretical and empirical Edgeworth expansions
for density-weighted averaged derivative estimates of
semiparametric index models.
Whittle pseudo-maximum likelihood estimates of parameters for stationary time series have been found to be consistent and asymptotically normal in the presence of long-range dependence. Generalizing the definition of the memory parameter d, we extend these results to include possibly nonstationary (.5 $\leq d <$ 1) or antipersistent (-.5 $< d <$ 0)...
The aggregation procedure when a sample of length N is divided into blocks of length m=o(N), m-->[infinity] and observations in each block are replaced by their sample mean, is widely used in statistical inference. Taqqu et al. (1995, Fractals, 3, 785-798), and Teverovsky and Taqqu (1997, J. Time Ser. Anal., 18, 279-304) introduced an aggregated va...
Semiparametric estimates of long memory seem useful in the analysis of long financial time series because they are consistent under much broader conditions than parametric estimates. However, recent large sample theory for semiparametric estimates forbids conditional heteroskedasticity. We show that a leading semiparametric estimate, the Gaussian o...
Semiparametric estimates of long memory seem useful in the analysis of long financial time series because they are consistent under much broader conditions than parametric estimates. However, recent large sample theory for semiparametric estimates forbids conditional heteroscedasticity. We show that a leading semiparametric estimate, the Gaussian o...
It is pointed out that two contradictory definitions of fractional Brownian motion are well established, one prevailing in the probabilistic literature, the other in the econometric literature. Each is associated with a different definition of nonstationary fractional time series. These various definitions have occasionally led to some confusion. T...
The concept of cointegration has principally been developed under the assumption that the raw data vector zt is I(1) and the cointegrating residual et is I(0), but is also of interest in more general, including fractional, circumstances, where zt is stationary with long memory and et is stationay with less memory, or where zt is nonstationary while...
There is frequently interest in testing that a scalar or vector time series is I(0), possibly after first-differencing or
other detrending, while the I(0) assumption is also taken for granted in autocorrelation-consistent variance estimation. We
propose a test for I(0) against fractional alternatives. The test is nonparametric, and indeed makes no...
The aggregation procedure when a sample of length N is divided into blocks of length m = o(N), m ? ? and observations in each block are replaced by their sample mean, is widely used in statistical inference. Taqqu, Teverovsky and Willinger (1995), Teverovsky and Taqqu (1997) introduced an aggregate variance estimator of the long memory parameter of...
Weak convergence to a form of fractional Brownian motion is established for a wide class of nonstationary fractionally integrated multivariate processes. Instrumental for the main argument is a result of some independent interest on approximations for partial sums of stationary linear vector sequences. A functional central limit theorem for smoothe...
We consider statistical inference in the presence of serial dependence. The main focus is on use of statistics that are constructed as if no dependence were believed present, and are asymptotically normal in the presence of dependence. Typically the variance in the limit distribution is affected by the dependence, and needs to be consistently estim...
A central limit theorem is given for certain weighted partial sums of a covariance stationary process, assuming it is linear in martingale differences, but without any restriction on its spectrum. We apply the result to kernel nonparametric fixed-design regression, giving a single central limit theorem which indicates how error spectral behavior at...
We introduce a nonlinear model of stochastic volatility within the class of “product type” models. It allows different degrees of dependence for the “raw” series and for the “squared” series, for instance implying weak dependence in the former and long memory in the latter. We discuss its main statistical properties with respect to the common set o...
Recently proposed tests for unit root and other nonstationarity of Robinson (1994a) are applied to an extended version of the data set used by Nelson and Plosser (1982). Unusually, the tests are efficient (against appropriate parametric alternatives), the null can be any member of the I(d) class, and the null limit distribution is chi-squared. The...
A central limit theorem is established for time series regression estimates which include generalized least squares, in the presence of long-range dependence in both errors and stochastic regressors. The setting and results differ significantly from earlier work on regression withlong-range-dependent errors. Spectral singularities are permitted at...
We introduce a nonlinear model of stochastic volatility within the class of ?product type? models. It allows different degrees of dependence for the ?raw? series and for the ?squared? series, for instance implying weak dependence in the former and long memory in the latter. We discuss its main statistical properties with respect to the common set o...
A general limit theorem is established for time series regression estimates which include generalized least squares, in the presence of long range dependence in both errors and stochastic regressors. The setting and results differ significantly from earlier work on regression with long range dependent errors. Spectral singularities are permitted at...
There is frequently interest in testing that a scalar or vector time series is I(0), possibly after first- differencing or other detrending, while the I(0) assumption is also taken for granted in autocorrelation-consistent variance estimation. We propose a test for I(0) against fractional alternatives. The test is non-parametric, and indeed makes n...
There exist several estimators of the memory parameter in long- memory time series models with the spectrum specified only locally near zero frequency. In this paper we give an asymptotic lower bound for the minimax risk of any estimator of the memory parameter as a function of the degree of local smoothness of the spectral density at zero. The low...
We discuss models that impart a form of long memory in raw time series xt or instantaneous functions thereof, in particular . on the basis of a linear or nonlinear model. The capacity of linear models for xt to imply long-memory in nonlinear functions of xt is discussed. Empirical observation motivates investigation of models which lead to short me...
New or modified methods for semiparametric analysis of fractional long memory in time series are described and applied to twenty-six stock prices and two stock indices. Evidence is found that some, but not all, of the stocks have long memory, while one of the indices exhibits mean reversion.
This paper considers spectral and autocovariance estimation for a zero-mean, band-limited, stationary process that has been sampled at time points jittered from a regular, equi-interval, sampling scheme. The case of interest is where the sampling scheme is near regular so that the jitter standard deviation is small compared to the sampling interval...
This paper discusses estimates of the parameter which governs the shape of the spectral density near zero frequency of a long memory time series. The estimates are semiparametric in the sense that the spectral density is parameterized only within a neighborhood of zero frequency. The estimates are based on averages of the periodogram over a band co...
We derive an optimal kernel K([lambda]) for spectral averaging in the neighbourhood of a spectral peak corresponding to long-range dependence. Unusually, K([lambda]) --> 0 as [lambda] --> 0.
Citations
... By combining autoregressive and moving average, proposed a functional autoregressive and moving average model. While these techniques are designed for analyzing short-memory stationary functional time series, Li et al. (2020Li et al. ( , 2021Li et al. ( , 2022 considered long-range dependent functional time series and proposed a functional autoregressive fractionally integrated moving average model. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/272451611394067-1441968934520_Q64/Degui-Li.jpg)
![[object Object]](https://i1.rgstatic.net/ii/profile.image/1106444233519105-1640808270033_Q64/Han-Lin-Shang.jpg)
... SEM addresses the spatial spillover effect caused by the lack of important variables or unobservable random shocks. The SAR model assumes that the explained variables will affect the economy of other regions through spatial interaction [50], whereas the SDM model considers the two types of spatial transmission mechanisms simultaneously. The SDM model also considers spatial interaction, that is, the urbanization level of a region is not only affected by the independent variables of the region but is also affected by the urbanization level and independent variables of the surrounding regions. ...
... By combining autoregressive and moving average, proposed a functional autoregressive and moving average model. While these techniques are designed for analyzing short-memory stationary functional time series, Li et al. (2020Li et al. ( , 2021Li et al. ( , 2022 considered long-range dependent functional time series and proposed a functional autoregressive fractionally integrated moving average model. ...
... Firstly, bootstrapping functional time series is still in its infancy, and we intend to extend the double bootstrap procedure to analyse stationary and weakly dependent functional time series (see, e.g., Zhu and Politis, 2017;Nyarige, 2016;Shang, 2018;Paparoditis, 2018;Paparoditis and Shang, 2020, for various single bootstrap procedures). Secondly, we aim to develop bootstrap procedures that can handle stationary long-memory functional time series (see, e.g., Li et al., 2020a) and non-stationary long-memory functional time series (see, e.g., Li et al., 2020b). ...
... Recently, Li et al. (2020b) applied a local Whittle estimator to estimate the long-memory parameter of a nonstationary functional time series. Their procedure, described in Section 4, constructs an estimate of the long-run covariance function. ...
... FPCA is the foundation of functional data analysis and most functional data analysis methods and developed on the basis of the FPCA. Scholars use long run covariance function which take the time correlation into account instead of covariance function in the FPCA, so that the existing analysis process can be followed (Horváth et al., 2016;Li et al., 2020;Rice et al., 2017). ...
... For recent development, seeSu and Jin (2010),Robinson (2011Robinson ( , 2012,,Robinson and Thawornkaiwong (2012),Su (2012),Malikov and Sun (2017),Robinson and Velasco (2018), andSun and Malikov (2018) among others. ...
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... We assume that γ 0 ∈ , where is a compact subset of R d γ . With d γ diverging, ensuring has bounded volume requires some care (see Gupta and Robinson, 2018). For a known function f (·), our aim is to test ...
... Linear AutoRegressive Conditional Heteroskedastic (LARCH) processes were introduced by Robinson (1991) to model the long-range dependence of volatility and leverage. They are studied for their stationarity and dependence properties in Robinson and Zaffaroni (1997), Giraitis et al. (2000), Berkes and Horváth (2003), and Giraitis et al. (2004). A LARCH(∞) process (X t ) t∈Z is defined by: ...
Reference: A new estimator for LARCH processes
... Porter-Hudak (1990) and Ray (1993 Arteche and Robinson (1999) presented an excellent discussion on the application, estimation and statistical inference of seasonal and cyclic long-memory processes. In the literature, long-memory processes with spectral singularities at the origin is a comprehensively studied research area, see Dahlhaus (1989), Fox and Taqqu (1986), Giraitis and Surgailis (1990), Heyde and Gay (1993) and Yajima (1985). ...
