Regularized Interval-valued Time Series Modeling*

Threshold autoregressive models for interval-valued time series data

Heteroskedasticity and autocorrelation consistent (HAC) covariance matrix estimation, or HAC estimation in short, is one of the most important techniques in time series analysis and forecasting. It serves as a powerful analytical tool for hypothesis testing and model verification. However, HAC estimation for long and high-dimensional time series is computationally expensive. This paper describes a novel pipeline-friendly HAC estimation algorithm derived from a mathematical specification, by applying transformations to eliminate conditionals, to parallelize arithmetic, and to promote data reuse in computation. We then develop a fully-pipelined hardware architecture based on the proposed algorithm. This architecture is shown to be efficient and scalable from both theoretical and empirical perspectives. Experimental results show that an FPGA-based implementation of the proposed architecture is up to 111 times faster than an optimised CPU implementation with one core, and 14 times faster than a CPU with eight cores.

Heteroskedasticity and autocorrelation consistent (HAC) covariance matrix estimation, or HAC estimation in short, is one of the most important techniques in time series analysis and forecasting. It serves as a powerful analytical tool for hypothesis testing and model verification. However, HAC estimation for long and high-dimensional time series is computationally expensive. This paper describes a pipeline-friendly HAC estimation algorithm derived from a mathematical specification, by applying transformations to eliminate conditionals, to parallelise arithmetic, and to promote data reuse in computation. We discuss an initial hardware architecture for the proposed algorithm, and propose two optimised architectures to improve the worst-case performance. Experimental systems based on proposed architectures demonstrate high performance especially for long time series. One experimental system achieves up to 12 times speedup over an optimised software system on 12 CPU cores.

The dynamic mode decomposition (DMD) algorithm is a widely used factorization and dimensionality reduction technique in time series analysis. When analyzing high-dimensional time series, the DMD algorithm requires extremely large amounts of computational power. To accelerate the DMD algorithm, we propose a quantum-classical hybrid algorithm that we call the quantum dynamic mode decomposition (QDMD) algorithm. Given a time series X ∈ R n × ( m + 1) with n ≫ m , the QDMD algorithm first executes quantum singular value decomposition on a matrix related to X and obtains a quantum state containing the main singular values and singular vectors of the decomposed matrix, then performs a low-sampling-frequency process on the obtained quantum state and computes the low-dimensional projection of the DMD operator through the sampling results. Finally, the algorithm computes the DMD eigenvalues and prepares the amplitude-encoding states of the DMD modes using the obtained classical information and X . Considering the main variables, the complexity of the QDMD algorithm is O ~ M m polylog n / ϵ , where M = O ~ m 3 / ϵ 2 denotes the number of samples. Compared with the classical DMD algorithm, which has complexity O ~ n m 2 log 1 / ϵ , the QDMD algorithm provides an exponential acceleration of n , at the cost of greater dependence on M and ϵ . We test the effects of M on the QDMD algorithm in the specific application scenarios of data denoising, scene background extraction, and fluid dynamics analysis. We determined that the QDMD algorithm requires only a small number of samples M in specific applications, further demonstrating the quantum advantage of the QDMD algorithm in high-dimensional data analysis.

SUMMARY The techniques of time series analysis are presented in this paper and are applied to certain non-stationary water pollution data, in a meaningful way, to obtain a more effective analysis of such data. The analysis results in models which may be used for forecasting and the sample spectra which describe the underlying processes which generated the data. The use of the results can be of great value in the understanding and in management aspects of our environmental problems. It is becoming ever more apparent that there must be more effective analysis and interpretation of water pollution data in order to be able to attain water quality. The development of more sophisticated measuring devices and techniques which will continuously record several variables of interest (for example, dissolved oxygen (DO), biochemical oxygen demand (BOD), sunlight intensity, and temperature), requires the use of dataanalysis techniques to obtain a more fundamental understanding of the processes which generated these data. By applying the techniques of time series analysis, maximum results are possible. The ultimate objective of time series analysis is the estimation and reconstruction of the underlying generating process from a given time series. From this analysis possible laws may be discovered or indicated that the variables may obey, and thus allow future values of the variables to be forecasted. Hence time series analysis affords us a way to gain maximum results from the pollution variables and thus can aid in the understanding and management of water quality. In this paper we shall make use of the techniques of time series analysis to analyze certain dissolved oxygen water quality data. The dissolved oxygen concentration which exists in a stream while it is receiving an organic waste load is the variable by which the waste assimilation capacity of a stream is measured. The dissolved oxygen data used in this analysis were originally presented by Thomann [1967] and were continuously recorded by monitoring stations situated along the Delaware Estuary. Thomann used the technique of Fourier analysis to analyze these DO data, but this approach requires the assumption of fixed amplitudes, frequencies, and phases, and breaks

We consider the problem of approximating sums of high dimensional stationary time series by Gaussian vectors, using the framework of functional dependence measure. The validity of the Gaussian approximation depends on the sample size $n$, the dimension $p$, the moment condition and the dependence of the underlying processes. We also consider an estimator for long-run covariance matrices and study its convergence properties. Our results allow constructing simultaneous confidence intervals for mean vectors of high-dimensional time series with asymptotically correct coverage probabilities. As an application, we propose a Kolmogorov–Smirnov-type statistic for testing distributions of high-dimensional time series.

In this article, the problem of estimating the autoregressive functions of certain processes by the nonparametric kernel method is investigated. Also, consistency properties are briefly presented under mixing conditions. It is indicated how one has to choose the smoothing parameter in practical situations, and two improvements on the usual cross-validation procedure are proposed. Finally, applications to the prediction of some economic data are presented.

Distributional approximations of (bi--) linear functions of sample variance-covariance matrices play a critical role to analyze vector time series, as they are needed for various purposes, especially to draw inference on the dependence structure in terms of second moments and to analyze projections onto lower dimensional spaces as those generated by principal components. This particularly applies to the high-dimensional case, where the dimension $d$ is allowed to grow with the sample size $n$ and may even be larger than $n$. We establish large-sample approximations for such bilinear forms related to the sample variance-covariance matrix of a high-dimensional vector time series in terms of strong approximations by Brownian motions. The results cover weakly dependent as well as many long-range dependent linear processes and are valid for uniformly $ \ell_1 $-bounded projection vectors, which arise, either naturally or by construction, in many statistical problems extensively studied for high-dimensional series. Among those problems are sparse financial portfolio selection, sparse principal components, the LASSO, shrinkage estimation and change-point analysis for high--dimensional time series, which matter for the analysis of big data and are discussed in greater detail.

Singular spectrum analysis based on the minimum variance estimator

A new technique for time series analysis, which is a combination of the maximum entropy method (MEM) for spectral analysis and the non-linear least squares method (LSM) for fitting analysis, is described. In this technique, the MEM power spectral density (MEMPSD) is calculated using a very large lag that could diminish the lag dependence of dominant periods estimated by the MEM analysis. The validity of this large lag is confirmed by the LSM, given that the ten dominant MEM periods are known quantities. To validate the MEM plus LSM technique, it is compared with autoregressive (AR) modelling, by analysing heart rate variability under pharmacological interventions (phenylephrine and trinitroglycerine), using 16 young males. The results indicate that the MEMPSD, when compared with the ARPSD, has numerous periods that could reproduce the original time series much more accurately, as revealed by the LSM analysis. However, both the low- and high-frequency powers with MEMPSD and ARPSDs shift in the expected directions in accordance with the pharmacological effects on the cardiovascular system. The implications of these results are discussed from the theoretical and practical standpoints of the MEM plus LSM technique, compared with AR modelling.

The past decade witnessed the rapid development of high dimensional statistics in deterministic design. High dimensional time series analysis, due to the time dependency, still faces several theoretical challenges. Among the time series models, the Vector Error Correction Model (VECM) is especially complicated because of the non-stationary components. The classical estimation strategies (e.g. Johansen's approach) fail to provide consistent estimates for dimensions larger than three. Moreover, it is impossible to apply existing statistical methods to determine VECM in high dimensions, i.e. when the dimension is allowed to increase with the number of observations or even larger than that. This dissertation aims at providing feasible regularized methods, which can determine and estimate high dimensional VECM with robust statistical properties. The detailed analysis is divided into three parts. First I develop new tailored Lasso-type methods to estimate VECM and prove their statistical properties, in a setting where the cointegration rank is fixed but unknown and the dimension is large but not increasing with sample size. Then this methodology is extended to cover also the high dimensional case with moving rank and dimension. For this the estimation strategy must be changed and the statistical analysis requires completely different high-dimensional techniques. Under specific assumptions, I propose the estimation strategy in the ultra-high dimensional case. From the application side, these techniques are highly valuable for appropriately treating complex potentially non-stationary systems not only in economics and finance but also in weather and climate systems. I also illustrate this for a portfolio of Credit Default Swaps in a banking-sovereign network and identify different risk clusters by measuring the interconnectedness over time, which is beyond the scope of previous methodologies. Moreover, I provide a detailed empirical study on a high-frequency portfolio where new high-dimensional time series techniques allow to account for liquidity effects through the Limit Order Book in a very detailed way. With this the new spillover channels in the system can be identified.

In recent years Singular Spectrum Analysis (SSA), a relatively novel but powerful technique in time series analysis, has been developed and applied to many practical problems across different fields. In this paper we review recent developments in the theoretical and methodological aspects of the SSA from the perspective of analyzing and forecasting economic and financial time series, and also represent some new results. In particular, we (a) show what are the implications of SSA for the, frequently invoked, unit root hypothesis of economic and financial times series; (b) introduce two new versions of SSA, based on the minimum variance estimator and based on perturbation theory; (c) discuss the concept of causality in the context of SSA; and (d) provide a variety of simulation results and real world applications, along with comparisons with other existing methodologies.

Forecasting the underlying potential governing the time series of a dynamical system

Do the propositions of mental accounting matter in investors’ portfolio manage- ment? In assisting the investors in their portfolio management, this study explores if the separation principle in mental accounting (MA) can be applied in their portfolio decision choices. It also examines if investors prefer an individual stock to the portfolio of stocks for investments. With the use of daily market prices data during a twenty-year time period for nine NSE Nifty sample stocks and their portfolio as well, this study applies the prospect theory (PT) decision references in the non-linear autoregressive distributive lag (NARDL) models. It also performs robustness checks with t-tests for differences between the coefficient magnitudes in the models. At PT implications with the variables for market premium, sys- tematic beta and isolation effects, the separation principle of the MA theory matters in the portfolio management for stocks’ returns and market return. Investors’ psychological effect is found to contribute to their preferences amongst the sample stocks for their inclusion in the portfolio. Ingenious applications of the PT views on MA in portfolio management reveal the presence of synchronicity in terms of long-memory and short-memory effects on returns of the stocks and portfolio as well. Even if the generalizability of the stated findings is sub- ject to its sample size, this empirical exploration with the Indian stocks market data shows original contribution in mental accounting and its use to explain the equity premium puzzle could enhance its applicative value.

High-dimensional time series analysis has become an active area of research, because of its increasing prevalence in applications as well as the unique challenges it presents. This thesis first explores a range of questions in this area, related to how high dimensionality invalidates conventional statistical methods. It seeks to find out how to extend the existing models to cope with increasing dimensionality; how to develop a reliable as well as efficient estimation approach under the complex structure of high-dimensional datasets; and how to gain some benefits from the current information explosion. We have addressed these problems through a theoretical study using M−estimation for high-dimensional time series linear models. One advantage of M−estimation is that it can act as a robust alternative for some classical statistical methods that are sensitive to outliers or distributional deviations. Furthermore, M−estimation features generality by covering a variety of estimation approaches, for example, the least squares estimation and quantile regression. Hence, the first objective of this thesis is to study the asymptotic properties of the M−estimator for analysing high-dimensional time series data. We also evaluate the finite sample performance of the proposed M−estimator by conducting a series of Monte Carlo experiments. Compared with the least squares estimator, the M−estimator has been found to perform much better, with high resistance to the presence of non-normal errors. Another central issue discussed in this thesis lies in panel data analysis. The increasing availability of panel datasets makes this an intriguing area to investigate. The advantage of panel datasets over univariate time series or cross-sectional data enables researchers to study the comprehensive interactions within socioeconomic networks. One remarkable feature of panel data is that cross-sectional units are likely to be correlated with each other, arising from spatial spillover, unobservable common factors and economic interactions among them. In the literature of late, econometricians have paid considerable attention to propose new methodologies for modelling the cross-sectional dependence, for instance, the multifactor error structure. However, the assumptions in the existing literature are overly restrictive (see, for example Bai 2009). Consequently, the second objective of this thesis is to develop the least squares estimator for a large panel data model with a multifactor error structure under a relaxed assumption framework. The estimator is shown to be consistent and asymptotic normal while its asymptotic distribution is non-centred. Hence, the closed expressions of the estimators for the asymptotic bias are given. Monte Carlo simulations not only investigate the small-sample performance of our estimator, but also show how well our bias estimators can centre the asymptotic distributions under different sample sizes and various strengths of dependence. In addition to large panels, short panels have also become pervasive in empirical studies, of which the cross-sectional dimension is relatively large compared with its temporal dimension. However, for the yearly data, fixed time periods can still span a rather long time span. The assumption of parameter stability becomes questionable due to the dramatically changing economic environment. A distance type estimator is introduced in this thesis for a short panel data model with a multifactor error structure, of which the asymptotic properties have been discussed. Monte Carlo simulations show that: (1) this structural break detection method performs well in terms of size and power; (2) it can also locate the break successfully; and (3) its setup incorporates the short panel data models with fixed effects as a special case. The proposed method, which uses a dataset consisting of more than 4,000 US financial institutions, has been further employed to provide empirical evidence of the well-known Gibrat’s ‘Law’.