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Determining a Piecewise Linear Trend of a Nonstationary Time Series Based on Intelligent Data Analysis. I. Description and Substantiation of the Method
Abstract
The authors propose considering the trend of a non-stationary time series as a linear regression with unknown switching points. The method of evaluating the switching points based on intelligent data analysis using statistical criteria is described and substantiated.
The article describes the results of the approbation of the method of constructing a piecewise linear trend, which can have breaks at the switching points as well as be continuous at these points, i.e., represent a linear spline. An example of applying the method for constructing a linear switching regression, which has two independent variables with a trend, is considered. The problems of spline approximation of the time series of logarithms of the number of people infected with COVID-19 in Ukraine are solved.
P.S.. Knopov, A.S. Korkhin. Statistical analysis of the coronavirus infection dynamics using stepwise switching regression.
It is proposed to model of the coronavirus infection dynamics using switching regression, the switching points of which are unknown. The step-by-step process of constructing a regression in time is described. The dynamics of the coronavirus infection in Ukraine has been studied.
Building upon the continuous record asymptotic framework recently introduced by Casini and Perron (2020a) for inference in structural change models, we propose a Laplace-based (Quasi-Bayes) procedure for the construction of the estimate and confidence set for the date of a structural change. It is defined by an integration rather than an optimization-based method. A transformation of the least-squares criterion function is evaluated in order to derive a proper distribution, referred to as the Quasi-posterior. For a given choice of a loss function, the Laplace-type estimator is the minimizer of the expected risk with the expectation taken under the Quasi-posterior. Besides providing an alternative estimate that is more precise—lower mean absolute error (MAE) and lower root-mean squared error (RMSE)—than the usual least-squares one, the Quasi-posterior distribution can be used to construct asymptotically valid inference using the concept of Highest Density Region. The resulting Laplace-based inferential procedure is shown to have lower MAE and RMSE, and the confidence sets strike a better balance between empirical coverage rates and average lengths of the confidence sets relative to traditional long-span methods, whether the break size is small or large.
The method is applicable to regressions whose variables are time series. The estimation method is based on the fact that these series are considered as observed values of continuous random functions of time. This property makes it possible to solve the estimation problem using gradient methods for optimization problems. Examples of using the proposed method are given.
Linear regression with switching in continuous time is considered. A method for estimation of switching points and switching regression parameters is described. Examples of its use are given.
Switching regression is considered in the case where switching points are unknown. A general method is described to estimate switching points and parameters of linear regression with switching. Examples of its use are given.
Algorithms of calculation of estimates of nonlinear regression parameters based on the linearization method are described. These algorithms allow us to solve ill-conditioned problems of the nonlinear least-squares method. Algorithms of calculation of an auxiliary problem that uses peculiarities of the problems of the least-squares method are proposed.
Having estimated a linear regression with p coefficients, one may wish to test whether m additional observations belong to the same regression. This paper presents systematically the tests involved, relates the prediction interval (for m = 1) and the analysis of covariance (for m > p) within the framework of general linear hypothesis (for any m), and extends the results to testing the equality between subsets of coefficients.
An outstanding introduction to the fundamentals of regression analysis-updated and expanded The methods of regression analysis are the most widely used statistical tools for discovering the relationships among variables. This classic text, with its emphasis on clear, thorough presentation of concepts and applications, offers a complete, easily accessible introduction to the fundamentals of regression analysis. Assuming only a basic knowledge of elementary statistics, Applied Regression Analysis, Third Edition focuses on the fitting and checking of both linear and nonlinear regression models, using small and large data sets, with pocket calculators or computers. This Third Edition features separate chapters on multicollinearity, generalized linear models, mixture ingredients, geometry of regression, robust regression, and resampling procedures. Extensive support materials include sets of carefully designed exercises with full or partial solutions and a series of true/false questions with answers. All data sets used in both the text and the exercises can be found on the companion disk at the back of the book. For analysts, researchers, and students in university, industrial, and government courses on regression, this text is an excellent introduction to the subject and an efficient means of learning how to use a valuable analytical tool. It will also prove an invaluable reference resource for applied scientists and statisticians.
This paper studies the least squares estimation of a change point in multiple regressions. Consistency, rate of convergence, and asymptotic distributions are obtained. The model allows for lagged dependent variables and trending regressors. The error process can be dependent and heteroskedastic. For nonstationary regressors or disturbances, the asymptotic distribution is shown to be skewed. The analytical density function and the cumulative distribution function for the general skewed distribution are derived. The analysis applies to both pure and partial changes. The method is used to analyze the response of market interest rates to discount rate changes. © 2000 by the President and Fellows of Harvard College and the Massachusetts Institute of Technolog
In a recent paper, Bai and Perron ( 1998 ) considered theoretical issues related to the limiting distribution of estimators and test statistics in the linear model with multiple structural changes. In this companion paper, we consider practical issues for the empirical applications of the procedures. We first address the problem of estimation of the break dates and present an efficient algorithm to obtain global minimizers of the sum of squared residuals. This algorithm is based on the principle of dynamic programming and requires at most least‐squares operations of order O ( T ² ) for any number of breaks. Our method can be applied to both pure and partial structural change models. Second, we consider the problem of forming confidence intervals for the break dates under various hypotheses about the structure of the data and the errors across segments. Third, we address the issue of testing for structural changes under very general conditions on the data and the errors. Fourth, we address the issue of estimating the number of breaks. Finally, a few empirical applications are presented to illustrate the usefulness of the procedures. All methods discussed are implemented in a GAUSS program. Copyright © 2002 John Wiley & Sons, Ltd.
Estimation and inference of linear trend slope ratios with an application to global temperature
- P Perron
- E Zorta
Economic and Statistical Models of Variable Structure
- B B Rozin
- V I Kotyukov
- M A Yagolnitser
Linear Regression Analysis
- J Seber