No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
DETERMINING A PIECEWISE LINEAR TREND OF A NONSTATIONARY TIME SERIES BASED ON INTELLIGENT DATA ANALYSIS. I. DESCRIPTION AND JUSTIFICATION OF THE METHOD
Abstract
The problem of identifying the trend of a non-stationary time series is often encountered in various applications. In the article, this trend is proposed to be represented as a linear regression with unknown switching points. Typically, such a regression is built using mathematical programming methods. Moreover, the desired variables are mixed variables, which significantly complicates the problem’s solution. The article proposes a different approach based on data mining using statistical criteria. The algorithms described in the article are used to solve a number of problems, including one practical problem. The calculations showed satisfactory accuracy. Keywords: linear regression, algorithm, time series, trend, methods, mathematical programming.
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 flow. An example of applying the method for constructing a linear regression with switches, which has two independent variables with a trend, is considered. The problem of spline approximation of the time series of logarithms of the number of infected people with COVID-19 in Ukraine is stated and solved. Keywords: trend, regression, switch point, spline, real-time calculation.
ABSTRACT
In this chapter, necessary and sufficient conditions for the existence
of a confidence interval of the minimum length l for a certain
parameter are obtained in the two most frequently used cases: (1) l is
proportional to the length of the probability interval of some random
variable; (2) l is proportional to the difference between the inverse
values of the ends of this interval. We have discussed the issues of
possible applications of this result. In particular, we describe how
to find the intervals for the degrees of freedom for the chi-square
and Fisher distributions, which allow for reducing the length of the
confidence interval on a certain (given) value. In other words, we
discuss the effectiveness of the usage of confidence intervals of
minimal length.
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.
We examine estimation and inference of the ratio of linear trend slopes between two time series. Our methodological results are motivated by recent empirical climate papers that estimate and test hypotheses about the relative warming rate in the lower-troposphere relative to surface warming, i.e. the amplification ratio. We analyze statistical properties of several estimators and test statistics that allow serial correlation. Relative merits of the estimators and test statistics depend on the magnitude of the trend slopes relative to the noise in the data. We make specific recommendations for practitioners. We revisit the empirical findings of Klotzbach et al. (2009, 2010) and obtain similar results using our recommended approach.
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.
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.
Economic-statistical models with variable structure
- B B Rozin
- V I Kotyukov
- M A Yagolnitser
Fundamentals of probability theory and mathematical statistics (for economists
- A Korkhin
- Z Przhebitsin