Biases of Correlograms and of AR Representations of Stationary Series

We derive the relation between the biases of correlograms and of estimates of auto-regressive AR(k) representations of stationary series, and we illustrate it with a simple AR example. The new relation allows for k to vary with the sample size, which is a representation that can be used for most stationary processes. As a result, the biases of the estimators of such processes can now be quantified explicitly and in a unified way.

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Generalized Levinson–Durbin Sequences and Binomial Coefficients

Integers ◽

10.1515/integ.2009.016 ◽

2009 ◽

Vol 9 (2) ◽

Author(s):

Paul Shaman

Keyword(s):

Stochastic Process ◽

Sample Size ◽

Mean Square Error ◽

Discrete Time ◽

Minimum Mean Square Error ◽

Stationary Stochastic Process ◽

Binomial Coefficients ◽

Stationary Processes ◽

Mean Square ◽

Special Case

AbstractThe Levinson–Durbin recursion is used to construct the coefficients which define the minimum mean square error predictor of a new observation for a discrete time, second-order stationary stochastic process. As the sample size varies, the coefficients determine what is called a Levinson–Durbin sequence. A generalized Levinson–Durbin sequence is also defined, and we note that binomial coefficients constitute a special case of such a sequence. Generalized Levinson–Durbin sequences obey formulas which generalize relations satisfied by binomial coefficients. Some of these results are extended to vector stationary processes.

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Construction of Algebraic and Difference Equations with a Prescribed Solution Space

International Journal of Applied Mathematics and Computer Science ◽

10.1515/amcs-2017-0002 ◽

2017 ◽

Vol 27 (1) ◽

pp. 19-32 ◽

Cited By ~ 1

Author(s):

Lazaros Moysis ◽

Nicholas P. Karampetakis

Keyword(s):

Difference Equations ◽

Time Horizon ◽

Polynomial Matrix ◽

Solution Space ◽

Elementary Divisor ◽

Polynomial Matrices ◽

Auto Regressive ◽

Finite Time Horizon ◽

Ar Representations ◽

Forward Operator

Abstract This paper studies the solution space of systems of algebraic and difference equations, given as auto-regressive (AR) representations A(σ)β(k) = 0, where σ denotes the shift forward operator and A(σ) is a regular polynomial matrix. The solution space of such systems consists of forward and backward propagating solutions, over a finite time horizon. This solution space can be constructed from knowledge of the finite and infinite elementary divisor structure of A(σ). This work deals with the inverse problem of constructing a family of polynomial matrices A(σ) such that the system A(σ)β(k) = 0 satisfies some given forward and backward behavior. Initially, the connection between the backward behavior of an AR representation and the forward behavior of its dual system is showcased. This result is used to construct a system satisfying a certain backward behavior. By combining this result with the method provided by Gohberg et al. (2009) for constructing a system with a forward behavior, an algorithm is proposed for computing a system satisfying the prescribed forward and backward behavior.

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Spurious OLS Estimators of Detrending Method by Adding a Linear Trend in Difference-Stationary Processes—A Mathematical Proof and Its Verification by Simulation

Mathematics ◽

10.3390/math8111931 ◽

2020 ◽

Vol 8 (11) ◽

pp. 1931

Author(s):

Zhiming LONG ◽

Rémy HERRERA

Keyword(s):

Sample Size ◽

Random Number ◽

Stationary Process ◽

Mathematical Proof ◽

Linear Trend ◽

Random Number Generator ◽

Ordinary Least Squares ◽

Stationary Processes ◽

Random Numbers ◽

Real Problem

Adding a linear trend in regressions is a frequent detrending method in economic literatures. The traditional literatures pointed out that if the variable considered is a difference-stationary process, then it will artificially create pseudo-periodicity in the residuals. In this paper, we further show that the real problem might be more serious. As the Ordinary Least Squares (OLS) estimators themselves are of such a detrending method is spurious. The first part provides a mathematical proof with Chebyshev’s inequality and Sims–Stock–Watson’s algorithm to show that the OLS estimator of trend converges toward zero in probability, and the other OLS estimator diverges when the sample size tends to infinity. The second part designs Monte Carlo simulations with a sample size of 1,000,000 as an approximation of infinity. The seed values used are the true random numbers generated by a hardware random number generator in order to avoid the pseudo-randomness of random numbers given by software. This paper repeats the experiment 100 times, and gets consistent results with mathematical proof. The last part provides a brief discussion of detrending strategies.

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