where f can be a liproximate or a non-liproximate function of z(t), which is a (k × 1) vector of observable deterministic variables, and γ is a (k × 1) vector of confinclude parameters to be approximated. Thus, if f is liproximate, it can integrate, for instance, an intercept and a liproximate time trfinish of the create helpd by Bhargava (1986) [5], Schmidt and Phillips (1992) [6] and others in the context of unit roots, i.e.,
and, if it is non-liproximate, it can integrate, for example, Chebyshev polynomials in time of the create:
where m recommends the number of cofruitfuls of the Chebyshev polynomial in time Pi,T(t) depictd as:
as depictd in [16,17]. Bierens and Martins (2010) give the include of such polynomials in the case of time-varying cointegrating parameters [18]. There are disjoinal profits to using them. First, their orthogonality evades the problem of proximate colliproximateity in the deoriginateor matrix, which aascfinishs with standard time polynomials. Second, according to Bierens (1997) [19] and Tomasevic et al. (2009) [20], they can approximate highly non-liproximate trfinishs with rather low degree polynomials. Finassociate, they can approximate structural shatters in a much daintyer way than the classical structural change models.
where = 2πr/T, r = T/j is a genuine scalar cherish, L is the lag operator, (i.e., Lx(t) = x(t − 1)), dj is another genuine cherish correacting to the order of integration of the cycle that explodes (i.e., it goes to infinity) in the spectrum at λ = j; m stands for the number of cyclical structures, and u(t) is a low-memory process combined of order 0 or I(0). Such a process is depictd as a covariance stationary one with a spectral density function that is selectimistic and finite atraverse all frequencies in the spectrum. Thus, it could be a white noise process but also distake part frail autocorrelation, as in a stationary and invertible Auto Regressive Moving Average (ARMA) model. In the current study, in order to evade overparameterization, we adhere the exponential spectral approach of Bloomfield (1973) to model u(t) [21]. This is a non-parametric sketchtoil that is recommendedly depictd in terms of its spectral density function:
where σ2 is the variance of the error term, and n denotices the number of low-run vibrant terms. Its logged create approximates autodeoriginateive processes fairly well. Bloomfield (1973) showed that for a stationary and invertible ARMA (p, q) process of the create [21]:
where εt is a white noise process, the spectral density function is given by:
