چکیده:
Stochastic, processes can be stationary or nonstationary.
They depend on the magnitude of shocks. In other words, in an
auto regressive model of order one, the estimated coefficient is
not constant. Another finding of this paper is the relation
between estimated coefficients and residuals. We also develop
a catastrophe and chags theory for change of roots from
stationary to a nonstationary one and vice versa
خلاصه ماشینی:
The Stationary-NonStationary Process and The Variable Roots' Difference Equations By: Hossein Abbasi-Nejad, Ph. D.
The rest of the paper is organized as follows: The next section debates about "Variable Root Differential Equations" (VRDE, henceforth) and stochastic unit root processes.
"z" is proxy for every variable such as shocks in stochastic difference equations or another variable depending on the model builder's view.
In Granger and Swanson ( 1997) a variable root difference equation (although not labeled VRDE) was used with the following form: Xt =atXt +Et Ei - i.
) 48 / The Stationary-NonStationary Process and The Variable Roots' Difference Equations Where e, , a Gussi an stationary series, has mean m, variance cr!
For the second term in the right hand of the equation, some parts will be zero and the others result in L E�-i = B i=O Therefore: E(y t) =AE(Yt-k-t )+B k var(y t) = var pk+l (L- s, )y r-x-: + var Li P i ( - Et) EH 1 1 L i=O This shows that the estimation of the VRDE models with OLS or ML are not only difficult but also strictly opposed to our fundamental opposed assumption of the variable roots.
5- Conclusion and further studies Economic series may be modeled bv variable root difference equations; such that for some observations, a root is proper which is improper for other observations (or periods with the same number of observations).