Unit Root

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A Unit Root is related with the existence of characteristic roots of a time series model.

References

2016

stochastic process]] has a unit root if 1 is a root of the process's characteristic equation. Such a process is non-stationary, but does not always have a trend.

If the other roots of the characteristic equation lie inside the unit circle — that is, have a modulus (absolute value) less than one — then the first difference of the process will be stationary; otherwise, the process will need to be differenced multiple times to become stationary. Due to this characteristic, unit root processes are also called difference stationary.
Unit root processes may sometimes be confused with trend-stationary processes; while they share many properties, they are different in many aspects. It is possible for a time series to be non-stationary, have no unit-root yet be trend-stationary. In both unit root and trend-stationary processes, the mean can be growing or decreasing over time; however, in the presence of a shock, trend-stationary processes are mean-reverting (i.e. transitory, the time series will converge again towards the growing mean, which was not affected by the shock) while unit-root processes have a permanent impact on the mean (i.e. no convergence over time).
If a root of the process's characteristic equation is larger than 1, then it is called an explosive process, even though such processes are often inaccurately called unit roots processes.
The presence of a unit root can be tested using a unit root test.
Definition
Consider a discrete-time stochastic process [math]\displaystyle{ \{y_t,t=1,\ldots,\infty\} }[/math], and suppose that it can be written as an autoregressive process of order p:
[math]\displaystyle{ y_t=a_1 y_{t-1}+a_2 y_{t-2} + \cdots + a_p y_{t-p}+\varepsilon_t. }[/math]
Here, [math]\displaystyle{ \{\varepsilon_{t},t=0,\infty\} }[/math] is a serially uncorrelated, zero-mean stochastic process with constant variance [math]\displaystyle{ \sigma^2 }[/math]. For convenience, assume [math]\displaystyle{ y_0 = 0 }[/math]. If [math]\displaystyle{ m=1 }[/math] is a root of the characteristic equation:
[math]\displaystyle{ m^p - m^{p-1}a_1 - m^{p-2}a_2 - \cdots - a_p = 0 }[/math]
then the stochastic process has a unit root or, alternatively, is integrated of order one, denoted [math]\displaystyle{ I(1) }[/math]. If m = 1 is a root of multiplicity r, then the stochastic process is integrated of order r, denoted I(r).
[math]\displaystyle{ z_t = z_{t−1} + a_t }[/math]
where [math]\displaystyle{ a_t }[/math] is a white noise process. In general, [math]\displaystyle{ at }[/math] can be a sequence of martigale differences, that is, [math]\displaystyle{ E(a_t|F_{t−1}) = 0, \;Var(a_t|F_{t−1}) }[/math] is finite, and [math]\displaystyle{ E(|at|^{2+\delta}|F_{t−1}) \lt \infty }[/math]for some [math]\displaystyle{ \delta \gt 0 }[/math], where [math]\displaystyle{ F_{t−1} }[/math] is the [math]\displaystyle{ \sigma }[/math]-field generated by [math]\displaystyle{ a_{t−1}, a_{t−2}, \cdots }[/math]. For simplicity, one often assumes that [math]\displaystyle{ Z_0 = 0 }[/math]. It will be seen later that this assumption has no effect on the limiting distributions of unit-root test statistics. This simple model plays an important role in the unit-root literature. The assumption that at is a martingale difference is the basic setup used in Chan and Wei (1988, Annals of Statistics) for their famous paper on limiting properties of unstable ARprocesses. However, this assumption can be relaxed without introducing much complexity. In what follows, we adopt the approach of Phillips (1987, Econometrica) with a single unit root and at is a stationary series with weak serial dependence. The case of unit roots with multiplicity greater than 1 or other characteristic roots on the unit circle can be handled via the work of Chan and Wei (1988).
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