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In probability theory and statistics, a unit root is a feature of some stochastic processes (such as random walks) that can cause problems in statistical inference involving time series models. A linear stochastic process has a unit root if 1 is a root of the process's characteristic equation .
In statistics, a unit root test tests whether a time series variable is non-stationary and possesses a unit root. The null hypothesis is generally defined as the presence of a unit root and the alternative hypothesis is either stationarity , trend stationarity or explosive root depending on the test used.
Which of the three main versions of the test should be used is not a minor issue. The decision is important for the size of the unit root test (the probability of rejecting the null hypothesis of a unit root when there is one) and the power of the unit root test (the probability of rejecting the null hypothesis of a unit root when there is not one).
In the former case of a unit root, stochastic shocks have permanent effects, and the process is not mean-reverting. In the latter case of a deterministic trend, the process is called a trend-stationary process , and stochastic shocks have only transitory effects after which the variable tends toward a deterministically evolving (non-constant) mean.
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms .
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 ...
In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function , then the characteristic function is the Fourier transform (with sign reversal) of the probability density function.
The n th roots of unity are, by definition, the roots of the polynomial x n − 1, and are thus algebraic numbers. As this polynomial is not irreducible (except for n = 1), the primitive n th roots of unity are roots of an irreducible polynomial (over the integers) of lower degree, called the n th cyclotomic polynomial, and often denoted Φ n.