In probability theory and statistics, a unit root is a property of certain stochastic processes (such as a random walk) that can create challenges for statistical inference in time series models. A linear stochastic process contains a unit root if 1 is a solution to its characteristic equation. Processes with a unit root are non-stationary, because they do not necessarily exhibit a deterministic 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. If there are d unit roots, the process will have to be differenced d times in order to make it 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, yet have no unit root and 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 sometimes 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 ( y t , t = 1 , 2 , 3 , … ) {\displaystyle (y_{t},t=1,2,3,\ldots )} , and suppose that it can be written as an autoregressive process of order p:
y t = a 1 y t − 1 + a 2 y t − 2 + ⋯ + a p y t − p + ε t . {\displaystyle y_{t}=a_{1}y_{t-1}+a_{2}y_{t-2}+\cdots +a_{p}y_{t-p}+\varepsilon _{t}.}
Here, ( ε t , t = 0 , 1 , 2 , … , ) {\displaystyle (\varepsilon _{t},t=0,1,2,\ldots ,)} is a serially uncorrelated, zero-mean stochastic process with constant variance σ 2 {\displaystyle \sigma ^{2}} . For convenience, assume y 0 = 0 {\displaystyle y_{0}=0} . If m = 1 {\displaystyle m=1} is a root of the characteristic equation, of multiplicity 1:
m p − m p − 1 a 1 − m p − 2 a 2 − ⋯ − a p = 0 {\displaystyle m^{p}-m^{p-1}a_{1}-m^{p-2}a_{2}-\cdots -a_{p}=0}
then the stochastic process has a unit root or, alternatively, is integrated of order one, denoted I ( 1 ) {\displaystyle I(1)} . If m = 1 is a root of multiplicity r, then the stochastic process is integrated of order r, denoted I(r).
Example The first order autoregressive model, y t = a 1 y t − 1 + ε t {\displaystyle y_{t}=a_{1}y_{t-1}+\varepsilon _{t}} , has a unit root when a 1 = 1 {\displaystyle a_{1}=1} . In this example, the characteristic equation is m − a 1 = 0 {\displaystyle m-a_{1}=0} . The root of the equation is m = 1 {\displaystyle m=1} . If the process has a unit root, then it is a non-stationary time series. That is, the moments of the stochastic process depend on t {\displaystyle t} . To illustrate the effect of a unit root, we can consider the first order case, starting from y0 = 0:
y t = y t − 1 + ε t . {\displaystyle y_{t}=y_{t-1}+\varepsilon _{t}.}
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