Unit root

In time series models in econometrics, a linear stochastic process has a unit root if 1 is a root of the process's characteristic equation. The process will be non-stationary. If the other roots of the characteristic equation lie inside the unit circle, then the first difference of the process will be stationary.Consider a discrete time stochastic process {$y\_\{t\},t=1,infty$}, and suppose that it can be written as an autoregressive process of order p:

:$y\_\{t\}=a\_\{1\}y\_\{t-1\}+a\_\{2\}y\_\{t-2\}\; +\; cdots\; +\; a\_\{p\}y\_\{t-p\}+varepsilon\_\{t\}$

Here, {$varepsilon\_\{t\},t=0,infty$} is a serially uncorrelated, mean zero stochastic process with constant variance $sigma^2$. For convenience, assume $y\_\{0\}\; =\; 0$. If $m=1$ is a root of the characteristic equation:

:$m^\{p\}\; -\; m^\{p-1\}a\_\{1\}\; -\; m^\{p-2\}a\_\{2\}\; -\; cdots\; -\; a\_\{p\}\; =\; 0$

then the stochastic process has a unit root or, alternatively, is integrated of order one, denoted $I(1)$. If "m" = 1 is a root of multiplicity "r", then the stochastic process is integrated of order "r", denoted "I"("r").

For example, the first order autoregressive model, $y\_\{t\}=a\_\{1\}y\_\{t-1\}+varepsilon\_\{t\}$, has a unit root when $a\_\{1\}=1$. In this example, the characteristic equation is $m\; -\; a\_\{1\}\; =\; m\; -\; 1$. The root of the equation is $m\; =\; 1$.

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 $t$. To illustrate the effect of a unit root, we can consider the first order case:

:$y\_\{t\}=\; y\_\{t-1\}+varepsilon\_\{t\}$

By repeated substitution, we can write $y\_\{t\}\; =\; sum\_\{j=1\}^t\; varepsilon\_\{j\}$. Then the variance of $y\_\{t\}$ is given by:

: $Var(y\_\{t\})\; =\; sum\_\{j=1\}^t\; sigma^2=t\; sigma^2$

The variance depends on t since $Var(y\_\{1\})\; =\; sigma^2$, while $Var(y\_\{2\})\; =\; 2sigma^2$. Note that the variance of the series is diverging to infinity with t.

Estimation in the presence of a unit root

Often, ordinary least squares (OLS) is used to estimate the slope coefficients of the autoregressive model. Use of OLS relies on the stochastic process being stationary. When the stochastic process is non-stationary, the use of OLS can produce invalid estimates. Granger and Newbold (1974) called such estimates 'spurious regression' results: high R² values and high t-ratios yielding results with no economic meaning.

To estimate the slope coefficients, we can
* assume the process is stationary (has no unit roots) and use OLS, or
* assume that the process has a unit root, and apply the difference operator to the series. OLS can then be applied to the resulting (stationary) series to estimate the remaining slope coefficients.

For example, in the AR(1) case, $Delta\; y\_\{t\}\; =\; y\_\{t\}\; -\; y\_\{t-1\}\; =\; varepsilon\_\{t\}$ is stationary.

In the AR(2) case, $y\_\{t\}\; =\; a\_\{1\}y\_\{t-1\}\; +\; a\_\{2\}y\_\{t-2\}\; +\; varepsilon\_\{t\}$ can be written as $(1\; -lambda\_\{1\}L)(1\; -\; lambda\_\{2\}L)y\_\{t\}\; =\; varepsilon\_\{t\}$ where L is a lag operator that decreases the time index of a variable by one period, $Ly\_\{t\}\; =\; y\_\{t-1\}$. If $lambda\_\{2\}\; =\; 1$, we can define $z\_\{t\}\; =\; Delta\; y\_\{t\}$ and then : $z\_\{t\}\; =\; lambda\_\{1\}z\_\{t-1\}\; +\; varepsilon\_\{t\}$is stationary. OLS can be used to estimate the remaining slope coefficient, $lambda\_\{1\}$.

If the process has multiple unit roots, the difference operator can be applied multiple times.

Properties and characteristics of unit-root processes

* Shocks to a unit root process have permanent effects which do not decay as they would if the process were stationary
* As noted above, a unit root process has a variance that depends on t, and diverges to infinity
* If it is known that if a series has a unit root, the series can be differenced to render it stationary. For example, if a series $Y\_t$ is I(1), the series $Delta\; Y\_t=Y\_t-Y\_\{t-1\}$ is I(0) (stationary). It is hence called a "difference stationary" series.

ee also

* Dickey-Fuller test
* Augmented Dickey-Fuller test
* Unit root test
* Phillips-Perron test (PP)
* Weighted symmetric unit root test (WS)
* Kwiatkowski, Phillips, Schmidt, Shin test, known as KPSS tests