Unit root

In time series models in econometrics (the application of statistical methods to economics), a unit root is a feature of processes that evolve through time that can cause problems in statistical inference if it is not adequately dealt with.

A linear stochastic process has a unit root if 1 is a root of the process's characteristic equation. Such a process is non-stationary. 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.

Definition

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 σ². For convenience, assume y₀ = 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).

Example

The first order autoregressive model, y_t = a₁y_{t − 1} + ε_t, has a unit root when a₁ = 1. In this example, the characteristic equation is m − a₁ = 0. 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, starting from y₀=0:

y_t = y_{t − 1} + ε_t.

By repeated substitution, we can write $y_{t} = y_{0} + \sum_{j=1}^t \varepsilon_{j}$. 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(y₁) = σ², while Var(y₂) = 2σ². Note that the variance of the series is diverging to infinity with t.

Related models

In addition to AR and ARMA models, other important models arise in regression analysis where the model errors may themselves have a time series structure and thus may need to be modelled by an AR or ARMA process that may have a unit root, as discussed above. The finite sample properties of regression models with first order ARMA errors, including unit roots, have been analyzed.

Estimation when a unit root may be present

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 called such estimates 'spurious regression' results: high R² values and high t-ratios yielding results with no economic meaning.

To estimate the slope coefficients, one should first conduct a unit root test, whose null hypothesis is that a unit root is present. If that hypothesis is rejected, one can use OLS. However, if the presence of a unit root is not rejected, then one should apply the difference operator to the series. If another unit root test shows the differenced time series to be stationary, OLS can then be applied to this series to estimate the slope coefficients.

For example, in the AR(1) case, Δy_t = y_t − y_{t − 1} = ε_t is stationary.

In the AR(2) case, y_t = a₁y_{t − 1} + a₂y_{t − 2} + ε_t can be written as (1 − λ₁L)(1 − λ₂L)y_t = ε_t where L is a lag operator that decreases the time index of a variable by one period: Ly_t = y_{t − 1}. If λ₂ = 1, the model has a unit root and we can define z_t = Δy_t; then

z_t = λ₁z_{t − 1} + ε_t

is stationary if | λ₁ | < 1. OLS can be used to estimate the slope coefficient, λ₁.

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 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 ΔY_t = Y_t − Y_{t − 1} is I(0) (stationary). It is hence called a difference stationary series.^{[citation needed]}

Unit root hypothesis

The diagram above depicts an example of a potential unit root. The red line represents an observed drop in output. Green shows the path of recovery if the series has a unit root. Blue shows the recovery if there is no unit root and the series is trend stationary. The blue line returns to meet and follow the dashed trend line while the green line remains permanently below the trend. The unit root hypothesis also holds that a spike in output will lead to levels of output higher than the past trend.

Economists debate whether various economic statistics, especially output, have a unit root or are trend stationary. A unit root process with drift is given in the first-order case by

y_t = y_{t − 1} + c + e_t

where c is a constant term referred to as the "drift" term, and e_t is white noise. Any non-zero value of the noise term, occurring for only one period, will permanently affect the value of y_t as shown in the graph, so deviations from the line y_t = a + ct are non-stationary; there is no reversion to any trend line. In contrast, a trend stationary process is given by

$y_t = k \cdot t + u_t$

where k is the slope of the trend and u_t is noise (white noise in the simplest case; more generally, noise following its own stationary autoregressive process). Here any transient noise will not alter the long-run tendency for y_t to be on the trend line, as also shown in the graph. This process is said to be trend stationary because deviations from the trend line are stationary.

The issue is particularly popular in the literature on business cycles. Research on the subject began with Nelson and Plosser whose paper on GNP and other output aggregates failed to reject the unit root hypothesis for these series. Since then, a debate—entwined with technical disputes on statistical methods—has ensued. Some economists argue that GDP has a unit root or structural break, implying that economic downturns result in permanently lower GDP levels in the long run. Other economists argue that GDP is trend-stationary: That is, when GDP dips below trend during a downturn it later returns to the level implied by the trend so that there is no permanent decrease in output. While the literature on the unit root hypothesis may consist of arcane debate on statistical methods, the implications of the hypothesis can have concrete implications for economic forecasts and policies.

See also

• Dickey–Fuller test
• Augmented Dickey–Fuller test
• Unit root test
• Phillips–Perron test
• Cointegration, determining the relationship between two variables having unit roots
• Weighted symmetric unit root test (WS)
• Kwiatkowski, Phillips, Schmidt, Shin test, known as KPSS tests