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Unit root

Unit root

Overview
In time series
Time series
In statistics, signal processing, and many other fields, a time series is a sequence of data points, measured typically at successive times, spaced at time intervals...

 models in econometrics
Econometrics
Econometrics is concerned with the tasks of developing and applying quantitative or statistical methods to the study and elucidation of economic principles. Econometrics combines economic theory with statistics to analyze and test economic relationships...

, a linear stochastic process
Stochastic process
In probability theory, a stochastic process, or sometimes random process, is the counterpart to a deterministic process...

 has a unit root if 1 is a root of the process's characteristic equation. The process will be non-stationary
Stationary process
In the mathematical sciences, a stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space...

. 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
Stochastic process
In probability theory, a stochastic process, or sometimes random process, is the counterpart to a deterministic process...

 {}, and suppose that it can be written as an autoregressive process of order p:
Here, {} is a serially uncorrelated, mean zero stochastic process with constant variance .
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Encyclopedia
In time series
Time series
In statistics, signal processing, and many other fields, a time series is a sequence of data points, measured typically at successive times, spaced at time intervals...

 models in econometrics
Econometrics
Econometrics is concerned with the tasks of developing and applying quantitative or statistical methods to the study and elucidation of economic principles. Econometrics combines economic theory with statistics to analyze and test economic relationships...

, a linear stochastic process
Stochastic process
In probability theory, a stochastic process, or sometimes random process, is the counterpart to a deterministic process...

 has a unit root if 1 is a root of the process's characteristic equation. The process will be non-stationary
Stationary process
In the mathematical sciences, a stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space...

. If the other roots of the characteristic equation lie inside the unit circle, then the first difference of the process will be stationary.

Definition


Consider a discrete time stochastic process
Stochastic process
In probability theory, a stochastic process, or sometimes random process, is the counterpart to a deterministic process...

 {}, and suppose that it can be written as an autoregressive process of order p:
Here, {} is a serially uncorrelated, mean zero stochastic process with constant variance . For convenience, assume . If is a root
Root (mathematics)
In mathematics, a root of a real-, complex- or generally vector-valued function ƒ is a member x of the domain of ƒ such that ƒ vanishes at , that is,...

 of the characteristic equation:
then the stochastic process has a unit root or, alternatively, is integrated of order
Order of integration
Order of integration, denoted I, is a summary statistic for a time series. It reports the minimum number of differences required to obtain a stationary series.- Integration of order zero :...

 one, denoted . 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, , has a unit root when . In this example, the characteristic equation is . The root of the equation is .

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 . To illustrate the effect of a unit root, we can consider the first order case:
By repeated substitution, we can write . Then the variance of is given by:
The variance depends on t since , while . 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
Ordinary least squares
In statistics and econometrics, ordinary least squares is a technique for estimating the unknown parameters in a linear regression model. This method minimizes the sum of squared distances between the observed responses in a set of data, and the fitted responses from the regression model...

 (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, is stationary.

In the AR(2) case, can be written as where L is a lag operator
Lag operator
In time series analysis, the lag operator or backshift operator operates on an element of a time series to produce the previous element. For example, given some time seriesthen for all...

 that decreases the time index of a variable by one period, . If , the model has a unit root and we can define and then
is stationary. OLS can be used to estimate the remaining slope coefficient, .

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 is I(1), the series is I(0) (stationary). It is hence called a difference stationary series.

Unit root hypothesis


Economists debate whether various economic statistics, especially output, have a unit root or are trend stationary. The issue is particularly popular in the literature on business cycles. Some economists argue that GNP has a unit root or structural break, implying that economic downturns result in permanently lower GNP levels in the long run. Other economists argue that GNP is trend-stationary. That is, when GNP 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.

See also

  • Dickey–Fuller test
  • Augmented Dickey–Fuller test
  • Unit root test
    Unit root test
    In statistics, a unit root test tests whether a time series variable is non-stationary using an autoregressive model. The most famous test is the augmented Dickey–Fuller test. Another test is the Phillips–Perron test. Both these tests use the existence of a unit root as the null...

  • Phillips–Perron test (PP)
  • Weighted symmetric unit root test (WS)
  • Kwiatkowski, Phillips, Schmidt, Shin test, known as KPSS tests