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In
probability theory Probability theory 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 ...
and
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, a unit root is a feature of some
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...
es (such as
random walk In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...
s) that can cause problems in
statistical inference Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properti ...
involving
time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Ex ...
models A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...
. A linear
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...
has a unit root if 1 is a root of the process's characteristic equation. Such a process is non-stationary but does not always have a trend. If the other roots of the characteristic equation lie inside the unit circle—that is, have a modulus (
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
) less than one—then the
first difference In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a paramet ...
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 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 ...
.


Definition

Consider a discrete-time
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...
(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_+a_2 y_ + \cdots + a_p y_+\varepsilon_t. Here, (\varepsilon_,t=0,1,2,\ldots,) is a serially uncorrelated, zero-mean stochastic process with constant variance \sigma^2. For convenience, assume y_0 = 0 . If m=1 is a
root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
of the characteristic equation, of multiplicity 1: : m^p - m^a_1 - m^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=a_1 y_+\varepsilon_t, has a unit root when a_1=1. In this example, the characteristic equation is m - a_1 = 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 ''y''0 = 0: :y_t= y_+\varepsilon_t. By repeated substitution, we can write y_t = y_0 + \sum_^t \varepsilon_j. Then the variance of y_t is given by: : \operatorname(y_t) = \sum_^t \sigma^2=t \sigma^2 . The variance depends on ''t'' since \operatorname(y_1) = \sigma^2 , while \operatorname(y_2) = 2\sigma^2 . Note that the variance of the series is diverging to infinity with ''t''. There are various tests to check for the existence of a unit root, some of them are given by: # The
Dickey–Fuller test In statistics, the Dickey–Fuller test tests the null hypothesis that a unit root is present in an autoregressive time series model. The alternative hypothesis is different depending on which version of the test is used, but is usually stationar ...
(DF) or augmented Dickey–Fuller (ADF) tests # Testing the significance of more than one coefficients (f-test) # The
Phillips–Perron test In statistics, the Phillips–Perron test (named after Peter C. B. Phillips and Pierre Perron) is a unit root test. That is, it is used in time series analysis to test the null hypothesis that a time series is integrated of order 1. It builds ...
(PP) # Dickey Pantula test


Related models

In addition to autoregressive (AR) and autoregressive–moving-average (ARMA) models, other important models arise in
regression analysis In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one ...
where the model errors may themselves have a
time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Ex ...
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 In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the ...
(OLS) is used to estimate the slope coefficients of the
autoregressive model In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...
. 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 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 ...
, whose
null hypothesis In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is ...
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, \Delta y_ = y_ - y_ = \varepsilon_ is stationary. In the AR(2) case, y_ = a_y_ + a_y_ + \varepsilon_ can be written as (1 -\lambda_L)(1 - \lambda_L)y_ = \varepsilon_ where L is a lag operator that decreases the time index of a variable by one period: Ly_ = y_ . If \lambda_ = 1 , the model has a unit root and we can define z_ = \Delta y_ ; then : z_ = \lambda_z_ + \varepsilon_ is stationary if , \lambda_1, < 1. OLS can be used to estimate the slope coefficient, \lambda_ . 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 \Delta Y_t=Y_t-Y_ 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. A unit root process with drift is given in the first-order case by :y_t = y_ + 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 economistsOlivier Blanchard
with the
International Monetary Fund The International Monetary Fund (IMF) is a major financial agency of the United Nations, and an international financial institution, headquartered in Washington, D.C., consisting of 190 countries. Its stated mission is "working to foster glo ...
makes the claim that after a banking crisis "on average, output does not go back to its old trend path, but remains permanently below it." argue that GDP has a unit root or
structural break In econometrics and statistics, a structural break is an unexpected change over time in the parameters of regression models, which can lead to huge forecasting errors and unreliability of the model in general. This issue was popularised by Da ...
, 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 hypothesis carries significant practical implications for economic forecasts and policies.


See also

*
Dickey–Fuller test In statistics, the Dickey–Fuller test tests the null hypothesis that a unit root is present in an autoregressive time series model. The alternative hypothesis is different depending on which version of the test is used, but is usually stationar ...
*
Augmented Dickey–Fuller test In statistics, an augmented Dickey–Fuller test (ADF) tests the null hypothesis that a unit root is present in a time series sample. The alternative hypothesis is different depending on which version of the test is used, but is usually stationar ...
* ADF-GLS test *
Unit root test 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 ...
*
Phillips–Perron test In statistics, the Phillips–Perron test (named after Peter C. B. Phillips and Pierre Perron) is a unit root test. That is, it is used in time series analysis to test the null hypothesis that a time series is integrated of order 1. It builds ...
* 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


Notes

{{DEFAULTSORT:Unit Root Regression with time series structure