Cointegration
Encyclopedia
Cointegration is a statistical property of time series
variables. Two or more time series are cointegrated if they share a common stochastic drift
.
(in the time series sense) but some linear combination
of them has a lower order of integration
, then the series are said to be cointegrated. A common example is where the individual series are first-order integrated (I(1)) but some (cointegrating) vector of coefficients exists to form a stationary
linear combination of them. For instance, a stock market index
and the price of its associated futures contract
move through time, each roughly following a random walk
. Testing the hypothesis that there is a statistically significant connection between the futures price and the spot price could now be done by testing for the existence of a cointegrated combination of the two series. (If such a combination has a low order of integration - in particular if it is I(0), this can signify an equilibrium relationship between the original series, which are said to be cointegrated.)
Before the 1980s many economists used linear regression
s on (de-trended
) non-stationary time series data, which Nobel laureate Clive Granger
and others showed to be a dangerous approach that could produce spurious correlation. His 1987 paper with Nobel laureate Robert Engle formalized the cointegrating vector approach, and coined the term.
The possible presence of cointegration must be taken into account when choosing a technique to test hypotheses concerning the relationship between two variables having unit root
s (i.e. integrated of at least order one).
The usual procedure for testing hypotheses concerning the relationship between non-stationary variables was to run ordinary least squares
(OLS) regressions on data which had initially been differenced. Although this method is correct in large samples, cointegration provides more powerful tools when the data sets are of limited length, as most economic time-series are.
In practice, cointegration is often used for two I(1) series, but it is more generally applicable and can be used for variables integrated of higher order (to detect correlated accelerations or other second-difference effects). Multicointegration extends the cointegration technique beyond two variables, and occasionally to variables integrated at different orders.
However, these tests for cointegration assume that the cointegrating vector is constant during the period of study. In reality, it is possible that the long-run relationship between the underlying variables change (shifts in the cointegrating vector can occur). The reason for this might be technological progress, economic crises, changes in the people’s preferences and behaviour accordingly, policy or regime alteration, and organizational or institutional developments. This is especially likely to be the case if the sample period is long. To take this issue into account Gregory and Hansen (1996) have introduced tests for cointegration with one unknown structural break
and Hatemi-J (2008) has introduced tests for cointegration with two unknown breaks.
Time series
In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...
variables. Two or more time series are cointegrated if they share a common stochastic drift
Stochastic drift
In probability theory, stochastic drift is the change of the average value of a stochastic process. A related term is the drift rate which is the rate at which the average changes. This is in contrast to the random fluctuations about this average value...
.
Introduction
If two or more series are individually integratedOrder 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 :...
(in the time series sense) but some linear combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
of them has a lower order of integration
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 :...
, then the series are said to be cointegrated. A common example is where the individual series are first-order integrated (I(1)) but some (cointegrating) vector of coefficients exists to form a 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...
linear combination of them. For instance, a stock market index
Stock market index
A stock market index is a method of measuring a section of the stock market. Many indices are cited by news or financial services firms and are used as benchmarks, to measure the performance of portfolios such as mutual funds....
and the price of its associated futures contract
Futures contract
In finance, a futures contract is a standardized contract between two parties to exchange a specified asset of standardized quantity and quality for a price agreed today with delivery occurring at a specified future date, the delivery date. The contracts are traded on a futures exchange...
move through time, each roughly following a random walk
Random walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...
. Testing the hypothesis that there is a statistically significant connection between the futures price and the spot price could now be done by testing for the existence of a cointegrated combination of the two series. (If such a combination has a low order of integration - in particular if it is I(0), this can signify an equilibrium relationship between the original series, which are said to be cointegrated.)
Before the 1980s many economists used linear regression
Linear regression
In statistics, linear regression is an approach to modeling the relationship between a scalar variable y and one or more explanatory variables denoted X. The case of one explanatory variable is called simple regression...
s on (de-trended
Trend estimation
Trend estimation is a statistical technique to aid interpretation of data. When a series of measurements of a process are treated as a time series, trend estimation can be used to make and justify statements about tendencies in the data...
) non-stationary time series data, which Nobel laureate Clive Granger
Clive Granger
Sir Clive William John Granger was a British economist, who taught in Britain at the University of Nottingham and in the U.S.A. at the University of California, San Diego. In 2003, Granger was awarded the Nobel Memorial Prize in Economic Sciences, in recognition that he and his co-winner, Robert F...
and others showed to be a dangerous approach that could produce spurious correlation. His 1987 paper with Nobel laureate Robert Engle formalized the cointegrating vector approach, and coined the term.
The possible presence of cointegration must be taken into account when choosing a technique to test hypotheses concerning the relationship between two variables having unit root
Unit root
In time series models in econometrics , 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....
s (i.e. integrated of at least order one).
The usual procedure for testing hypotheses concerning the relationship between non-stationary variables was to run ordinary least squares
Ordinary least squares
In statistics, ordinary least squares or linear least squares is a method for estimating the unknown parameters in a linear regression model. This method minimizes the sum of squared vertical distances between the observed responses in the dataset and the responses predicted by the linear...
(OLS) regressions on data which had initially been differenced. Although this method is correct in large samples, cointegration provides more powerful tools when the data sets are of limited length, as most economic time-series are.
Test
The three main methods for testing for cointegration are:- The Engle-Granger two-step method (null: no cointegration, so residual is a random walk).
- The Johansen procedureJohansen testIn statistics, the Johansen test, named after Søren Johansen, is a procedure for testing cointegration of several I time series. This test permits more than one cointegrating relationship so is more generally applicable than the Engle–Granger test which is based on the Dickey–Fuller test for...
. - Phillips-Ouliaris Cointegration Test available with RR (programming language)R is a programming language and software environment for statistical computing and graphics. The R language is widely used among statisticians for developing statistical software, and R is widely used for statistical software development and data analysis....
(null: no cointegration)
In practice, cointegration is often used for two I(1) series, but it is more generally applicable and can be used for variables integrated of higher order (to detect correlated accelerations or other second-difference effects). Multicointegration extends the cointegration technique beyond two variables, and occasionally to variables integrated at different orders.
However, these tests for cointegration assume that the cointegrating vector is constant during the period of study. In reality, it is possible that the long-run relationship between the underlying variables change (shifts in the cointegrating vector can occur). The reason for this might be technological progress, economic crises, changes in the people’s preferences and behaviour accordingly, policy or regime alteration, and organizational or institutional developments. This is especially likely to be the case if the sample period is long. To take this issue into account Gregory and Hansen (1996) have introduced tests for cointegration with one unknown structural break
Structural break
A structural break is a concept in econometrics. A structural break appears when we see an unexpected shift in a time series. This can lead to huge forecasting errors and unreliability of the model in general...
and Hatemi-J (2008) has introduced tests for cointegration with two unknown breaks.
See also
- Granger causalityGranger causalityThe Granger causality test is a statistical hypothesis test for determining whether one time series is useful in forecasting another. Ordinarily, regressions reflect "mere" correlations, but Clive Granger, who won a Nobel Prize in Economics, argued that there is an interpretation of a set of tests...
- Error correction modelError correction modelAn error correction model is a dynamical system with the characteristics that the deviation of the current state from its long-run relationship will be fed into its short-run dynamics....
- Stationary Subspace AnalysisStationary subspace analysisStationary Subspace Analysis is a blind source separation algorithm which factorizes a multivariate time series into stationary and non-stationary components.- Introduction :...
External links
- Cointegration Primer by David R. Stockwell, 2010
- A drunk and her dog: An illustration of cointegration and error correction by Michael P. Murray, 1994. An intuitive introduction to cointegration.