Cointegration Measure, CI(d,p)

A Cointegration Measure, CI(d,p), is a statistical measure of a set of two or more nonstationary time series.

• Context:
  • It can be defined as:

the time series [math]\displaystyle{ x_t }[/math] and [math]\displaystyle{ y_t }[/math] are said to be co-integrated of order CI(d,p) if:

1. [math]\displaystyle{ x_t }[/math] and [math]\displaystyle{ y_t }[/math] are both integrated of order [math]\displaystyle{ d }[/math]

2. There exist a parameter [math]\displaystyle{ \alpha }[/math] such that [math]\displaystyle{ u_t=y_t-\alpha_t }[/math] is a stationary process

the time series [math]\displaystyle{ u_t }[/math] is said to be integrated of order [math]\displaystyle{ d-p }[/math].

• • It can be tested by a Cointegration Test, such as a Johansen Test.
  • …
• Counter-Example(s):
  • Autocorrelation;
  • Unit Root;
  • Correlation Matrix.
• See: Order of Integration, Johansen Test, Unit Root, Order of Integration, Linear Combination.

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References

2018

• (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/cointegration Retrieved:2018-12-14.
  • Cointegration is a statistical property of a collection (X₁, X₂, ..., X_k) of time series variables. First, all of the series must be integrated of order d (see Order of integration). Next, if a linear combination of this collection is integrated of order zero, then the collection is said to be co-integrated. Formally, if (X,Y,Z) are each integrated of order d, and there exist coefficients a,b,c such that aX + bY + cZ is integrated of order 0, then X, Y, and Z are cointegrated. Cointegration has become an important property in contemporary time series analysis. Time series often have trends — either deterministic or stochastic. In an influential paper, Charles Nelson and Charles Plosser (1982) provided statistical evidence that many US macroeconomic time series (like GNP, wages, employment, etc.) have stochastic trends — these are also called unit root processes, or processes integrated of order . They also showed that unit root processes have non-standard statistical properties, so that conventional econometric theory methods do not apply to them.

2016

• https://github.com/statsmodels/statsmodels/
  • Main Features
    • Time Series Analysis: models for time series analysis.
      • Hypothesis tests for time series: unit root, cointegration and others

2005

• (Sørensen, 2005) ⇒ Sørensen, Bent E. "1 Cointegration." (2005). http://www.uh.edu/~bsorense/coint.pdf
  • QUOTE: The definition in the simple case of 2 time series [math]\displaystyle{ x_t }[/math] and [math]\displaystyle{ y_t }[/math], that are both integrated of order one (this is abbreviated I(1), and means that the process contains a unit root), is the following:

Definition: [math]\displaystyle{ x_t }[/math] and [math]\displaystyle{ y_t }[/math] are said to be cointegrated if there exists a parameter [math]\displaystyle{ \alpha }[/math] such that

[math]\displaystyle{ u_t = y_t − \alpha x_t }[/math]

is a stationary process.

(...) A stochastic process is said to be integrated of order p, abbreviated as I(p), if it need to be differenced p times in order to achieve stationarity. More generally [math]\displaystyle{ x_t }[/math] and [math]\displaystyle{ y_t }[/math] are said to be co-integrated of order [math]\displaystyle{ CI(d,p) }[/math] if [math]\displaystyle{ x_t }[/math] and [math]\displaystyle{ y_t }[/math] are both integrated of order d; but there exist an [math]\displaystyle{ \alpha }[/math] such that [math]\displaystyle{ y_t − \alpha x_t }[/math] is integrated of order [math]\displaystyle{ d-p }[/math]. For the rest of this chapter I will only treat the CI(1,1) case, which will be referred to simple as cointegration. Most applications of cointegration methods treats that case, and it will allow for a much simpler presentation to limit the development to the CI(1,1) case.

For many purposes the above definition of cointegration is too narrow. It is true that economic series tend to move together but in order to obtain a linear combination of the series, that is stationary one may have to include more variables. The general definition of co-integration (for the I(1) case) is therefore the following:

Definition: A vector of I(1) variables [math]\displaystyle{ y_t }[/math] is said to be cointegrated if there exist at vector [math]\displaystyle{ β_i }[/math] such that [math]\displaystyle{ \beta'_{i}y_t }[/math] is trend stationary. If there exist r such linearly independent vectors [math]\displaystyle{ \beta_i,\; i =1,...,r }[/math], then [math]\displaystyle{ yt }[/math] is said to be cointegrated with cointegrating rank r. The matrix <math>\beta = (\beta_1, ...\beta_r) is called the cointegrating matrix.

1999

• http://www.eco.uc3m.es/~jgonzalo/teaching/EconometriaII/cointegration.HTM
  • QUOTE: Definition: If there exists a stationary linear combination of nonstationary random variables, the variables combined are said to be cointegrated. ...

    The notion of cointegration arose out of the concern about spurious or nonsense regressions in time series. ... For there to be a long run relationship between the variables they must be cointegrated.

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