Integrated Time Series

An integrated time series is an unsteady time series , the differences of a certain order from which are a stationary time series. Such series are also called differential-stationary (DS-series, Difference Stationary) . An example of an integrated time series is the random walk , often used in modeling financial time series.

Content

Definition

To determine the integrated time series, it is necessary to determine the class of time series, called series stationary with respect to the trend ( TS- series, trend stationary). Row $x_{t}$ ${\ displaystyle x_ {t}}$ is called a TS- series if there exists some deterministic function f (t) such that the difference $x_{t}-f(t)$ ${\ displaystyle x_ {t} -f (t)}$ is a stationary process. In particular, all stationary rows belong to TS-series. However, many TS series are non-stationary. TS series also includes, for example, a linear (deterministic) trend model $x_{t}=a+bt+\varepsilon _{t}$ ${\ displaystyle x_ {t} = a + bt + \ varepsilon _ {t}}$ where the model error is a stationary process (usually white noise).

Temporal $X_{t}$ ${\ displaystyle X_ {t}}$ the series is called integrated of order k (usually write $X_{t}\sim I(k)$ ${\ displaystyle X_ {t} \ sim I (k)}$ ) if the differences of a series of kth order $\vartriangle ^{k}x_{t}$ ${\ displaystyle \ vartriangle ^ {k} x_ {t}}$ - are stationary, while differences of a lower order (including zero order, that is, the time series itself) are not TS-series . In particular, I (0) is a stationary process.

Example

Consider an example - a random walk process with drift (drift) - an integrated first-order process $I(1)$ ${\ displaystyle I (1)}$

$x_{t}=a+x_{t-1}+\varepsilon _{t}$ ${\ displaystyle x_ {t} = a + x_ {t-1} + \ varepsilon _ {t}}$

where the random error of the model is white noise . The first time series differences are obviously stationary. Imagine the model in a slightly different form:

$x_{t}=a+x_{t-1}+\varepsilon _{t}=a+a+x_{t-2}+\varepsilon _{t-1}+\varepsilon _{t}=a+a+a+x_{t-3}+\varepsilon _{t-2}+\varepsilon _{t-1}+\varepsilon _{t}=...=x_{0}+at+\sum _{i=1}^{t}\varepsilon _{i}$ ${\ displaystyle x_ {t} = a + x_ {t-1} + \ varepsilon _ {t} = a + a + x_ {t-2} + \ varepsilon _ {t-1} + \ varepsilon _ {t} = a + a + a + x_ {t-3} + \ varepsilon _ {t-2} + \ varepsilon _ {t-1} + \ varepsilon _ {t} = ... = x_ {0} + at + \ sum _ {i = 1} ^ {t} \ varepsilon _ {i}}$

Thus, a random walk with a drift looks like a linear trend model with one very significant difference - the variance of the model error $V(\sum _{i=1}^{t}\varepsilon _{i})=t\sigma ^{2}$ ${\ displaystyle V (\ sum _ {i = 1} ^ {t} \ varepsilon _ {i}) = t \ sigma ^ {2}}$ proportional to time, that is, with time tends to infinity. Moreover, the mathematical expectation of a random error is zero. Even if we apply a linear (deterministic) trend exclusion procedure to a time series, we will still get a non-stationary process - a stochastic trend.

Integration and Unit Roots

The concept of an integrated time series is closely related to unit roots in autoregressive models . The presence of unit roots in the characteristic polynomial of the autoregressive component of the time series model means the integration of the time series. Moreover, the number of unit roots coincides with the order of integration.

Literature

Ayvazyan S.A. Applied statistics. Fundamentals of Econometrics. Volume 2. - M .: Unity-Dana, 2001 .-- 432 p. - ISBN 5-238-00305-6 .
Magnus Ya.R., Katyshev P.K., Peresetskiy A.A. Econometrics. Beginner course. - M .: Case, 2007 .-- 504 p. - ISBN 978-5-7749-0473-0 .
Econometrics. Textbook / Ed. Eliseeva I.I. - 2nd ed. - M .: Finance and statistics, 2006. - 576 p. - ISBN 5-279-02786-3 .