Unit root

From a unit root is called in econometrics, especially in the time series analysis if 1 is a root of the characteristic polynomial. A stochastic process, which has such a unit root is not stationary, one also speaks of a stochastic trend. This is especially important because many statistical estimation methods such as the method of least squares assume stationary data and produce nonsensical results, if the underlying series are non-stationary, for example in the case of spurious regression.

For AR (p ) process

Consider the example of a discrete-time stochastic process { }. Assume in addition that this process can be represented as an autoregressive process of order p:

In the case { } is a non- autocorrelated stochastic process with zero mean and constant variance. For simplicity, assume that. If a zero of the characteristic polynomial

, then the stochastic process has a unit root or, in other words, it is integrated with order one, written. If a multiple zero of order, then the stochastic process is integrated with order.

For AR ( 1) process

A more specific example is the autoregressive first order model. This has a unit root, though. In this case, the characteristic polynomial. The solution of the equation. If the process has a unit root, then the time series is not stationary. This means that the moments of the stochastic process depend on. This arises as follows. In the case of a unit root process is therefore

So repeatedly substituting results. Consequently, the variance is

However, because the variance depends on.

Differentiating an integral with first-order time series times, one obtains a stationary process. Can be optionally present cointegration.

Perception of processes with unit root

William Feller 's chances at it, what time shares a random walk that starts at zero, spends in the positive or in the negative numbers and points from this example of a possible miscalculation. The probability that predominantly positive or predominantly negative numbers are accepted, is much bigger than the probability for a balance: If listed, the number of coats of arms and the number of figures, for example, when twenty times repeated toss of a coin, the probability that over time either crest or number at least 16 times in leadership is about 68.5 %. In contrast, the probability that at the end of coats of arms and number have equal number of times in the lead, only about 6%. The intuitive assumption that a doubling of the considered time interval results in a doubling of passes through zero with it is wrong.