A root equal to 1 in the characteristic equation of a stochastic process.

What does $I(r)$ indicate?

A process with a unit root of multiplicity $r$; generally, $r$ differences are needed to make it stationary.

When does an AR(1) model have a unit root?

For $y_t=a_1y_{t-1}+\varepsilon_t$, when $a_1=1$.

How is a random walk written?

As $y_t=y_{t-1}+\varepsilon_t$, where the current value is the previous value plus an innovation.

How do shocks behave in a unit-root process?

A shock can have a permanent effect on the series' level.

EconometricsLesson 5 of 5 in the course Time series

Unit Roots: Integration, Testing, Differencing, and Shock Persistence

5 flashcards · 4 quiz questions · 5 match pairs · 3 fill-in sentences · English

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Summary

A unit root is a feature of a time-series process that helps explain why its behavior can change over time and why ordinary statistical methods may not apply directly. This lesson introduces the definition of a unit root and shows how it makes a series non-stationary.

$y_t=a_1y_{t-1}+a_2y_{t-2}+\cdots+a_py_{t-p}+\varepsilon_t.$

The characteristic equation describes the process's roots. A process has a unit root when 1 is a solution to that equation. For the AR( $p$ ) form above, the equation is $m^p-a_1m^{p-1}-a_2m^{p-2}-\cdots-a_p=0$ . If $m=1$ is a root with multiplicity one, the process is integrated of order one, written $I(1)$ . If the root 1 has multiplicity $r$ , the process is integrated of order $r$ , written $I(r)$ .

All flashcards

What is a unit root?: A root equal to 1 in the characteristic equation of a stochastic process.
What does $I(r)$ indicate?: A process with a unit root of multiplicity $r$ ; generally, $r$ differences are needed to make it stationary.
When does an AR(1) model have a unit root?: For $y_t=a_1y_{t-1}+\varepsilon_t$ , when $a_1=1$ .
How is a random walk written?: As $y_t=y_{t-1}+\varepsilon_t$ , where the current value is the previous value plus an innovation.
How do shocks behave in a unit-root process?: A shock can have a permanent effect on the series' level.

Quiz questions

What can happen when ordinary least squares is applied directly to non-stationary data?Answer: It can produce apparently strong but misleading regression results.
How do temporary shocks affect a trend-stationary series?Answer: They affect deviations from the trend, which tend to fade.
What does a characteristic root greater than 1 indicate?Answer: An explosive process.
What role does the drift term $c$ play in $y_t=y_{t-1}+c+e_t$ ?Answer: It adds a systematic average change over time.

Based on the Wikipedia article Unit root, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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