Unit Root Testing in Finance

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What can happen if two variables are trending over time and a regression of one on the other is performed?

The regression will always have a low R2 if the variables are unrelated

The regression will not be affected by the trending variables

The regression will not be valid due to non-stationarity

The regression could have a high R2 even if the two variables are totally unrelated (correct)

What happens to the persistence of shocks for nonstationary series?

The persistence of shocks will be zero

The persistence of shocks will be infinite (correct)

The persistence of shocks will decrease over time

The persistence of shocks will converge to a constant value

Which model characterizes non-stationarity as a random walk with drift?

$y_t = \mu + y_{t-1} + u_t$ (correct)

$y_t = \alpha + \beta t + u_t$

$y_t = \alpha + t + u_t$

$y_t = \mu + y_{t-1} + \varepsilon_t$

What happens to the standard assumptions for asymptotic analysis if the variables in a regression model are not stationary?

The standard assumptions for asymptotic analysis will not be valid Signup and view all the answers

In the model $y_t = \mu + \phi y_{t-1} + u_t$, what does it mean if $\phi > 1$?

Shocks to the system are not only persistent through time, they are propagated so that a given shock will have an increasingly large influence. Signup and view all the answers

What is the impact of shocks in an AR(1) process with no drift, $y_t = \phi y_{t-1} + u_t$?

Given shocks become more influential as time goes on. Signup and view all the answers

What is the correct treatment for inducing stationarity in a trend-stationary series?

Detrending Signup and view all the answers

What is required to induce stationarity in a stochastic non-stationary series?

"Differencing once" Signup and view all the answers

What structure is introduced into the errors if a trend-stationary series is differenced once?

MA(1) structure Signup and view all the answers

What happens if we try to detrend a series which has stochastic trend?

We will not remove the non-stationarity. Signup and view all the answers

What does it mean if a non-stationary series needs to be differenced d times before it becomes stationary?

It is said to be integrated of order d. Signup and view all the answers

What does an I(0) series indicate?

A stationary series Signup and view all the answers

What characteristic differentiates an I(2) series from an I(0) or I(1) series?

An I(2) series contains two unit roots and requires differencing twice to induce stationarity. Signup and view all the answers

What did Dickey and Fuller's test aim to test for in time series data?

The presence of a unit root in the data. Signup and view all the answers

What is the null hypothesis being tested in Dickey and Fuller's test?

$\phi = 1$ in $y_t = \phi y_{t-1} + u_t$ Signup and view all the answers

What is the impact of non-stationarity on regression analysis?

It can lead to high R2 even if the variables are unrelated Signup and view all the answers

What characterizes non-stationarity as a random walk with drift?

$y_t = \mu + y_{t-1} + u_t$ Signup and view all the answers

What happens to the standard assumptions for asymptotic analysis if the variables in a regression model are not stationary?

The usual t-ratios will not follow a t-distribution Signup and view all the answers

What is the impact of $ ext{AR}(1)$ process with no drift, $y_t = ext{ϕ} y_{t-1} + u_t$?

Shocks become more influential as time goes on for $ ext{ϕ} > 1$ Signup and view all the answers

What characteristic differentiates an $I(2)$ series from an $I(0)$ or $I(1)$ series?

An $I(2)$ series contains two unit roots and requires differencing twice to induce stationarity Signup and view all the answers

What does it mean if a non-stationary series needs to be differenced $d$ times before it becomes stationary?

It is said to be integrated of order $d$, denoted as $I(d)$ Signup and view all the answers

What happens if we try to detrend a series which has a stochastic trend?

The non-stationarity will not be removed Signup and view all the answers

What structure is introduced into the errors if a trend-stationary series is differenced once?

$MA(1)$ structure Signup and view all the answers

What did Dickey and Fuller's test aim to test for in time series data?

The presence of a unit root in the autoregressive model Signup and view all the answers

Which model characterizes non-stationarity as a random walk with drift?

$y_t = ext{μ} + ext{ϕ}y_{t-1} + u_t$ Signup and view all the answers

What is required to induce stationarity in a stochastic non-stationary series?

Differencing once Signup and view all the answers

Non-stationary series can strongly influence its behavior and properties - e.g. persistence of shocks will be infinite for ______ series

nonstationary Signup and view all the answers

The random walk model with drift: $y_t = \mu + y_{t-1} + u_t$ and the deterministic trend process: $y_t = \alpha + \beta t + u_t$ are used to characterize ______

non-stationarity Signup and view all the answers

If two variables are trending over time, a regression of one on the other could have a high R2 even if the two are totally unrelated. This is known as a ______ regression

spurious Signup and view all the answers

In the model $y_t = \mu + \phi y_{t-1} + u_t$, if $\phi > 1$, it characterizes non-stationarity as a ______ with drift

random walk Signup and view all the answers

An I(0) series is a ______ series

stationary Signup and view all the answers

An I(1) series contains one unit root, e.g. yt = yt-1 + ut

[blank] Signup and view all the answers

An I(2) series contains two unit roots and so would require differencing twice to induce stationarity. I(1) and I(2) series can wander a long way from their mean value and cross this mean value ______.

rarely Signup and view all the answers

The general case of an AR(1) with no drift: yt = yt-1 + ut can be written as: ______ = (yt-2 + ut-1) + ut

yt Signup and view all the answers

If a non-stationary series, yt must be differenced d times before it becomes stationary, then it is said to be integrated of order ______. We write yt I(d).

d Signup and view all the answers

The two will require different treatments to induce stationarity. The second case is known as deterministic non-stationarity and what is required is ______.

detrending Signup and view all the answers

The early and pioneering work on testing for a unit root in time series was done by Dickey and Fuller (Dickey and Fuller 1979, Fuller 1976). The basic objective of the test is to test the null hypothesis that  =1 in: yt = yt-1 + ut against the one-sided alternative  ______.

> 1 Signup and view all the answers

We say that we have induced stationarity by “differencing ______”

once Signup and view all the answers

If we take the model yt = yt-1 + ut and subtract yt-1 from both sides, we get: yt = ______ + ut

μ Signup and view all the answers

The first case is known as stochastic non-stationarity. If we let yt = yt - yt-1 and L yt = yt-1 so (1-L) yt = yt - L yt, then we can write yt - yt-1 =  + ut as: yt = ______ + ut

μ Signup and view all the answers

The explosive case is ignored and we use  = 1 to characterize the non-stationarity because –  > 1 does not describe many data series in economics and finance. –  > 1 has an intuitively unappealing property: shocks to the system are not only persistent through time, they are propagated so that a given shock will have an increasingly large ______.

influence Signup and view all the answers

Unit Root Testing in Finance

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Questions and Answers

What can happen if two variables are trending over time and a regression of one on the other is performed?

What happens to the persistence of shocks for nonstationary series?

Which model characterizes non-stationarity as a random walk with drift?

What happens to the standard assumptions for asymptotic analysis if the variables in a regression model are not stationary?

In the model $y_t = \mu + \phi y_{t-1} + u_t$, what does it mean if $\phi > 1$?

What is the impact of shocks in an AR(1) process with no drift, $y_t = \phi y_{t-1} + u_t$?

What is the correct treatment for inducing stationarity in a trend-stationary series?

What is required to induce stationarity in a stochastic non-stationary series?

What structure is introduced into the errors if a trend-stationary series is differenced once?

What happens if we try to detrend a series which has stochastic trend?

What does it mean if a non-stationary series needs to be differenced d times before it becomes stationary?

What does an I(0) series indicate?

What characteristic differentiates an I(2) series from an I(0) or I(1) series?

What did Dickey and Fuller's test aim to test for in time series data?

What is the null hypothesis being tested in Dickey and Fuller's test?

What is the impact of non-stationarity on regression analysis?

What characterizes non-stationarity as a random walk with drift?

What happens to the standard assumptions for asymptotic analysis if the variables in a regression model are not stationary?

What is the impact of $ ext{AR}(1)$ process with no drift, $y_t = ext{ϕ} y_{t-1} + u_t$?

What characteristic differentiates an $I(2)$ series from an $I(0)$ or $I(1)$ series?

What does it mean if a non-stationary series needs to be differenced $d$ times before it becomes stationary?

What happens if we try to detrend a series which has a stochastic trend?

What structure is introduced into the errors if a trend-stationary series is differenced once?

What did Dickey and Fuller's test aim to test for in time series data?

Which model characterizes non-stationarity as a random walk with drift?

What is required to induce stationarity in a stochastic non-stationary series?

Non-stationary series can strongly influence its behavior and properties - e.g. persistence of shocks will be infinite for ______ series

The random walk model with drift: $y_t = \mu + y_{t-1} + u_t$ and the deterministic trend process: $y_t = \alpha + \beta t + u_t$ are used to characterize ______

If two variables are trending over time, a regression of one on the other could have a high R2 even if the two are totally unrelated. This is known as a ______ regression

In the model $y_t = \mu + \phi y_{t-1} + u_t$, if $\phi > 1$, it characterizes non-stationarity as a ______ with drift

An I(0) series is a ______ series

An I(1) series contains one unit root, e.g. yt = yt-1 + ut

An I(2) series contains two unit roots and so would require differencing twice to induce stationarity. I(1) and I(2) series can wander a long way from their mean value and cross this mean value ______.

The general case of an AR(1) with no drift: yt = yt-1 + ut can be written as: ______ = (yt-2 + ut-1) + ut

If a non-stationary series, yt must be differenced d times before it becomes stationary, then it is said to be integrated of order ______. We write yt I(d).

The two will require different treatments to induce stationarity. The second case is known as deterministic non-stationarity and what is required is ______.

The early and pioneering work on testing for a unit root in time series was done by Dickey and Fuller (Dickey and Fuller 1979, Fuller 1976). The basic objective of the test is to test the null hypothesis that  =1 in: yt = yt-1 + ut against the one-sided alternative  ______.

We say that we have induced stationarity by “differencing ______”

If we take the model yt = yt-1 + ut and subtract yt-1 from both sides, we get: yt = ______ + ut

The first case is known as stochastic non-stationarity. If we let yt = yt - yt-1 and L yt = yt-1 so (1-L) yt = yt - L yt, then we can write yt - yt-1 =  + ut as: yt = ______ + ut

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