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Questions tagged [unit-root]

A unit root is a property of a non-stationary time series which can lead to spurious regressions and wrong inference.

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Dickey-Fuller Unit Root test, Deterministic Terms

I'm learning about unit root tests and i'm trying to implement these tests in python. I have some doubts about the theory underlying the deterministic terms inside the regression. Some textbooks ( ...
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Intuition behind unit roots in practice

One area where the application of unit roots to time series modelling seems very intuitive is in climate change: carbon dioxide stays in the air, so past shocks (size of flow) have a cumulative effect ...
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Unit root stationarity and modelling AR(p) process

I'm reading through Introduction to Econometrics by Gary Koop. I'm a little confused on the process for modelling AR(p) processes. Hopefully someone can help clarify things for me. Let me set out my ...
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Characteristic Polynomials for AR(p) Processes with intercepts

If we have AR(1) process with no intercept like: $$ x_t=\phi x_{t-1}+w_t, $$ it has a unit root when $|\phi|=1$. If we have an AR(p) process with no constant $$ x_t=\phi_1 x_{t-1}+\phi_2 x_{t-2}+\...
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Unit-Root Asymptotics

I am using the book "Time-series-based econometrics" by Hatanaka to learn about asymptotic theory of unit roots. However, it is quite technical, so I am also using Hamilton's "Time ...
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Is it legit to estimate an AR(1) model for non-stationary time series?

Suppose ${X_{t}}$ is a non-stationary process. The goal is to estimate the following AR(1) model: $$X_{t}=\alpha +\beta X_{t-1}+\epsilon_t.$$ From classical time series analysis, we know that ...
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Unit root testing in python

I have a time series (ploted below) which I want to test for the presence of unit root. The way I understand it, a time series with unit root have persistent, lasting effects after an unexpected ...
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Regarding explosive AR processes and stationarity

I often see this: If we have an $\text{AR}(1)$ process,$x_t=\phi x_{t-1}+w_t, $ $x_t$ is: stationary if $|\phi|<1$ an unit root (nonstationary) if $|\phi|=1$ explosive (and nonstationary) if $|\phi|...
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Missleading ADF Results

I am trying to use the adfuller test to determine whether the following timeseries is mean reverting (stationary) From the chart we can see that its obviously (at least my non-expert eye) non ...
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How to deal with different orders of integration between explained and explaining variables?

Is there a standard, or at least a valid, regression approach if you are trying to regress a dependent variable with a unit root against a set of stationary independent variables? I know I could ...
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Rewriting the characteristic polynomial using that the inverse roots are invertible

I am looking at an I(1) process for an AR(3) model: $$x_t=\theta_1x_{t-1}+\theta_2x_{t-2}+\theta_3x_{t-3}+\varepsilon_t$$ I rewrite it using the lag-operator: $$(1-\theta_1L-\theta_2L^2-\theta_3L^3)...
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How to derive the (partial) maximum likelihood estimator for a simple autoregressive model

I am trying to derive two maximum likelihood estimators which I have seen in a statistics book, but I am unable to derive them and would really like some help. It goes like this: Consider the simple ...
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Multiple comparison problems for Unit Root testing

Lets say I have a multivariate time series and I will run unit root tests for each variable to figure out if the variable is stationary and how many differencing are necessary to make it stationary. ...
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ADF & KPSS Test Reporting Series as I(19)

I am currently attempting to determine the order of integration for a nonstationary time series of the Federal Funds Rate (https://fred.stlouisfed.org/series/FEDFUNDS - Monthly, 1990 to 2004, 168 obs.)...
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How to Handle a Trend-Stationary Dep. Variable and Stationary and Non-Stationary (Unit-Root) Ind. Variables?

I am trying to determine the best way to proceed when one has a mix of stationary, trend-stationary and non-stationary variables with unit roots. My dependent variable $Y_t$ is a trend-stationary ...
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ARIMA(1, 1, 0) and explosive process

What is the difference between ARIMA(1, 1, 0) whose AR coefficient is smaller than 1 (e.g. 0.3) after being differenced and an explosive process whose AR coefficient is 1.3? Are these two identical? ...
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Zivot Andrews unit root test: Why is the p-value 0.162 nonsignificant but the test statistic -20 is highly significant?

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Do monthly sales of a retail shop have a unit-root?

Simple question to understand if by nature monthly sales from a retail shop on the high street are likely to have a unit-root in its timeseries or not?
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What happens if you difference a IMA?

When we over-difference a time serie, we are introducing units roots in its moving average component, hence obtaining an IMA. My question is: IMA is an integrated time serie, hence it has unit root ...
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The difference of a variable has a unit root, but the variable itself doesn't

I am trying to build an ARDL model with the six following variables: lnNO2,lnP,lnGDP,...
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Unit root test and cointegration

maybe someone can help me with my data. I analyse how macroeconomic indicators affect stock index. For this analysis I prefer VAR model.In my case data of all variables are non-stationary - I have ...
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Unit root test on growth variable?

When constructing an ADF unit root test, is it Ok that input data is a growth variable? I was thinking about when you compute growth rate $\left(\frac{Y_t - Y_{t-1}}{Y_{t-1}}\right)$ and when ...
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ARDL Unit root df

When applying ARDL I used Dickey-Fuller unit root test, and found variables integrated at I(1) and I(0). For running the ARDL model do I use my original data (not differenced) or do I use the ...
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Augmented Dickey–Fuller test, Test Model

I don't understand how we obtain the model that we test for a unit root in the ADF test. Let me explain better. ADF test is used to test if a time series has a unit-root. We assume that the Data ...
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Prove that OLS estimator of AR(1) coefficient for an AR(p) process with unit root converges to 1

Suppose that the DGP is an AR(p) with a unit root. When we fit, using OLS, an AR(1) model $x_t=\alpha x_{t-1}+u_t$ to the data, we get $\alpha=1$, indicating, correctly, that this is a unit root ...
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how to confirm a trend is deterministic or stochastic?

So I got this basic concept of deterministic and stochastic trend. $$\begin{align*}\text{Deterministic trend (DT)} & : y_t = \beta t +\varepsilon_t\\ \text{Stochastic trend (ST)} & : y_t = \...
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7 votes
2 answers
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Why, exactly, is a unit root a problem?

Suppose that we have observations $x_1,\dots,x_n$ for some process. We want to fit an AR(k) model to these observations. I do not understand why the naïve OLS approach to estimate our AR(k) ...
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Can a process with a unit root still be stationary?

It is quite clear, by expanding the inverse lag polynomial, that a process without unit roots will be weak-stationary. However, I'm not finding anywhere a proof of the other implication: to be weak ...
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Trend stationary data series

I have a trend stationary data series that does not have a unit root. The data are hourly with about five years of data. I have controlled for the apparent trend in the data using a series of binary ...
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Need help in panel data tests and steps

Greetings! I am analyzing the impact of climate change of wheat crop across 4 districts for the last 30 years. It is imperative to check the stationarity of the panels. I performed LLC and IPS tests. ...
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ADF test in R and Gretl - Why are the results different?

I am working on a time series-based study on the Czech Republic. I have macroeconomic data from 1993 to 2021. I tested my time series for stationarity using both R (function ...
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ECM long-run elasticity with one I(2) variable

I am estimating a 2SLS using a time series simultaneous equation ECM. The purpose is to estimate the price elasticity of electricity demand. Assume that both price and demand are I(1) variables, ...
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Does the AR-parameters sum to 1 violate stationarity?

Use AR(2) model as an example of my question: $$x_t = \phi_1 X_{t-1} + \phi_2 X_{t-2} + \varepsilon_t$$ if $\phi_1 = 0.5$ and $\phi_2 = 0.5$, then sum of AR-parameters equals to 1. Does it violate the ...
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Sign of first level differenced variable changes sign in regression

I have first level differenced an independent variable in order to combat unit root in panel data. My panel data consists of N = 487 and T = 1491. When comparing the coefficient of the original ...
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Investigating the presence of unit root in the following $X_t$

I am given a model and need to calculate the unit-root of $X_t$ but it seems that there is no unit-root. The model is given: $X_t = (x_{1t},x_{2t})'$ $$\Delta X_t = \alpha \beta ' X_{t-1} + \epsilon_t ...
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Interpreting Phillips Perron Test results using R

I performed pptest in R for corn variables and am unsure how to decipher the p-value and what that means for the hypothesis.
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Ng and Perron (2001) unit root test in R [closed]

I am looking for a function in R, which executes Ng and Perron (2001) unit-root test and computes all 4 test statistics like in eViews: So far I have read vignettes of packages bootUR and CADFtest, ...
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Process is NOT integrated but autocorrelated. Can I use linear regression models?

I'm carrying out a stationarity analysis over a process Y that looks like this: From the autocorrelation function the process doesn't look strictly stationary since the decay is quite low, even if it ...
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Understanding Dickey-Fuller-Result

I am currently looking into the dickey-fuller test but seem to not understand something fundamental. To understand the test better, I created 4 time series: cons, lin, r1 and r2. How these are ...
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5 votes
1 answer
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Deriving ADF unit root test form for the time series with quadratic deterministic trend

I have the following time series process $y_t $ $$\Delta y_t = \delta + \gamma t + \epsilon_t$$ where $e_t$ is white noise process with the variance of $\sigma^2$. I guess that whereas $\Delta y_t$ is ...
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Unit root testing in AR(2) simulated data

I'm teaching a course in time series analysis, and using a lot of R simulation. I've come across the following situation. Consider an AR(2) process with $x_t = x_{t-1} - x_{t-2}+w_t$ where $w_t$ is ...
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Does the Augmented Dickey-Fuller test only consider first-order effects?

The original Dickey-Fuller test provides a test for the H0: no unit root, so $\gamma \neq 0$ in a simple AR(1) model: $$X_t = \rho X_{t-1} + \varepsilon_t$$ $$ \Delta X_t = \gamma X_{t-1} + \...
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How to interpret the results of mean and variance to determine the stationarity of time series

After breaking the series into 3 parts and then compute mean and variance for each part. I need to understand what is the acceptable difference between the 3 parts to say this series is stationary? ...
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Order of integration for a time series with constant mean and increasing variance

I am trying to find the order of integration of a time series. I checked for stationarity using the ADF and KPSS tests. Both the tests indicated non-stationarity, so I differenced the series once and ...
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Testing by using KPSS

When use KPSS to test the stationarity of data series I get this warning .Does it affect the final results? ...
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4 votes
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The meaning of Test Unit Roots: unanswerable and answerable questions

In the section 15.4 of Hamilton's book Time Series Analysis ( 15.4: The meaning of Test Unit Roots) the author says: Although it might be very interesting to know whether a time series has a unit root,...
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Is it possible to describe a AR(1) process as stable in terms of a stability triangle?

Is it possible to describe AR(1) as stable in terms of a unit triangle as in the explanation below or not because a lack of a second degree characteristic polynomial? Consider the equation $$ \lambda^...
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purtest and psacf: How to test it in R?

I am working with balanced panel data (annual country level data: 37 countries and 11 years) and would like to test stationarity. I am using fixed effects for both country and year. Now, I am checking ...
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statsmodels.adfuller weird behaviour of usedlag value

I'm learning about time-series analysis and have two series on which I'm performing an Augmented Dickey-Fuller test in order to check for stationarity. I'm trying to understand why I get very ...
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Examining stationarity, mean and variance of time series

Hi guys, I encounter this question for a Business Forecasting module and I am very confused by it. Firstly, this looks like an autoregressive model of order 1. From the looks of it, the φ coefficient ...
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