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31 votes

What is the difference between a stationary test and a unit root test?

The Concepts and examples of Unit-root tests and stationarity tests Concept of Unit-root tests: Null hypothesis: Unit-root Alternative hypothesis: Process has root outside the unit circle, which is ...
Ferdi's user avatar
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22 votes
Accepted

How is the augmented Dickey–Fuller test (ADF) table of critical values calculated?

I am not sure an easy answer is possible here. As can be found in many textbooks, the limiting null distribution of the "Dickey-Fuller t-statistic" is that of a nonstandard random variable which may ...
Christoph Hanck's user avatar
14 votes

What is the difference between a stationary test and a unit root test?

I don't totally agree with the accepted answer: the null hypothesis of the KPSS test is not stationarity, but trend stationarity, which is quite a different concept. To summarize: KPSS test: Null ...
Jonathan Zimmermann's user avatar
13 votes

Why is the dickey fuller test different from a simple t-test

You are right that the test statistic is just a standard t-statistic. It, however, follows a different null distribution, i.e., using critical values from the t or normal distribution would lead to ...
Christoph Hanck's user avatar
12 votes

Contradictory results of ADF and KPSS unit root tests

Please have a look at my answer to the following question. What is the difference between a stationary test and a unit root test? Here is the most important part of the answer: If you have a time ...
Ferdi's user avatar
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9 votes
Accepted

OLS - Non stationary variables but stationary residuals - is this OK or not?

...the dependent variable (Y) and the independent variables (X1, X2, X3, ...) are non-stationary. But the residuals are found to be stationary. Does this mean my regression is OK?... This suggests ...
Michael's user avatar
  • 3,328
7 votes

Unit root tests ambiguous - is time series stationary?

ADF and PP tests address a specific form of nonstationarity, i.e. unit-root nonstationarity. Apparently, you reject that form. However, unit-root nonstationarity it is not the only possible form of ...
Richard Hardy's user avatar
7 votes

Must a time series be stationary if it has no unit root?

No, roots outside the unit circle gives asymptotic stationarity, not strict stationarity: For a standard ARMA time-series model, the recursive formula for a model locks in the autocorrelation of the ...
Ben's user avatar
  • 126k
7 votes

Roots within the unit circle and non-stationarity

Not restricted to time-series analysis, characteristic equations (CE) are used in many applications or problems, such as differential/difference equation solving, signal processing, control systems ...
gunes's user avatar
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7 votes

Why, exactly, is a unit root a problem?

Adapting this from Bauwens & Lubrano (1999), the part of the statistical procedure that "breaks down" in the presence of unit roots is asymptotic normality of the (OLS) estimator. For a ...
Durden's user avatar
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6 votes
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Manual calculation of ARIMA(1,1,0) forecast

The ARIMA(1,1,0) model is defined as follows: $$ (y_t - y_{t-1}) = \phi (y_{t-1} - y_{t-2}) + \varepsilon_t \,, \quad \varepsilon_t \sim NID(0, \sigma^2) \,. $$ The one-step ahead forecast is then (...
javlacalle's user avatar
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6 votes
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what means to be outside unit circle?

The roots in this case are roots of a polynomial, and they can be (and often are) complex numbers. That means they have coordinates, in this case called the real part and the imaginary part. As an ...
kjetil b halvorsen's user avatar
6 votes
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Dickey-Fuller test significant => series stationary?

You are not misunderstanding anything as far as I can tell. You are presenting a solid story with logical supporting arguments. I could stop here and I think your question would be answered, but let ...
Richard Hardy's user avatar
6 votes

Influence of Seasonality on Unit root tests

A few days after I posted this question, I attempted simulations to test my intuition that strong seasonality can hide the underlying stochastic trend in the usual ADF test. Consider the following ...
Dayne's user avatar
  • 2,641
6 votes
Accepted

Trend stationary data series

A linear trend term has no unit root. A unit root arises in models like $y_t=y_{t-1}+\epsilon_t$, where the characteristic polynomial $1-z=0$ has solution 1. A linear trend model correspondings to ...
Christoph Hanck's user avatar
5 votes

Dickey-Fuller unit root test with no trend and supressed constant in Stata

As for your first question, yes, the applied researcher should think about the correct specification of the deterministic part. It is definitely not appropriate (something you did not suggest, to be ...
Christoph Hanck's user avatar
5 votes

Criticism of augmented Dickey-Fuller test

There are various criticisms, some of which, in my opinion, are more pertinent than others: Lack of power: When testing $\rho=1$ against $|\rho|<1$, it is argued that many macroeconomic time ...
Christoph Hanck's user avatar
5 votes

Choosing the maximum lag length in the augmented Dickey-Fuller test

This is can be a tricky one. These Zivot Notes discuss a slightly more advanced way to select lags for the ADF. That being said, it is good to remember that purpose of including lags is to control ...
Jacob H's user avatar
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5 votes

Interpreting R's ur.df (Dickey-Fuller unit root test) results

I found Jeramy's answer pretty easy to follow, but constantly found myself trying to walk through the logic correctly and making mistakes. I coded up an R function that interprets each of the three ...
Hank Roark's user avatar
5 votes

Are seasonal differencing and polynomial trends interchangeable?

I do not have general answers, but here are some thoughts that will hopefully be helpful. First, can a repeated seasonal differencing filter also remove a polynomial trend? In which conditions? I ...
Richard Hardy's user avatar
5 votes

Can Dickey-Fuller be used if the residuals are non-normal?

Yes, that is not a necessary condition. Recall that all we know about the null distribution of the Dickey-Fuller test is its asymptotic representation (although the literature of course considers many ...
Christoph Hanck's user avatar
5 votes

Trend stationary and difference stationary simultaneously

It certainly is possible that we may have a process that has both a deterministic time trend and a unit root, for example: $$y_t = \delta t + y_{t-1} + u_t \tag{1}$$ with $u_t$ being white noise, $...
Alecos Papadopoulos's user avatar
5 votes
Accepted

How to show the inconsistency of the OLS estimator for unit-root AR(1) processes by simulation?

You will not be able to show this result (by simulation or otherwise) because it does not hold. When the true AR parameter is unity, the OLS estimator is superconsistent, not inconsistent. See for ...
Chris Haug's user avatar
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5 votes
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What is an integrated time series?

Consider the first difference $\Delta u_t$ of a linear process (a fairly general way of stating that something does not have a unit root) $u_t=\sum_{j=0}^\infty\psi_j\epsilon_{t-j}$ with $\psi_0=1$ ...
Christoph Hanck's user avatar
5 votes

Normalization of time series data

You seem to be confusing two separate concepts: Unit root tests are used to determine whether a specific type of time series model, namely auto-regressive models, are stationary or not. It is only a ...
Skander H.'s user avatar
5 votes
Accepted

Making an AR(3) model weakly stationary

You can't make your process stationary. It is simply not stationary, because it has a unit root. However, there is a related process which has no unit root, and is stationary. Your process can be ...
Chris Haug's user avatar
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4 votes

Interpreting R's ur.df (Dickey-Fuller unit root test) results

More info in Roger Perman's lecture notes on unit root tests See also table 4.2 in Enders, Applied Econometric Time Series (4e), which summarizes the different hypotheses to which these test ...
user3096626's user avatar
4 votes
Accepted

Understanding Fisher unit root test in Stata: trend, demean

Are you using the regular dfuller or the xtunitroot? Fisher has nothing to do (at least directly) with the regular ...
Richard Hardy's user avatar
4 votes

Interpreting adfTest results in R

Both tests reject their respective null hypotheses (hint: low $p$-value associated with the test statistic points to a rejection) in favour of the alternative. Alternative hypothesis in both cases ...
Richard Hardy's user avatar
4 votes
Accepted

Granger Causality test pre-conditions

If you have two time series, then the univariate unit-root test results coincide nicely with the cointegration test results. Both indicate that the series are stationary. Moreoever, this is as ...
Richard Hardy's user avatar

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