A stationary process (or time series) is one whose joint distribution is constant over time. A weakly stationary process or series is one whose mean and covariance function (variance and autocorrelation function) are constant over time.

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Modelling stationary and integrated time series in one system

I am currently investigating commodities and their impact on the oil price. I have 8 variables of different stationarities $y$ = dependent variable (oil price) is non-stationary I(1); three ...
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A time series which cannot be made stationary

I have a series of data points, of days between successive earthquakes. I am trying to model time between earthquakes, in order to predict waiting times. Using Minitab I have tried differencing, ...
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Rules of integrated series and balanced regressions

Background There are various rules of linear combinations of integrated series. Let's just consider the $I(0)$ and $I(1)$ cases. For example, if $x_{t} \sim I(1)$, $y_{t} \sim I(0)$, then $ax_{t} + ...
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Vector autoregression for mix of stationary and nonstationary variables

I am currently investigating the impact of certain indicators such as GDP and inflation on the stock market. However some of my variables are non-stationary and some stationary in levels. All ...
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16 views

Example and simulated path of strict(ly) stationary process

There are many question on stationary process but very few related to strict stationary process. I am just looking for an example and simulated path of strict stationary process and how strict ...
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23 views

Augmented Dickey-Fuller test and Differencing (R)

I'm using the ADF test to check for stationarity of two variables, using the ndiffs function in R. ...
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35 views

Is there an unambiguous stationarity test for time series?

It seems to me that the time series plot, the correlograms (ACS, PACS) and the "autocorrelation check for residuals test" can all be subject to interpretation. (I am using SAS 9.4) Is there an ...
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66 views

SARIMA models: Good fit and stationarity

I'm doing a study where I try to fit two different data series to two different SARIMA models. Both series includes trends and seasonality (daily and quartely, I think) and I'm having problems to make ...
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23 views

How to interpret linear filter formula

I am taking my first course in time-series analysis, and I recently encountered the so-called linear filter for the first time. I thought I could just skip this section. Apparantly though, this ...
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126 views

Forecasting technique for daily data with monthly and day of week seasonality

I have daily data for 3 years. This sales data is of seasonal nature as business has spikes and downfall by month. Also, sales differ by each day of the week. for example, monday in general in a month ...
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132 views

Convert double differenced forecast into actual value

I have already read Time Series Forecast: Convert differenced forecast back to before difference level and How to "undifference" a time series variable None of these unfortunately gives ...
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65 views

Why can correlograms indicate non-stationarity?

I'm reading about correlograms, and how they can be used to detect non-stationarity. Supposedly, if the autocorrelation constant is significant, and/or declines slowly, we would deem the time series ...
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36 views

Interpretation of ADF(Augmented Dickey-Fuller) and KPSS (Kwiatkowski–Phillips–Schmidt–Shin) tests for time series

Can anyone please clarify for me the differences between ADF (Augmented Dickey-Fuller) and KPSS (Kwiatkowski–Phillips–Schmidt–Shin) tests in testing the stationarity of a time series? I tested my ...
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56 views

Interpretation of ADF(Augmented Dickey-Fuller) and KPSS(Kwiatkowski–Phillips–Schmidt–Shin) tests for time series [duplicate]

Can anyone please clarify for me the differences between ADF(Augmented Dickey-Fuller) and KPSS (Kwiatkowski–Phillips–Schmidt–Shin) tests in testing the stationarity of a time series? I tested my time ...
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18 views

deterministic time trend vs stationarity

Sorry for the newbie inquiry but I'm having a little trouble making sense of stationarity and how a the presence of a time trend impacts this. I'm working on a model for operating margins and as a ...
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28 views

How should I run a regression on cointegrated variables?

The variables are non-stationary for levels but become stationary for first differences, in other words they are $I(1)$. Also they have been already further checked for cointegration using Johansen ...
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54 views

When is an “ARIMA process” stationary?

I'm sorry if this is a duplicate, but I can't seem to find the answer to this. If $Z_t$ is a white noise process and $X_t$ satisfies $$ \phi(B) X_t = \theta(B) Z_t $$ (where $B$ is the lag ...
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Stress testing modeling

I try to make an analysis of time series to develop an econometrics model and then estimate the prob. of default with a stress scenario. About the model, i have some question about the step on which ...
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25 views

Can someone explain Metropolis Hastings algorithm with simplest possible example? [duplicate]

I am just beginning to learn about Metropolis Hastings algorithm and MCMC techniques. I have a basic understanding of Markov chains and stationary distributions and need for the Metropolis Hasting but ...
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12 views

Second Differences and Quadratic Trends

I know that any second differenced model has a quadratic trend, but how do you know if its best modelled with stationary or non stationary deviations around this?
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37 views

How will ARMA be affected if a non-stationary time series is not made stationary?

Which components or statistics of ARMA will be affected - and how - if the stationarity condition is violated?
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I have two time series recorded on a weekly basis that have a quadratic trend. Should I difference the series twice with lag 1?

I have two time series on google Trends weekly data for the search of the terms 'machine learning' and 'data science'. The data can be found here ...
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48 views

Stationarity Tests in R, checking mean, variance and covariance

This question was first asked on Stackoverflow, but as no one was able to answer, I wanted to ask it here. The question is: is there a test for stationarity that is both able to identify ...
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If time series $A, B$ and $A, C$ are cointegrated, is $B, C$ also cointegrated? And with similar confidence?

While I am interested in the general case, let's consider the case of three series. So, suppose time series $A, B, C$. If I can show that $A, B$ and $A, C$ are cointegrated with a test statistic $\ge ...
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Bootstrapped p-values calculation

I'm testing whether technical trading rules can deliver superior returns in contrast to a benchmark, the risk free rate. As performance measurement, denoted by φ, I use the annualized Sharpe ratio. I ...
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Which test to apply if variables are a mixture of I(0), I(1) and I(2)?

I have annual data with 25 observations. 1 dependent variable and 4 independent variables. I used to methods to test stationarity. First I de-trended data and used ADF test and found out that all my ...
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Is AR(1)-ARCH(1) covariance stationary?

Say I have the following model: $$ y_t = c+\phi y_{t-1} +\epsilon_t \,, \epsilon_t|\Omega_{t-1} \tilde{} WN(0,\sigma_t^2 ) $$ $$ \sigma_t^2=\alpha_0+\alpha_1\epsilon_{t-1}^2 $$ $$ |\phi|<1 \,, ...
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75 views

Conflicting results from $p$-value and $t$-value: Should I ignore the $p$-value in the ADF test?

I'm pretty new to the concepts of stationarity/cointegration. I am using the "urca" package in "Rstudio" to run my tests. I have been trying to run cointegration tests, but the frustrating thing is ...
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51 views

Stationary vs Stability

I am searching for an example of an unstable VAR($p$) process (its reverse characteristic polynomial has no roots inside and on the complex unit circle) which is stationary. I come up with this ...
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53 views

Stationarity of MA models in time series

I am aware that MA models are always stationary, however I was wondering if this meant that they are always weakly stationary? Can a MA model being strongly stationary or are they always only weakly ...
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83 views

Unit Root Test - Order of Integration (Johansen)

Let's assume we have 2 variables and test each of them for a unit root with the ADF test. When plotting the data, we can see that it has some up/down movements but is overall clearly trending upwards. ...
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Solution of GARCH model

We consider the GARCH model $$ \begin{align} \varepsilon_t &= \sigma_ {t} Z_t, \\ \sigma_t^2 &= \omega + \alpha \varepsilon_ {t-1}^2 + \beta \sigma_{t-1}^2. \end{align} $$ Questions: ...
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Stable VAR($p$) procress: Is there an easy way to do this?

Assume a $K$-dimensional VAR($p$) process given by $$y_t=\nu+A_1y_{t-1}+\ldots+A_py_{t-p}+u_t$$ This process is called stable if the roots of the reverse characteristic polynomial are bigger than 1 in ...
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38 views

Proving a time series is stationary

I am trying to prove a time series is stationary and I think I am misunderstanding something. My question is $$\ X_t=U_1 sin(yt) + U_2cos(yt) $$ with $E[U_1]=E[U_2]=0$ and ...
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28 views

Checking if a time series is weak-sense stationary

I just learned that a time series is called weak-sense stationary if: 1 $\mathbb{E}[X_t] = \mu$ is independent of $t$. 2 $\operatorname{Cov}(X_{t+h},X_t)$ is independent of $t$ for each $h$. I am ...
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What's the name for a time series with constant mean?

Consider a random process $\{X_t\}$ for which the mean $\mathbb{E}(X_t)$ exists, and is constant, for all times $t$, i.e. $\mathbb{E}(X_t)=\mathbb{E}(X_{t+\tau})$ for all times $t$ and time shifts (or ...
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Can a time series be stationary if the formula for the mean level depends on $t$?

I just started working on time series with the book from Brockwell and Davis. I'm still not that familiar with stationary time series. The book says that a time series is stationary if: 1 ...
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xtunitroot fisher interpretation

I would like to test if one of my variables is stationary (unemployment) with the xtunitroot fisher test. First, I do NOT include "trend" in the command and then I include "trend" - the results ...
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Why is the kernel - $(1-|\gamma_i-\gamma_j|) \ k_{SE}$ - non-stationary?

Stationary kernel which only depends upon the lag between the inputs and not on the absolute values of the inputs. Having read this guideline, I was wondering why the following kernel mentioned ...
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Problems with taking the first difference of a stationary series

Suppose you have sufficient observations for running a time-series regression, but you do not have sufficient observations to accurately test for the stationarity of the residuals. If you do take the ...
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how does Augmented Dickey Fuller results help to make the data stationary?

I have done an Augmented Dickey Fuller test on a variable without differencing it (or without making any transformations). Attached is my output. Now my question is how will this result (considering ...
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Cointegration - stationary time series in R

I have 2 time series that are stationary (according to ADF, PP, KPSS tests) so I cannot use the Johansen cointegration test. Can you recommend what model should I use in this case? I will try to ...
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1answer
110 views

Is a model with a sine wave time-series stationary?

Today I was introduced to the definition of stationarity being that the marginal distribution of a process does not change over time, and the mean and variance remain constant over time. I questioned ...
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22 views

Need a deterministic nonstationary function

Can anyone give me an example of a 1D deterministic function that is nonstationary? I want to try fitting a stationary Gaussian Process to data generated from a nonstationary function in hopes of ...
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Which ARIMA parameters fit my data?

I have the following data; ...
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Stationarity of time series, what if we reject stationarity?

My question is simple, the answer probably won't be. If I reject stationarity of an ARIMA model, what are the consequences regarding Confidence intervals for parameters Prediction of future values ...
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Panel estimation - all variables required to be stationary?

I am using fixed and Tobit estimator using panel data. As a result of the panel stationary test, one of the independent variables seems to have a unit root though its first difference is stationary. I ...
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Why is this time-series stationary?

I am using python for time-series analysis of count data and came across a problem where I have a time-series that to me looks non-stationary but the Augmented Dickey-Fuller test (implemented in ...
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Autocovariance Estimation and Stationary Processes

I am going to work on a project involving time series and therefore I am trying to understand some basic definitions. I am currently trying to grasp the autocovariance estimation procedure. When we ...