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Questions tagged [stationarity]

A strictly stationary process (or time series) is one whose joint distribution is constant over time shifts. A weakly stationary (or covariance stationary) process or series is one whose mean and covariance function (variance and autocorrelation function) do not change over time.

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Is the uniform distribution stationary? [on hold]

Is the uniform distribution stationary or non-stationary? The normal distribution is stationary in all cases? The normal distribution is non-stationary in which cases? or in all cases else normal? -...
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Stationarity of ADL(p, q) with heteroskedasticity

Suppose I have the model $$y_t = \alpha_0 + \alpha_1 y_{t-1} + ... + \alpha_p y_{t-p} + \beta_0 x_t + ... + \beta_q x_{t-q} + \epsilon_t,$$ where $\{x_t\}$ is a stationary process and $\epsilon_t$ has ...
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Confused: The data length of covariations in a nonstationary GEV

I have established a non-stationary GEV model and expressed the location parameter μ(x) as a function of two covariates (x1, x2) to reflect changing conditions, while the scale and shape parameters ...
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Existence of weakly stationary process for given mean and covariance

I know that if a process is weakly stationary, the mean of the process will be time-independent and the covariance will be a function of the time difference. My question: if I have a time-...
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Do measurable maps preserve stationary ergodicity?

In a recent effort to establish stationary ergodicity for a certain stochastic process, I just happened to come across a statement, which I find to be little bit confounding. Given two measurable ...
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How should I select the lag for Zivot Adrews Test (ur.za) in R? [closed]

What is the best way to implement the lag order selection of the Zivot & Andrews Test in R?
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Biased estimates of Hurst exponent in R/S analysis

I've used the standard R/S algorithm for estimating the Hurst exponent in Mathematica*, and tested it on fBm and fGn for $H\in\{0.05,0.1,\ldots,0.95\}$, generating 1000 time series for each $H$. The ...
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Do we need ergodic-stationarity of the response variable in OLS spline regression?

I was wondering if we need the response variable to be ergodic stationarity when estimating an OLS spline regression. My intuition tells me that it's not needed but I would like to have a confirmation ...
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Bayesian hierarchical temporal models

I have been asked recently to transform an already existing Bayesian hierarchical model into an non-stationary model by making the input and the latent variable time dependent(or non stationary). let ...
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Stationarity test for autocorrelation

can we use unit root test of residual to detect autocorrelation in a time series model? Are stationarity of the residual means there is no autocorrelation?
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How can I choose correct variant of ADF test?

Sorry for this question, but I am not sure in this problem. Can I make decision according to AIC, BIC and so on?
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Evaluating if time series need differencing

I am a total beginner with time series analysis. I use R. I understand that time series data need to be stationary for analyses like cross-correlation or modeling. I am, however, struggling with ...
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Am I doing the correct data transformation for Granger causality tests

I have seven sets of time series, below is my process flow, am I doing the correct thing here? especially step 4. Raw data transform and test stationary with unit root test (ADF), with level, first ...
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strategy for using stationarity tests

I have a long (~20,000 points) time series that I tested for stationarity. I am following this strategy: I started by plotting the series and determining visually whether a drift / trend exists or ...
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Can discrete time series be non-stationary?

We all know about time series that grow over time, but it seems like we only ever see continuous values such as the plot shown below. But is there such thing as a time series that might start out ...
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Is it possible to extend the near-optimal finite horizon average reward to infinite horizon discounted reward in RL?

Is it possible to extend the near-optimal finite horizon average reward to near-optimal infinite horizon discounted reward in RL (for example, in the context of Q-learning)? If yes, how? I believe ...
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VAR Model using Stata

I'm relatively new to the VAR model and have been using Sean Becketti's 'Introduction to Time Series Using State' as reference and wanted to check if I am on the right track. As of now, I have 5 ...
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Why do we add $\delta_i \Delta Y_{t-i}$ in order to make an ADF test?

Let's say we have the ADF test for no constant and no deterministic trend. So the regression model is $$\Delta Y_t=\delta Y_{t-1}+\sum_{i=1}^{p-1} \delta_i \Delta Y_{t-i}+Z_t$$ according to https://en....
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R auto.arima with transformed covariates

I have a non-stationary output time-series (oil prices) that is to be forecasted with 20 different input time series. The series are all non-stationary. I am considering two approaches. Approach 1) ...
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What is local stationarity and how to detect it in time series

What is local stationarity time series and how to detect? and what is the difference between it and between global stationarity?
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Why would you introduce a constant in a moving average model, when you already have the option of differencing?

In the online forecasting book of Hyndman (https://otexts.com/fpp2/MA.html) firstly the use of differencing is explained. After that he shows the formula for a moving average model: $$y_t = c + \...
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Verify if $y_t=\phi y_{t-1}+\epsilon_t$ is second order stationary

Here$\;\{\epsilon_t\}\;$is a sequence of identically distributed Student t random variables with d.f.=2. Is this process second order stationary? I am confused since $\;Var(\epsilon_t)\;$ tends to ...
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nonlinear regression with time series error

I have a question about data analysis. I fitted my data to non linear regression by using nls function in R. Then I plot the residuals. The residuals are non ...
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Is this Time Series stationary?

The adf test makes me think it is indeed stationary, since p<0.01, we can reject the null hypothesis and so it would be reasonable to think it is stationary, but that peak is kind of out of place ...
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How can we generate stationary and non-stationary time series data in python?

I need custom simulated stationary and non-stationary(trending) time series data for one of my python projects. I searched all the web for it but didn't find anything for python. Then I came across ...
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Stationarity with some disruptive values

I have a time series with a huge disruption in a pair of its values, which if I understand well makes squared residuals not independent. The ADF test suggests the process is stationary, and there ...
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ARCH testing and Stationarity

I have a time series y and I need to build the "best" autoregressive model for it i.e. y(q*). What I do: 1. Start with y(1) and test for serial correlation in the errors until I find an AR lag -lets ...
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Why is this non-stationary series diagnosed as stationary by adfuller test in python?

I simulated a time series model x_t = -1.1x_t-1+e_t in python. By visually checking the time series plot, I think that it should be non-stationary. Because the variance increases as time going by. ...
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Is a test of stationarity important before fitting GARCH?

Is it important (and if it is why?) to test for stationarity of a time series before fitting a GARCH model/ ARMA-GARCH to a financial time series?
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What is an overall procedure for cointegration test?

I'm working on a set of macroeconomic variables form 1992M01 to 201407. They are PPI, CPI, industrial production, stock price index and exchange rate. I know that I should run a cointegration test for ...
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ARMA process forecasts and maximum likelihood parameters

I have some trouble understanding the forecasting/inference process of ARMA models. From Hamilton (which I am reading now), we can obtain forecasts at $Y$ from any linear process with r.v. values $X$...
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Stationary and ergodic r.v: relation between error and independent variable

In time series often hold the condition that a r.v. is stationary and ergodic, allowing the application of the law of large number. If in a model as: Y= a + bX + u ...
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Removing Variance in Time Series After Applying Log Transformation

I'm trying to look at natural gas prices from 2003-2018. The issue is after applying log transformation and then diffrencing data by 1, I still seem to get an increase in variance from mid 2014-2018. ...
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VAR/VEC models: checking stationarity during cross-validation

I am attempting to derive a single multivariate/vector autoregressive (VAR) model from a large dataset (6-minutes sampled at 250Hz in total w/ 50 vars) using cross-validation (CV) to optimize model-...
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1answer
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VAR model for a stationary and a nonstationary series

I have a stationary and a non stationary time series. For estimating a VAR model, both time series should be differenced or only the second?
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Is this interpretation of the results of unit-root and stationarity tests considered acceptable?

I am studying the following time series (annual freq., sample 1960-2020): Casual inspection reveals (to my eyes) the elephant in the room: there's an outlier around 1974 after 1994 the variability ...
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Stationarity tests

It is generally accepted that ADF is the most common test to check whether a time series has a trend or not. My question is: What if we just regressed the values of the dependend variable on their ...
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Generate Gaussian process with squared exponential covariance function

In a (stationary) Gaussian Process, values which are closeby are more similar than values far away from each other. The correlation function tends to zero as distance increases. Often, one models the ...
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Stationary processes that do not satisfy Gordin's central limit theorem

We are doing an assignment for our Advanced Econometrics course for which we are trying to illustrate Gordin's Central Limit Theorem by simulation. We used an AR(1) process to show that if the ...
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1answer
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Time series analysis via generalized additive models: model assumptions and stationarity

I have settled on building a generalized additive mixed model using mgcv::gamm, on data and for purposes I have described in more detail here. In a nutshell, I want ...
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Stationarity in a ARIMA Model with External Regressors

I have a question about when we apply an ARIMA model with external regressors. Just a quick note of my understanding on the topic, an ARIMA model with external regressors is when we apply a Regression ...
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1answer
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Is standard Brownian motion (AKA a Wiener process) weakly or strictly stationary?

Question Let $B(t)$ be a standard Brownian motion (AKA a Wiener process). Is $B(t)$ weakly or strictly stationary, particularly as defined here? My Thoughts We know, by definition, that its ...
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1answer
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Does forecasting with ARIMA lose non-stationary components?

Suppose I have a time series $Y$. I have read that an ARIMA model consists as an ARMA model of a stationarized version of $Y$. If I try to predict $n$ ticks ahead with an ARIMA forecast model (with $...
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How to test for constant mean over time with time series data

I have a data set that looks like it may not be stationary. As a test, I ran a linear regression of the data against an x variable that was an index of time, 1:400 periods. I saw that the slope ...
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Law of Large Numbers for Covariance Stationary Processes… Difference and Relationship between LLN and Ergodicity

We have a covariance stationary time series. We must assume that the time series was produced by an ergodic process if we are to make the bridge between the realization of the time series that we ...
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Spatial-grid data over time: check whether time series grids are temporally stationary

I have time-series spatial grid data, represented either as matrix or rasters. And I would like to assess whether they are temporally stationary or not. Do you know any test or R package that could ...
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Converting nonstationary process

I am looking at the hourly load demand profile for an entire year. See image below. However, from my understanding of stationarity, this process is non stationary as it has seasonality as well as a ...
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Fitting a NN model on one-to-many function

Given $f(x) = y$ as a surjective (many-to-one) function, we know that $f^{-1} (y) = x$ is a one-to-many mapping for function $f^{-1}$. In my application, $x$ is a spatial data represented by a 2D ...
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Thomas Sargent's intuition as to why every covariance stationary series has an infinite-order Wold representation

In his classic book "Time Series Analysis", James Hamilton references Thomas Sargent (["Dynamic Macroeconomic Theory"], 1987, pp. 286-290) as a "nice sketch of the intuition behind this result [Wold ...
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What guarantees the existence of a finite representation of the Wold decomposition? Mechanics and Intuition

Every covariance stationary process can be written as a linear, infinite distributed lag of white noise. In other words, every covariance stationary process has a Wold representation. Then we go on to ...