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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 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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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 ...
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Autocorrelation Functions/Autocorrelograms and the assumption of Weak Stationarity?

Does it make sense to even speak of the autocorrelation function or of the autocorrelogram for a non-weakly stationary series? Anytime we see an autocorrelation function or an autocorrelogram can we ...
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Why Does Second Order Weak Stationarity include Statement on Covariances in addition to Statement on Mean and Variance?

A stochastic process is second order weakly stationary if all random variables have same mean (first moment), and same variance (second moment?), and covariances that are time-invariant (second moment ...
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Timeseries Analysis: Stationary Process - How can you predict

So I've read a lot on stationary processes and how we need to make sure the time series is stationary. Below I've linked the resources I've been going through. I get that you can then predict a ...
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Selection of dynamic regression model differencing based on cross-validation

I have got confused a lot. Suppose I have a time series, which is non-stationary with high probability (I cannot plot it, since there are 1000s of them). I need to fit regression with error being ...
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How to correctly transform seasonally (lagged) differenced long forecast?

I have problem to make proper inverse transformation of a seasonally differenced forecast. For example, I have the time series with the two periods ...
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Stationary time series input to ARMA-GARCH

I’m thoroughly confused as I have been using ARMA-GARCH to model a conditional mean and conditional variance so I can effectively remove them by taking the residuals and be left with an i.i.d process. ...
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Dealing with non-stationarity in panel data

I am using panel data for my analysis. My dependent variable is non-stationary while all my explanatory variables are stationary. Also, my dependent variable is a bounded variable (an index variable ...
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stationarity and fractional differencing

This is a methodology question. I would like to make the data stationary but not transform it "too much" (information loss), before it is fit for statistical/ML purposes such as regression or PCA. ...
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Modeling quarterly default rate (non stationary, autorregressive time series)

I am a student writing my thesis on default rate modeling. My major is finance, so I'm not really experienced in econometrics. I'm trying to create a model for quarterly corporate default rates (...
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Model identification from Eviews Dickey-Fuller test output

So our econometrics professor gave us the following Eviews outputs. First, it asked if it was the right decision to differentiate twice, which, according to what I interpret from the Eviews display, ...
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Determining whether a model with random walk errors is stationary

If we have a model like an AR(1) except the errors are a random walk (i.e. not iid), then is the model itself stationary? So the model is: $$ x_t=kx_{t-1}+\epsilon_t $$ where $k$ is constant and $0<...
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How to deal with both stationary and non-stationary time series

I'm a bit frustrated since the time series I am trying to analyse right now has definitely non-stationary curve but it's last values differ greatly from the mean making the time series stationary. ...
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Stationary vs. Trend-Stationary Time Series: Auto.Arima difference parameter

I have the following time series training_ts that looks like this: It appears to be stationary or "trend-stationary". When I analyze the stationarity using ...
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Proving whether a series is stationary

I want to prove whether the following equation is stationary or not: $$ x_t = (x_{t-1} + \epsilon_t) (1+k(x_{t-1}+\epsilon_{t})^2)^{-1/2} $$ Also written like: $$ x_t = (x_{t-1} + \epsilon_t) \frac{1}...
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How valuable is a regression result with serially correlated residuals?

I performed a linear regression with two time series variables. First, I checked stationarity of both time series via ADF/PP/KPSS tests and all three indicated non-stationarity. However the test of ...
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PCA's eigenvector with low variance, why people think they are 'noise'?

When we do a textbook PCA decomposition, get a series of eigenvalue $\lambda$ and eigenvector $v$ that fulfill: $ Av= \lambda v $ we can sort these eigenvalues (together with the corresponding eigen ...
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Making a time series stationary by demeaning

Tom's answer https://stats.stackexchange.com/a/7848/49691 to question How to make a time series stationary? highlights that: "If a series exhibits level shifts (ie change in intercept) the ...
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Johansen cointegration test doesn't have any value in the eigenvalue column

After running Johansen cointegration test on time series data of 9 variables covering 37 years using Stata, the following were the results (I have also included a screen capture of the results both ...
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temporal stationary process?

An extended question to this old post: Does applying ARMA-GARCH require stationarity? I suspect my data (long data with a few hundred thousand points) is not completely stationary. But some segments ...
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Unit Root - Real Interest Rate

I want to find out whether the real interest rate of different countries are non-stationary. The real interest rate is defined as the difference between the nominal interest rate and the inflation ...
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What to do if time series are non-stationary? [closed]

Data: I have a time series data of 2528 daily observations for OMXS.30 (Stokholm) closing price. The aim is to fit proper ARCH/GARCH models and use for forecast daily Value at Risk. Here is a plot of ...
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Stationarity restriction of a TGARCH process?

What is the stationarity/convergence restriction for a threshold GARCH model, TGARCH? I know that for a GARCH model: $\alpha+\beta<1$, but I'm guessing it's not that simple for a TGARCH model. ...
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How to relate roots of AR and MA to unit circle

I'm working on these problems and think I figured out most of the steps, but am stuck near the end as I don't understand how to relate my roots back to the unit circle in order to determine ...
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Solve for inequality of AR model

I was working through my textbook and found this problem that I got stuck at: Consider the AR(2) Model $$X_t = \phi_1X_{t-1}+\phi_2X_{t-2}+\epsilon_t$$ We can assume $\phi_2 > 0$, so the roots of ...
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Determine if AR(p) model is causal stationary or invertible

I was going through these problems and think I figured out most of them both, but am having some troubles at one of the last steps. The question is for each of the following models: Express them ...
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calculating mean and variance of non-stationary time dependent samples

I know normal procedure for calculating mean and variance which assumes that samples are iid.. I first want to know, how to calculate these two parameters if samples are time dependent but time ...
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ADF test showing stationary for a non stationary series

I am running an ADF test in R on the following series: This to me is clearly non-stationary, but when I run the ADF test: ...
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Is the time series $Y_t = \frac{1}{2} Y_{t-1} + \frac{1}{2} Y_{t-2} - \frac{1}{3} \epsilon_{t-1} + \epsilon_t$ stationary?

How can I tell if the series $Y_t = \frac{1}{2} Y_{t-1} + \frac{1}{2} Y_{t-2} - \frac{1}{3} \epsilon_{t-1} + \epsilon_t$ is stationary?
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Expected Value of an AR(1) process

I saw the answer on this post and got confused about a couple things in its explanation. Mainly, I am unsure of How the poster immediately knows the process $X_t = c+\phi_1 Y_{t-1} + \epsilon_t$ is ...
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Determining Polynomial Trend, Stationarity, and Covariance of a Process

I'm given $$ Y_t = p(t) + \epsilon_t $$ where $\epsilon_t$ is a stationary series with covariance $\gamma_t$. Also given $$ p(t) = \sum_{r=0}^kK_rt^r $$ where the $K$s are constants, for the ...
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Showing the covariance and autocorrelation functions of a stationary time series are symmetric around 0

I need to show that the covariance and autocorrelation functions of a stationary time series are symmetric around zero. From my understanding, this entails $$ \gamma(h) = \gamma(-h) $$ $$ \rho(h) = \...
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Mean Square Convergence of a Linear Process Defined in Terms of a Stationary Time Series

I am following Brockwell and Davis (Introduction to Time Series and Forecasting, 3rd Edition). Chapter 2, Proposition 2.2.1 claims the following. If $\{Y_t\}$ is a stationary time series with mean ...
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Difference between independence and stationarity tests in time series

This is meant to be a general question, aiming to clarify the topic for a beginner in TSA, as I haven't found any clear introductory explaination yet. Suppose I am working with some data which ...
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Covariance stationarity - Example

I am studying time series by myself. I've just faced this process: $y_t = \epsilon_t \epsilon_{t-1}$, where $\epsilon_t$ is Gaussian white noise, with zero mean and variance equal to one. Is this ...
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State space models with non stationary/unit root factors

state space model , I am trying to implement is as follows $$ Y_t= CY + FF* X_t + Ve_t$$ $$(X_t-m0)= GG (X_{t-1}-m0) +W\eta_t$$ I am enforcing GG to be to be diagonal for the base case. I am getting ...
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Example of a first order stationary process which is not second order stationary

I need an example of a process that is stationary to order 1 but not to order 2.
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How to show that second order stationarity implies first order stationarity?

If a process is second order stationary i.e. joint pdf is independent of absolute time, how can it be shown that it is first order stationary as well i.e. First order pdf is independent of time origin?...
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generalized additive model with nonstaionary dependent variable (time-series analysis)

I am trying to use gam model to analyze pm2.5 effect on death. However, the dependent variable 'death number' is not stationary. (the data is Time Series data) What I know is for Time Series data I ...
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Why is this process not WSS?

Reading Hayes's Statistical Digital Signal Processing and Modeling section 3.3.4, they define Wide Sense Stationary (WSS) and provide a few examples in the text (on the top of page 83 in my version) ...
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Johansen test - full rank however variables are I(1)

I have a situation in which my Johansen cointegration test results indicate a full rank, rejecting both that there is no cointegrating vector as well as that there is at most one. I am working with ...
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How Enforce inevitability and Enforce stationarity works in time series?

statsmodels.tsa.statespace.sarimax.SARIMAX () the function here, have 2 parameters that are "enforce_stationarity" and "enforce_invertibility " How do they enforce these 2 properties after ...