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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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Interpretation of regression coefficients: second-order differencing

Main Question: How are the coefficients of the second-order differenced explanatory variables to be interpreted? (See the attached screenshot of my result.) Analysis framework: I examine the ...
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Correlation of Aggregate Data on a monthly to quarterly basis

I am working with a data set of 42 countries of monthly migration. I want to extract factors using PCA, and find non stationary errors of my model, so I am working in first difference. However there ...
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Interrupted time series analysis of data with deterministic trend and seasonality

I am trying to evaluate the impact of an intervention on a selected outcome variable using interrupted time series data. I have aggregated a five-year data into monthly values to create a data-set 0f ...
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Understanding the infinite sum of random variables

I am doing a course on time series analysis, and am struggling with this definition: We call a weakly stationary process $\{X_t\}$ invertible with respect to a white noise $\{\epsilon_t\}$ if ...
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White noise stricly stationary proof

How do you show that the white noise process is strictly stationary? Let's consider the i.i.d. white noise process $a_t$: \begin{align} E[a_t] &= 0\\ Var[a_t] &= \sigma_a^2 \end{align} The ...
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Stochastic process $X_t=\varepsilon_t\varepsilon_{t-1}$ is not white noise, where $\varepsilon_t \stackrel{iid}{\sim} N(0,\sigma^2)$

Let $\{\varepsilon_t\}$ satisfy $\varepsilon\stackrel{iid}{\sim} N(0,\sigma^2)$ Let $X_t$ be defined as $$X_t=\varepsilon_t\varepsilon_{t-1}$$ Is $\{X_t\}$ stationary? Is it white noise? ...
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Generate a non stationary series by differencing a stationary series?

Is this possible? If so, under what circumstances does this happen?
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Stationarity of AR(p) process

I'm looking for a proof for the stationarity of an AR(p) process, I know a stationary process $Yt$ must fulfill the following conditions: $(1)$-$E(Y_t)=m$ for all values of $t$. $(2)$-$Var(Y_t)=\...
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Spikes impact on time series stationarity

I have a demand time series that are highly impacted by promotion (spikes). Do spikes violate the assumptions of stationarity? Can we apply the KPSS test or ADF to test whether the series is ...
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differencing in sARIMA models

Im currently trying to fit unemployment data to a sARIMA model. Unemployment has usual yearly seasonal trends so a seasonal difference is given. Log transformation is applied to minimize the errors ...
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Stationary time series but the mean is not constant

I am trying to do time series analysis in R (I am new to the concepts). So first I tried t geenrate my own synthetic data and check if I am well using the functions in R. I generated a time seres ...
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Why does an AR(p) process require the largest eigenroot to be <1/ characteristic roots lie outsied the unit circle to be stationary?

For reference, this is the paragraph in Wikipedia I'm struggling with. https://en.wikipedia.org/wiki/Autoregressive_model I do understand the simpler proof for under which conditions an AR(1) ...
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VARX model on non -stationary timeseries

I am going through the lecture notes on VARX by Dr Tsay Pg 11-22 Link Plot of endogenous and exogenous series shows that these are not stationary. Pg 15 shows lag 2 VAR model is fit at level. Not ...
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return level in non-stationary case using rlevd function

I am fitting a GP distribution to a non-stationary series with the fevd function. However, the results of the return periods (...
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Existence of a common term if time series are pairwise cointegrated

Let's suppose that we have $n$ time series that are integrated of order one: $y_t^i\sim I(1)$ for $i=1, 2, \dots, n$ The difference between any two series is stationary: $y_t^i-y_t^j\sim I(0)$ for $...
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Roots of lag polynomial and stationarity

I encountered the following theorem: If all the roots of the AR lag polynomial of an ARMA process lie outside the unit circle, the process is stationary. I noticed that this is only an ...
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Distribution of sample variance for dependent variables

In several references I have found distribution of sample mean $\hat{\mu}$ for dependent data. Here, for example. But can someone give expression (preferably with derivation) for distribution of ...
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stationary vs. non-stationary GARCH process

I estimated a GARCH(1,1) model and the sum of the ARCH paramter alpha and GARCH paramter beta equals 1.7. This points to an undefined unconditional variance and it follows that the conditional ...
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Explosive AR(MA) processes are stationary?

According to Theorem 8.8 in Time Series A.W. van der Vaart an ARMA process $$\phi (L)X_t=\theta(L)\epsilon_t$$ has a unique stationary solution $X_t=\psi(L)\epsilon_t$ with $\psi=\theta/\phi$ if $\...
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Threshold variable: Stationary?

Say I have an Logistic Smooth transition model (or even a "TAR" model): \begin{equation} \tilde{y}_{t}=x_{t}^{\prime }\beta _{1}(1-{g}(z_{t};\gamma ,\delta ))+x_{t}^{\prime }\beta _{2}{g}(z_{t};\gamma ...
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How do I analyze time series with less variation in values?

I know i am asking a very generic question but this is something that i encountered in one of my projects. I am working on churn prediction for a bank and one of the features that i was using average ...
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How to deal with non-stationary spatial autoregressive model

I wonder what to do in case a spatial autoregressive parameter is found to be unity. If in time series econometrics a unit root is detected, differencing the process helps. Is there something similar ...
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How to determine sampling size requirements for forecasting of non stationary system that is locally stationary

I'm evaluating forecasting ocean wave height for the near future using auto regressive techniques, where the model is estimated every time step. I am constantly troubled by how to define forecast ...
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Are explosive ARMA(1, 1) processes stationary?

I was reviewing time series textbooks recently and have been left confused since. In particular I have looked into the book of Brockwell and Davis (Introduction to Time Series and Forecasting, Second ...
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Not testing for stationarity in a panel data set

Currently, I am analyzing a panel data set (different individuals over time) using panel data models with the lagged dependent variable as an explanatory variable. In this case the stationarity ...
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Stationary processes for AR, MA, ARMA

Depending on the parameters, the AR, MA and ARMA can be either stationary or non-stationary. For instance for an AR(1) process, if $|\phi|<1$, the process is stationary and else it is non-...
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Does this decomposition theorem of stationary process exist?

I have this vague impression that there was a theorem on Wikipedia about stationary process, saying that if $(x_k)$ is a strictly stationary process, then there exists a decomposition in the form of $...
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What is rolling mean and standard deviation in terms of stationarity?

I would like to know what a rolling mean and rolling S.D means in terms of achieving stationairty concerning a time series? I ran an ADF test and it told me my time series was stationary however, by ...
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47 views

Roots within the unit circle and non-stationarity

I am quite new to time series analysis and I am delving for the first time into stationary processes. I don't seem to understand the concepts of non-stationarity and the presence of roots within the ...
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1answer
33 views

Stationary and non-stationary variables in time series - how to difference?

I want to predict a multivariate daily time series, the target output is the volume of packages that is send and the covariates are day specific information as weather, the distance to holidays but as ...
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33 views

First difference stationary process

I have a question regarding an I(1) process. Suppose we have the following equation: $$ Y_0 = \alpha_1Y_{-1}+\alpha_2Y_{-2}+\alpha_3Y_{-3}+...+\alpha_4Y_{-4} $$ $$ with \sum\limits_{i=1}^4 \alpha_{...
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problem regarding time series modeling using R

I have this time series data and my aim is to fit a time series model. When i plot the time series data , it seems to be data is not stationary. These are the plots based on the first difference, ...
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Why do the Dickey-Fuller test and Lo-MacKinlay Variance Ratio test yield such different outcomes?

It is my belief that a unit root implies a random walk, but not vice versa. Therefore, would one not expect the Dickey-Fuller test to find non-stationarity in the same cases as the Lo-MacKinlay ...
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39 views

Forecasting Methodology

Suppose I have only 1 variable (data on export, monthly, non-seasonally adjusted) from Jan 1960 till Mar 2019. My task is to obtain forecasts of this series for the coming year (i.e. Apr 2019 - Mar ...
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57 views

Show that $Y_t$ is an AR(2) model

Let: $$Y_t=\beta_1Y_{t-1}+\epsilon_t$$ $$\text{with}$$ $$\epsilon_t=\beta_2\epsilon_{t-1}+u_t, ~~~~~~~~~~\epsilon_t \sim i.i.d.(0, \sigma^2)$$ $$\text{where}$$ $$\beta_1 \ne 0, ~~~~~~~~~ |\beta_2|<...
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33 views

Are trend-stationary series I(0)?

I have time-series of different interest rates. Graphs of all series show existence of trend. For some of these series ADF-test with constant rejects null hypothesis. For others, null hypothesis is ...
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Testing for (covariance) stationarity specifically (vs. testing for dependence)

I'm using Wooldridge's textbook as my guide for a time-series cross-sectional project. Wooldridge, unlike many, distinguishes between (covariance) stationarity and weak dependence. He says that both ...
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Autoregressive process with random walk perturbation (with drift)

Suppose we have an autoregressive process, $$y_t=\phi y_{t-1} +u_t$$ where $|\phi|<1$. If $u_t$ is an i.i.d random variable this process is stationarity. What if $$u_t=u_{t-1}+g+\epsilon_t$$ where $...
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Detect stationary section in an overall non-stationary time series

I'm working with a time series data from neuron firing rate. This time series is usually overall non-stationary due to experimental noise. However, I want to extract stationary section in this overall ...
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How can i make constant variance for ARIMA model?

I want to fit ARIMA model to univariate time series. For this i took log then difference twice but the variance of the series is not constant. i also try with box.cox transformation but variance is ...
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Method of analyze returns for nonstationary time-series

guys. Let's have 2 types of time series: TrendStationary (or TS-stationary) and DifferenceStationary (or DS-stationary) time series. So, what's the best strategy of analyzing and forecasting time ...
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1answer
32 views

What should I conclude if a Unit Root Test gives contradictory results with & without a trend?

I have a data set for a variable, for which I have run some unit-root tests: ADF (constant/without trend): t-stat=-1.0816, p-val=0.7218 - DNR ADF (constant & trend): t-stat=-4.5203, p-val=0.0021 -...
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37 views

SARIMA- determine order of difference

I have a question about seasonal ARIMA model to determine when to use seasonal difference only or use difference of seasonal difference. For example, I have some data that have both trend and seasonal ...
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29 views

Regression with non-stationary data

I am doing time series regression (the form I prefer is what SAS calls regression with AR error a form of GLS that runs OLS on the residuals and that has various names in the literature). The problem ...
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31 views

non-stationary time series for VAR model forecasting

I'm working with a VAR model to do forecast involving two non-stationary time series (quarterly frequency). The literature indicates to verify if there is cointegration and, otherwise, to use the ...
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Using non-stationary time series in cross-correlation analysis

I have modelled organism dynamics and abiotic factors time series in order to understand their seasonal oscillation and trend over time. Now I want to identify if there are any correlation between ...
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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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1answer
20 views

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 ...