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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Showing that random process is stationary

Suppose i have $x_t, \bar{x_t}, t\in \mathbb{Z_+}$ independent 2-states $\{0, 1\}$ Markov chains with positive transition probabilities. Initial states are $x_0 = 0; \bar{x}_0 = 1$. For which positive ...
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Converting random non stationary series to stationary series [closed]

Is it possible to convert non stationary random series, for instance the price of BTC, to stationary? While messing around with some code, I subtracted the open from OHLC values of BTC and got a ...
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Seasonal and trend adjustment for irregularly spaced time series

I know of different methods that exist to remove seasonality and trend in the data to make it stationary. However, that exists only for regular time series; that is, a series that follows a fixed ...
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Is stationarity of variables neccessary condition for Bayesian VAR?

I am trying to run a BVAR on 5 variables. Four out of five are non-stationary. So shall I do the first difference of the non-stationarity variables or take them in level for running the BVAR? And what ...
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Inclusion of year and seasons as variable for regression with non-stationary response?

The common knowledge is that OLS only makes sense if both the response and explanatory variables are stationary (ignoring exceptions like cointegration), as otherwise, there could be effects of ...
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Establish convergence to normal area

I will try and make my question abstract, since I have two problems of the same overall type. I am given a time series $Y_t$, t=1,...,T. I know that this time series has a (slowly) changing ...
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Stationary time series having unusual ACF and PACF plots

I'm analysing a highly stationary time series and while plotting ACF and PACF I noticed a strange bump at a later lag very close to the 0.5 threshold level. Does it affect the AR degree=2 and AM ...
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How do I interpret my autocorrelation and partial autocorrelation of COVID-19 time series data?

Note: I've edited my question based on comments from @whuber. I have a time series of the USA's COVID-19 daily deaths smoothed to mitigate massive, weekly data dumps. My two-plus years of data comes ...
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Can I use non-stationary variables in forecasting problem

I want to build survival analysis model (Cox PH) with time-varying covariates. Time-varying covariates are macroeconomic variables. Therefore, they are same for each individual at the same calendar ...
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Why is this linear combination of random variables from a white noise stationary?

I'm looking at the following definition of a causal AR(p) (autoregressive) model: An AR(P) model $\phi(B)x_t= \epsilon_t$ is said to be causal if it has a stationary solution $$x_t=\epsilon_t +\sum^{\...
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Time series data in regression analysis

I'm making a regression analysis in Python to find out the dependence between the stock price and several variables : my dependent variable - share price of company, independent variables - price of ...
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Regression analysis with time series data

I'm completely stuck. I'm making a regression analysis in Python : my dependent variable - share price of company, independent variables - price of steel, price of coal and changing in local currency. ...
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Johansen Cointegration Test at Levels or First Differences in a VAR Model

I have multiple variables that I am trying to perform a VAR model with, but all my variables are non-stationary at levels as they fail the Augmented Dickey Fuller test. Having taken the first ...
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Covariance of a convolution between a gaussian random walk and white noise [closed]

I want to compute the covariance of $$U_t:=\sum_{l=-L}^{l=L} (X_l-X_{l-1})X_{l-t}$$ with $X_t$ defined as : \begin{align*} X_0&=0 \\ X_t&=X_{t-1}+\epsilon_t \end{align*} $t=1,2,...$...
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Are all dummy variables stationary?

If a time series model contained only dummy variables as dependent and independent variables. Is it always stationary?
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1 Stationary time series, 1 non stationary: do I need to transform BOTH, OR can I use VAR with 1 transformed and 1 stationary variable?

I am doing a time series forecast using VAR. I have 2 time series, "orders" and "calls" The orders time series is stationary The calls time series is non-stationary Let's say I use ...
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Fixed Effects Model with non stationary variables

I'm in a bit of a pinch. I want to make a panel regression model, with N=22 and T=28. I guess this makes it a moderate panel (if not a long one). Not too big a N, but a fairly large T for Panel Data ...
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Which model should I use for Time series data?

I need to regress one dependent variable (dummy variable), against several other independent variables (dummy and non dummy variables). (FYI : I'm not using the past performance of dependent variable, ...
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How to deal with trend of independent variables in panel data regression? (if differencing does not work)

I am running a pooled OLS regression on the following model: Running this regression results in the following results: The R-squared seems so high that I get the suspicion that something goes wrong. ...
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Examining stationarity, mean and variance of time series

Hi guys, I encounter this question for a Business Forecasting module and I am very confused by it. Firstly, this looks like an autoregressive model of order 1. From the looks of it, the φ coefficient ...
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Augmented Dickey-Fuller test says its stationary, but it doesn't look like so (IMHO)

This is my TS: As I apply the adf.test(myts), see below the result: ...
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How does covariance stationarity even exist?

I've been wondering recently about covariance stationarity. Say we have a stationary series with statsmodels' ADF and KPSS results: ...
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Estimating stationarity of evolving distribution

Consider a probability distribution $P_t$ at time $t \in \mathbb{N}$ which evolves in time, but is assumed to become stationary at some unknown time $t_0$, i.e $P_{t_0+1} = P_{t_0}$ To make matters ...
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The correct pre-processing of time-series data when using LSTM

I have a time-series data that I would like to use for forecasting the data trend using LSTM. I followed https://towardsdatascience.com/the-complete-guide-to-time-series-analysis-and-forecasting-...
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Stationary time series and ARMA process

Based on the shown tests ADF, ACF and PACF is the differenced time series stationary and could it be characterized by an ARMA process? Thanks!
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Can do smoothing after making time-series data stationary

Can do smoothing after making time-series data stationary? And is it useful to do smoothing before making data stationary?
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Why does my data look the same after log, root, Box-Cox transformations?

I have to forecast the amount of cars sales for the next 12 months. The data I have gathered are from 2013-2021 (108 months). This is what the plot of my data looks like using Rstudio and its ...
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Does the Absence of a Unit Root Imply Wide Sense Stationarity?

I'm taking a course on time series currently and have been slightly confused about the interplay between unit roots and stationarity in a question I've been attempting to answer. The question set up ...
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Stationarity and ergodicity of a process conditional on a finite trajectory

Let us say we are interested in a single time series, e.g. the daily closing share price of Tesla. We can model it as a realization of a stochastic process $\{Y_t(\omega)\}$. It corresponds to a ...
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What happens if one of my series in the VAR is not stationary?

I have a VAR model that comprehends 6 series. Only one of them is non-stationary even after taking the 1st difference. Do I need to take the 2nd difference? My concern is to approximate too much the ...
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Difference between stationarity and ergodicity

I am currently studying time series. However, I have trouble understanding how ergodicity and stationarity differ. Could someone clarify the difference between the two concepts in simple terms and ...
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'ar' part of model in not stationary error

I have been trying to plot a time series model for the equation $X_t = 0.5 X_{t-1} + 0.5 X_{t-2} + \epsilon_t$ ...
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How to change some parameters of a process to make it WSS?

Consider the R.P., $X(t)=Au(t-T)$ where, $$A \sim N(\mu,\sigma^2)$$ $$T \sim Exp(\lambda)$$ and $u(t)$ is the unit step function. If $A$ and $T$ are independent we'll have, $$E(X(t))=\mu(1-\exp\{-\...
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Which criteria should I use to evaluate a time series for stationarity?

I have been given two criteria to evaluate the stationarity of a time series: A stationary series will have no sample acf (SAC) or sample pacf (SPAC) values outside of confidence bounds on the chart. ...
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Phillip-Perron test - sample size requirement?

I understand that both Phillips-Perron and Augmented Dickey Fuller tests are for the presence of a unit root in a time series. It is my understanding that these test conduct those tests differently, ...
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How to get around the weaknesses of stationarity tests like ADF test?

I have a large search space of ~600 time series, where I am constantly scanning for stationarity (I have a "rolling window" for all the time series where I run a stationarity test at every ...
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Is this statement of a stationary density function correct?

I'm planning to use a discrete-time stochastic process defined in the following paper: Nicolau, J. (2002). Stationary Processes That Look Like Random Walks—The Bounded Random Walk Process in Discrete ...
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VAR with trend-stationary variables

I'm currently trying to estimate a VAR with 3 variables - consumption, investment and a credit spread. I have inspected the variables and run ADF tests to determine that they are in-fact trend-...
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Forward/Backward Iteration and Stationary/Stability

Suppose I have an AR(1) process of the form: $$y_t = \phi y_{t-1} + \epsilon_t$$ where $\epsilon_t$ is a white noise process with mean zero and variance $\sigma^2$. If $|\phi| < 1 $ , the model is ...
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How to visually interpret Augmented Dickey-Fuller test results between two time series?

Time-series #1 and #2 are daily reported disease cases in the Central vs. Southern regions, respectively. The p-values from the Augmented Dickey-Fuller test is less than 5% for #1 and more than 5% for ...
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derivation for the standard deviation of autocorrelation

The standard deviation of the estimated autocorrelation coefficient is given as (1/sqrt(sample size). Can someone derive this?
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Cointegrated regression with stationar

For several years, I have been thinking about cointegration regression involving stationary variables as explanatory variables. I am looking for comments on whether the following procedure is ...
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why are there many different ways to achieve stationarity?

In time series analysis, I've read two main ways to achieve stationarity. difference (or log difference) normalize with mean 0 and std 1 What are the main differences between the two and how should ...
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How is this series possibly stationary (by ADF Test)? [duplicate]

Here is an attached image of a time series that I am testing for stationarity using the Augmented Dickey-Fuller test. Here is the command I ran in R: ...
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Autocovariance of Explosive AR(1) model with $|\phi|>1$ expressed as a stationary process

I am working through the book called Time Series Analysis and Its Applications by Shumway and Stoffer. I am stuck deriving an equation given in example 3.4 in the book (page 80 for the fourth edition),...
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Logit with non-stationary predictors

I have a logit with two predictors that are I(1). Can I meaningfully test for cointegration between the predictors. If so, what would I save for the logit? I am assuming that cointegration between the ...
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How to make quarterly population data stationary?

So, I am trying to build the Time Series model for the quarterly population estimate for Ontario provided by Stats Canada (https://www150.statcan.gc.ca/t1/tbl1/en/tv.action?pid=1710000901). As seen ...
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Current best practices for detecting Unit Root in time series data?

I'm trying to get an overview of what are the current best practices for detecting unit roots in time series data. The main approach I came across is the Augmented Dickey Fuller (ADF) test, which ...
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Renewal process: Stationary and independent increments?

For a renewal process, we know that the inter-arrivals are independent but not exponentially distributed, as opposed to the Poisson process for which the inter-arrivals are exponential. We also know ...
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In a time series forecasting, should we apply differencing on entire dataset if one or two features are non stationary?

I'm working on a time series forecasting model using VAR (Vector Autoregression). I have 6 features, out of which 2 features are not stationary. If I apply first-order differencing on those features, ...
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