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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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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Chow Test: Do you need stationarity to model a time-series for identifying structural breaks?

What am I trying to achieve? I am trying to test whether there is a structural break in a time-series of proportions at a known break date (21 Dec 2019) . Below is a plot of the original time-series (...
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179 views

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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423 views

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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73 views

Questions regarding geodesics in Adler and Taylor's “Random Fields and Geometry”

I'm working through some calculations in Adler & Taylor's Random Fields and Geometry. $f$ is a real, scalar, zero-mean random field parametrized by $x^i$ (elements of some topological space $T$). ...
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662 views

Stationarity of independent variables in ARIMAX

I am running an ARIMA model with exogenous variables. Do all my exogenous variables have to be stationary or is it okay if one of my exogenous variable is non-stationary?
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Stationarity tests for time series

I am currently working on time series modeling, especially on stationarity tests. For this purpose, I am extensively using Pfaff's book "Analysis of integrated and cointegrated time series with R" and ...
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427 views

Can a time series be stationary and still have seasonality?

Would there be a case that a time series does have seasonality but, ADF test fails to point it out. I want to be sure of it being stationary so that I can use it in a regression and be sure that the ...
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1answer
246 views

If linear combination of two time series processes is non-stationary does it mean one of the two series is non-stationary

Suppose I have 2 time-series processes. If they are jointly weakly stationary then the linear combination is weakly stationary. If the linear combination is non-stationary does it mean at least one ...
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275 views

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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165 views

Stationary Distribution of Multiplicative Autoregressive Model

I know for the additive autoregressive model the stationary distribution of $\{X_t\}$ can be found, if it exists, in the following way: \begin{align} X_t &= \alpha X_{t-1} + \epsilon_t\\ \...
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1answer
72 views

What model to fit to a time series data with abnormal fluctuation?

The plot of the time series data I have: I can not understand how should I model this kind of dataset. Try: To estimate the deterministic part of the time series, I have fitted a cubic spline. Then ...
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180 views

Tests of stationarity in irregularly (unevenly) spaced time series

I need to do check if my time series data is stationary or not. However, the data is so irregular that cannot be transformed into evenly spaced. Any suggestion?
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368 views

Regressing nonstationary on stationary variable

I am trying to empirically estimate the coefficient for the Okun's law as a relationship between output growth and unemployment. I am using the simple gap version, where I regress real output growth (...
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205 views

How do I solve this stochastic differential equation?

So I have a second order stationary process $Y(t), \infty < t < \infty$ which has a continuous sample function, mean $\mu_Y = 1$ and covariance function $r_Y(t) = e^{-|t|}, -\infty < t < \...
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349 views

Determining parameters in AR model for non-stationary time series

I am currently trying to fit an AR model to some financial data. The time series $Y_t$ in levels is clearly non-stationary; however it appears the first differences $dY_t$ are stationary (and this is ...
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528 views

Time Series: Seasonality and trend

I am interested in financial time series and I have a small question regarding the use of the forecast package. The time series I am interested in is a monthly one and present clear evidences of ...
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1answer
1k views

Is a trend-stationary variable I(1) or I(0)?

I am trying to do cointegration analysis between two variables. I first used the standard Dickey-Fuller and Phillips-Perron tests; they concluded my variables were I(1). I then did cointegration and ...
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Most general definition of cointegration

I am looking for a rigorous and general treatment of cointegration. Unfortunately, many of the econometrics textbooks and papers I have found in this area either place a lot of restrictions on the ...
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30 views

Time series - Stationarity and invertibility?

Sometimes when I take material from time series to study, it appears out of nowhere "for a process to be stationary it is necessary for the roots of the characteristic polynomial to fall outside ...
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25 views

why arima uses differencing transform not the log transform to make data stationary?

I am currently working on time series project and i am naive. I would like to ask, there exist strict stationary, differencing stationarity. If i understood correct the first order differencing ...
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To difference to not difference, that is the question

Definitions Skip if you know what are second order stationary processes and what we mean by integrated processes. Def.1: Suppose that we have a set of $K$ input variables and a target variables $(...
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3answers
455 views

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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35 views

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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191 views

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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330 views

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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103 views

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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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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355 views

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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658 views

Awful performance of LSTM on noisy time series after stationarisation

Note. The post is quite long because I added some thought process for the sake of seeing the big picture. So grab a coffee and indulge yourself. For tldr the actual question on the bottom. I put my ...
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634 views

Augmented Dickey-Fuller - mismatch between R packages

The Augmented Dickey-Fuller test is used to check whether a series has any detectable trend or drift. It is commonly used as a test of stationarity (the alternative hypothesis). According to ...
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1answer
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Why is the optimal policy non-stationary in the case finite-horizon problems, whereas it is stationary in the case of infinite-horizon problems?

I have difficulty understanding the meaning of stationary policy in the RL (MDP) setting. Specifically, let's assume stationary dynamics $$P(s_{t+1}=j|s_t=i,a) = P (s_{k+1}=j|s_k=i,a) \ \forall t,k,i,...
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Do I need to use stationary time series data when building linear model with AR error terms?

I have a linear model $y=\beta*x+\epsilon$. However, the error term is modeled as an AR term, where $\epsilon(t)=p*\epsilon(t-1)+w(t)$, $w(t)$ is white noise. Y and X are both time series economic ...
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43 views

While selecting model order for VAR models is it sound to stop increasing when a root outside the unit circle is found?

Basic question I guess. I'm fitting VAR models (and derivatives), and I've tried my hand on model order selection based on regularization but now I'm back to informative criteria (IC). Thing is my ...
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190 views

Correlation of nonstationary time series (levels vs differences)

I wonder what the relationship between the empirical sample correlation of two time series in levels and the one of the differenced series is. I know that for nonstationary variables, it makes little ...
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177 views

Steps to find optimal transformation for wide-sense stationarity

I've been trying to automate the procedure of choosing the best transformation for a non-stationary process (in R). For lack of a better term, "best transformation" here refers to the quality of ...
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1answer
151 views

References Request (Least-Squares Estimates for non i.i.d. Processes)

I am interested in suggestions concerning possible applications/problems within applied statistics with respect to estimates of least-squares for non-stationary designs. In particular, I would like to ...
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2answers
607 views

Does correlation between residuals of ARIMA models for two time series tell anything about how they fluctuate together?

I wonder whether one can judge strength of coupling between fluctuations of two time series by looking at correlation between residuals of ARIMA models for these two series. Let's say I have two ...
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234 views

How is the Ornstein-Uhlenbeck process related to the error of an exponential moving average?

Is anyone aware of a direct relationship between the residual of an exponential moving average and the Ornstein-Uhlenbeck process? For example, assume a series, $Y_{t}$, that follows a geometric ...
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39 views

Stationarity of a variable measured at irregular intervals

I have data collected on a meeting-by-meeting basis, where the change in time between two meetings is not constant, i.e., $\Delta t\ne1$. Are ordinary Augmented Dicky Fuller and Phillips-Perron tests ...
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86 views

predicting the future in a stationary stochastic process

Let's say I have a strictly-stationary stochastic process with known PSD (power spectral density). The process has been running, and I have all the data from time $t=-\infty$ to $t=0$. I want to ...
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69 views

Stationarity in the Almon lag model

I have a quick question regarding the Almon approach (Shirley Almon) as presented in chapter 17 of Gujarati's Basic Econometrics. In an example given in the textbook, they use non-stationary data ...
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372 views

Understanding stationarity in stochastic processes and time series

I am having trouble fully grasping the concept of stationarity in time series. Here is what I have gathered so far. A stochastic process is a collection of random variables with mean $\mu$ and ...
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231 views

Fitting a non-linear model where observations at each time are random variables drawn from a different (non-Gaussian) distribution

I have a non-linear (and not clearly linearizable) function of a few parameters that models a response over an independent variable (time): $$ f(t;\lambda_1,\lambda_2,\lambda_3). $$ The function $f$ ...
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946 views

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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What is a test that I can use to determine if a time series is first-order stationary?

I need to test that one of the time series in my analysis has a constant mean over time. Is there a standard test I can use to help me determine this? I know that I can use a nonparametric procedure ...
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114 views

Averaging time series to improve stationarity - loss of power?

Short version When averaging over a presumed stationary time series and calculating statistics (e. g. normalized mean square error) to compare to a simulation (atmospheric turbulence model) of the ...
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979 views

White noise for level, log and log differences data sets

I am using eviews 7 and I have 3 data sets for DAX stock market index: level (dax), log (ldax), and log differences (dldax). I need to check whether the error terms of these data sets are white noise ...
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Non-causal AR models

There is a theorem which states that for a non-causal AR(MA) process, you can produce another equation for $X_t$, with a different (but related) white noise sequence, which is causal. (See the clipped ...
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What is meant by this idea that a covariance function of a Gaussian process “induces” properties? And what is the connection to stationarity?

I am currently studying the textbook Gaussian Processes for Machine Learning by Carl Edward Rasmussen and Christopher K. I. Williams. Chapter 1 Introduction says the following: The specification of ...

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