# 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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### Fitting a VAR model with nonstationary data and possible cointegration

I have time series data from multiple realisations (trials) of the same process in the shape (n_trials, n_sensors, n_times) which I would like to fit a vector ...
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### Linear Returns vs Log Returns to make a Forex time series stationary

I have a GBPUSD Forex time series. I am preparing the series as an input to LSTM. As a best practice for LSTM I am making the series stationary and I have tried both linear returns and log returns for ...
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### How to assess stationarity when kpss and dickey-fuller test give conflicting results?

I have used R to test the residuals of my time series data, I have used the tseries function for the kpss and dickey fuller test. Both gave conflicting results where dickey fuller said the data was ...
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### Should I always convert time-series data to stationary before forcasting?

I am trying to predict how much revenue a store will generate in next month based on revenues of previous months. I was doing simple regression for forecasting before, but I have recently read about ...
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### Stationarity of time series when there is a clear trend

Hi I'm new to time series analysis and analyzing mobility level time series these days. The mobility level time series show a clear trend (increasing over time); however when I run Augmented Dickey ...
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### When to include drift and trend in a VAR

If I am building a VAR model with three variables and one of them is trend stationary while the two others are just stationary (to be exact two are stationary after taking the first or second ...
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### Confused about stationarity and ARIMA processes

So I am quite confused about stationarity in ARIMA processes. For example, a Random Walk is an ARIMA process with order (1,0,0). Does this mean that a Random walk is stationary? Stationarity implies ...
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### $E(f(X_0)f'(X_l)) = E(f'(X_0) f(X_l))$ for a stationary process?

I have a stationary process $\{X_t\}_{t\in\mathbb N}$ on $\mathbb R$ and two functions $f, f': \mathbb R \rightarrow \mathbb R$. Does the equation $$E(f(X_0)f'(X_l)) = E(f'(X_0) f(X_l))$$ hold? Do ...
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### Structural break test for non-stationary time series

I recently found out that Bai and Perron test for identifying structural breaks using the package strucchange can't be used in case time series is non stationary. ...
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### what should I do about a non-stationary variable in a panel-data interaction?

We have panel data on immigration stocks, immigration flows, and immigration policy for 30 countries and 10-30 years. We would like to test the theory that the effect of immigration flows (i.e., ...
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### Forecasting with VAR model

Suppose we want to build a VAR model with two non stationary historical series $\{X_t\}$ and $\{Y_t\}$ . Let us further suppose that in orded to get stationarity I should tranform the series ...
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### Problem with differencing trend-stationary processes?

I have read somewhere (possibly in Hayashi's Econometrics text) that it can be problematic to work with differenced trend-stationary processes. I have not been able to find the exact source again and ...
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### What is my 'p' and 'q' terms for below ACF and PACF

I was plotting the acf and pacf of a time series in order to find p and q values.Here is the ...
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### Is it possible to convert all non-stationary time series to stationary ones?

According to this question Is every non stationary series convertible to a stationary series through differencing I already know that differencing isn't enough to make every non-stationary series ...
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### Covariance of a non-stationary AR(2) process

I have the following AR(2) process: $(1+B^{2})X_t=Z_t$ where $X_0=X_1=0$ and $t=1,2,3...$ This is clearly not stationary since the roots are i,-i and therefore have modulus of 1 (i.e on the boundary ...
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### Pre-Whitening results not conducive to white noise property [closed]

I have these two time series(df_errror,df_booking) (both are non-stationary, seasonal) that I want to prewhiten and then find cross correlation: I used this code for auto_arima for df_booking: ...
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### When does $\lim_{j\to \infty} \mathbb{E}_t[X_{t+j}]$ exist?

Consider a univariate stochastic process (time series) $X_t$. I am interested in conditions under which $\lim_{j\to \infty} \mathbb{E}_t[X_{t+j}]$ exists. For example if $X_t$ is a stationary process ...
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### Non-Sensible Estimates for MLE of AR Processes

I am taking a course on Time Series Econometrics and I am solving a problem set that requires students to explicitly write maximum likelihood functions for, as an example, AR processes and estimate ...
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### Autocovariance of a non-stationary process

I'm just going to apologize first thing, because I know my understanding of these topics is very lacking. I'm reading some lecture notes from what appears to be an econometrics course, and they are ...
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### Is it enough for the ACF to be defined at $1$ for it to be defined at every other natural

I've read that a time series should be a weakly stationary for the Autocorrelation function (ACF) to make sense. The definition for weakly stationary series that I have is that all the observations ...
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### ARMAX stationarity - Do the exogenous variables need to be stationary too?

It concerns the ARMAX time series modelling. In order to have a stationary time series, does the exogoneous part need to be stationary too ? Or it does not have any influence ? Thank you all !
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### Comparing simulated and real time series of order parameter

I am currently working on a model that tries to mimic a certain macroscopic feature observed in experimental data. I use an order parameter $P$ to study the stability of ordered phases ($P$ close to 1 ...
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### PCA for non-stationary time series

I would like to use principal component analysis (PCA) to identify common components/factors of some time series. However, the data is not stationary, so I took the first difference to make sure the ...
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### Standardized GARCH-residuals, distributions and AIC

So, I have been wondering about an interesting observation. My data contains 1006 log-returns of the SP500-index and I've estimated a GARCH(1,1)-process with Gaussian quasi-maximum likelihood - ...
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### Confusion on the connection between causality and stationarity and possible implications

Let $(x_{t})_{t\in \mathbb Z}$ be a causal AR(p) process with operator $\phi$ such that $\phi(L)=\phi_{0}-\phi_{1}L-...-\phi_{p}L^{p}$ and $(\epsilon_{t})_{t \in \mathbb N_{0}}$ white noise sequence: ...
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### About the power spectral density of poly-cyclostationary process

For a cyclostationary process (CS), I know that the time-averaged power spectral density (PSD) is derived by the Fourier transform of the zeroth-order cyclic autocorrelation function. How about poly-...
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### Local Length Scale Kernel in Gaussian Processes (Nonstationary)

I've been working on Gaussian Processes and came across a rather nice local length scale kernel proposed in "Nonstationary Gaussian Process Regression Using Point Estimates of Local Smoothness" from ...
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### ARIMA forecasts with autocorrelated residuals

I have a time series on consumer price index (CPI) and want to forecast inflation which is in my case the first difference of the log of CPI: ...
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### VAR Model: Non-stationarity of variables

I am currently working on an empirical analysis in R. To give you some background information: I want to estimate a VAR-model to subsequently develop IRFs from it (using cholesky decomposition). My ...
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### If I split a stationary ARMA process into two parts, are they also stationary?

Considering an Auto-Regressive Moving Average (ARMA) model, \begin{equation*} y_k = \phi_0 + \sum_{j=1}^{p} \phi_j y_{k-j} + \sum_{l=1}^{q} \theta_l \varepsilon_{k-l}+ \varepsilon_k, \qquad \text{for}...
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### Stationary seasonal model

Given Box-Jenkins seasonal model on p.310 $$\phi_p(B)\Phi_P(B^s)\triangledown^d\triangledown_s^Dz_t =\theta_q(B)\Theta_Q(B^s)a_t$$ I want to show that the $z_t$ process is stationary (or not). How ...
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### Can someone explain the importance of mean stationarity in time series?

In regular regression, the expected value of Y | X is allowed to change. In fact we generally use regression when we want to model this change in conditional mean. I am not understanding why in time ...
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### Invertibility after differencing

Given a non-invertible MA model $$Y_t = e_t - \theta_1e_{t-1} - \theta_2e_{t-2}$$ where $\theta_1$ and $\theta_2$ are provided (known) parameters, for which values of these parameters can I take a ...
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### Should stationarity be checked for *both* time series models (ARIMA, ECM…) and causal models with explanatory variables?

I read this paper that discusses if Time Series models or Causal models are the best for forecasting GDP. I am familiar with unit root tests (ADF, PP) and stationarity test (KPSS) applied to a time ...
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### Bernoulli process with nonstationary probability

Say we have a process $X_t\vert P_t\sim \mathrm{Bin}(n,P_t)$ where $X_t$ is observable but $P_t$ is not. Also, the success probability $P_t$ might vary over time and I don't assume some ...
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### If $X(t)$ is wide sense stationary (WSS), and f(.) is a real-valued monotonic function, then is $f(X(t))$ and X (t) jointly WSS?

If $X(t)$ is a continuous wide sense stationary (WSS) process, we know that E(X(t)) is independent of time R_xx (t1,t2) depends on t1-t2 only. Now if we have a function f(z) where z is any real ...
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### Relationship between strict stationarity and 2nd order stationarity

I've been reading my GARCH class notes and I found some discussions about stationarity. "In Finance, stationarity of order 2 is often considered as more restrictive than the strict stationarity, ...
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### Augmented Dickey Fuller Test Procedure

So I am trying to understand the inner-workings of the Augmented Dickey Fuller Test. In the limited amount of novice information I could find, it seemed to be executed slightly differently. The best ...
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### Check stationary assumption after parameter estimation in ARMA model

I understand that given time series data, say $\{X_t: t=1,\ldots,T\}$, we usually use the Augmented Dickey–Fuller test to verify the stationarity assumption before conducting parameter estimation. ...
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### Reason for I(1) integration order limit in ARDL regression

What is the reason for I(1) integration order limit of independent (or dependent) variable in ARDL regression?, to be specific I(2) variable will 'break' the ARDL model/estimation. Fast thinking I ...
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### Math behind Differencing: Is White Noise Stationary?

I'm just starting to learn the math behind stationarity and differencing, so I apologize if this is a silly question. Lets say I have a non-stationary time series process (pure random walk) defined by:...
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### Should stationarity be checked also for non-linear regression?

Achen (2000) and Keele and Kenny (2006) show that, as long as stationarity holds, including lagged values of the dependent variable may bias the other regression parameters toward zero. This is fine ...
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### ARIMA Model Non-Stationary Time Series

Suppose that the data generated process is the following: Y(t) = 1.2*Y(t-1) + 0.2 The process is clearly non-stationary. My question is why we can't fit an AR(1) model and make predictions?
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### Unit root test (e.g. ADF) vs Autocorrelation test (e.g. Breusch-Godfrey)

I am confused with the purpose of unit root tests in relation to serial correlation tests. I know they test for different things, but ultimately tests if a time series is stationary or not. From what ...
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### Does it always make sense to make a time series stationary?

As stated in the question title, I can't understand the logic behind making a time series stationary. I do understand that stationarity is a necessity if we want to do forecasting because we need a (...