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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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Dickey-Fuller test statistical significance

I recently read about the Dickey-Fuller test. Firstly about the transformation from $$ y_t=\rho y_{t-1}+\epsilon_t $$ to: $$ y_t-y_{t-1}=(\rho-1) y_{t-1}+\epsilon_t $$ I assumed it is to get the ...
Tomer Gigi's user avatar
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Signal fusion between different sensors

I have a 30-year time series of variable (soil water content at three depths) constrained between maximum and minimum values. During this period, three different types of sensors were used to record ...
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Stationary Check (ADF, KPSS and ACF PACF Plot) for SARIMAX model

I'm still confused about whether my data is stationary in terms of variance and mean or not, because based on the ADF test and KPSS test the data already shows stationary (because it rejects H0) but ...
Farah Pangesti's user avatar
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Covariance stationarity for an AR(1) with squared terms?

I have a simple, but surprisingly mind-numbing, problem. I am familiar with determining stationarity for an AR(p) process: look at the roots from the characteristic equation. What if we had higher-...
Danny Klinenberg's user avatar
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how to determine if there is changes in the process or the process dynamics

I have two groups of data representing a filtration process- Group1 and Group2. The underlying principle of both the groups are the same. However, in Group1 there are 11 filters doing the filtration ...
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general procedure with panel data? [closed]

I've read that the general approach for modeling with panel data (in my case, 18 countries x 18 years) is as follows: identify and collect data on dependent variable, main independent variable and ...
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Granger causality test with I(1) and I(2) variables

I have two time series variables for daily financial data: "A" is I(1) and "B" is I(2). After log-differencing once and twice, respectively, to make them stationary, I obtained &...
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Characteristic Polynomials for AR(p) Processes with intercepts

If we have AR(1) process with no intercept like: $$ x_t=\phi x_{t-1}+w_t, $$ it has a unit root when $|\phi|=1$. If we have an AR(p) process with no constant $$ x_t=\phi_1 x_{t-1}+\phi_2 x_{t-2}+\...
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How does KPSS test for stationarity if it is about trend stationarity vs unit roots?

ADF Test: Null Hypothesis: The series has a unit root. Alternate Hypothesis: The series has no unit root. KPSS test: Null Hypothesis: The process is trend stationary. Alternate Hypothesis: The ...
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Estimating an ARMAX model using an I(2) exogenous variable

Is it valid to estimate an ARMAX model using I(1) and I(2) variables, which are made stationary after first and second differencing, respectively? For instance, I have an I(1) stock price variable, ...
Pepe Frog's user avatar
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Stationary temperature relationships from non-stationary data?

In stable operating conditions, the temperature $T$ of an engine can be thought as a monotone increasing function of its angular speed $\omega$. However, if $\omega$ changes abruptly, the temperature ...
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Testing time series data stationarity

I am working with time series and want test different forecasting methods but first I need to test if my time series (sales) data is stationary or not. So I have been learning about KPSS and Dickey-...
monique's user avatar
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Non-stationary time series: what are the advantages of doing analysis in levels instead of differences?

Suppose we want to analyze some non-stationary time series, x(t) and y(t). For simplicity, assume they are I(1). We can analyze them in levels (using cointegration tests) or in differences. What are ...
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Brockwell/Davis seem to say more persistence implies better predictability---do I have a counterexample?

Brockwell/Davis, Introduction to Time Series and Forecasting, p. 40, write (notation slightly adapted; please refer to screenshot below) The best linear predictor $l(Y_{T})=aY_{T}+b$ for a stationary ...
Christoph Hanck's user avatar
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Simulation of Random Processes to Check Stationarity

I am wondering if this is a valid approach: I want to validate that certain random processes are not weakly stationary (constant mean, covariance depends on the lag, finite variance) through ...
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Are stationary processes non-predictable, and non-stationary ones predictable?

I am reading A canonical analysis of multiple time series by Box and Tiao (1977). In the abstract of the paper, the authors mention: The least predictable components are often nearly white noise ...
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Importance of stationarity for ARIMA/ARIMAX/SARIMAX for predictive purposes

I am doing a forecasting project right now and I could use some understanding of why stationarity is importance when forecasting in general. Especially for the SARIMAX model. I know the problem of ...
Mathias Nissen's user avatar
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Non-stationarity and panel data

Spurious regression happens when we regress two independent non-stationary time-series: $$y_t = y_{t-1} + u_t \qquad x_t = x_{t-1} + v_t \qquad \operatorname{corr}(y_t, x_t) = 0$$ Then if we do the ...
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Contradictory Sources on Seasonality being a nonstationarity

I have been trying to figure out whether seasonality means nonstationarity, and the answers from many (often reliable) sources seem to be contradicting. (lets define stationarity as weakly ...
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Conditional and unconditional mean in GARCH(1,1) model

Say I have a stationary time series and want to fit a GARCH(1,1) model. Does this mean that the conditional mean, which is used in GARCH, would always be the same as the unconditional mean of the ...
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Is stationarity important when using boosting models?

I've studied time series for the past months and I've seen mainly two ways of building a forecasting model: Using ensemble algorithms and making the time series look like a cross-sectional data, in ...
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Why do we make a time series stationary if the ARIMA, AR and other models are clearly working with the dependence of lags?

When we run a AR model, we are using a linear combination of its lags to predict the current value. So this means that the lags are related to each other (at least t-1, t-2, ..., t-n are related to t0)...
Andrew Joplh's user avatar
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Interpretation of ADF test output with R

I Run ADF test with R for 3 different models include,1.No deterministic terms, 2. with constant and 3.constant and trend based on the below code where all model specifications employ a lag-order of p =...
Neda Fathi's user avatar
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Difference transformation and Stationarization of Moving Average

I have a temperature sensor data, I want to denoise it. The first thing that came to my mind was to take the moving average, it was very smooth but it is still not stationary. If I take the log ...
Clankk's user avatar
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Choosing number of lag AND model form for Augmented Dickey-Fuller test

Before realising an Augmented Dickey-Fuller (ADF) test, one has to answer 2 questions, how many lags p to include in the model, AND which model to choose among the following: No constant, no trend ...
cp123456's user avatar
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MD, tICA and (random, stationary) processes

I am asking myself wether PCA and tICA mandatorily need: 1) 2) Random data as input, i.e. the values sampled per each feature need to have "no memory" of the other ones; Indeed, I was ...
Jacopo's user avatar
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2 answers
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VAR regression between I(1) and I(0)

I am considering two time series and I would like to to a VAR regression between them. The ADF test rejected stationarity in only one of them, so the time series would be I(0) and I(1). I understand ...
dleal's user avatar
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4 votes
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Is it legit to estimate an AR(1) model for non-stationary time series?

Suppose ${X_{t}}$ is a non-stationary process. The goal is to estimate the following AR(1) model: $$X_{t}=\alpha +\beta X_{t-1}+\epsilon_t.$$ From classical time series analysis, we know that ...
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Unit root testing in python

I have a time series (ploted below) which I want to test for the presence of unit root. The way I understand it, a time series with unit root have persistent, lasting effects after an unexpected ...
FeCostaPa's user avatar
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The importance of stationarity for the oracle property of Elastic Net Regression?

I've been on the lookout for a while, but unfortunately, I'm still coming up empty-handed in my search for papers or books that dive into the theoretical derivation or simulation of the impact of non-...
Joe94's user avatar
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Analyse and compare non-stationary time-dependent signals

I need a sanity check. I am working with a data set of force-time, displacement-time, and velocity-time signals. All of my signals are non-stationary, and they contain trends (low frequency variations)...
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Stationary Bootstrap Block Size Impact on Portfolio Simulation Results

I'm analyzing simulated portfolios generated using the stationary bootstrap method proposed by Politis et al. (1994). This method is expected to be robust to the choice of average block size, as it ...
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Invariant event defined in terms of stationary stochastic sequence

In "Almost Sure Convergence" by Stout, there is indicated that the concept of invariant event (and further, the concept of ergodicity) can be defined in terms of given stationary stochastic ...
Mentossinho's user avatar
2 votes
1 answer
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Is Changepoint Detection valid if the process is not piecewise stationary?

I have seen these two definitions: Let us consider a multivariate non-stationary random process $y=\left\{y_1, \ldots, y_T\right\}$ that takes value in $\mathbb{R}^d(d \geq 1)$ and has $T$ samples. ...
pandashelp's user avatar
3 votes
1 answer
167 views

Regarding explosive AR processes and stationarity

I often see this: If we have an $\text{AR}(1)$ process,$x_t=\phi x_{t-1}+w_t, $ $x_t$ is: stationary if $|\phi|<1$ an unit root (nonstationary) if $|\phi|=1$ explosive (and nonstationary) if $|\phi|...
da7666's user avatar
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2 votes
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What to do in Box-Jenkins framework when time series has deterministic trend and seasonality?

I'm self-studying time series and I'm puzzled by apparent lack of consistency between : the "classical" decomposition of time-series and the Box-Jenkins methodology. Concerning the ...
Johannes Konrad's user avatar
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Non constant Feature Importance [closed]

I have a financial dataset which has 10 years worth of data. The aim is to build a regressor capable of predicting next year sales. So, if I want to predict sales for 2024, I could use data from 2023, ...
Nick's user avatar
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FAVAR model, stationarity and Toda & Yamamoto

To overcome the problems of non-starionarity and cointegration between variables, Toda and Yamamoto (1995) suggested to estimate a VAR with a number of lags sufficient to avoid the problem of ...
Ricter's user avatar
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2 answers
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Approach to Handling Stationarity in Multi-dimensional Time Series Forecasting with AutoARIMA

I am working on a time series forecasting project for a meal delivery service that operates in multiple cities. The company has several fulfillment centers across these cities for dispatching meal ...
user172500's user avatar
1 vote
0 answers
41 views

Stationarity issue in Factor Analysis

I'm applying Exploratory Factor Analysis (EFA) on 20 variables to identify latent factors. The normalized data (Z-scores) is not stationary (Augmented Dickey-Fuller test). First differencing makes the ...
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Does cointegration test of exogenous variable with Y variable make sense when doing ARIMAX/SARIMAX?

The cointegration test between two time series variable is generally relevant from my understanding when you are performing a regression model. In terms of ARIMA model the approach is straightforward ...
Sayooj Balakrishnan's user avatar
3 votes
1 answer
62 views

Wold's decomposition theorem for stationary processes

The posts How come the deterministic part of Wold decomposition does not violate stationarity? More about the deterministic part of Wold decomposition express some concerns as regards the "...
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How would you argue against the bold statement that stationary is needed for sensible predictions

We talked about a (time series) process generating observations. Somebody gave the bold statement: "Regression or predictions for general processes only make sense (generalize well) when the ...
Ggjj11's user avatar
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2 votes
1 answer
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Is it legit to compute/define moving average series for a stationary process?

Suppose we have $X_{1}, X_{2}, ..., X_{n}$ sequence of $iid$ random variables with mean $\mu$ and standard deviation $\sigma$. By definition, the time series $x_{1}, x_{2}, ..., x_{n}$ is a stationary ...
Sane's user avatar
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2 votes
1 answer
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More about the deterministic part of Wold decomposition

This is a follow-up on this question of mine. Wold's representation theorem states that every covariance-stationary time series $\{Y_t\}$ can be written as the sum of two time series, one ...
Richard Hardy's user avatar
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Why is mean of innovations restricted to zero in definition of VAR process?

The VAR process is defined as: $$\begin{align} \mathbf{y}_t = A_1\mathbf{y}_{t-1} + \dots + A_d\mathbf{y}_{t-d} + \boldsymbol{\epsilon}_t, \quad t \in \mathbb{Z} \end{align}$$ where $\boldsymbol{\...
Dylan Dijk's user avatar
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Valid forms of exploratory data analysis for time series that don't assume stationarity?

Lets say we are given a time series sample and want to try to create a model to forecast future values of said time series When trying to build a model to forecast time series data, many statistics ...
QMath's user avatar
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Stationarity and moving standard deviation

Suppose $\{X_t\}$ is stationary process. We observe a sample of $N$ observations from the process, i.e., $x_1, x_2, ..., x_N$. The stationarity property implies that the distribution doesn't change ...
Sane's user avatar
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2 votes
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Proving Monotonic Decrease of Kullback-Leibler Divergence in Iterative Method for Stationary Distribution Estimation

Introduction Consider a well-behaved Markov chain with desirable properties (irreducible, aperiodic, positive recurrent), characterized by a transition matrix $P$ and a stationary distribution $\pi$. ...
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Missleading ADF Results

I am trying to use the adfuller test to determine whether the following timeseries is mean reverting (stationary) From the chart we can see that its obviously (at least my non-expert eye) non ...
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