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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13 views

Can I model a trend and seasonal component for a stationary time series?

I modelled quarterly german inflationdata in a state space model with a stochastic level and stochastic seasonal. But now I recognized that I need a stationary time series because I have to compare it ...
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Addition or subtraction of lag terms in the autocorrelation expectation formula?

I have some confusion about autocorrelation. In my notes I have defined, $$r[k] = E[y[n]y^*[n-k]]$$ Is this the standard way of writing autocorrelation? What are conditions such that $r[k] = r[-k]$?...
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co-variance of strictly stationary process

How to prove mathematically that co-variance is dependent on time-lag(k) for strictly stationary process? Given that distribution function is time invariant.
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Can a ratio variable be trend stationary?

Can a ratio variable, e.g. the wage share of factor incomes, really be trend stationary? It is bounded between 0 and 1 and moves in between during long periods, acting like a non-stationary variable. ...
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Interpretation of time series residual and stationarity

Does it mean that time series model is well generalized (or is a better model) if residual from time series data and model prediction is stationary? Or it means something else?
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Modelling transient effects together with permanent effects [closed]

I have the time series with transitory non-seasonal shocks (possibly, with exponential decay, but I need to obtain the shape explicitly). In particular, I have trading data and need to predict price ...
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17 views

constant variance for strictly stationary process

How to prove mathematically that variance is constant for strictly stationary process? Given that distribution function is time invariant. It is intuitive but not sure where to start to prove it.
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12 views

Can adf test and kpss test contradict?

I have a time series data for 18 months. To check for stationary I conducted adf test, to which my p value is 0.8. And kpss test has a p value of 0.1 , so at 95% confidence level I fail to reject null ...
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How to confirm if the series is stationary?

Visually, the below plot doesn't seem to be stationary. However, on differencing, I am not very sure if the time series is stationary. Visually, it's not. Please suggest any statistical techniques ...
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What would be an adequate test for a multivariate distribution to test for stationarity?

I have monthly returns of a certain portfolio with $d$ assets and I want to check it for stationarity; More precisely , I want to test whether the multivariate distribution of $$(X^t_{1}, \ldots, X^...
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Stationarity in LSTM multivariate prediction

I am constructing a multivariate LSTM NN to predict a financial time series that is non-stationary at 1% significance level from an ADF test. However, the test rejects H0 of non-stationarity at 5%. ...
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What conditions to check before regressing y against x in terms of practical applications?

As far as I understand, non-stationarity leads to spurious regressions. Keeping this in mind, is running ADF test sufficient? I see a seasonality in x (visually), however both y and x pass ADF tests. ...
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35 views

How to work with multi-step forecasting on differenced time series

I have a financial time series that I wish to make 5 step ahead (t+5) forecasts on. As the series is non-stationary, I have differenced the series. For every time step t, the response variable is ...
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35 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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Dynamic Time Warping and non-stationarity of time series

Dynamic Time Warping (DTW) computes the optimal alignment between points of two time series. If we consider two non-stationary time series (e.g. the first proportional to the second), is the DTW ...
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Which Impulse Response Function To Use And In What Order?

In Stata I used the varbasic function to estimate a VAR model (with 6 variables and 2 lags, all variables are stationary). For the interpretation I still need the Impulse Response function. If I now ...
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Time series: non-stationary in the short term, but stationary in the long term

I am working two time series y and x, and I try to fit the linear model y ~ x. y is stationary, x is not during the modeling period (most recent 50 quarters). But ADF test on the entire series of x, ...
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37 views

What is the distribution of the initial point of a stationary VAR(1) process?

I want to generalize the answer here to this case of a VAR(1) model. Suppose that $X_t \in \mathbb{R}^n$ and that $\Lambda \in \mathbb{R}^{n \times n}$. If we have the stochastic process $\left\{ ...
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59 views

Stationairy Time Series (Conditional moments)

I am reviewing the fundamentals of time series modelling and came across the following question regarding the concept of stationarity: A time series is strictly stationary if its unconditional ...
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Time Series Analysis and ARIMA

Why it is important to convert any time series problem data to stationary before applying ARIMA .can anyone please tell me intuition behind it ? ( I know if i dont convert it to stationary it will ...
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for which variables in panel data i should do stationary test?

i have a panel data and my dependent variable was stationary according to ADF test.now i dont know should i test it for my independent variables too or not? btw does it make sense to take stationary ...
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70 views

Can I perform cross correlation on non-stationary data?

I have level data which need different level of integration to be stationary. I would like to know how to perform a correlation analysis. Can I perform this on raw data or must the data be stationary?
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how does R ensure stationarity and invertibility when fitting an ARIMA model?

In the Forecasting: principles and practice book they claim that: R ensures the fitted model is both stationary and invertible I checked and indeed - for example the following model fits, even ...
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Should I make my time-series stationary to apply dynamic time warping?

I have two time-series Y and X, they both are univariate and integration of order one. After cointegration analysis, I concluded that these series are not cointegrated. Therefore, I cannot use the ...
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44 views

show that z(t) = A cos(bt) + B sin(bt) is second order weakly stationary process

I know that in order for a stochastic process to be a second-order weakly stationary process Then for every t, the following conditions should hold: E(Z(t)) = µ, D(Z(t)) = σ and cov(Z(t), Z(t+p)) = ...
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Johansen Cointegration and ADF Example

I'm trying to reproduce the Example 3.2 from "Likelihood-based Inference in Cointegrated Vector Autoregressive Models" by Søren Johansen. The example proposes the following processes: $X_{1t} = \sum_{...
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Generating dependent strictly stationary random processes

for some nonlinear prediction experiments I'd like to generate a nonlinearily dependented, strictly stationary process. Does anyone know a resource explaining how to do this? One idea would be to ...
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When should ADF, SPSS and LJUNG-BOX test be used?

I have several different time series data. I am using ARIMA for prediction provided that the data is stationary. To test for stationarity I am confused as to which test to use. Is there any guideline ...
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GEV distribution: nonstationary location sign changes when adding nonstationary scale parameter in MLE

I am trying to estimate the location, scale and shape for a nonstationary GEV distribution with block maxima. HRindex is the yearly maximum daily rainfall. I used following code in R to achieve ...
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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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42 views

Example of random process with negative variation

I study about random processes. Let us have $\{X_1, X_2, \dots X_n\}$ observations. I learned, that in stationary time series the sample autocovariance function is defined as $$ \widehat{γ}(h)= \...
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How to check whether a given ARIMA (p, d, q) process is stationary or not?

I know that a finite MA process $X_t = \Theta(B)Z_t$ is always stationary. Also, whether an AR(p) process is stationary or not can be verified by checking the roots of $\Phi(B)=0$ where the process ...
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form of the model when using backshift operator

Be $Y_t=X_t + \epsilon_{1,t}$, in which $X_t = X_{t-1} + \epsilon_{2,t}$ and $E[\epsilon_{1,t}\epsilon_{2,s}] = 0 \forall t,s$. How could I say why this process is related with a model on the form $(1-...
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101 views

Is this time series stationary? What would be your approach to forecasting it? [closed]

I've been working on the time series prediction of a signal and came across a small misunderstanding. The signal is depicted below: Apparently it looks like there are several stationary local areas ...
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34 views

Show that the autocovariance function of stationary process {${X_t}$} is positive definite

Show that the autocovariance function of stationary process {${X_t}$} with mean $\mu_X$ and variance $\gamma_X (0) > 0$ is positive definite, i.e., $\begin{equation} \sum^n_{t=1} \sum^n_{t'=1} ...
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Stationarity for an AR(2) process

How can I show that the following AR(2) process is stationary $X_t = X_{t-1} + cX_{t-2}+Z_t$, provided -1 < c < 0 ? I represented the series as $\Phi(B)X_t = Z_t$ and then tried to find out ...
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ML preprocess to achieve stationarity

I would like to use Machine learning models on top of multivariate time series data to forecast long horizons (for example 400 items and their historical sales in the last year & content features)...
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28 views

Weak stationarity of the stochastic process and the impact of the lagged white noise

I am struggling with following exerises: Consider following discrete-time stochastic processes $Y_t$: $$Y_t=\frac {5}{3} Y_{t-1}-\frac{2} {3} Y_{t-2}+\epsilon_t$$ $$Y_t=1.3 Y_{t-1}-0.4 Y_{t-2}+\...
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48 views

Statsmodel ADF interpretation

I'm regressing two time-series against one another, and I'm struggling with how to interpret an ADF test: does a low value indicate that a series, once detrended, would be stationary; or does it show ...
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Variable does not follow a normal distribution, can I trust its p-value from a Unit Root test? [closed]

If a variable does not follow a normal distribution, I should not trust in the p-value from a Unit Root test, right?
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Unit Roots in Short Horizon

I have a series that is stationary in the long run. However, in the model development sample - which is a short horizon - the same series is trending. Now, should I consider this series as non-...
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19 views

How can I estimate autoregression when non-stationary?

I have a series that I believe has one autoregression characteristic under condition A (example: positive) and another under condition B (example: negative). Is there a way (hopefully in Python) to ...
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24 views

modelling on differenced data

I have a time-series data that I want to model using machine learning models like Lasso Regression, Ridge, elastic net, etc. However, in order to make it stationary, I difference the output variable, ...
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Stationary distribution of AR(1) process with AR(1) shocks

I am trying to find the stationary distribution of an AR(1) process, where the shock terms themselves are an AR(1) process. That is, the process moves subject to the following 2 equations: \begin{...
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1answer
56 views

Detecting change point in a time series

I'm dealing with time series from satellite imagery, where I have a sudden change (drop), that I can see from the plot, but I need a statistical test to detect it. I already checked for stationarity ...
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Questions about the stability (and stationarity) of a system and state space representations

I'm pretty new to the topic and I'm trying to understand how to determine the stability of a process. I'm giving this discrete-time stochastic system: $$ \cases{ s_t = 2s_{t-2} + 3w_{t-2} \\ y_t = ...
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What's the intuition behind re-parameterization in the Dickey-Fuller test?

In text books and lecture slides, people often explain that the normal t-test of, say, the AR(1) parameter in the Dickey-Fuller test does not follow the usual distribution. It also explain that after ...
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logic of stationary property?

I am extremely puzzled... In textbooks I read that the "stationary property" is having the same statistical properties, in two chunks of my time series. I do not understand... How can I possibly ...
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Time fixed effects and stationarity data

Currently, I am working with a panel data with which I try to explain households real expenditure on health (hre) from the occurrence of natural disasters (nd). In particular, unit root tests suggest ...
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Examining Cointegration Before or After Deseasonalizing

I am trying to build a VAR model with two time series, one of which is I(0) and one of which is I(1). I originally tested these for cointegration with a Johansen Test and found them cointegrated with ...