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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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What to do if time series are non-stationary? [on hold]

Data: I have a time series data of 2528 daily observations for OMXS.30 (Stokholm) closing price. The aim is to fit proper ARCH/GARCH models and use for forecast daily Value at Risk. Here is a plot of ...
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Stationarity restriction of a TGARCH process?

What is the stationarity/convergence restriction for a threshold GARCH model, TGARCH? I know that for a GARCH model: $\alpha+\beta<1$, but I'm guessing it's not that simple for a TGARCH model. ...
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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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1answer
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Solve for inequality of AR model

I was working through my textbook and found this problem that I got stuck at: Consider the AR(2) Model $$X_t = \phi_1X_{t-1}+\phi_2X_{t-2}+\epsilon_t$$ We can assume $\phi_2 > 0$, so the roots of ...
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1answer
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Determine if AR(p) model is causal stationary or invertible

I was going through these problems and think I figured out most of them both, but am having some troubles at one of the last steps. The question is for each of the following models: Express them ...
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calculating mean and variance of non-stationary time dependent samples

I know normal procedure for calculating mean and variance which assumes that samples are iid.. I first want to know, how to calculate these two parameters if samples are time dependent but time ...
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1answer
38 views

ADF test showing stationary for a non stationary series

I am running an ADF test in R on the following series: This to me is clearly non-stationary, but when I run the ADF test: ...
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1answer
38 views

Is the time series $Y_t = \frac{1}{2} Y_{t-1} + \frac{1}{2} Y_{t-2} - \frac{1}{3} \epsilon_{t-1} + \epsilon_t$ stationary?

How can I tell if the series $Y_t = \frac{1}{2} Y_{t-1} + \frac{1}{2} Y_{t-2} - \frac{1}{3} \epsilon_{t-1} + \epsilon_t$ is stationary?
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1answer
18 views

Expected Value of an AR(1) process

I saw the answer on this post and got confused about a couple things in its explanation. Mainly, I am unsure of How the poster immediately knows the process $X_t = c+\phi_1 Y_{t-1} + \epsilon_t$ is ...
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How to calculate the Var and Covariance of exp(X_{t})+2X_{t-1}?

How to calculate the Variance and Covariance of $exp(X_{t})+2X_{t-1}$? where X is an i.i.d. normal random variables with mean zero and variance one?
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Determining Polynomial Trend, Stationarity, and Covariance of a Process

I'm given $$ Y_t = p(t) + \epsilon_t $$ where $\epsilon_t$ is a stationary series with covariance $\gamma_t$. Also given $$ p(t) = \sum_{r=0}^kK_rt^r $$ where the $K$s are constants, for the ...
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1answer
43 views

Showing the covariance and autocorrelation functions of a stationary time series are symmetric around 0

I need to show that the covariance and autocorrelation functions of a stationary time series are symmetric around zero. From my understanding, this entails $$ \gamma(h) = \gamma(-h) $$ $$ \rho(h) = \...
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Mean Square Convergence of a Linear Process Defined in Terms of a Stationary Time Series

I am following Brockwell and Davis (Introduction to Time Series and Forecasting, 3rd Edition). Chapter 2, Proposition 2.2.1 claims the following. If $\{Y_t\}$ is a stationary time series with mean ...
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2answers
76 views

Difference between independence and stationarity tests in time series

This is meant to be a general question, aiming to clarify the topic for a beginner in TSA, as I haven't found any clear introductory explaination yet. Suppose I am working with some data which ...
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24 views

Covariance stationarity - Example

I am studying time series by myself. I've just faced this process: $y_t = \epsilon_t \epsilon_{t-1}$, where $\epsilon_t$ is Gaussian white noise, with zero mean and variance equal to one. Is this ...
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16 views

State space models with non stationary/unit root factors

state space model , I am trying to implement is as follows $$ Y_t= CY + FF* X_t + Ve_t$$ $$(X_t-m0)= GG (X_{t-1}-m0) +W\eta_t$$ I am enforcing GG to be to be diagonal for the base case. I am getting ...
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Example of a first order stationary process which is not second order stationary

I need an example of a process that is stationary to order 1 but not to order 2.
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20 views

How to show that second order stationarity implies first order stationarity?

If a process is second order stationary i.e. joint pdf is independent of absolute time, how can it be shown that it is first order stationary as well i.e. First order pdf is independent of time origin?...
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generalized additive model with nonstaionary dependent variable (time-series analysis)

I am trying to use gam model to analyze pm2.5 effect on death. However, the dependent variable 'death number' is not stationary. (the data is Time Series data) What I know is for Time Series data I ...
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Why is this process not WSS?

Reading Hayes's Statistical Digital Signal Processing and Modeling section 3.3.4, they define Wide Sense Stationary (WSS) and provide a few examples in the text (on the top of page 83 in my version) ...
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Johansen test - full rank however variables are I(1)

I have a situation in which my Johansen cointegration test results indicate a full rank, rejecting both that there is no cointegrating vector as well as that there is at most one. I am working with ...
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How Enforce inevitability and Enforce stationarity works in time series?

statsmodels.tsa.statespace.sarimax.SARIMAX () the function here, have 2 parameters that are "enforce_stationarity" and "enforce_invertibility " How do they enforce these 2 properties after ...
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Is the Dickey-Fuller tests is a seasonality test as it tests the existence of the unit root?

Knowing that the "Dickey-Fuller tests" tests if the times series is stationary or not by testing the existence of the unit root, after fitting the time series to AR(1) process. Does "Dickey-Fuller ...
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Clarifying Elder & Kennedy 2001 Case 3, how to test for the presence of intercept?

I am attempting to follow the unit root testing strategy for Case 3 as per [Elder & Kennedy 2001]. After failing to reject there is no unit root, I wish to test for the presence of an intercept. ...
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1answer
43 views

Vector Autoregressive Regression in Levels

Various textbooks suggest that it is essential to test the variables used for stationarity before a VAR anaylse. If the tests give an indication of I(1) variables, these variables should be ...
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28 views

Divergent Dickey Fuller, KPSS, and Phillips-Perron tests

I'm fairly new to both CrossValidated/StackExchange and I'm no expert on Time Series Analysis, so please forgive any 'newbie' errors here. I've got an issue of divergent Unit Root tests results ...
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1answer
75 views

ARIMA Time series analysis forecasting [closed]

I am having a small project on Time series analysis for that I have hourly sales data for that I need to forecast hourly sales for the next 1 month, i.e around next 720 hours I am exploring ARIMA for ...
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1answer
26 views

Non-stationarity of macroeconomic y/y growth rates

Often we are taught that growth rates are stationary. Or at least, this is what I have often been taught. But I've a set of macroeconomic variables, in levels, which I transform and take natural log ...
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17 views

Wind (speed) - a non-iid random variable?

Over a time period of 1 year, half hourly wind speed data is collected. Is it false to state that the collected data is not-iid? I know that wind is seen as a non-stationary process, but does it also ...
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1answer
53 views

How to interpret autocorrelation plot?

I'm having trouble making sense out of this ACF plot According to an ADF test, the series is definitely stationary. Also, the presence of autocorrelation is explained by the order 1 lag, as evidenced ...
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1answer
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Forecasting non-stationary time series using MLP

I noticed that in many tutorials with neural networks people difference their time series prior to training/forecasting. Suppose that we have a window model with many autoregressive terms (say 365 ...
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1answer
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How to check if a process has constant variance?

I am using KPSS test to verify if my process has constant variance around the mean, but I am not sure if this is the correct test for my case. In KPSS the null hypothesis is that the process is ...
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How to approach SSM models for time series forecasting in general?

I have worked on SSM model using KFAS package (https://cran.r-project.org/web/packages/KFAS/KFAS.pdf) in R. Package suggests me to use one of the Box_Jenkins method to implement SSM. So we convert ...
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1answer
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Stationarity independent variables in GARCH

I was wondering if included independent variables in a GARCH model (either in the mean- or conditional variance) need to be stationary. For example, I have interest rate data, which in itself is not ...
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Stationarity condition of AR processes

For a stationary $AR(p)$ model, $\theta(B)X_t = w_t$, how can I show that $\theta(z) \neq 0$ for all $|z| = 1$. I tried it as: $\theta(z) = (1-\frac{1}{\lambda_1}z)(1-\frac{1}{\lambda_2}z)...(1-\...
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Textbook covering multiple linear regression with stohastic regressors for time-series

I am looking for a textbook or some other resource where the following multiple linear regression problem is considered. The model is: $Y_n = \beta X_n + a+ \epsilon$ Where $\{X_n\}_{n\in\{0,...,T\}...
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1answer
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Confidence interval for the mean of a bounded non stationary random variableb given N samples

I have a random variable bounded on [0,1]. The mean can be modeled as a random process. Assume the change of the mean |mu_i - mu_{i+1}| is bounded with a known bound. Given N (N is not large) samples (...
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Are there any other models besides ARMA models that require stationarity?

Every now and then I come across a discussion of forecasting methods that mentions the topic of stationary time series vaguely without specifying that it is a question mainly in the context of ARMA ...
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What is the best algorithm(s) for binary classification on data that changes over time?

I have been building models using a data set of over 100k observations of customer data (demographics, mobile/desktop user, etc.) over a period of one year. I am having trouble getting accurate ...
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clustering non-stationary data

I'd like to partition a set of n-dimensional data points into clusters. Each data point is associated with a timestamp. A few challenges for my domain are: 1) The data are non-stationary, such that ...
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Conditions on stationary distribution for continuous cases

Above is from my Bayesian notes, I have questions as: I know for discrete case, the stationary distribution $p(\theta)$ is defined as $$p(\theta) = A p(\theta)$$ where $A$ is the Markov Chain. But ...
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Moment of stationary gaussian process

Is there any formula to let me express the k moment of a centered gaussian stationary random variable with respect to the k moment of a standard gaussian random variable
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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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1answer
24 views

A sufficient condition $p(\theta)$ to be the stationary distribution is the reversibility

I was reading my notes, it says that: ''A sufficient condition $p(\theta)$ to be the stationary distribution is the reversibility: $$\sum_{\theta}p(\theta)p(\theta^*|\theta)=\sum_{\theta}p(\theta^*)...
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How to add correlation structure in mixed model lme() in R with non-stationary time series?

.. After reading many of these posts, here is my first question (!): I have data of nutrient concentrations measured daily in 10 streams (variable Site) during 365 days. I want to explain global ...
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1answer
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Question about ARMA order in the univariate GARCH model specification

Would it be correct to say that the series is stationary in the below code, since only ARMA order is specified? ...
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1answer
31 views

Some convenient way to transform time series into stationary one

In this Machine Learning Mastery post we read that in order to predict time series using LSTM network, it is good to make the data stationary first and then scale it to the interval $(-1, 1).$ In ...
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23 views

How to interpret a VAR model without significant coefficients?

I am trying to investigate the relationship between some Google Trends Data and Stock Prices. I performed the augmented ADF Test and KPSS test to make sure that both time series are integrated of the ...
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KPSS Test Trend/Level value

I have just been introduced to the KPSS test and am not quite sure I understand what the KPSS Trend value is that is given in the output. I understand that, based on these results, reject the null at ....
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1answer
58 views

The distribution of the initial point of an AR process

Consider a stochastic process $\{X_t, t = 1, 2, \ldots\}$ following the model $$X_t = \alpha X_{t-1} + e_t,$$ where $e_t \thicksim f$. Can I say that the distribution of the initial point, $X_1$, ...