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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What guarantees the existence of a finite representation of the Wold decomposition? Mechanics and Intuition

Every covariance stationary process can be written as a linear, infinite distributed lag of white noise. In other words, every covariance stationary process has a Wold representation. Then we go on to ...
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345 views

Nonstationary solutions for stationary ARMA equations

By "stationary" I mean "weakly stationary". Consider a "stationary" AR(1) equation: $$X_t=\varphi X_{t-1}+\varepsilon_t,$$ where $t\in\mathbb{Z}$ are discrete time moments, $\varepsilon_t$ a zero-...
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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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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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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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493 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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1k views

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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226 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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168 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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136 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
51 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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124 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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421 views

Difference between the Wold Decomposition and MA representation

The Wold Theorem states that any (weakly) stationary process $(x_t)_{t=-\infty}^{+\infty}$ with zero mean can be decomposed into $$ x_t=\sum_{j=0}^{\infty} b_j\epsilon_{t-j}\ + \eta_t , $$ where the ...
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103 views

Non-stationary time series model

I recently have a task related to non-stationary model in time series but I'm running out of ideas how to solve it, may be someone can help? I am tasked to find the auto-covariance for the model $$ ...
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1k views

Identifying lagged effects / Distributed Lag Model

I would like to create a linear distributed lag model in order to do some forecast and also being able to interpret the results. Unfortunately I'm a bit confused with the process I should follow....
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357 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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203 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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317 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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521 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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58 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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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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146 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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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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75 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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469 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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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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149 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
7k views

Variance of a stationary AR(2) model

I have two questions: 1) When one says an ARMA process is 'stationary,' do they mean strongly stationary or weakly stationary? 2) Is there a quick way to find the variance of a stationary AR(2) ...
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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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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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278 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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222 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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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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110 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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916 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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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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2answers
95 views

Identifying Early Indicators Time Series Analysis

I have a time series representing demand for a product which looks as follows: Clearly, this time series shows an upward trend and it's variance does seem non-stationary as well. Further, I have a ...
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Is stationarity a requirement when using neural networks for time series forecasting?

I'm getting conflicting information on whether stationarity is a requirement when using neural networks for time series forecasting: In this lecture, the speaker says it isn't a requirement. In this ...
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29 views

Does stationarity have to be satisfied for a time series regression problem?

I am recently trying to build a time-series regression model to predict for the output flow rate of a reactor given some upstream conditions. The reactor behaves quite cyclically because the ...
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40 views

Not testing for stationarity in a panel data set

Currently, I am analyzing a panel data set (different individuals over time) using panel data models with the lagged dependent variable as an explanatory variable. In this case the stationarity ...
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2answers
203 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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1answer
260 views

Removing Variance in Time Series After Applying Log Transformation

I'm trying to look at natural gas prices from 2003-2018. The issue is after applying log transformation and then diffrencing data by 1, I still seem to get an increase in variance from mid 2014-2018. ...
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79 views

How to test for constant mean over time with time series data

I have a data set that looks like it may not be stationary. As a test, I ran a linear regression of the data against an x variable that was an index of time, 1:400 periods. I saw that the slope ...
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40 views

Autocorrelation Functions/Autocorrelograms and the assumption of Weak Stationarity?

Does it make sense to even speak of the autocorrelation function or of the autocorrelogram for a non-weakly stationary series? Anytime we see an autocorrelation function or an autocorrelogram can we ...
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1answer
222 views

Modeling quarterly default rate (non stationary, autorregressive time series)

I am a student writing my thesis on default rate modeling. My major is finance, so I'm not really experienced in econometrics. I'm trying to create a model for quarterly corporate default rates (...
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256 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 ...