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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How come the deterministic part of Wold decomposition does not violate stationarity?
Wold's representation theorem states that every covariance-stationary time series $\{Y_t\}$ can be written as the sum of two time series, one deterministic and one stochastic:
$$
Y_t=\sum_{j=0}^\infty ...
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Stationary distribution of Markov chain with continuous state space [migrated]
I'm considering a Markov process with a continuous state space. Let $V(x)$ be a differentiable function, $\Delta t$ a fixed time step, and, at every step, set
$$x_{n+1}= x_n-\alpha \frac{dV}{dx}\Big|_{...
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What should we consider when visually evaluating non-stationary data before making assertions?
I often see charts displaying multiple time series plotted together and they are often accompanied by explanations relating one series to the other. For example, in the below chart, one might say less ...
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Stationarity and Chow-Lin disaggregation
My goal is to have higher frequency data (quarterly) for the GDP of my region (yearly data) through quarterly auxiliary data. Therefore my goal is to perform a Chow-Lin regression using the package <...
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ADF & KPSS Test Reporting Series as I(19)
I am currently attempting to determine the order of integration for a nonstationary time series of the Federal Funds Rate (https://fred.stlouisfed.org/series/FEDFUNDS - Monthly, 1990 to 2004, 168 obs.)...
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Differencing and Stationarity in Panel Data Analysis
As far as I know, the source of randomness is different in cross-sectional and time-series analysis.
The following is my understanding.
When we use a cross-sectional data set $\{X_i\}_{i=1}^{N}$, the ...
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Do we need stationarity for Bayesian network modelling?
Most of the Bayesian network packages in R dealing with continuous data require data to be Gaussian. Does this necessitate the data should also be stationary in order to run the model?
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How to Handle a Trend-Stationary Dep. Variable and Stationary and Non-Stationary (Unit-Root) Ind. Variables?
I am trying to determine the best way to proceed when one has a mix of stationary, trend-stationary and non-stationary variables with unit roots.
My dependent variable $Y_t$ is a trend-stationary ...
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Calculate price forecasts from forecasted returns
I have a question which makes me so hurt.
Let's have a price time series $y_{t}$ for the same asset (for example, daily S&P 500 values) $y_{t}$
It can be trendy (trend stationary or difference ...
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Stationary spatial random filed
I am very new to the spatial model. My question is, what does a stationary spatial random field mean? If I divide the data, say, "meuse" dataset in "sp" package in"R" ...
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Testing Trend Stationarity against Stationarity
I am trying to find a test where null hypothesis is that the series is trend stationary. I can assume that the trend is linear if that is going to help. So the series for null hypothesis is given by:
$...
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Strongly vs. weakly stationary ergodic process
I'm reading a paper where a process is assumed as strong stationary ergodic. And I have read some materials where an ergodic process is defined for a weak stationary process, like
$$\lim_{n\rightarrow\...
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Mincer-Zarnowitz test with cointegrated time series
The Mincer-Zarnowitz test of forecast optimality regresses forecast errors $e_{t+h|t}$ on the forecasts $\hat y_{t+h|t}$,
$$
e_{t+h|t} = \gamma_0 + \gamma_1\hat y_{t+h|t} + u_t \tag{1}
$$
or in ...
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Lasso for non-stationary time series
Does it make sense to use Lasso to find an explanatory variable x to predict my variable y, assuming both y and x are non-stationary? (I'm using both variables as levels, not differences). If I find a ...
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A time series which doesn't look stationary but pass the ADF (augmented Dicky-Fuller) test [duplicate]
My time series look like this, but it passed the ADF test. Does it make any sense?
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Input/References on time series modeling/prediction without assuming stationarity
I am trying to find out what methodologies there are for analyzing/modeling time series without assuming the time series is stationary/trying to make the time series stationary. I am wanting to do ...
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Higher moments of Vector Auto-Regressive (VAR) process
If we have a VAR process:
$\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}$
With the stability condition ...
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Stationarity in panel data analysis
I'm analyzing a monthly district-wise dataset of 10 years to determine factors influencing dengue incidence. The dataset is not stationary and the dengue incidence shows a seasonal pattern. My ...
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ADF tests: phi2 and phi3 explodes when differencing
ADF tests are used to inform on the order of integration of a time series. With the function ur.df, three different specifications can be used, called "trend&...
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Can We Use Mutual Information to Determine if a Time Series has a Difference Trend or a Time Trend?
Let's say we have a finite real-valued time series (finite subsequence of a realization of a real-valued stochastic process), $X_t$.
To address my question, we make no initial stationarity assumptions....
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Are non-constant polynomial means a special case of seasonality?
In this video, it is said that an otherwise-stationary time series with non-constant linear mean is analyzed by taking the first difference of the time series to produce a new, stationary time series. ...
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Cointegration in quantile regression
I used quantile regression for my research. My variables were significant but my pseudo r was low. So I tested for cointegration.
All my variables are I(1) and when I run the models with raw (...
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Estimation of AR model with Durbin-Levinson
Let $X_t$ be a stationary time series. We have computed $\rho_1 = 0.8, \rho_2 = 0.5, \rho_3 = 0.4$. If we assume that an AR(3) model without a constant is appropriate, get estimates of $\phi_1$, $\...
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Showing stationarity thanks to integrals
Let $X_t$ be a stationary process with spectral representation $$X_t = \int_{\left[-\frac12 , \frac 12\right]}e^{2 i \pi f t}\,dZ(f).$$
Assume that $E\lvert dZ(f)\lvert^2=\lvert G(f) \rvert^2\,dF(f)$ ...
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Find spectral density of a process
Let $X_t = C \cos(2πw t) + D \sin(2πwt) + U_t$ where $C, D$ and $U_t$ are Gaussian random variables with unit variance, and where all the variables are independent, and $U_t$ is white noise. Determine ...
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Monte Carlo simulation to get stationary distribution of a complex system
I was wondering if I could get some help with my problem.
I have a complex Markov chain where I cannot track its transition analytically. Instead, I decided to simulate $N$ numbers of particles for $T$...
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Stationary if and only if Toeplitz matrix
It is known that a stationary process (mean and autocovariance do not vary with respect to time and the 2nd moment is finite for all times) has a Toeplitz covariance matrix.
But is the converse true ? ...
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Gradient vs differences to remove non-stationarity in time series?
When dealing with non-stationary time series (for instance, in auto-correlation analysis), differencing (computing absolute differences between consecutive samples/observations) is often regarded as ...
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How to test for statistical independence on non-stationary time series?
I have multiple time series on which I want to identify statistically significant (if any) trends.
To that end, I started by conducting the Augmented Dickey Fuller (ADF) test to identify which series ...
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For checking stationarity for applying Granger Causality Test, should the range of both the time series be same?
I have 2 time series - one is for scores of people in a survey and the other for the sales of a product for 10 distinct countries. The survey data is categorized into 3 categories for each country. So,...
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Serial correlation requirements for Mann-Kendall test
I want to apply the Mann-Kendall (MK) test to identify relevant trends in multiple time series, however literature on the matter tells me one hard requirement is for no serial correlation to exist in ...
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Geostatistics: Covariance vs Semivariance
I am confused by the following page in Geostatistics for
Environmental Scientists, Webster & Oliver:
My understanding
Given locations specified by a vector $\mathbf{x}$, we assume an underlying ...
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Can an ADF-test distinguish between stationarity and trend stationarity?
I am learning Time series analysis and I am studying the augmented Dickey Fuller test. My understanding of the null hypotheses is clashing with all the explanations I find online and in books. I refer ...
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Rejection of ADF-test for log returns and AIC selected ARIMA(0,0,0) and ARIMA (0,0,0) with a drift?
I use monthly log returns for some stock portfolios and rejects the null of the ADF-test for both. Hereafter I use AIC to select best fitting models using auto.arima in R. The selected models are ...
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First difference of logs of negative numbers causes trouble
I have the following problem: I have a timeseries with the prices for a few futures, which is non-stationary (according to ADF test). If I apply first difference of logs, ADF shows stationarity.
But I ...
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Stationarity and AR(p) representation
I know that covariance stationary processes can always be represented as an $MA(\infty)$ process, i.e.
$$ Y_t = \alpha + \sum_{s=0}^{\infty} \beta_{s} \varepsilon_{t-s}. $$
with uncorrelated ...
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Stationarity of ARMA-like time series
It is well known that for $X_t \sim ARMA(p,q)$ where $\phi(B)X_t = \theta(B)Z_t, Z_t\sim WN(0, \sigma^2)$, if $\phi(z)\neq0$ in the unit circle, $\{X_t\}$ is stationary.
Now assume $\{Y_t, t=0, \pm1, ....
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Differencing, taking logs or squaring does not fix nonstationarity. What to do?
I want to test for correlation in a time series model. However, all four independent variables and the dependent variable are non-stationary. I tried taking first, second and third differences but my ...
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Can a VAR(d) process be strictly stationary?
If a stochastic process is generated by a vector autoregressive process of order $d$, can it be strictly stationary. I know that under the stability condition, that this is weakly stationary process.
$...
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Stationarity in an interrupted time series
I am using proc autoreg in SAS to conduct an ITS analysis and I have a question about stationarity. Proc autoreg is able to ...
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Cross-sectional dependence and second generation unit root tests
Could someone explain me in simpler terms what a cross sectional dependence and panel unit root tests does in practice? How is panel unit root tests different from any other unit root tests such as ...
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ACF and PACF vs Ljung Box test
I have a time series with realized sales prices on monthly basis in a large European city which comes as an index and I would like to do 1 period ahead forecasting.
I have run ADF and KPSS for unit ...
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How to calculate heteroskedastic standard errors
I'm doing curve fitting, but my error is non-stationary. The variance decreases:
I'm looking for a signal in the noise (In this case at x=90, y=50).
I'd like to calculate the "standard error&...
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Why is $\sum_{i=1}^{n} \sum_{j=1}^{n} Cov(X_i,X_j) = \sum_{i-j=-n}^{n} (n-|i-j|)\gamma(i-j)$ for a stationary time series
My book presents the following derivation of the variance of the mean estimator $\bar{X_n} = \frac{1}{n} \sum_{i=1}^{n} X_{i}$ for a stationary process $(X_t)_{t}$ with autocovariance function $\gamma(...
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When is an AR(1) process strictly stationary?
Suppose I have an AR(1) process $X_t=aX_{t-1}+e_t$, where $e_t$ is a white noise with zero mean and finite variance. Under what conditions do I have $\{X_t\}$ being strictly stationary
in the sense ...
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If a strictly stationary process is also independent, does this imply i.i.d.?
Suppose I have a time series process $\{X_t\}$ that is strictly stationary in the sense that the joint distribution of $[X_{t_1},...,X_{t_k}]$ and $[X_{t_1+a},...,X_{t_k+a}]$ are the same for any set ...
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Does $ARMA(p,q)$ process need to be invertible and have a causal stationary solution to be written in $MA(\infty)$ representation?
Does $ARMA(p,q)$ process need to be invertible and have a causal stationary solution to be written in $MA(\infty)$ representation?
And if you write the process in terms of $Z_t$ instead of $X_t$, then ...
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Existence of stationary process with a given ACF
Consider the sequence
$$\gamma(\tau) = \begin{cases}
1 & \text{if } |\tau| ≤ K \\
0 & \text{if }|\tau| > K
\end{cases}$$
where $K$ is a positive integer. Is $\gamma(\tau)$ an auto ...
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What a stochastic process with constant variance is called?
Consider a stochastic process $\left( x_t \right)_{t \in \mathbb{R}}$ with auto-covariance function $k(t, t') = \mathbb{E}[(x_t - \mu_t) (x_{t'} - \mu_{t'})]$ where $\mu_{t} = \mathbb{E}[x_t]$.
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compute the Wold representation of this process
I want to find the Wold representation of $y_t = e_t + \alpha e_{t-1} e_{t-2}$, but I'm having difficulties with the product.
For instance, I tried:
$$
y_t = e_t + \alpha e_{t-1}e_{t-2} \pm e_{t-1}^2 ...
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