A stationary process (or time series) is one whose joint distribution is constant over time. A weakly stationary process or series is one whose mean and covariance function (variance and autocorrelation function) are constant over time.

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Understanding stationarity with Inflation

I am looking at the link between inflation and insolvencies for an econometrics project. I have the raw quarterly insolvency data and raw quarterly CPI data for the UK (roughly 100 samples) from ...
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When an ARMAX model is stationary? Why stationarity or invertibility is needed?

Let $y_t$ a stochastic process and $\tau_t$ presents the time duration between the $t$ and $t-1$ event.The ARMA(p,q,r) with exogenous variables is defined as: $$ y_t = \varepsilon_t + ...
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How to test for wide-sense stationarity with only one sample path of the process?

I have a univariate time series consisting of 70,000 observations (power consumption of a building) over equal time increments (15 minutes). How do I check whether this realization is wide-sense ...
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Lag order in ur.df or adf test [R]

I want to test stationarity using adf test or ur.df function on R ur.df(y, type = c("none", "drift", "trend"), lags = 1,selectlags = c("Fixed", "AIC", "BIC")) My question is when using adf.test the ...
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28 views

Non-Stationarity, first differences and Panel Data

I have build a sentiment index and am now validating its statistical properties and significance. I stumbled upon two problems. (1) dfglsindicates that my ...
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84 views

Does covariance stationarity lead to mean stationarity necessarily?

Traditionally a weak stationary process is also called covariance stationary, but those 3 properties are exposed: $$E[Xt] = μ , \forall t$$ $$var(Xt) = \sigma^2, \forall t$$ $$cov(Xt, Xt−j) = ...
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16 views

Test for stationarity in an unbalanced panel

I have an unbalanced panel model and I need to check it for stationarity. So, I need to perform a Unit Root test (I think I will use a Fisher Type Test?). But I am a bit confused whether (1) I need ...
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44 views

Time series and stationnarity tests

I perform some time series fitting with the help of the forecast and urca packages. I have a question regarding the correspondance between results coming from statistical test such as KPSS, ADF or ...
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35 views

When is the autocorrelation function of a stationary process decreasing/nonincreasing? Markovian?

When is the autocorrelation function of a stationary process strictly decreasing or nonincreasing? Can being Markovian make it true? When is the autocorrelation function of a stationary process ...
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11 views

Dickey-Fuller for 2D signals?

I would like to run a stationarity assessment test on a 2D signal. Is there a suitable test, e.g. a 2D DF or ADF, etc? Thanks. Edit: A 2D signal is a signal defined on a lattice, i.e. instead of a ...
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24 views

Checking whether a given sample is stationary

Let $x[n]$ be some time series in 1D or $x[m,n]$ in 2D, of length $N$ (resp. $N^2$) How can I assess whether it is stationary? At least in the weak sense. I can check whether the stdev remains ...
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How to tell stationarity from a sample path?

Given a sample path, we can roughly tell whether the mean changes over the time, and, when it doesn't, whether the deviation from mean changes over the time. (Correct me if I am wrong.) But that is ...
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Can nonstationarity be told from the autocorrelation function?

Here "stationarity" means the first and second moments don't change over time. From a page of Time Series: Theory and Methods, by Peter J. Brockwell, Richard A. Davis In this chapter we shall ...
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101 views

How to simulate only stationary AR(1) with φ = 0.9?

I am interesting in simulating AR(1) processes with 0.9 parameter and n = 10. The itterations should be 10000. When I was trying to run the program it gave me an error in the estimation procedure. ...
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55 views

What are the stationarity requirements of using regression with ARIMA errors for inference?

What are the stationarity requirements of using regression with ARIMA errors (dynamic regression) for inference? Specifically, I have a non-stationary continuous outcome variable $y$, a ...
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31 views

Estimator of autocovariance in a wide-sense stationary process

From Wikipedia http://en.wikipedia.org/wiki/Ergodic_process One can discuss the ergodicity of various properties of a stochastic process. For example, a wide-sense stationary process $x(t)$ has ...
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30 views

A question on transfer modeling for the intervention analysis of time series data

When reading the section of intervention analysis of time series, I have one question regarding the following descriptions. The following graph defines several response patterns for step function ...
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14 views

Is k-th difference method suitable for making the data stationary?

I have a huge data set which is non-stationary. (Checked non-stationarity with unit root test). I'm wondering how can I make my data stationary. I need stationary series because I want do some ...
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101 views

Pros and Cons: Methods for Detrending Time Series Data

My memory is fuzzy on the advantages and disadvantages of various methods for detrending time-series data. I'm looking for a succinct summary of why and when one should or should not use the ...
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55 views

Relationship between average and characteristic function of a Gaussian process

I'm having trouble understanding an equality given in a book ("Speckle Phenomena in Optics" by Joseph Goodman p.145) for a zero mean, stationary Gaussian process: $\overline{\exp(i ...
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60 views

Stationary Process in Plain English

How would you describe stationary process in plain English to someone with no mathematical background, using real life examples? The target audience is adults with reasonable intelligence, but most ...
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106 views

Implementing Neural Network for time series

I am currently working on neural networks for time series forecasting, my doubt is do we need to account for issues like trend,non stationarity and seasonality while using neural networks as opposed ...
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171 views

How can I check if two numbers are equal (with some allowed error)?

I have a lot (about 10000 or more) measurement results. I measured the performance of different algorithms (it doesnt really matter which algorithms for now on). I want to check if my measurement ...
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126 views

Simple Average Method, calculating moving average and moving variance, how can I say if stationary or not?

Lets say I have a sample data (here is just 10 numbers, in real I have about 10000 measurement results). Then, I want to check if the data is stationary or not using Simple Average Method. For ...
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Which one of these looks stationary?

Step 1. To answer "Final Question" ( linked: "THE FINAL QUESTION : Order of differencing, to achieve stationary and interpretation of arima() , acf, pacf?") Expecting to find correct order of ...
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133 views

Weak stationarity and ARMA-ARCH/GARCH models?

I am slightly irritated about weak stationarity in connection to ARCH/GARCH models. I do not know the answer and I am not sure about it: The basic question is: Do we have to test weak ...
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Differentiate a ARMA(p,q)-GARCH(p,q) or not?

I'm confused about the stationarity condition. I'm fitting a ARMA to a time series that might not have a constant mean. When I fit a ARMA model to it the residuals looks stationary. Do I have to ...
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86 views

What is the real meaning of null hypothesis in unit-root test for a AR(p) process?

There are functions in R (e.g., PP.test and adf.test) which have null hypothesis of unit-root in the process ($H_0$: there is a ...
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128 views

Is non-stationary AR(p) process constant in mean?

A non-stationary $AR(1)$ process, which is a random walk, is constant in mean, but not constant in variance. How about the other $AR(p)$ processes with the order $p>1$? Are they constant in mean?
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72 views

How to explain this unit root process?

I have a time series $X_t$ (shown below) with a structure break. The stationary test kpss.test() says it has a unit root. How to explain this? Why does $X_t$ have a ...
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229 views

Unit root tests: how to decide if to include a trend and/or a constant

Applying a test to univariate time series data for checking if the series has a unit root or not, one is faced with a decision if one would like to test if the series is stationary around a constant ...
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208 views

Non-Stationary Time Series Forecasting

Suppose I have a non-stationary limited data. Do I have to make it stationary before making forecasts? Can I use exponential smoothing, moving averages or even Holt Winters methods without making my ...
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65 views

Is this a stationary time series?

I have a wait-time time series for 10 weekdays(2 weeks) with 10 minutes intervals. I'm having hard time to interpret this? Is this stationary? I also applied Philips-Perron Unit root test and I got ...
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56 views

Stationarity in OLS time series and asymptotic properties

I think I lack somewhat deeper understanding of this topic, but I thought stationarity is required in order for OLS to have asymptotic properties. "But stationarity is not at all critical for OLS to ...
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82 views

Handling stationarity issues in proc ucm/state space time series models

Hope I'm able to find someone who can answer this question. The previous one didn't get answered! Proc ucm is the SAS implementation (using state space concepts) to isolate the unobserved trend, ...
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22 views

Is is logical to have AR terms in stationary series?

Ok so the theory says that ARMA should be applied to stationary series only. By definition stationary process is one with constant mean and variance. However if we use AR terms to model the series ...
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51 views

ADF Test on Volatile Series. Expected results?

I tried performing the Augmented Dickey-Fuller Test on SP500 data from MASS library and got the following results: ...
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94 views

$R^2$ from a regression of two trend-stationary processes, $Y_t$ and $X_t$

In Estimation and Inference in Econometrics, by Davidson and MacKinnon, p.671, they claim that $R^2$ from a regression of $Y_t$ on $X_t$, where both time series are trend stationary, tends to 1 as $n$ ...
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Problem defining ARIMA order

This is a long post so I hope you can bear with me, and please correct me where I'm wrong. My goal is to produce a daily forecast based on 3 or 4 weeks of historical data. The data is 15 minute ...
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38 views

Stationarity of $\nabla x_{t}$

How should I prove that for a given series $x_{t}=\frac{3}{2}x_{t-1}-\frac{1}{2}x_{t-2}+w_{t}$ that is non-stationary, $y_{t}=\nabla x_{t}$ is stationary? I attempted to get an equation based on the ...
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200 views

Explaining augmented Dickey-Fuller regression output

I have monthly returns data going back to 1991 and I'm trying to work out if the data has a tendency to mean revert over time. In order to work this out I've used the Augmented Dickey Fuller test on ...
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133 views

Downsampling stationary time series data, effect on variance

Suppose I have stationary time series data, like this: The time series has Gaussian noise around a true mean. If I then take windows of N samples across the series and average them to generate a ...
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26 views

What methods can be used to compare the consistency of the distributions sampled from independent Markov Chains?

I have a set of independent random walkers on a graph that produce a distribution of the nodes in that graph. I would like to test whether the random walkers have produced a distribution which is ...
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197 views

How to determine correlation between stationary and non-stationary time series

I have three time series of economic data based on quarterly observations; A, B and C, and I would like to ascertain the correlation (or not) between A and C as well as the correlation between B and ...
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82 views

Stationarity and homoscedasticity in break tests

Let me start by saying that I'm really not an expert in statistics/econometrics. Now to my questions: I have a data set containing weekly prices of a stock. I want to check whether a structural break ...
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202 views

How to interpret the result of Fisher's unit root test [duplicate]

Following are the results from Fisher-type unit-root test for RDI (dependent variable). How do you interpret it? ...
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207 views

Why is “stationarity” assumed in time series data?

A stochastic process is composed of a sequence of random variables ordered by time, and a time series is just a realization of such a process. The book that I'm reading says: "if we assume ...
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Stationarity - assumptions and examination

I am examining rodent captures on six permanent rodent trapping grids measuring 150 x 150 meters and consisting of 121 trap stations evenly spaced 15 meters apart. There are six such trapping grids ...
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Time series: one I(1) and one I(0) variable, should I use VAR/VEC, test for cointegration?

Like the title says, I've got two time series, one is stationary to begin with and thus has no unit root, the other time serie is stationary after one-time differencing. I want to create a model out ...