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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Identify the stationary time series

Identify the stationary time series for which $$ \gamma(h) =(-1)^{|h|}+\cos \left(\frac{\pi}{4}h\right)$$ is ACVF. This is a homework problem. Stuck at first level. Please give some hints. Thanks in ...
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Unit Root Test and rolling data

Please could you kindly advise on the implication of testing for unit root (using the augmented dickey fuller approach) on rolling data. My feeling is that this wouldn't make much sense given the non ...
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Cointegration between two stationary processes

I have two stationary time series. I would like to check for cointegration between them. Does this make sense, and can I just use Engle-Granger Test (two step) for Cointegration for this?
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How to stationarize profit and loss data with an increasing variance and large negative values for time series analysis?

PnL can take large negative values, and its variance increases over time as the firm grows. If we do differencing, an increasing variance remains. If we take log, negative values cannot be defined. ...
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Mix of I(0) and I(1) in VECM

Is there an official academic journal I could use as reference which states that a mix of I(0) and I(1) variables can be used in constructing a VECM?
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Retreiving Integrated Fitted Data from Stationary Fitted Data

Note that this is a simplified example: I have some time series that I made stationary by differencing twice. Then I ran arima on it, and set d = 0 to prevent ...
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34 views

Which models require stationarity?

Regression models (at least up to GLMs) do not traditionally require stationarity (although the requirement for the residuals is even stronger than stationarity). ARMA-style time series models seem ...
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Can I difference after fitting a time series regression model?

Suppose that I have a time series that exhibits a notable trend, and I want to test a hypothesis that a second variable is related to that trend. I fit a linear regression model with that second ...
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39 views

Inverse Differencing and ARIMA Model Equivalence

I've developed a ARIMA model with exogenous variable. Before fitting the model, I made every time series stationary by differencing (each variable had a different order of integration). For ...
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128 views

How to compare time series with cyclical data, and describe any changes or trends

I have a bunch of time series where the data has a natural (known) cycle, for example daily or annual (or both). Here is an example (this is 6 years worth of temperature data sampled hourly): ...
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88 views

Difference between series with drift and series with trend

A series with drift can be modeled as $y_t = c + \phi y_{t-1} + \epsilon_t$ where $c$ is the drift(constant), and $\phi=1$ A series with trend can be modeled as $y_t = c + \delta t + \phi y_{t-1} + ...
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Transforming time series of different time horizon to stationary

I have a list of monthly time series data with different time periods and different order of integration. I want to transform them all to stationary and a same time period. I noticed that the order ...
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Stationary and ADF for unit root

If I calculate the returns in my time series as follows, $\ln(P_t / P_{t-1})$, where $P_t$ is the price at time t and $P_{t-1}$ is the price at $t-1$ then I do the ADF test, my $H_0$ is rejected. ...
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what if I know my time series are cointegrated over a long period but not over a short period?

I am regressing time series on time series. I have tested for cointegration on the entire time sample (3 years) and the series are cointegrated. I need to make a rolling window of regressions (to ...
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226 views

How can I show that a random walk is not covariance stationary?

How can I show that a random walk ($y$ follows a random walk) is not covariance stationary? I tried to work on the formula below (with no results) Could you give me just a hint on how to proceed ...
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137 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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43 views

GARCH(1,1) regression in Eviews

I'm having a problem in doing a GARCH(1,1) regression. I'm trying to regress gold prices serie on stock returns series as in the following equation in eviews: $$ r = c(1) + c(2) \cdot s + c(3)\cdot ...
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101 views

Where can I find ADF test library or source code from c#

I would like to test for stationarity in cointegration. I intend to use an augmented dickey fuller test. However, I need one for c# - either a library or the source code. Or is your have source in a ...
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22 views

What does it mean to build a so-called mean equation?

A series is denoted by Xt. How do we build a time-series model for Xt that is the "mean equation" of Xt? Is this another way of asking to build a stationary time-series?
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condition for a ARMA process to be wide-sense stationary

For a ARMA process, some (e.g. in Tsay's Financial Time Series) said: it is wide-sense stationary, iff all the roots of its AR characteristic polynomial are greater than 1 in magnitude. This is ...
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58 views

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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50 views

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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84 views

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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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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98 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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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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70 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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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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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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29 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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161 views

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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203 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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186 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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62 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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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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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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176 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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84 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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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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157 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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1answer
190 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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445 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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244 views

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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254 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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107 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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201 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?