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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Stationarity achieved now what?

My data series got stationary at first difference. Now I need to make a new variable with Yt-Yt-1 but how do I accommodate the Trend and intercept as my series got stationary only when these two were ...
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14 views

Can we conclude anything about two time series which have the same order of integration?

Two time series, when tested for stationarity, were found to be of the same order of integration. Can we conclude anything about the two series?
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18 views

raw data stationary but still can see trend and seaonality is stl

So I am looking at unit sales data. I am doing a univariate time series analysis. My data is weekly sales numbers figures, spanning 2012- 2014 (obviously no till end 2014). I first ploted my response ...
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21 views

KPSS test in R interpretation

So in the R package tseries's kpss.test function, it seems I can specify whether the null hypothesis is trend stationary or level stationary(default). ...
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23 views

Unbiased Estimator of Product

Suppose there are stationary times series $\{A_i\}_{i=1}^{T},\{B_i\}_{i=1}^{T},\{C_i\}_{i=1}^{T},\{D_i\}_{i=1}^{T}$, which may not necessarily be independent processes. We know that ...
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43 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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22 views

Is x(t) = ax(t-1) + e stationary or not? [duplicate]

Is $x(t) = ax(t-1) + e$ a stationary process or not? E.g. with $a=0.8$, I would imagine it is not, but if I graph it with random normal error terms it looks stationary.
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10 views

Variance of the second order stationary process's mean

So i have a second order stationary process with the following covariance function $r_X(t) = \alpha e^{- \beta |t|}, -\infty < t < \infty$ Now, $\bar{X} = \frac{1}{T} \int_{0}^{T} X(t) dt$ ...
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45 views

Skewed posterior distribution on constrained parameter space for Bayesian inference of MCMC. Advice on what to do?

I am running a fully Bayesian MCMC procedure to estimate some time series models, and my model has a lot of parameter estimates. In particular, one of these parameters, $\phi$, is $\in [-1,1]$. The ...
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14 views

Are these nonstationary variables?

If I have understood correctly, computing the correlation of two nonstationary variables can lead to spurious results. For example, computing the correlation of two stock price time series would lead ...
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44 views

Stationarity ⇒ homoscedasticity? [duplicate]

If my data is stationary, can I also write that it is homoscedastic? Does stationarity imply homoscedasticity of the data?
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41 views

proving the asymptotic distribution of the mean

Let ${X_t} = \mu + \sum\limits_{j = - \infty }^{ + \infty } {{\psi _j}{\varepsilon _{t - j}}}$ with $\varepsilon$ is a white noise iid with variance $\sigma^2$ , $\sum\limits_{j = - \infty }^{ + ...
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29 views

Stationarity tests in the frequency domain for regression

Strict stationarity is the strongest form of stationarity. It means that the joint statistical distribution of any collection of the time series variates never depends on time. So, the mean, variance ...
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1answer
44 views

Auto correlation function of AR(p) process

I am doing a time series course and in the theory part there are few things I don't understand.In obtaining auto correlation function for AR(p) process it is done as: AR(p)=$X_t = α_1X_{t−1} + ...
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1answer
69 views

Definition and proof of Strict Stationarity

The definition of strict stationarity I'm using is the following: $(X_1,...,X_n)=^d(X_{1+h},...,X_{n+h})$, for any integer h, and positive integer n. I'm trying to prove that ...
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56 views

Should I use stationarity test before OLS regression

I need to know if conducting a stationarity test on the variables, such as the Dickey-Fuller test, is important before doing any regression like OLS? if so, if the variable is stationary after ...
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46 views

mixed results for stationarity tests and structural breaks

Following situation: I want to forecast a time series of the number of trucks on the motorway in some country. Here how the regular week looks like: I have data for 4 years and divide the huge time ...
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21 views

“Iterating”? For MA and AR processes

I am not sure what is being done here, but I keep seeing statements like Given $X_t - \phi X_{t-1} = Z_t$ $...(1)$ then $$X_t = -\phi^{-1}Z_{t+1} + \phi^{-1}X_{t+1}$$ $$ = ... $$ $$= ...
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43 views

What is the source of nonstationarity in this VAR model?

I am trying to forecast a VAR model, which consists out of 5 variables with a monthly frequency. The problem is that the VAR model produces an unstable forecast and I am not sure what the source of ...
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3answers
166 views

How do I detrend time series?

How do I detrend time series? Is it ok to just take first difference and run a Dickey Fuller test, and if it is stationary we are good? I also found online that I can detrend the time series by ...
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1answer
27 views

Stationary function

I am reading Karl Rasmussen's book on Gaussian processes and in the introductory chapter he has the following statement: ...
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1answer
113 views

stationarity in time series

I'm learning a Time series course and I have a few questions. Strictly stationary is a process if the joint distribution of $X_{t1},X_{t2},...,X_{tm}$is the same as the joint distribution of ...
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47 views

A proof for the stationarity of an AR(2)

Consider a mean-centred AR(2) process $$X_t=\phi_1X_{t-1}+\phi_2X_{t-2}+\epsilon_t$$ where $\epsilon_t$ is the standard white noise process. Just for sake of simplicity let me call $\phi_1=b$ and ...
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130 views

How to write an AR(2) stationary process in the Wold representation

I managed to write an AR(1) process in the Wold representation with help from the geometric series. I am having trouble with a stationary AR(2). How could I do?
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1answer
37 views

What is a stationary function?

Snoek et al, have a recent paper "Input Warping for Bayesian Optimization of Non-Stationary Functions" (http://arxiv.org/abs/1402.0929) which mentions "stationary functions". I understand what a ...
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45 views

Panel data models when some regressors are non-stationary - I(1)

My model is $$ Y_{it}=X_{it}'\beta+\varepsilon_{it} $$ where $Y_{it}$ is a vector of weekly observations of a dependent variable and $X_{it}$ is a vector of explanatory variables (also weekly) with ...
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124 views

Does Stationarity for Time Series extend to Independent Variables?

There have been many questions about the importance of stationarity and also its means of calculation here on CV, but one question that I have not seen an answer to is whether or not stationarity (in ...
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27 views

Is it appropriate to run the Error Correction Model on data which are not I(1)?

I have intraday data (frequency = 1 min.) for 6 stocks and 1950 observations per each time series. I checked stationarity for the level data and first difference and it appears that: 5 stocks's ...
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28 views

Perron's test which allows testing for the broken trend stationarity in R: the code and application?

I am trying to run the Perron's test in R, which allows testing for the broken trend stationarity. I was trying to find the package and code which is running this test, but without success. While ...
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16 views

Unit root for bounded variable

I have a time series which is discreet and bounded (between 0 and 100). It’s actually also increasing over time, from low to high. When I test for a unit root, I find it. And yet I wonder if unit ...
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19 views

AR models on non stationary data

i am currently reading Diebold and Li's 2006 paper: Forecasting the term structure of government yields where the authors fit, albeit simple, AR(1) models on clearly non stationary data. Why is this ...
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27 views

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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15 views

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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29 views

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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57 views

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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10 views

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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39 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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45 views

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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1answer
42 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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163 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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328 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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38 views

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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55 views

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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1answer
51 views

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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487 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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179 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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120 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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156 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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24 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?