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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Weakly stationary time series - what type of model is this?

Say I have the following model: $y_t = 0.5y_{t−1} +x_t +v_{1t}$, and $x_t = 0.5x_{t−1} +v_{2t}$, where both $v_{1t}$ and $v_{2t}$ follow IID normal distribution ∼ (0,1). How would I go about showing ...
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78 views

Weakly stationary time series questions

Say I have the following model: $y_t = 0.5y_{t−1} +x_t +v_{1t}$, and $x_t = 0.5x_{t−1} +v_{2t}$, where both $v_{1t}$ and $v_{2t}$ follow IID normal distribution ∼ (0,1). The following statements ...
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Stationarity of the TGARCH

I'm going through "GARCH models" by Francq and Zakoian (2010). They define the TGARCH(1,1) as $$\sigma_t = \omega + \beta_1 \sigma_{t-1} + \alpha_{1,+}\epsilon_{t-1}^+ - \alpha_{1,-}\epsilon_{t-1}^- ...
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1answer
13 views

Issue with forecasting a time series that is a white noise using ARMA

I am working on stock prices and stock returns and I'm supposed to do some forecast on these data. The stock prices series is not stationary and even if the stock returns series is, it is a white ...
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31 views

Stationarity of subsample

Consider that I have a weakly stationary series for the period 2003M1-2014M12. I want to make a VAR model for the subsample 2007M1-2014M12. Should I reconsider the weak stationarity of my series, so ...
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1answer
37 views

Interpreting the Dickey Fuller test

I can't seem to make sense of the following results. The time-series looks more non-stationary than stationary, and when I fit an ARMA(1, 0, 0) it estimates the AR(1) term to be very close to unity ...
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19 views

Should I use a VAR or VECM to model a time series in which one variable is stationary?

I was wondering which of the above two models would be most suitable. I'm modelling the relationship between interest rates, unemployment, CPI, GDP and London house prices (all UK data). I was going ...
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23 views

What do you think of this correlogram?

do you think this weekly data is stationary? Unit-root test indicates rejects null of non-stationary (rejects null of unit root). Thanks for your input.
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41 views

Correlation & Stationarity

In a purely historical or backward-looking, descriptive context, is it incorrect to naively compute the covariance $$ \mathbf{E}(XY) - \mathbf{E}(X)\mathbf{E}(Y) $$ and Pearson correlation coefficient ...
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1answer
24 views

Stationarity Testing on different time series data

For linear regression modeling, I have macroeconomic data that goes from 1985-2016 which i will use as my independent variable. My dependent variable data ranges from 2002-2016. My question is for ...
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44 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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Engle and Kozicki (1993)'s Serial Correlation Common Feature (SCCF) test

I have two auto-correlated stationary time series of I(o). I want to look for a common feature in them as per Engle and Kozicki (1993). Specifically I want to see if there is a linear combination of ...
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1answer
44 views

Does stationarity imply absense of upward or downward trend?

I wonder if a time series being stationary implies that there can be no upward or downward trend. It appears to me that such an implication should hold, since in order to be stationary a time series ...
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1answer
42 views

Does autocorrelation imply stationarity?

Let $$ \begin{aligned} y_t &= a + bx_t + u_t, \\ u_t &= \phi u_{t-1} + e_t \end{aligned} $$ where $ e_t$ follows a White Noise process. Let Breusch-Godfrey LM test statistic be strictly ...
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First difference of non-stationary - does the prediction accumulate the errors?

I am modeling a non-stationary process (I(1) actually), it looks like this: I have 146 data points (monthly data). The ideal model in my case should have: Macro-variables sensitivity Predict the ...
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23 views

How to code dummy variables for structural breaks in VAR

This question is really 2-in-1: 1) How do I code dummy variables for the following series that has 2 structural breaks in trend; an initial upward trend, then a much flatter upward trend, then ...
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26 views

Unit root tests ambigous - is time series stationary?

I am testing a time series (quarterly) for stationarity. However, using the KPSS test, the ADF test and PP test, I get different results (ADF and PP reject non-stationarity, KPSS rejects stationarity, ...
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32 views

Help discussing stationarity using correlograms? ARMA/ARIMA modelling

I am currently trying to understand how to use correlograms to examine stationarity and analysis the appropriate models. Please can you advice, below I have included my ACFs and PACFs, and I am trying ...
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30 views

Johansen Test - Full rank but variables are not I(0)

I have a model of four implied exchange rates which are based on stock prices denominated in two currencies. They should all be cointegrated based on the law of one price. Each of the series is ...
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16 views

augmented dickey fuller statistic

I am trying to write code for an ADF test in C++. As I understand it, the Dickey-Fuller statistic is a modified version of the t-statistic. Currently, I have functioning code that calculates a t score ...
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1answer
66 views

How can this time series be stationary?

I have the following irregularly spaced time series. The related autocorrelogram is: and I run the following tests: ...
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45 views

Find a time series representation in terms of {Xt} for a process {Yt} whose spectral density is fY (·) = 2f(·)

Suppose you are given a set of data $x_1, \dotsc , x_n$ from a time series process $\{X_t\}$ that has spectral density $f(\cdot)$. Find a time series representation in terms of $\{X_t\}$ for a ...
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27 views

Find all possible values of c that makes the following an autocovariance function?

p(h)= { 1 at h=0; c at h= 2 and -2 ; and 0 otherwise I'm not sure how to go about this because this is all we are given, any tips would be extremely helpful!
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71 views

Writing an undergraduate Econometrics project on Okun's Law - omitted variable bias?

I'm writing my first empirical piece on the validity of Okun's Law in the US and UK over a 40 year period back to 1975, using annual time-series data. I am still in the planning stage of my project, ...
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17 views

Expressing a non-unique stationary distribution for a markov chain?

I am working ahead of my stochastic processes class, so some of what is written below may be inaccurate. I am working a problem that asks me to compute a stationary distribution for a Markov Chain, ...
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67 views

How can I represent exogenous I(0) variable in VECM / Cointegrating relationship?

So I am doing an econometric study on Bitcoin's USD price. As an exogenous regressor, I have the total number of bitcoins in circulation (nbtc). The graph looks like this: This variable is: ...
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22 views

How do I detrend this (non-linear) time series before differencing?

I have a variable that looks like this (after taking natural log): I can't run stationarity tests as there are clearly structural changes in the data. How can I get around this – am I supposed to ...
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1answer
14 views

Stationarity GARCH(p,q) model

Why is it that a GARCH(p,q) model is second-order stationary iff the sum of the coefficients $\alpha_1, ..., \alpha_q, \beta_1, ..., \beta_p$ are less than 1 while, as against the usually stronger ...
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17 views

ARMA stationarity conditions

Are ARMA(p,q) model stationarity conditions are the same as for AR(p) or some additional requirements must be met?
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44 views

How to show that the signal $x_n = Acos(\omega n)$ can be fully predicted by a system with two weights $w_1,w_2$

I am trying to solve the following exercise: Show that the signal $x_n = Acos(\omega n)$ can be fully predicted by a system with two weights $w_1,w_2$ (i.e. $x_n = w_1 x_{n-1} + w_2 x_{n-2}$). ...
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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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Non stationary 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 ...
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44 views

Testing for stationarity

We know that the definition of stationarity (either weak or strong) of a random time series involves having the same joint distribution or statistic (like mean or variance) for "any" set of time ...
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55 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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25 views

How can I make these time-series data stationary?

I'm working on eventually getting to a Granger-causality (or VAR) test on my independent variable(s), but first I need to sort out my dependent variable, which is Bitcoin's USD exchange rate ...
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78 views

online detection of plateaus in time series

I need to detect plateaus in time series data online. The data I am working with represents the magnitude of acceleration of a tri-axis accelerometer. I want to find a reference time window that I can ...
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19 views

Non Stationary highly correlated lagged data vs stationary uncorrelated lagged data

I'm doing a time series forecast, and I kind have reached a roadblock. This is kind of urgent so even small inputs will be greatly appreciated. So, I have data of transactions, I take first ...
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18 views

Confused by definition of stationary stochastic process

Borrowing heavily from definition of stationary stochastic process here, I am having a hard time understanding why $$F(X_{n_1}, X_{n_2},...,X_{n_k}) = F(X_{n_1+n},...,X_{n_k+n}),$$ for every $k \ge ...
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13 views

simulating a stationary markov chain approach

I have hard time simulating a markov chain stated here: Consider a set of N transformations T= {t1,...,tn} . Let S={s1,...,sL} ...
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49 views

Does ARMA-EGARCH require a stationary time series?

For modelling the conditional variance of a real exchange rate by an ARMA-EGARCH model, should the real exchange rate series be stationary?
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1answer
73 views

Is a Markov chain with a limiting distribution a stationary process?

From Wikipedia: a stationary process (or strict(ly) stationary process or strong(ly) stationary process) is a stochastic process whose joint probability distribution does not change when ...
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103 views

Transforming a time series so it is stationary

I am not sure if I am undertaking the following steps correctly. I am trying to make this time series stationary: as you can see, it is decreasing: In the beginning I tried taking log, but it is ...
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4answers
196 views

Time series analysis: since volatility depends on time, why are returns stationary?

I run Dickey Fuller test in order to know if stock returns are stationary. I get that no matter which stock I take, his return is stationary. I don't know why I get this result since it is clear that ...
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1answer
74 views

Modelling stationary and integrated time series in one system

I am currently investigating commodities and their impact on the oil price. I have 8 variables of different stationarities $y$ = dependent variable (oil price) is non-stationary I(1); three ...
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40 views

A time series which cannot be made stationary

I have a series of data points, of days between successive earthquakes. I am trying to model time between earthquakes, in order to predict waiting times. Using Minitab I have tried differencing, ...
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Rules of integrated series and balanced regressions

Background There are various rules of linear combinations of integrated series. Let's just consider the $I(0)$ and $I(1)$ cases. For example, if $x_{t} \sim I(1)$, $y_{t} \sim I(0)$, then $ax_{t} + ...
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Vector autoregression for mix of stationary and nonstationary variables

I am currently investigating the impact of certain indicators such as GDP and inflation on the stock market. However some of my variables are non-stationary and some stationary in levels. All ...
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20 views

Example and simulated path of strict(ly) stationary process

There are many question on stationary process but very few related to strict stationary process. I am just looking for an example and simulated path of strict stationary process and how strict ...
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36 views

Augmented Dickey-Fuller test and Differencing (R)

I'm using the ADF test to check for stationarity of two variables, using the ndiffs function in R. ...
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1answer
41 views

Is there an unambiguous stationarity test for time series?

It seems to me that the time series plot, the correlograms (ACS, PACS) and the "autocorrelation check for residuals test" can all be subject to interpretation. (I am using SAS 9.4) Is there an ...