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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Trend or stationary data [on hold]

I would to ask about acf data pattern that has shown from minitab output. Can I determine that the data pattern has trend? I'm still confused about how many lag should be a criteria for trend or ...
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19 views

Disadvantageous of non stationary process

If the process is not stable, as i konw the prediction will be impossible and the mean and variance will be instable or even infinite, and may be there is a correlation between variables. Are there ...
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Stationarity of independent variables in ARIMAX

I am running an ARIMA model with exogenous variables. Do all my exogenous variables have to be stationary or is it okay if one of my exogenous variable is non-stationary?
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1answer
72 views

TSO error - non-finite value supplied by optimum [closed]

I'm trying to detect outliers within a financial time series which represents the ratio of cash distributions to equity holders as a percentage operating earnings for the period. Visual inspection ...
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1answer
30 views

When would the data set be stationary?

I am not a statistician or mathematician and I need some help. I did three experiments as I labeled on the figure, 3,4,5, each experiment has x and y results. The markers are the real data and the ...
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1answer
30 views

Causality and stationarity of AR models

Studying AR models, I found that there are two properties that these models can have stationarity and causality. For what concerns stationarity, I have studied that this condition is satisfied if ...
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33 views

predicting the future in a stationary stochastic process

Let's say I have a strictly-stationary stochastic process with known PSD (power spectral density). The process has been running, and I have all the data from time $t=-\infty$ to $t=0$. I want to ...
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26 views

Contradiction in the ADF (Augmented Dickey-Fuller) and KPSS (Kwiatkowski–Phillips–Schmidt–Shin) tests for financial time series

I use the ADF and KPSS to test for stationarity / non-stationarity of price increments in financial time series. The two test applied provide different results for low lags, but the same result for ...
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17 views

Cointegration test of stationary variables with structural breaks

I am trying to implement test of cointegration (Johansen trace and eigenvalue) between 2 variables that are stationary at first difference but after Zivot-Andrews test of unit root I found they have ...
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R: Stationarity/non-Stationarity: implementing a solution

I have an Augmented Dickey Fuller Test in R on leg_totalbills that shows I can not reject the null hypothesis: Unit Root. The ...
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1answer
30 views

R: Box.test vs adf.test vs kpss.test

I stuck in checking my Time Seies for stationarity with several tests: ...
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7 views

Fractional Gaussian noise, the KPSS test, and stationarity

Fractional Gaussian noise (fGn) is characterized by the mean ($\mu$), the standard deviation ($\sigma$), and the Hurst index ($H$). It's my understanding that it is stationary, for the simple reason ...
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1answer
28 views

How does trend stationary recovers from shocks in long run?

I was trying to understand difference between drift and trend wherein I came across concepts of unit roots and trend stationary. (I haven't read any books on time series, just going through web). ...
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25 views

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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80 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 are ...
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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
18 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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1answer
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
42 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 (0....
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22 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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27 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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42 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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1answer
57 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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45 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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45 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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28 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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2answers
36 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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34 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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33 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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19 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
67 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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82 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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1answer
70 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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25 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
17 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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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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45 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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46 views

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 zero-...
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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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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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27 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 (http://...
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4answers
86 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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20 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 ...