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 of a variable measured at irregular intervals

I have data collected on a meeting-by-meeting basis, where the change in time between two meetings is not constant, i.e., $\Delta t\ne1$. Are ordinary Augmented Dicky Fuller and Phillips-Perron tests ...
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1answer
27 views

Seasonal data deemed stationary by ADF and KPSS tests

I have got two time series and I want to evaluate a VAR model. For this, it is necessary that both time series are stationary. Using R, I have found periodicity ...
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12 views

Question about trend stationarity

I am using Matlab in order to test stationarity of macroeconomic time series such as GDP, PCI or population. My first question is the following: I start by testing the stationarity of the data: I ...
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23 views

Does this series have a unit root?

I am trying to figure out whether my time series is stationary or not. In order to do so I run four different tests: the three test from the Dickey-Fuller test (standard, drift and trend) given by the ...
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1answer
32 views

Can we use Box-Ljung as a stationarity test for time series?

It's all in the title, I know that we usually use Box-Ljung to test the randomness in a time series (independence of residuals), but I found this post about how to tell if a time series is stationary ...
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90 views

Identifying lagged effects / Distributed Lag Model

I would like to create a linear distributed lag model in order to do some forecast and also being able to interpret the results. Unfortunately I'm a bit confused with the process I should follow....
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10 views

Minimum sample size to check second order stationarity

I can understand second order stationarity refers to weakly stationary processes in which mean, variance, and autocovariance do not depend on time. But is it possible to check weak stationarity/second ...
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15 views

When does the law of large numbers hold for RVs from a distribution with infinite variance?

Under what conditions does the law of large numbers hold (or fail, if that is easier to describe) for independent identically distributed random variables drawn from a distribution with a finite mean ...
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20 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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25 views

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
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 ...
3
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1answer
34 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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37 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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32 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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22 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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45 views

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
41 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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8 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
29 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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26 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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81 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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17 views

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
22 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
32 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
52 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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26 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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29 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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43 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
81 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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18 views

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
48 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
47 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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18 views

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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44 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
40 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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36 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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44 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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20 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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28 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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83 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
79 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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28 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
22 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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23 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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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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48 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 ...