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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Where is the dominated convergence theorem being used?

I am trying to fully understand the proof of a theorem, I only have a problem with the application of the dominated convergence theorem. For the sake of completeness I will upload the whole statement ...
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53 views

Property of the autocovariance function in time series

In the framework of time series analysis Why does $\lim_{n \rightarrow \infty} n^{-1} \sum_{|h| <n} |\gamma(h)| = \lim_{n \rightarrow \infty} 2|\gamma(n)| $? The LHS (left hand side) sequence of ...
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30 views

When is a Ljung-Box test significant?

I have trouble understanding the output of the Ljung-Box test due to conflicting information: The R documentation doesn't actually say how to interpret the output. This site states that small ...
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11 views

What is the relationship between correlation and coherence?

What is the relationship between correlation (autocorrelation, cross-correlation, etc) and coherence?
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5answers
307 views

Is it always required to achieve stationarity before performing any time-series analysis?

For example, I know that for ARIMA models stationarity needs to be achieved. What about Exponential Smoothing? Is it also required?
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1answer
48 views

Explosive processes, non-stationarity and unit roots, how to distinguish?

I understand that if we have a simple model such as: $Y_t$=$\rho$$Y_{t-1}$+$\epsilon_t$ where $\rho$ is less than one in absolute value then we have a stationary process. If $\rho$ equals one then ...
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41 views

Which is the best criterion for DF-GLS lag selection?

When you have an output such as this in Stata for dfgls: ...
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1answer
25 views

Is there an optmal lag choice in the KPSS test?

Is there an optimal lag choice in the KPSS test in Stata? For instance, in my example below, for some lags (less than 7) you reject the null for any level of significance. But afterwards, that does ...
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1answer
28 views

How do I test the stationarity of data using minitab? [closed]

I am working on a time series and trying to fit ARIMA to predict future values.However, I am facing trouble with finding out whether the data is stationary or not.
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2answers
50 views

Stationarity after differencing

I have the following two processes: \begin{align} x_t &= x_{t-1} + u_t \tag{1} \\ x_t &= {\beta}_0 + {\beta}_1t + u_t \tag{2} \end{align} Differencing once leads to: \begin{align} \Delta ...
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2answers
90 views

Interpreting results of KPSS test in R

I've been trying to create an ARIMA model however, I'm not sure how to determine if the data is stationary or not. I preformed a KPSS test in R using kpss.test from ...
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1answer
55 views

Interpretation of VAR and causality

I have two time series(X1 and X2) each having 900 records. I wanted to establish relationship between them and put it in ...
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36 views

Does a stationary process necessarily have to be mean-reverting?

I wonder about if a stationary process is by definition mean-reverting too. I know the formal definition of a stationary process, but I'm not sure about the definition of a mean-reverting process. ...
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29 views

How to find if there is a trend in a time series and stationarity

I would like to conclude on a given time series that if it has Trend or not. I have carried out a cox-stuart test in R and have decomposed to inspect the series visually but still a bit confused on if ...
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2answers
41 views

Intuition for auto-correlation for mean reverting process

How should my auto-correlation plot look like for a mean reverting process? From what I have recently learned, auto-correlation should be low and should decay fast enough. But when I run the following ...
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1answer
31 views

Stationary dependent variable

When running a time series, the Dickey-Fuller test of the dependent variable is statistically significant, meaning that it is stationary (which is also confirmed by looking at a plot of the variable). ...
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45 views

How can I determine if a time-series is statistically stable?

I have time-series data that tracks the number of sydromics records my organization receives each week. The number of records had been steadily increasing as more organizations started sending us data ...
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38 views

Why and when stationarity is achieved by decomposition rather than differencing in ARIMA model

I would like to understand relationships between variables by which cross-correlation function, that means what is the extent one variable influence the other one. ARMA model is used to fit two ...
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23 views

Good TS fit but no stationarity

I have yearly time-series that I want to predict, and for that I fitted an ARX (auto-regressive with a exogenous input) model to previous years (training set) and test it for the last year. My ...
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53 views

Does stationary data need to be normal?

So I already ran some tests to make my data stationary. Differencing and box-cox transformation in particular. According to the augmented-dickey fuller test, after performing the above mentioned ...
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24 views

Does the pre-test assignment of values to the pre-sample periods have any negativity on ADF test (that uses common sample)?

Reproducible example added: Tech info: In all of the lag selection procedures in econometrics, same sub-sample must be used to determine the correct optimal minimum lag. The question ...
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1answer
64 views

Regression in levels vs. regression in differenced form

I want to compute the following regression using R. lm(EurOis3~EurepOis3+Vstoxx+log(Open.Market.Operations)+CDS). I am using daily data(i.e. I have 5 observations ...
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39 views

Regression with differenced variables

This is my data frame containing some interest rates as well as the amount outstanding of open market operations. Now I want to regress EurOis3 on ...
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54 views

Unit roots and order of differencing

I'm studying the stationarity with unit root tests and the order of integration in time series $\ln(x)$ and $\ln(y)$ found here. I'm using Dickey-Fuller test with constant but no trend. From what I ...
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49 views

The constant term after 1st differencing

My instructor stated that when the dependent variable is 1st differenced, the constant term represents the deterministic change or trend in the dependent variable. When I search for information ...
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1answer
48 views

ADF test results confusion

When I ran ADF test with my data set, I got following results. I am confused about why "alternative hypothesis" is always (even for real non-stationary series) showing as "stationary"? ...
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1answer
50 views

Difference between random walk and process integrated of order one?

I know that an $I(1)$ process becomes stationary after differencing once. However, I somehow always equated that to its being a random walk because say having a unit root process like \begin{eqnarray} ...
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1answer
43 views

Stationarity requirements of using regression with ARIMA errors - ROUND 2

Referring to this Post: What are the stationarity requirements of using regression with ARIMA errors for inference? I would like to seek a confirmation of the below practice: The situation is as ...
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1answer
43 views

ACF values in identifying non-stationarity

I have used NIST data to calculate ACF in excel which worked fine and coded in our programming language (NOT R). Here is the plot of ACF: Now my questions are: 1) From this ACF series how can I ...
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1answer
97 views

How to detect if Ergodicity, Stationarity and Martingale. dif. sequence?

I'm not sure, but I think I've read somewhere that because the Classical Linear Regression model assumes to have a random sample, when researchers they might not be in presence of a sample with that ...
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27 views

the decision of being White noise on e-view

And for example, let's take SMA(2) model in this table does there exist white noise ? Which value I observe to decide the existance of white noise? Please explain it. Thank you
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1answer
38 views

Stationarity of AR(1) process whose autoregressive parameter could change over time

Imagine an AR(1) has an autoregressive parameter which could change in time. $y_t-\mu=\phi_t (y_{t-1}-\mu)+\varepsilon_t\,$, where $\phi_t$ is not always constant but still lies inside the usual ...
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1answer
81 views

Is a Brownian motion non-stationary?

This Wiki-Article quotes "a Brownian motion process, is non-stationary" I dont see why this is the case? A stationary process means that the distribution of ...
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1answer
34 views

Stationarity and seasonality of residuals

Why is it necessary to evaluate stationarity and seasonality of model residuals? Or is it? The model in question is an OLS model that represents a relationship between Y and a bunch of economic ...
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1answer
48 views

Are linear processes stationary?

I am reading Soren Johansen's book on cointegration and I'm wonder about the following definition: Definition 3.1. A linear process is defined by $Y_t=\sum_{i=0}^\infty C_i\epsilon_{t-i}$, $t=0, ...
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47 views

Transforming Series with 2nd Moment Nonstationarity

I am trying to induce stationarity in this series. I have graphed a range-mean plot to detect 1st and 2nd moment nonstationarity. Can anyone suggest a transformation that will remedy the 2nd moment ...
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3answers
126 views

Is my time series stationary?

I am using R and have found that both KPSS ( Kwiatkowski-Phillips-Schmidt-Shin ) and the adf (Dickey-Fuller) tests indicate stationarity, having a p-value of 0.01. Here is a plot of the original ...
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The rejection of ADF test can indicate the covariance stationarity?

I am curious that if the ADF test indicates the time series data has no unit root, can we conclude that the time series is stationary (time-invariant mean, variance and covariance)? Here is a small ...
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51 views

Expectation of output of an LTI system w.r.t. a WSS random process

Let $X(t)$ be a wide-sense stationary random process―i.e., its expectation is a constant and its autocorrelaton function is a function only of time differences―and let $Y(t) = X(t) * h(t)$ where ...
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2answers
68 views

Stationarity of Detrended and Deseasoned time series

I removed trends and seasons from given time series and plotted the residual time series. I would like to know if there is any way that this plot could suggest that residual series is stationary? What ...
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32 views

Time Series Stationarity and Histograms

In a paper I am reading, the author discusses stationarity and plots the histogram of returns for the time series he is studying. I was wondering if there was any relationship between stationarity of ...
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1answer
66 views

Time-series and autocorrelation inequality

I am having problems proving for a weakly stationary process $\{X_t : t\in T\}$: $\rho_X(2)\geq 2 (\rho_X(1))^2-1$ where $\rho_X(j)=corr(X_t, X_{t+j})$. So far I have shown that $-1\leq ...
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1answer
32 views

Augmented Dickey Fuller Test for Stationarity

How do I interpret this? ADF t-value = -4.76 Critical Values -2.57823 -2.883037 -3.47937 ADF t-value = -1.23 Critical Values -2.57823 -2.883037 -3.47937
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Averaging time series to improve stationarity - loss of power?

Short version When averaging over a presumed stationary time series and calculating statistics (e. g. normalized mean square error) to compare to a simulation (atmospheric turbulence model) of the ...
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105 views

Dependent variable is non-stationary and independent variable is stationary - residual series?

I ran a regression model where dependent variable is non-stationary (I know this is wrong) and my independent variable is stationary...I find that the residual series are stationary... how is it ...
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1answer
62 views

How to make series stationary when dependent variable is log(y)

I need some help in understanding the following: I have a time series data (y) that I am using to run regression models. However, my dependent variable is log(y). Should I test for stationarity of ...
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1answer
124 views

Help understanding how the cointegration equation for VECM models are derived

I am learning about Vector Error Correction Models from Sean Becketti's "Introduction to Time Series using Stata". While I know how to run the Stata commands to estimate the VECM, I have no idea why ...
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Defining the probability distribution of a Random vector given the probability over a “sub-vector”

Suppose I want the probability distribution over a random vector $X={X_1 ,X_2 ... X_n }$. What I already have with me is the distribution over a subvector $X_i , X_{i+1}...X_m$, $m<n$ which I ...
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57 views

Regressing nonstationary on stationary variable

I am trying to empirically estimate the coefficient for the Okun's law as a relationship between output growth and unemployment. I am using the simple gap version, where I regress real output growth ...
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Stationarity at different confidence levels

Everyone! My question is, when looking for cointegration between two variables, I need to make sure that they are not stationary in levels. However, one of the variables is not stationary at the 5% ...