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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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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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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25 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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39 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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28 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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66 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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24 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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33 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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66 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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24 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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38 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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35 views

Is the stationarity condition necessary for estimating logit/probit model?

I'm going to estimate both a logit and a probit model. Since both the models contain lagged explanatory variables, I want to know if the stationarity condition for this variables has to be verified. ...
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46 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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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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25 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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64 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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29 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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60 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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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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50 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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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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83 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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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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35 views

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% ...
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Does a seasonal time series imply a stationary or a non stationary time series

If I have a time series that has got seasonality, does that automatically make the series non stationary? My intuition (probably off) is that it does not. Seasonality means that the series goes up ...
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78 views

Does stationarity under ADF test imply mean, variance and covariance stationary?

Newbie question. I am reading about stationary series and understand that it has many forms: mean stationary variance stationary covariance stationary If I run an augmented dicky fuller test and ...
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Spectral density of sampled process

A real valued time-continuous process $X(t)$ is described by the spectral density according to the top of the image below. The process is sampled with the sampling distance $d=1/40$. The question is ...
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118 views

How to “undifference” a time series variable

I need to "undifference" or "integrate" a time series variable. In its current state, it is twice-differenced (a money market, cash return proxy variable that was I(2) to achieve stationarity). I ...
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40 views

Stationarity of an AR(1) process

I have problems with answering problem (b) and (c) in the following exercise. Can any of you people help me out? Let N={0,1,2,...} denote the set of natural numbers, $\{\epsilon_t \}_{t \in N} \sim ...
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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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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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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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9 views

Stationarity & seasonality in R [duplicate]

I have a weekly number of items sold from 2012 to 2014. 2014 not being complete. So bassically two periods. I am looking at seasonality and stationarity of the response variable (# of items sold) I ...
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85 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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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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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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Variance of the second order stationary process's mean

I have a second order stationary process with the following covariance function: $$r_X(t) = \alpha e^{- \beta |t|},\quad -\infty < t < \infty$$ Now, $$\bar{X} = \frac{1}{T} \int_{0}^{T} X(t) ...
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71 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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16 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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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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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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81 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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68 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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105 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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67 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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60 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 ...