Tagged Questions

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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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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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 ...
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19 views

“Iterating”? For MA and AR processes

I am not sure what is being done here, but I keep seeing statements like Given $X_t - \phi X_{t-1} = Z_t$ $...(1)$ then $$X_t = -\phi^{-1}Z_{t+1} + \phi^{-1}X_{t+1}$$ $$ = ... $$ $$= ...
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30 views

What is the source of nonstationarity in this VAR model?

I am trying to forecast a VAR model, which consists out of 5 variables with a monthly frequency. The problem is that the VAR model produces an unstable forecast and I am not sure what the source of ...
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118 views

How do I detrend time series?

How do I detrend time series? Is it ok to just take first difference and run a Dickey Fuller test, and if it is stationary we are good? I also found online that I can detrend the time series by ...
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25 views

Stationary function

I am reading Karl Rasmussen's book on Gaussian processes and in the introductory chapter he has the following statement: ...
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87 views

stationarity in time series

I'm learning a Time series course and I have a few questions. Strictly stationary is a process if the joint distribution of $X_{t1},X_{t2},...,X_{tm}$is the same as the joint distribution of ...
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A proof for the stationarity of an AR(2)

Consider a mean-centred AR(2) process $$X_t=\phi_1X_{t-1}+\phi_2X_{t-2}+\epsilon_t$$ where $\epsilon_t$ is the standard white noise process. Just for sake of simplicity let me call $\phi_1=b$ and ...
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How to write an AR(2) stationary process in the Wold representation

I managed to write an AR(1) process in the Wold representation with help from the geometric series. I am having trouble with a stationary AR(2). How could I do?
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What is a stationary function?

Snoek et al, have a recent paper "Input Warping for Bayesian Optimization of Non-Stationary Functions" (http://arxiv.org/abs/1402.0929) which mentions "stationary functions". I understand what a ...
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28 views

Panel data models when some regressors are non-stationary - I(1)

My model is $$ Y_{it}=X_{it}'\beta+\varepsilon_{it} $$ where $Y_{it}$ is a vector of weekly observations of a dependent variable and $X_{it}$ is a vector of explanatory variables (also weekly) with ...
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Does Stationarity for Time Series extend to Independent Variables?

There have been many questions about the importance of stationarity and also its means of calculation here on CV, but one question that I have not seen an answer to is whether or not stationarity (in ...
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Is it appropriate to run the Error Correction Model on data which are not I(1)?

I have intraday data (frequency = 1 min.) for 6 stocks and 1950 observations per each time series. I checked stationarity for the level data and first difference and it appears that: 5 stocks's ...
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26 views

Perron's test which allows testing for the broken trend stationarity in R: the code and application?

I am trying to run the Perron's test in R, which allows testing for the broken trend stationarity. I was trying to find the package and code which is running this test, but without success. While ...
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Unit root for bounded variable

I have a time series which is discreet and bounded (between 0 and 100). It’s actually also increasing over time, from low to high. When I test for a unit root, I find it. And yet I wonder if unit ...
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15 views

AR models on non stationary data

i am currently reading Diebold and Li's 2006 paper: Forecasting the term structure of government yields where the authors fit, albeit simple, AR(1) models on clearly non stationary data. Why is this ...
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27 views

Identify the stationary time series

Identify the stationary time series for which $$ \gamma(h) =(-1)^{|h|}+\cos \left(\frac{\pi}{4}h\right)$$ is ACVF. This is a homework problem. Stuck at first level. Please give some hints. Thanks in ...
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Unit Root Test and rolling data

Please could you kindly advise on the implication of testing for unit root (using the augmented dickey fuller approach) on rolling data. My feeling is that this wouldn't make much sense given the non ...
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Cointegration between two stationary processes

I have two stationary time series. I would like to check for cointegration between them. Does this make sense, and can I just use Engle-Granger Test (two step) for Cointegration for this?
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How to stationarize profit and loss data with an increasing variance and large negative values for time series analysis?

PnL can take large negative values, and its variance increases over time as the firm grows. If we do differencing, an increasing variance remains. If we take log, negative values cannot be defined. ...
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8 views

Mix of I(0) and I(1) in VECM

Is there an official academic journal I could use as reference which states that a mix of I(0) and I(1) variables can be used in constructing a VECM?
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Retreiving Integrated Fitted Data from Stationary Fitted Data

Note that this is a simplified example: I have some time series that I made stationary by differencing twice. Then I ran arima on it, and set d = 0 to prevent ...
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37 views

Which models require stationarity?

Regression models (at least up to GLMs) do not traditionally require stationarity (although the requirement for the residuals is even stronger than stationarity). ARMA-style time series models seem ...
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Can I difference after fitting a time series regression model?

Suppose that I have a time series that exhibits a notable trend, and I want to test a hypothesis that a second variable is related to that trend. I fit a linear regression model with that second ...
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40 views

Inverse Differencing and ARIMA Model Equivalence

I've developed a ARIMA model with exogenous variable. Before fitting the model, I made every time series stationary by differencing (each variable had a different order of integration). For ...
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148 views

How to compare time series with cyclical data, and describe any changes or trends

I have a bunch of time series where the data has a natural (known) cycle, for example daily or annual (or both). Here is an example (this is 6 years worth of temperature data sampled hourly): ...
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144 views

Difference between series with drift and series with trend

A series with drift can be modeled as $y_t = c + \phi y_{t-1} + \epsilon_t$ where $c$ is the drift(constant), and $\phi=1$ A series with trend can be modeled as $y_t = c + \delta t + \phi y_{t-1} + ...
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Transforming time series of different time horizon to stationary

I have a list of monthly time series data with different time periods and different order of integration. I want to transform them all to stationary and a same time period. I noticed that the order ...
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51 views

Stationary and ADF for unit root

If I calculate the returns in my time series as follows, $\ln(P_t / P_{t-1})$, where $P_t$ is the price at time t and $P_{t-1}$ is the price at $t-1$ then I do the ADF test, my $H_0$ is rejected. ...
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45 views

what if I know my time series are cointegrated over a long period but not over a short period?

I am regressing time series on time series. I have tested for cointegration on the entire time sample (3 years) and the series are cointegrated. I need to make a rolling window of regressions (to ...
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295 views

How can I show that a random walk is not covariance stationary?

How can I show that a random walk ($y$ follows a random walk) is not covariance stationary? I tried to work on the formula below (with no results) Could you give me just a hint on how to proceed ...
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160 views

Time Series: Seasonality and trend

I am interested in financial time series and I have a small question regarding the use of the forecast package. The time series I am interested in is a monthly one and present clear evidences of ...
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GARCH(1,1) regression in Eviews

I'm having a problem in doing a GARCH(1,1) regression. I'm trying to regress gold prices serie on stock returns series as in the following equation in eviews: $$ r = c(1) + c(2) \cdot s + c(3)\cdot ...
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134 views

Where can I find ADF test library or source code from c#

I would like to test for stationarity in cointegration. I intend to use an augmented dickey fuller test. However, I need one for c# - either a library or the source code. Or is your have source in a ...
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23 views

What does it mean to build a so-called mean equation?

A series is denoted by Xt. How do we build a time-series model for Xt that is the "mean equation" of Xt? Is this another way of asking to build a stationary time-series?
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condition for a ARMA process to be wide-sense stationary

For a ARMA process, some (e.g. in Tsay's Financial Time Series) said: it is wide-sense stationary, iff all the roots of its AR characteristic polynomial are greater than 1 in magnitude. This is ...
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Understanding stationarity with Inflation

I am looking at the link between inflation and insolvencies for an econometrics project. I have the raw quarterly insolvency data and raw quarterly CPI data for the UK (roughly 100 samples) from ...
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58 views

When an ARMAX model is stationary? Why stationarity or invertibility is needed?

Let $y_t$ a stochastic process and $\tau_t$ presents the time duration between the $t$ and $t-1$ event.The ARMA(p,q,r) with exogenous variables is defined as: $$ y_t = \varepsilon_t + ...
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How to test for wide-sense stationarity with only one sample path of the process?

I have a univariate time series consisting of 70,000 observations (power consumption of a building) over equal time increments (15 minutes). How do I check whether this realization is wide-sense ...
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116 views

Lag order in ur.df or adf test [R]

I want to test stationarity using adf test or ur.df function on R ur.df(y, type = c("none", "drift", "trend"), lags = 1,selectlags = c("Fixed", "AIC", "BIC")) My question is when using adf.test the ...
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82 views

Non-Stationarity, first differences and Panel Data

I have build a sentiment index and am now validating its statistical properties and significance. I stumbled upon two problems. (1) dfglsindicates that my ...
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103 views

Does covariance stationarity lead to mean stationarity necessarily?

Traditionally a weak stationary process is also called covariance stationary, but those 3 properties are exposed: $$E[Xt] = μ , \forall t$$ $$var(Xt) = \sigma^2, \forall t$$ $$cov(Xt, Xt−j) = ...
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Test for stationarity in an unbalanced panel

I have an unbalanced panel model and I need to check it for stationarity. So, I need to perform a Unit Root test (I think I will use a Fisher Type Test?). But I am a bit confused whether (1) I need ...
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80 views

Time series and stationnarity tests

I perform some time series fitting with the help of the forecast and urca packages. I have a question regarding the correspondance between results coming from statistical test such as KPSS, ADF or ...
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50 views

When is the autocorrelation function of a stationary process decreasing/nonincreasing? Markovian?

When is the autocorrelation function of a stationary process strictly decreasing or nonincreasing? Can being Markovian make it true? When is the autocorrelation function of a stationary process ...
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Dickey-Fuller for 2D signals?

I would like to run a stationarity assessment test on a 2D signal. Is there a suitable test, e.g. a 2D DF or ADF, etc? Thanks. Edit: A 2D signal is a signal defined on a lattice, i.e. instead of a ...
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Checking whether a given sample is stationary

Let $x[n]$ be some time series in 1D or $x[m,n]$ in 2D, of length $N$ (resp. $N^2$) How can I assess whether it is stationary? At least in the weak sense. I can check whether the stdev remains ...
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How to tell stationarity from a sample path?

Given a sample path, we can roughly tell whether the mean changes over the time, and, when it doesn't, whether the deviation from mean changes over the time. (Correct me if I am wrong.) But that is ...
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Can nonstationarity be told from the autocorrelation function?

Here "stationarity" means the first and second moments don't change over time. From a page of Time Series: Theory and Methods, by Peter J. Brockwell, Richard A. Davis In this chapter we shall ...