Questions tagged [stationarity]

A strictly stationary process (or time series) is one whose joint distribution is constant over time shifts. A weakly stationary (or covariance stationary) process or series is one whose mean and covariance function (variance and autocorrelation function) do not change over time.

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Information Criteria and Sample size

In the estimation of the univariate time series model, we need to determine the correct order. For the general ARMA (p,q) model, we can determine the true order by information criteria $$AIC(p,q) = ln(...
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ADF Test and KPSS Test contradicting, differencing wont make the time series stationary

My time series is on Life expectancy at birth from 1960 to 2018. Obviously, it has an increasing trend and ACF also supports this because the ACF values dampen so slowly. However, ADF test p-value is ...
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Checking the first difference of a process which is already stationary

Knowing that a process is already stationary (i.e. I(0) and ADF test for presence of unit root has been rejected), is there any need to check the first difference? If yes, what is the rationale/...
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Joint distributions of a strictly stationary time series model

A strictly stationary time series model $\{X_t\}$ is by definition a stochastic process where $$(X_1,\ldots,X_n) \overset{d}{=} (X_{1+h}, \ldots, X_{n+h}).$$ Why does this imply that $(X_t,X_{t+h}) \...
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(def. of strongly stationary) Difference between same joint distribution compared to same distribution everywhere [duplicate]

I'm learning about strongly stationary processes with the definition: This implies that they have the same distribution for every $t$, I don't really understand what more "same joint ...
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Markov property on non-stationary time-series

I am working with a dataset with several time-series, and I want to test the Markov property on all of them. However, some of them are non-stationary and following the definitions on the text "...
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Why must the non-zero eigenvalues in the Johansen test be between 0 and 1?

Why are the non-zero eigenvalues in the matrix $\Pi$ in the Johansen test between 0 and 1? Why can't they be greater than 1 or less than zero? My lecturer just dropped in that's its because "the ...
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Time Series Forecasting Arima family methods

I'm fairly new to time series analysis and forecasting. I'm using the uci househould power consumption dataset to build a model to forecast energy consumption. The dataset measures the power (kW) ...
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Question on Adfuller test results variation

I have a series on which I am trying to run linear regressions and determine the stationarity. The test for stationarity changes from being stationary to non-stationary when I move the window to the ...
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XGBoost for Time-Series Forecasting - Issues with Stationarity Transformations

I'm trying to forecast daily Covid vaccinations in Germany, especially focussing on using tree-based ensemble methods. One issue with tree-based methods for forecasting is extrapolation, when the time-...
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Why does ACF of stationary time series drop to 0 quickly?

A 'weakly stationary' time series has the property that the autocorrelation is only dependent on the lag and its structure does not change over time apart from the properties of constant mean and ...
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Strict stationarity with dependent variables

Strict stationarity means that the joint distribution of any subset of random variables within a time-series is time-invariant. Can you think of any practical (or real-life) example where this applies ...
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Stationarity test for count data time series

can someone give me an overview of stationarity test for time series of count data? For instance, I would like to know if there exists a test similar to the Augmented Dickey Fuller or how this test ...
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When non stationarity is a problem?

So in the literature one can find many times that ML works on the assumption that data distributions are stationary. Now I can do multiple tests on my dataset to show the violation on the non ...
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ARIMA Correlogram Classification - Insignificant P-Values (Reposted w/ specific questions) [duplicate]

I am trying to apply ARIMA forecasting to a stock's market close price. It is daily data with 129 observations. I used Augmented Dickey-Fuller Test and a Correlogram to confirm the data is non-...
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How to handle a structural break of only one variable in a VAR model?

I am estimating a VAR-model with two variables: GDP real and Investments. Investments has a structural break, GDP real not. As I understood it is only possible to add a dummy variable to both ...
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Find the order of integration of an ARIMA model from a specified equation

Suppose I'm given an equation: $$ Y_t = 1.7 Y_{t-1} - 0.7 Y_{t-2} + e_t $$ If I'm given the data, I can use the Box-Jenkins methodology coupled with stationarity tests like the ADF test to find the ...
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How to select ARIMA model with cyclic ACF?

My annual time series has following ACF/PACF structure. Based on what ARIMA model should be selected here? Exponential decreasing of ACF --> AR(4) probably? Or because of periodical ACF maybe SMA? ...
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Find the correlation function of stochastic process given differential equations

Assume two systems for which the following differential equations hold between their input and output signals. $$a \dfrac{dv(t)}{dt}+b v(t)=x(t)$$ $$\dfrac{dy(t)}{dt}=v(t)u(t)$$ Also, assume that the ...
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Stationarity check in ARX

How to formally check stationarity condition in a regression in the form: $$ y_{t}=\alpha_{1}y_{t-1}+\alpha_{2}y_{t-2}+\beta_{1}x_{t}+\varepsilon_{t} \ ? $$ In the case of AR(2) there are some ...
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Strong stationarity and Markov property for an AR(1) process

Suppose I have an AR(1) process of the form $$X_t=\phi X_{t-1}+\epsilon_t,$$ where $\epsilon_t$ is a Gaussian white noise. Suppose that $X_t$ is weakly stationary check if $X_t$ is strongly ...
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Intuition of Random Walk having a constant mean

I am very new to time series analysis. A random walk is defined as $Y_t=\phi Y_{t-1}+\varepsilon_t$, where $\phi=1$ and $\varepsilon_t$ is white noise. It is said that process is non-stationary for ...
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AR(1) - Stationarity condition

Consider the well-known AR(1) model: $$x_t = \phi X_{t-1} + \epsilon_t$$ where, as usual, $\epsilon$ is an independent white noise process. I have read many sources. All of them get away saying that ...
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Regressing I(1) variable on I(0) variable

I am dealing with time series regression, where I have stationary and nonstationary variables. Can I regress nonstationary I(1) variable on stationary variable when controlling for the lag of the ...
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Create stationary dataset with log, sqrt and cube root layering

I have a time variant dataset with highly non stationary data. The dickey-fuller p value always at the 0.90 to 0.99 area. Even after single transformation, still data generates with p value about 0.80....
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Does a time series clustering that uses cross-correlation as proximity measure require stationarity?

I have a set of time series which exhibits no autocorrelation but the variance is not constant. I remember that one of the requirements of cross-correlation to be meaningful is weak stationarity (we ...
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Time series analysis: Why am I getting a reciprocal condition number when trying to estimate VAR-Model? /w reproducible example

I'm currently trying to identify an appropriate VARMA(p, q)-representation for a multivariate time series using the MTS::-package in R. The series comprises n = 126 ...
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Covariance of a Stationary Process

Let $Y_t$ be a stationary process such that $Y_1 = a_1$ and $Y_2 = \theta a_1 + a_2$, where $\theta$ is a parameter and $a_t$ is the white noise process with mean 2 and variance $\sigma^2_a = 0.5$. ...
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How to show that $var(\hat{\mu}) < var(\bar{X}) $for a stationary process ${X_t}$, where $X_t = \mu + Z_t + Z_{t-1} $?

If ${X_t}$ is a stationary time series with mean $\mu$ then the usual estimator for $\mu$ is the sample mean $\bar{X} = \frac{X_1+...+X_n}{n}$. Assume we have $X_t = \mu + Z_t + Z_{t-1}$, where ${Z_t}...
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In stationary time series, how much time needed to go back to its mean?

Question I have a time-series data which is stationary as below. If an observation at t time point is far from the mean, how much time do I need, on average, for the timeseries to go back to its ...
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Stationarity of variables for Local Projections

Should we care about the stationarity of the time series when doing Local Projections? In his website Jorda (the author of Local Projections method) provides a code for doing local projections : https:...
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Predicting stationarity of a time series

I have a time series, for example equity prices. This series is weakly stationary (for instance an AR process that does not violate the stability condition). How can I make predictions on how likely ...
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Probability of next item in normal distribution series given a series of values?

Let' say that I have a normal dist data with x as time and y as the dependent variable. For all purposes, the data is normally distributed i.e. stationary. Now it would be clear that all my data ...
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Are there tests for unit root in respondent-driven sampling referal chains?

The assumption of independence in respondent-driven sampling is likely to be violated when there is homophily (individuals are more likely to recruit new study participants who share characteristics ...
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What are the stationarity conditions for an AR(4) process?

The stationarity conditions for an AR(2) process are: $$a_1 + a_2 < 1$$ $$a_2 - a_1 < 1$$ $$a_2 > -1$$ And the stationarity conditions for an AR(3) process are: $$a_1 + a_2 + a_3 < 1$$ $$...
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Is it fine to choose 0 lag for adf test in my data?

The level values weren't stationary so I took percent changes $(P_t-P_{t-1})/P_{t-1}$. Here's the data: These are the PACFs to determine lags. I think the lag can be 0, or 7 or 11 in case of GDP and ...
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Interpreting ACF and PACF plots on trading volume

I'm a fairly new one to time series analysis. I was analyzing the daily trading volume of stock derivatives for the past year and trying to see if there is a seasonality pattern. I tried to make the ...
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Making an AR(3) model weakly stationary

I have a model as:$$r_t=0.05+\frac{7}{6}r_{t-1}+\frac{1}{6}r_{t-2}-\frac{1}{3}r_{t-3}+a_t$$ When checking for stationarity: $$1-\frac{7}{6}x-\frac{1}{6}x^2+\frac{1}{3}x^3=0$$ I get $x\in \{-2,1,1.5\}$....
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Non stationarity and forecasting

Let's assume we have estimated a linear regression model on a dataset from 2000 to 2017. The data were stationary. What happens if the data are no longer stationary in the next years? Do the forecasts ...
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How to interpret regression results when the data have been detrended?

I am planning to build a linear regression model where I explain flight ticket demand with airfares, lagged airfares, GDP etc. based on monthly data from the past 15 years. This is my first time ...
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Do Recurrent Neural Networks assume stationarity or just a general kind of sequential dependence?

Just when I thought I had convinced myself that RNNs make no other assumption about a sequence other than that there are dependencies between the inputs and that (in the case of monodirectional RNNs) ...
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Normality of tau-statistics ($\tau_{\mu}$ and $\tau_{\tau}$) in presence of unit roots

The original Dickey-Fuller (1979) paper, considers three regressions ($(1.1), (2.1)$ and $(2.2)$) but only two DGP ($1.1$ and $2.1$), while deriving the limiting distributions. The paper defines three ...
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Autocorrelation AR(1) process

I am doing a self-study question. I need to find the autocorrelation $\rho(2)$ for the following AR(1) process: $y_t = y_{t-1} + \epsilon_t;\\ \epsilon_t \sim (0,\sigma_\epsilon^2) $ For that I need ...
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HAC variance to construct standard errors

I am facing some difficulties understanding this question. It hasn't been long since I started with econometrics, so I'm new to all of this. Suppose we have a function $$E[c_t|y_t,c_{t-1},y_{t-1},c_{t-...
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Recognizing the Seasonal Effects from a Time Plot

From the first plot, I have determined that there is a seasonal pattern of period $L = 12$. However, in my ACF plot (a), it appears that the period is $L = 6$. Am I misinterpreting? The data had a ...
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Second-order and strictly stationary time series is weakly stationary - proof

I keep reading that second-order and strictly stationary time series has constant mean, variance and its autocovariance is time independent, but I can't find proof of that. My definition of such time ...
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What are the characteristics of a trend and break stationary process?

I have a time series with around 380 data points (day-wise data acquired from instruments). I want to model these in ARIMA. It is my understanding that first I'll have to check for stationarity of the ...
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Is AR timeseries always invertible?

I have just started learning time-series and the foremost thing that I read was that in my book it is explicitly written that an AR process is always invertible. But why is that ? If XT= a X(T-1) + et ...
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Interpretation of several I(1) variables ADL regression

I see a lot of weird interpretation of coefficients, when working with a time-series model with two (or more) variables. Specifically I am thinking of two series that are I(1) and are then log-diffed ...
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Time Series Stationary or Not

Here is my time series plot of some data. There appears to be a constant variance, but I don't believe that the mean is constant (e.g., big dip around time $t=17$ and big increase around time $t = 57$)...

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