# Questions tagged [autoregressive]

The autoregressive (AR) model is a stochastic process modelling time series, which specifies the value of the series linearly in terms of the previous values.

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### Variance of a stationary AR(2) model

I have two questions: 1) When one says an ARMA process is 'stationary,' do they mean strongly stationary or weakly stationary? 2) Is there a quick way to find the variance of a stationary AR(2) ...
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### Simulate AR(1) process in R with specified nonzero mean and AR coefficient

I need to simulate an AR(1) process with the following equation in R: $$X_{t} = 5 + 0.5X_{t-1}+Z_t$$ Where $Z_t$ ~ White Noise(0,1) and $T=500$. I know I should be using the ...
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### Autocorrelation of a stationary AR(2) process

Consider the stationary AR$(2)$ process of the form: $y_{t} = \alpha + \phi_{1} \ y_{t-1} + \phi_{2} \ y_{t-2} + u_{t}$ where $u_{t}$ is i.i.d. white noise. Just as a head's up, we have not covered ...
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### Where is $|\theta|<1$ used in recursive method derivation of invertibility of MA(1)?

Silly question: From what I understand, an MA process is invertible when it can be represented as an AR($\infty$) process. When using lag operator, it is somewhat clear that $|\theta|<1$ is ...
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### How to write variance covariance matrix of AR(1) process in R

I'm trying to write autocovariance matrix of AR(1) process in R and I'm having difficulty. The autocovariance matrix that I'm using in my project takes the form as shown in the picture: I also ...
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### Cross correlation influenced by self auto correlation

I have two stationary time series ts1, ts2, I wanna find the cross correlation ($\textrm{CCF}$) between them. As a result, it ...
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### Can there exist a unit root series that’s Granger-caused, or better predicted with a model other than the AR process we tested using ADF?

If a series has a unit root, then it is a function of random white noise. Therefore, it follows a random walk process. Is it then possible for: Some other series to Granger-cause the unit root series?...
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### Are all $AR(p)$ processes for which $|a_1|,....,|a_p| < 1$ stationary?

For an $AR(p)$ process $Y_t = a_1Y_{t-1}+a_2Y_{t-2}+...+a_qY_{t-q}$ : Is having the coefficients $|a_1|,....,|a_p| < 1$ just a necessary condition for stationarity, or is it sufficient as well?
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### Determining standard error of the mean from a correlated, stationary time series using known autocorrelation without block averaging

I'd like to determine the SEM of measurements taken from a stationary time series. SEM calculation using all measurements isn't accurate because adjacent measurements may be highly correlated, so the ...
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### Skipping lags in autoregressive modeling?

is it possible to skip immediately preceding time points? So that, if, for example, you are using model order 2, that is, two time points, but not the two immediately previous time points, but rather, ...
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### Proof of contemporaneous exogeneity, and its implications for an AR(1) model

It can be shown by contradiction that exogeneity fails to hold for an AR(1) model. Is there any proof that contemporaneous exogeneity does not fail to hold? All I've come across is assuming it does ...
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### If an auto-regressive time series model is non-linear, does it still require stationarity?

Thinking about using recurrent neural networks for time series forecasting. They basically implement a sort of generalized non-linear auto-regression, compared to ARMA and ARIMA models which use ...
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### What is the difference between deterministic and stochastic model?

Simple Linear Model: $x=\alpha t + \epsilon_t$ where $\epsilon_t$ ~iid $N(0,\sigma^2)$ with $E(x) = \alpha t$ and $Var(x)=\sigma^2$ AR(1): $X_t =\alpha X_{t-1} + \epsilon_t$ where $\epsilon_t$ ~...
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### Variance of a smoothed AR(1) process

The query I have relates to calculating the variance of AR(1) processes that are smoothed with a simple moving average. So: In an AR(1) process of the form: $$X_t=c+\varphi X_{t-1}+\varepsilon_t,$$...
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### Random walk estimation with AR(1)

When I estimate a random walk with an AR(1), the coefficient is very close to 1 but always less. What is the math reason that the coefficient is not greater than one?
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### What's a stationary VAR?

What is a stationary VAR (vector autoregression)? Can a VAR with non-stationary variables be stationary? How do you test whether a VAR is stationary or non-stationary? (Example in ...
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### Why is the dickey fuller test different from a simple t-test

I am trying to understand why should there be different distribution for t-statistic, in case of AR model, Dickey-Fuller test For e.g. Say, the model is $Y_t = \beta_lY_{t-1} + \varepsilon_{t}$. ...
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### Poisson with an autoregressive term

I want to fit a fairly "standard" Poisson model, but with an autoregressive term. $N_i \sim \mathrm{Pois}( \lambda_i E_i)$ with $\log \lambda_i = X_i \beta + \delta$ $\delta \sim AR(1)$ $X_i$ is a ...
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### Invertibility of AR(p) model

Notation: $\dot{Z}_t = Z_t - E(Z_t)$, so that it is centered at 0. $a_t$ stands for the residual and we assume the $a_t$ are independent and normally distributed with mean 0 and constant standard ...
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### Can autoregressive coefficient values be greater than 1?

I am using multivariate autoregressive (MAR) models to fit my long-term dataset of species abundances and environmental variables but when I use only the data from a specific period of the year (e.g.,...
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### Brockwell/Davis seem to say more persistence implies better predictability---do I have a counterexample?

Brockwell/Davis, Introduction to Time Series and Forecasting, p. 40, write (notation slightly adapted; please refer to screenshot below) The best linear predictor $l(Y_{T})=aY_{T}+b$ for a stationary ...
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I want to know (mathematically) how the following expression changes as $M$ increases but still have no clue after thinking about it for a while. Any suggestions or comments will be much appreciated. \$...