# 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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### Solve for inequality of AR model

I was working through my textbook and found this problem that I got stuck at: Consider the AR(2) Model $$X_t = \phi_1X_{t-1}+\phi_2X_{t-2}+\epsilon_t$$ We can assume $\phi_2 > 0$, so the roots of ...
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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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### Using ar function in R to fit a AR(p) with predetermined exact order p. [closed]

I am playing with R's sunspot data and I have $X$ as the yearly sum of sunspots(hence $X$ is a vector of length 235). I want to fit different AR models with different orders, $p = 1, 2, ..., 20$. I ...
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### 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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### what is the intuition behind stationarity condition for AR(p) process?

i get that you have to find the roots of the characteristic polynomial but can someone explain the intuition behind the roots must be outside the unit circle? what is a unit circle? before anyone ...
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### Why use Granger causality instead of autoregression?

I'm working on an analysis on GDP growth. I want to test whether regional growth in GDP in a time frame can be explained by the growth of air traffic in a preceding period (and controlling for some ...
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### Expand a power of the difference operator in terms of time series $z_t$

I am trying to use excel to plot different time series. I have the equation $(1-L)^2 * z_t$ I know that $(1-L)*z_t$ is equal to $z_t-z_{t-1}$ Can I just expand $(1-L)^2$ using basic algebra and ...
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### AR.OLS isn't matching to an OLS on the autoregressive lags, Why?

I am using R and running ar.ols() on some data. And trying to compare to a more "manual" method of computing an AR model by doing lm() using the autoregressive lags as my independent variables. ...
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### Breusch-Pagan Test for ARIMA Model in R [closed]

I am testing my model using the Breusch-Pagan Test, but have not been able to find anything online regarding how to calculate it for an ARIMA Model. My AR1 Model is: ...
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### How to decide the optimal AR-model order?

I'm trying to create AR-model on wheather data and I wondered is there a method or algorithm which can find the optimal order for an AR-model? I'm using Matlab for my data-analysis, is there a ...
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### Test for independence of random variables

I have a time series of data (about 300-750 elements, depending on the sample) and a model that has some random residues. I used the Kolmogorov–Smirnov test to make sure that the normality hypothesis ...
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### what R function fits a smoothing spline regression model with correlated errors?

I want to fit a smoothing spline regression model with correlated errors (it's a time series) using R. All I could find is function ssr, from library ...
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### What does “AR(p) filtered series” mean?

I guess this means that omitting some variables in a certain interval, say, $(x_1, x_2, x_3, x_4, x_5) \to (x_1, x_5)$ in AR(4) model. Is it right? Or does this means eliminating autocorrelations ...
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### Vector autoregression - number of lags

I am constructing a Vector autoregression model and I have used AIC to find how many lags I should use. Does 7 lags seem unreasonable? I am trying to find the impact the property market has had on the ...
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### Sampling from an AR(1) process using normal samples

This is probably a very straight forward question but I want to verify how I should sample from an AR(1) process in R using just the rnorm() function in R (or any ...
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### Solver for the true auto-covariance function in AR(p)

Suppose I have the following $AR(p)$ model. $$X_t = \sum_{i=1}^{p} \phi_i X_{t-i} + \epsilon_t\,,$$ where $\epsilon_t$ has mean 0 variance $\sigma^2$. I am in the situation where the $\phi$s are ...
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### The relationship between autoregressive model and distributed-lag model

The autoregressive models (koyck model, adaptive expectation model, potential adjustment model) I have learned so far are all derived from distributed lag models. And intuitively it makes sense since ...
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### AR process stationarity

For $X[n] =aX[n-1]+W[n]$ When $W[n]$ is iid. One can say that $X[n]$ is the output of $W[n]$ thrown into an LTI system. So how can it be that $X[n]$ is not necessarily WSS, if we know that a WSS ...
Backgrounds I have a time series, and I fitted an AR(p) model with trend of $t^2$, with the help of auto.arima, in R package <...