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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913 views

Variance of AR(1) process using lag operator

Suppose for the AR(1) model, $$Y_t=\phi_1Y_{t-1}+e_t$$ I want to find the variance $Var(Y_t)$ using lag operator: $$Y_t=(1-\phi_1L)^{-1}e_t$$ My way is simply taking the variance, $$Var(Y_t)=(1-\...
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2answers
52 views

In AR(1), why is $X_i|X_{i-1}=x_{i-1}\sim N(\alpha x_{i-1},\sigma^2)$?

The AR(1) model starting with $X_0=0$ is $$X_i=\alpha X_{i-1}+\epsilon_i, \ i=1,...,n, \ -1<\alpha<1$$ where $\epsilon_i\sim N(0,\sigma^2)$ are independent error terms. Why then is $$X_i|X_{i-...
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2answers
2k views

Confusion related to the calculation of autocovariance

I have a confusion related to the calculation of autocovariance Suppose $X_t = \phi X_{t-1} + \epsilon_t$ Then how the autocovariance $E(X_{t+n}X_t) - \mu^2 = \frac{\sigma_{\epsilon}^2}{(1-\phi^...
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3answers
2k views

predictions for AR(1) model

I don't understand how predictions can trace the actual data so closely (see the code below)? Does that make sense? The model is $Y_t = \theta Y_{t-1} + Z_t$ where $Z_t$ is random noise. Hence the ...
2
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1answer
205 views

OLS estimation of intercept in AR($p$) in R

I investigate the performance of the OLS estimator of an AR($3$) model given by $$ X_t=\mu+\phi_1X_{t-1}+\phi_2X_{t-2}+\phi_3X_{t-3}+\varepsilon_t $$ for $t\in\mathbb Z$ using the following code: <...
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1answer
85 views

Autoregressive process with random walk perturbation (with drift)

Suppose we have an autoregressive process, $$y_t=\phi y_{t-1} +u_t$$ where $|\phi|<1$. If $u_t$ is an i.i.d random variable this process is stationarity. What if $$u_t=u_{t-1}+g+\epsilon_t$$ where $...
2
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1answer
57 views

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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1answer
5k views

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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1answer
7k views

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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1answer
70 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 ...
2
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2answers
726 views

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 ...
2
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1answer
619 views

How can I convert a Gauss-Markov process to i.i.d. Gaussian process?

I am wondering is there any straight forward approach to convert a Gauss-Markov process, i.e., a First order autoregressive process with i.i.d. Gaussian input, with the covariance matrix $K=Toeplitz(1,...
2
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1answer
237 views

Augmented Dickey-Fuller Test and Lag Length

In R, using the package tseries, one uses the command adf.test for the Augmented Dickey-Fuller Test. However, this assumes a ...
2
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1answer
295 views

time series model with additional, time-independent regressors?

How does one introduce time-independent regressors into a time-series model? Let's say that you want to model house prices based on mortgage valuations from the past 5 years AND based on additional ...
2
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1answer
464 views

Moments of an AR(1) Process

Definition of an AR(1) process In an Autoregressive Process, a time series can be generated based on a stochastic difference equation. \begin{align} X_t = c + \phi \, X_{t-1} + \epsilon \end{align} ...
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1answer
1k views

Gaussian distribution of AR(1) model

This is very basic, but I have been stuck here for a while. Consider an AR(1) model $Y_t = c+\phi Y_{t-1} +\epsilon_t$, where $c$ is a constant. If $\epsilon_t \sim i.i.d. N(0, \sigma^2),$ then $...
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1answer
625 views

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 ...
2
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2answers
39 views

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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1answer
1k views

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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2answers
4k views

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: ...
2
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1answer
119 views

Stochastic Volatility Model

In Kim et al. (1998) stochastic volatility model is specified as: $y_t=\beta\exp({\frac{h_t}{2}})\varepsilon_t,\quad t\geqslant1$ $h_{t+1}=\mu+\phi(h_t-\mu)+\sigma_\eta\eta_t$ $h_1\sim N(\mu,\frac{...
2
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1answer
249 views

Proving for an AR(2) process that $E[X_t | F_{t-1}]=E[X_t | F_{t-2}]=E[X_t | F_{t-3}]$

An exercise states: Using the law of iterated expectations applied to an AR(2) process, verify that $E_{t−k} . . . E_{t−1} (X_t ) = E(X_t |F_{t−k} ) $ for $ k = 1, 2, 3 $ where $ E_{t−k} (X_t ) = E(...
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2answers
2k views

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 ...
2
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2answers
458 views

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 ...
2
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1answer
38 views

Autoregressive model with observable noise

The classical autoregressive model is a linear model for the dynamic variable $x$, where the added noise $\epsilon$ is directly affecting the dynamics of the model $$x_{t} = \sum_i \alpha_i x_{t-i} + \...
2
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1answer
24 views

What does it mean “analyze sample time series data when only a single series is available”?

Since the book says, it will use time series to mean either realization of a process or a process, I have no idea how to interpret the following sentence. "This notion, called weak stationary(i.e....
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1answer
56 views

Feasibility of running mixed-effects poisson/logistic regression with correlation structure such as AR(1), Toeplitz

I'm not aware of any R package that lets me use specify the covariance pattern model such as in the package nlme and run the mixed effects poisson/logistic ...
2
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1answer
305 views

Population autocovariance goes to zero, assuming covariance stationary

In time series context, let $\gamma_j=E[(y_t-\mu)(y_{t-j}-\mu)]$ denote population autocovariance, where $\mu$ is population mean of $y_t$, assuming covariance-stationary. Then, $\gamma_j$ goes to $0$ ...
2
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1answer
27 views

Is there any theory on the order of Autoregression model for periodic time series? [closed]

Say M periodic signals, then one can safely say using AR-M model can achieve the perfect prediction. But how about further, in a more general sense, is there any publications on this? Update: Here ...
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1answer
485 views

Approximate AR(p) with a product of AR(1) and AR(2)

Literature suggests that any AR(p) ARIMA model can approximated as a combination of AR(1) and AR(2) processes. For example, one book suggests that an AR(3) model with the following coefficients: ...
2
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1answer
104 views

Bayesian autoregressive model with second peak at 1 in posterior distirbution of AR parameter

I am trying to run a Bayesian hierarchical AR1 model for a set of fairly short time series. In some of the series I get a second peak around 1 in the posterior distribution of the AR1 parameter. ...
2
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2answers
983 views

When is ARMA(p,q) strictly (strongly) stationary and what is its unconditional distribution?

It seems that the discussions on ARMA are always focused on weak (second-order) stationary, but what about strong stationary? What are the conditions on the coefficients for it to be strictly ...
2
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1answer
300 views

Does stationarity of AR(p) imply innovations are i.i.d.?

My lecture notes give the following definition: A stochastic process $(X_t)_{t\in\mathbb{Z}}$ is called autoregressive of order $p$ if it satisfies: $$X_t=\phi_1X_{t-1}+...+\phi_{t-p}X_{t-p}+W_t.$$ ...
2
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1answer
467 views

Can I use a spatially lagged Dependent Variable while using Spatial Error Model?

I have following issue: I run spatial diagnostics on dependencies for my Log-Log Transformed regression model. LM Tests (including Robust) are highly significant. Since I am using GeoDa, I cannot ...
2
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1answer
671 views

What's the meaning of the expansion coefficient of the AR model?

I am trying to understand the meaning of the phi parameter of the AR modeling. A bit of background: I am digging into statistical parametric mapping (SPM) and the prewhitening method, used to get rid ...
2
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1answer
135 views

AR(2) process: are leptokurtic residuals OK?

I have a time series of logarithmic returns. After inspection of the ACF and PACF plots, I tried to fit AR(2), MA(2) and ARMA(1,1) models and eventually found out that the AR(2) version can possibly ...
2
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1answer
5k views

conditional vs unconditional forecast variance in AR(1)

I have trouble showing that conditional forecast error of AR(1) has smaller variance than the unconditional one. I can show that cond. forecast error is: $$ Y_{T+1}=aY_{T}+\epsilon_{T+1} $$ $$ \...
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1answer
301 views

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 ...
2
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1answer
472 views

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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1answer
685 views

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 ...
2
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1answer
33 views

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 ...
2
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1answer
89 views

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 ...
2
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1answer
608 views

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 ...
2
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1answer
31 views

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 ...
2
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1answer
182 views

How to simulate AR(p) model with trend

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 <...
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1answer
20 views

AR(1) Finding $\gamma_l$

I have $\gamma_l = Cov(r_t, r_{t-l})$ as a definition in my notes and now I need to find $gamma_l$ for a series $r_t- m = p(r_{t-1} - m) + a_t$ where $r_t$ is a linear time series with expected value $...
2
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1answer
878 views

Temperature time series forecasting predictions converging to a certain value

I am trying to forecast the value of the ambient temperature based on given data on Python. The data frequency is 15 minutes. In order to predict future values, I am using a simple autoregressive ...
2
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1answer
3k views

Expected Value of an AR(1) process

I saw the answer on this post and got confused about a couple things in its explanation. Mainly, I am unsure of How the poster immediately knows the process $X_t = c+\phi_1 Y_{t-1} + \epsilon_t$ is ...
2
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1answer
140 views

Parameters in Autoregressive representation of an ARCH model

Suppose we have a $0$ mean time serie representing stock index returns about a title, $r$. I also know it follows an $ARCH(p)$ model with parameters $\omega$ and $\alpha$, specified in the following ...
2
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1answer
141 views

A first order autoregressive process with lag-1 auto-correlation 0.5

I was reading my professor's notes by myself, it says that: ''A first-order autoregressive process with lag-1 auto-correlation 0.5 can be generated from $\theta^{(t+1)}|\theta^{(t)} \sim N(0.5\theta^{...

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