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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Determining order of AR model

Suppose that we have the following model $$ y[t] = A_1\sin(\omega_1 t+\phi_1)+A_2\sin(\omega_2 t + \phi_2)+ \cdots + A_p \sin(\omega_pt + \phi_p) + z(t) . $$ Let us call this signal as B. Then in ...
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35 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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140 views

Differencing a time series

I am looking to find the ACF of a time series, but after it is differenced. $y_t = a_1y_{t-1} + \epsilon_t , \mid a_1 \mid < 1$ I understand that to find the ACF this process needs to be ...
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19 views

AR(2) simulation problem

Take covariances $Cov[X_{t-2},X_{t}]$, $Cov[X_{t-1},X_{t}]$ and $Cov[X_t,X_t]=Var[X_t]$ and calculate the parameters for the AR(2) process ($a_1$, $a_2$ and $\sigma^2$ (the variance of the error ...
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27 views

How to interpret the characteristic roots of moment equation of a AR(2) model?

I am learning the financial time series using the book 'Analysis of financial time series' by Ruey Tsay. In chapter 2, they introduced AR(2) models. The moment equation (which is the function between ...
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20 views

Autoregressive Markov chain simulation and the likelihood ratio test for Markov property

I am trying to estimate a Markov chain of second order (Markov chain that fulfills $P[X_t|X_{t-1},X_{t-2}]=P[X_t|X_{t-1},X_{t-2},...,X_{t-p}]$) using an AR(2) process. Once I have simulated the ...
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96 views

How to fit log-linear poisson autoregressive mixed model?

I have time-series count data $N_{i,j}$ (population sizes in site $i$ and year $j$) and I want to correlate year-to-year changes with the environmental conditions $x_{i,j}$. For this, I want to fit ...
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18 views

ARDL, Lag Terms and Singularity

I am interested in fitting an ARDL model that has 4 lags for each explanatory variable. However, when I fitting the model in R. R says that coefficients are not defined because of singularities. Is ...
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20 views

Computing Standard Errors in EM algorithm

I'm applying the EM to a hidden markov chain (the $\mathbf{Z}=\{Z_1,...,Z_n\}$ variable), with observations(the $\mathbf{Y}=\{Y_0,...,Y_n\}$ variable) dependent not only on the hidden markov chain, ...
2
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1answer
30 views

Mixture of normals, dependent on past

I have the following probability model: $(X_k|\text{PastHistory}_{k-1}, \theta_0,\theta_1,\theta_2) \sim (\pi\cdot N(\theta_1+\theta_0\cdot X_{k-1},1)+(1-\pi)\cdot N(\theta_2+\theta_0\cdot ...
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1answer
57 views

Conditional expectation in AR(1) process

Suppose we have a stationary AR(1) process: $Y_{t+1}=a+ \rho Y_{t} + \epsilon_{t+1}$ where $\epsilon_{t+1}$ is white noise with probability density function $\phi(.)$. Now say we have a equation ...
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34 views

Fit Negbin glm model with autoregressive correlation structure

I am attempting to estimate the effect of various variables on the time-series of counts of reported cattle stillbirths. We investigate the effect of day-of-week, month, holidays etc…and also the ...
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23 views

Vector autoregression with gradient descent

I am no expert in statistics, but I have been asked to implement a VAR model using gradient descent in R. I have written a code that, from what I have been told, it makes sense. However, the estimated ...
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25 views

Is a Stationary VAR Process with Zero Mean Gaussian Innovations a Gaussian Stationary Process?

Consider the stationary VAR process $${\bf X}_t = \sum_{\tau = 1}^{L} A_\tau {\bf X}_{t-\tau} +{\bf \epsilon}_t$$ If the innovations $\epsilon_t \sim MVN({\bf 0},\Sigma)$ then is ${\bf X}_t$ a ...
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2answers
123 views

Is non-stationary AR(p) process constant in mean?

A non-stationary $AR(1)$ process, which is a random walk, is constant in mean, but not constant in variance. How about the other $AR(p)$ processes with the order $p>1$? Are they constant in mean?
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51 views

Estimating a VAR model with variable coefficients

I want to estimate a VAR model based on the Dufour and Engle paper "Time and the Price Impact of a Trade" (2000). There, the parameter $ b_{i} $ of the endogenous variable $ x_{i} $ is dependent on ...
2
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1answer
66 views

Determining the amount of lag in an autoregressive model

I have done a lot of work in regression (time-invariant) but I am just now studying forecasting. My question is about determining the amount of lag to use in an autoregressive model. I assume that ...
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1answer
123 views

Auto-Regressional & Moving Average Model Formula Properties

I seeking help in understanding specific values underlying the formula's for the MA(p) model & the AR(q) model. I am attempting to implement the models (building up to the combined ARIMA model) in ...
2
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1answer
170 views

How to plot spectra of an AR(2) process

I am stuggling with this problem and was hoping to find some guidance to answer it. Let $y_t=\phi_1y_{t-1}+\phi_2y_{t-2}+\epsilon_t$, with $\epsilon_t\sim N(0,1)$. Now, I want to plot the spectra ...
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27 views

Determining the posterior distribution for an Autoregressive or order 1 model

Question: For this question, note that the notation $y_{1:T} = (y_1, y_2, \cdots, y_T)$, ie, a vector of random variables. Consider the following AR(1) model: \begin{align*} y_{t+1} = \phi y_t + ...
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116 views

Backshift operator applied to a constant

This questions is two part: 1) What happens when you apply the backshift operator to a constant? For example, if I have the AR process $$(1-\phi B)(y_t-\mu)=\epsilon_t$$ does that equal ...
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1answer
75 views

How to do Univariate Heteroscedasticity Test

I just wanted to know how to do Heteroscedasticity Test on a Univariate Model? ex: an univariate autoregressive model ex: an univariate ARCH/GARCH model If it is possible, how does one do that in ...
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78 views

Auto regressive process, maximum likelihood estimator

A first-order autoregressive process, $X_0,\dots,X_n$, is given through the following conditional distributions: $X_i | X_{i-1},\dots,X_0 \sim \mathcal{N}(\alpha X_{i-1},1)$, for $i = 1,2,\dots,n$ and ...
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39 views

Tips when regressing growth rates

I have 20 months of Year over Year growth rates for a X independent variable and Y dependent variable. The correlation between these two variables is 0.72. I would like to predict Y using X for ...
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54 views

Can I get an univariate ARMA(2,1) representation from a bivariate VAR process?

Suppose the VAR is on (x,y) and I want to get an ARMA(2,1) expresion for x, how can i do that? For example, $\left[ \begin{array}{l} x_t\\ y_t \end{array} \right] = \left[ \begin{array}{l} ...
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281 views

Meaning of output of function “ar” in R

How should I read the output of the function ar in R. For example, take this VAR model: ...
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154 views

Is this explanation of the Box-Jenkins approach correct?

Can someone please tell me what they think of my explanation of the Box-Jenkins approach to forecasting time series? Do you have anything to add (in particular to my explanation as to the intuition ...
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175 views

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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151 views

Time series: correcting the standard errors for autocorrelation

I have performed a number of tests to detect any presence of autocorrelation in my monthly return series. The test results confirm that the standard errors are not independent. A Durbin-Watson test ...
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284 views

AR(1) coefficient is correlation?

Is the ar1 coefficient from an AR(1) model the "first order correlation of the noise" of a time series? I'm using R's aws package and one of the arguments of the ...
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264 views

AR(1) process with heteroscedastic measurement errors

1. The problem I have some measurements of a variable $y_t$, where $t=1,2,..,n$, for which I have a distribution $f_{y_t}(y_t)$ obtained via MCMC, which for simplicity I'll assume is a gaussian of ...
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1answer
210 views

How to estimate a model with fixed and random effects for a long panel dataset?

NOTE: I am using Stata for doing this. I have a long panel dataset, meaning my N is much smaller than my T. I have N = 5, T = 61. I tried to estimate my model, but I get an error related to the ...
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1answer
134 views

Residuals in double seasonal exponential smoothing

I have a time series with muliple seasonal cycles, which are 24 and 168 hours for my case. I would like to use Double Seasonal Exponential Smoothing method to forecast, which was published by James W. ...
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62 views

What is geometric autoregressive process?

Can anyone give a definition for Geometric Autoregressive Process? Any specific properties? And, in what fields is this mostly applied? To add some context to the question, here is a section of the ...
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21 views

Number of areas in conditional autoregressive models

This is a simple question on Bayesian spatial modelling via conditional autoregressive modelling. What is, according to your judgement (and possibly some methodological source), the minimum number ...
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2answers
258 views

How to understand SARIMAX intuitively?

I'm trying to understand a paper about electric load forecasting but I'm struggling with the concepts inside, specially the SARIMAX model. This model is used to the predict the load and uses many ...
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112 views

Lagged dependent variables and dummy interaction terms interpretation

Quick question about interpreting lagged variables. It is my understanding that if I have a regression with 2 lagged dependent variables, the marginal effect of the lagged dependent variable is the ...
2
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1answer
184 views

AR(1) parameter estimation

Given a time series, I'd like to estimate the parameters of an AR(1) model for it. As explained on wikipedia, there are different ways for doing that. What may be called a naive method is to compute ...
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2answers
64 views

Method to remove bad values in time series (bad values known to take on a particular value)

This sounds easy, but I don't know of a good statistical method for it. I have a time series that has (good) data points that range from ~3.5 to 30. The data are collected by an automated sensor. ...
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50 views

Regressing coffee beans in US to coffee beans in EU

We're trying to model two time series: a random walk (independent variable) vs. the sum of this random walk and a mean-reverting process. For example: coffee bean 100kg prices (EU) vs. coffee bean ...
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50 views

asymmetric vector autoregression

When fitting a vector auto-regressive model to a few time series, all the lags up to a certain pre-specified number will be retained in the model. This is true even if the coefficients for those lags ...
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48 views

Prediction of $X_{t+2}$ of an AR(2) process.

I want to find the best linear predictor, in MSE sense, of $\hat{X}_{t+2}$ in terms of $X_s'$s where $s \le t$ and $$X_t = \phi_1X_{t-1}+\phi_2X_{t-2} + Z_t\,,\, Z_t \sim WN(0,\sigma^2)$$ $X_t$ is ...
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86 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 ...
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716 views

Exogeneous regressors in auto.arima and using them in forecast function in R

I'm trying to forecast a seasonal time series based on its historical values, and also two more time series (that are seasonal themselves.) I'm trying to use an auto.arima, and I'm going to input ...
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297 views

Why are MA(q) time series models called “moving averages”?

When I read "moving average" in relation to a time series, I think something like $\frac{(x_{t-1} + x_{t-2} + x_{t-3})}3$, or perhaps a weighted average like $0.5x_{t-1} + 0.3x_{t-2} + 0.2x_{t-3}$. ...
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65 views

Question on autoregressive equation

I was given a reduced form VAR model, where the dependent variable is inflation, and independent variables include inflation lagged by four periods (L.Inflation) and other exogenous variables. The ...
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240 views

Testing for autocorrelation of the residuals

I'm trying to test for autocorrelation in the residuals of an AR(p) model in stata using the command varlmar. The stata output is: "the exogenous variables may not be collinear with the dependent ...
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181 views

STATA - Procedure for properly estimating an AR(p)

I'm trying to estimate an autoregressive process AR(p). Following the literature: 1) I checked if the series is stationary or not running the augmented Dickey-Fuller test (as I expected, the ...
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49 views

Stata - Correction for finite sample

I need to run an autoregressive model with correction for finite sample coefficients, following the model proposed by Shaman & Stine in 1989. I usually use Stata and MATLAB for my analysis, but I ...
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156 views

Need a clear and simple auto-regressive model example

This may be hard to find, but I'd like to read a well-explained auto-regressive model example that: uses minimal math extends the discussion beyond building a model into using that model to forecast ...