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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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323 views

Nonstationary solutions for stationary ARMA equations

By "stationary" I mean "weakly stationary". Consider a "stationary" AR(1) equation: $$X_t=\varphi X_{t-1}+\varepsilon_t,$$ where $t\in\mathbb{Z}$ are discrete time moments, $\varepsilon_t$ a zero-...
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What guarantees the existence of a finite representation of the Wold decomposition? Mechanics and Intuition

Every covariance stationary process can be written as a linear, infinite distributed lag of white noise. In other words, every covariance stationary process has a Wold representation. Then we go on to ...
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598 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, ...
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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 ...
4
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0answers
40 views

Autocorrelation in Elo ratings

FiveThirtyEight uses the following formula for their NFL Elo ratings: $$ R_i^{k+1} = R_i^k + K \cdot M(z) \cdot A(x) \cdot (S_{ij} - \sigma(x)) $$ where $z$ is the game's margin of victory, $x=R_i^k - ...
4
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127 views

Stationary Distribution of Multiplicative Autoregressive Model

I know for the additive autoregressive model the stationary distribution of $\{X_t\}$ can be found, if it exists, in the following way: \begin{align} X_t &= \alpha X_{t-1} + \epsilon_t\\ \...
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97 views

Identification Problem in Minimum Distance Estimation

I have the following problem with a system of minimum distance equations I want to solve. The objective is to estimate the parameters of the random variables in the following DGP: $$ x_t= \phi_t(\...
4
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1answer
472 views

Model selection and estimation for pseudo out-of-sample forecasting

I have quarterly data on inflation from 1990 Quartal 1 to 2016 Quartal 3. If I want to perform the pseudo out-of-sample forecasting one quarter ahead with an autoregressive function, do I have to ...
4
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242 views

Describe AR process with additive white noise using ARMA process

Disclaimer: This is a homework problem This is a problem from "Adaptive Filter Theory" by Haykin. Problem 2.10 (2nd edition). Problem A discrete-time stochastic process $\{x(n)\}$ that is real-...
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2k views

What is the exact log-likelihood of an AR(2) model?

Let's say we have the following AR(2) model: $y_t=\phi_0+\phi_1y_{t-1}+\phi_2y_{t-2}+e_t, \; e_t\sim N(0,\sigma^2_e)$ with T observations in total. Working out the conditional log-likelihood is ...
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310 views

Determining parameters in AR model for non-stationary time series

I am currently trying to fit an AR model to some financial data. The time series $Y_t$ in levels is clearly non-stationary; however it appears the first differences $dY_t$ are stationary (and this is ...
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449 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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400 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 ...
3
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1answer
44 views

When to use AR and when to use MA model?

When to use an AR model and when to use an MA model to model time-series data. What aspects of data are modelled by the AR process which can't be done by MA and vice-versa?
3
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1answer
70 views

First difference of AR(1) process

Given AR(1): $$X_t - \mu = \phi(X_{t-1}-\mu) + \epsilon_t$$ where $$ \mu = 0.85 \\ \phi=0.59 $$ and $$ W_t = X_t - X_{t-1} $$ Compute $$ Corr(W_t,W_{t-1})=-0.205 \\ Cov(W_t,W_{t-4})=-0.43 \\ Corr(...
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129 views

Dealing with autocorrelation using Generalized Least Squares

I have a time series data set where the auto correlation of the residuals follow an exponential decay. I was wondering how I should deal with this? I would like to fit a linear model and have tried ...
3
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68 views

Thomas Sargent's intuition as to why every covariance stationary series has an infinite-order Wold representation

In his classic book "Time Series Analysis", James Hamilton references Thomas Sargent (["Dynamic Macroeconomic Theory"], 1987, pp. 286-290) as a "nice sketch of the intuition behind this result [Wold ...
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simulate autoregressive data that is also multivariate normal

I am trying to simulate data that is correlated to varying degrees. However, the data itself will have a degree of autocorrelation as well. I can get the first part of the problem with mvrnorm ...
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48 views

Result of an ADF-Test compared with an estimated AR (p) model

I am currently investigating the inflation persistence for different countries by using R. I took data from the OECD for Sweden (1993-2017) and checked first if the series is stationary with the ur....
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65 views

Stationary density of $\{X_t\}$ as a solution of integral equation

For the model, $X_t = \alpha X_{t-1} + \epsilon_t$, we find the integral equation related to stationary distribution in the following way: Let $X_{t-1}\thicksim f$ and $X_t|X_{t-1}=x \thicksim q(y|x)$...
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60 views

Fit ARMA model to ACF

If I have the autocovariance function $\gamma_\tau$ (numerically over a given set of lags $\tau = 0 \ldots n - 1$) of a stationary linear stochastic process, is there an efficient way to determine the ...
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258 views

Assumptions on Neural Networks (NNETAR)

Are there any assumptions that must be covered when fitting an NNETAR model? non-correlation, normality, or something? I've already saw Rob Hyndman post where he says NNETAR doesn´t need stationarity, ...
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57 views

Nonparametric Quantile Regression for AR(1)-ARCH(1) process

I would like to estimate the conditional scale function $(\sigma_\tau(X_t))$ in a QAR-QARCH model represented by: \begin{equation} Y_t = \mu_\tau(X_t) + \sigma_\tau(X_t)\epsilon_t,\, t = 1,2,\ldots \...
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708 views

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.,...
3
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256 views

Ljung-Box and Breusch-Godfrey for univariate time series

I am testing autocorrelation of a raw time series, not the residuals with Ljung-Box and Breusch-Godfrey tests. The problem is that I am getting contradicting p-value results for some series: Ljung-...
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765 views

Why do we use prediction error decomposition for the derivation of the likelihood function for AR(p)

A good example of deriving a likelihood function is the normal distribution: The PDF of the normal distribution is: $$ f(x;\mu, \sigma) = \frac{1}{\sigma\sqrt{2\pi}}exp[\frac{(x - \mu)^2}{2\sigma^2}]...
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173 views

MLE estimation of Autoregressive Conditional Poisson model

The density of an Autoregressive Conditional Poisson ACP(p,q) model is defined as $$ f(x | \lambda_{t}) = \frac{\lambda_{t}^{x}\exp[-\lambda_{t}]}{x!},$$ where $$\lambda_{t} = \omega + \sum_{j = 1}...
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65 views

Stationarity in the Almon lag model

I have a quick question regarding the Almon approach (Shirley Almon) as presented in chapter 17 of Gujarati's Basic Econometrics. In an example given in the textbook, they use non-stationary data ...
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540 views

Derivation of conditional expectation and variance of the AR(1) process

I have a question regarding the AR(1) process. I want to derive the conditional expectation $E(X_{(t)}| X_{(0)})$ and the variance $\operatorname{Var}(X_{(t)}|X_{(0)})$ of the AR(1) process: $$X(t)=aX(...
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564 views

Longitudinal data analysis where meaning and metric of response variable varies over time

Determining what factors predict change over time is a topic of investigation in many fields and there are a variety of readily implemented methods for analysing repeated measures in the same metric. ...
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753 views

The inverse of AR correlation matrix

I want to find the inverse of the following matrix: $$ R_{k-1}=\begin{pmatrix} 1 &\rho &\rho^2 &\cdots &\rho^{k-2} \\ \rho &1 &\rho &\cdots &\...
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164 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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255 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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292 views

(Quantile regression) AR(1) variable in the design matrix

I'm not doing a pure QAR (quantile auto regression) but I do have a lagged dependent variable (AR(1)) as a predictor. I'm using the quantreg package in ...
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15 views

when fitting a regression model to a time-series, can I use lagged values of the time-series itself?

I'm fitting a regression model $y_t$ to a time series $x_t$ (not a dynamic model involving ARMA terms!). I saw that useful predictors to put in my model are $t$, seasonality variables and lagged ...
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31 views

Context in which an AR(1) error term can be considered a random effect?

We have the following situation: \begin{aligned} y_t &= f(x_t)+u_t, \\ u_t &= au_{t-1}+\epsilon_t, \\ \epsilon_t &\sim N(0,\sigma^2). \end{aligned} To make it simple, let's assume $f$ is ...
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What happens when using Durbin-Watson Test for AR(2)?

In my textbook, it says Durbin-Watson Test can be used only for AR(1) because d-statistic becomes biased if error term isn't follow AR(1) process. I'm curious why d-statistic gets bias when using DW ...
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What is the difference between the results using different AR(p) estimation methods?

There are three different ways to do AR(p) estimation. OLS MLE Yule=Walker Equation What are the differences between the results using these three methods? // http://www2.econ.osaka-u.ac.jp/~...
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1answer
52 views

General formula for AR($p$) auto-regressive time series

I'm trying to find a reference (including the full formula) for the following. If $X_n = a_1 X_{n-1} + \cdots a_p X_{n-p} + e(n)$ where $\{e(n)\}$ is a white noise, then $$ X_n=g(e_0,e_1,\ldots,e_n)+\...
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29 views

On the stationary density of an autoregressive model of order 2

Consider a stochastic process $\{X_t, t = 1, 2, \ldots\}$ following a stationary AR(2) model $$X_t = \theta_1 X_{t-1} + \theta_2 X_{t-2} +e_t,$$ where $e_t \thicksim N(0, \sigma^2)$. I want to find ...
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90 views

How many lags to use in ADF test?

So I've ran a ADF test on my data multiple times with different lags and all up to a lag of 4 have a p-value below .05. So in this case how many lags do you decide to use? Could this also provide a ...
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40 views

Do the assumptions for linear regression apply to AR(p) models?

If we have a stationary time series and we want to model it as an AR(p) process, what conditions must hold besides the stationarity itself? Are they the same a the assumptions for linear regression: ...
2
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0answers
48 views

Regressing across multiple different time series using exogenous variables?

To make this situation clear, I'll use a somewhat silly, but conceptually simple example. Imagine I record teams of movers carrying furniture down the block. I measure the furniture's position/speed ...
2
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148 views

What's the variance of an AR(1)/ARCH(1)

The main question is: an AR(1)/ARCH(1) process has the variance of the ARCH(1)? I've tried to compute the unconditional variance of an AR(1)/ARCH(1) model, so an AR(1) in which the noise is modelled ...
2
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0answers
109 views

Markov Chain order 1 vs. AR(1) … Difference and Implication for Parameter Estimation

As other posts on this site indicate, the difference between a time-homogeneous Markov Chain of order 1 and an AR(1) model is merely the assumption of i.i.d. errors, an assumption that we make in AR(1)...
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225 views

How to relate roots of AR and MA to unit circle

I'm working on these problems and think I figured out most of the steps, but am stuck near the end as I don't understand how to relate my roots back to the unit circle in order to determine ...
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0answers
124 views

Convergence of predictions of an autoregressive model

I have performed a simple autogregressive model with lag 2 on a time series data. After obtaining the coefficients, I have computed the predictions. Since the lag is 2 in model, the first prediction $\...
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57 views

Forecasting autoregressive model. What's the best linear predictor?

Obviously if $X_t = \phi X_{t-1} + Z_t$, then the best linear predictor of $X_t$ given $X_{t-1}$ is $X_t = \phi X_{t-1}$. But if $\phi$ is unknown, one may attempt to substitute $\phi$ by a Yule-...
2
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0answers
69 views

divergence of beta estimates between OLS and regression with ARIMA error

I have physiological time-series data: ~60k observations per channel, ~100 Hz sampling. I will model individual channels with ~20 regressors. Under OLS, given temporal autocorrelation in the data, ...
2
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0answers
428 views

LASSO in AR-Models

I couldnt find such a post here. I am highly interested in applying the lasso to different situations. However, I am actually dealing with time series models of high order. I have found some research ...