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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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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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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477 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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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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1answer
213 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?
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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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158 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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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
518 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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276 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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577 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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339 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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484 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 ...
4
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437 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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ACF of differenced MA(p) process

I have an MA(4) process applied to the first order seasonal difference of $Y_t$ as follows: $(1-B^s) Y_t = (1+\theta_1B+\theta_2B^2+\theta_3B^3+\theta_4B^4) Z_t$ where $Z_t \sim N(0,\sigma^2)$ This ...
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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: ...
3
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193 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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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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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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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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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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335 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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62 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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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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284 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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909 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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197 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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66 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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622 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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792 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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178 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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269 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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301 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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27 views

Cumulative Effect

Let $X_t$ be a causal AR($p$) model. If we have $n$ observations of this time series and fit an AR($m$) model with $m\gt p$ to the data; that is $$X_t = \phi_1 X_{t-1} + \cdots + \phi_m X_{t-m} + W_t,...
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Can we express an AR(1) process as follows?

If $X_t$ follows an AR(1) process as follows \begin{equation} X_t=\rho X_{t-1}+\varepsilon_t \end{equation} Would it be correct to express the above as \begin{equation} X_t=\mathbb{E}\{X_t\mid X_{t-...
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53 views

How to estimate a single AR model for multiple time series?

I have ACF of a time series data, I want to interpolate the ACF using AR model for which I need to calculate the coefficient of the AR model The problem is I have (6000x6000) matrix of ACF and I cant ...
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0answers
36 views

Standard errors with delta method

Trying to recreate other author's results. E.g. this paper. Introduction to the model is on page 10, while table with results is presented on page 13. Under the table there's a small note that SE were ...
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0answers
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Cross-lagged model with more than two variables (SEM)

I was wondering if anyone could point me to some literature discussing cross-lagged structural equation models with more than two variables: all the materials I found keep it very simple, and I ...
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0answers
144 views

Is there a library to fit a Threshold Autoregressive Model (TAR) in Python?

I tried to look into Statsmodel but I couldn't find it. I know that in R there is the TAR package. I would like to find something similar for Python. My entire project is written in Python and I've ...
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0answers
39 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 &\overset{iid}{\sim} N(0,\sigma^2). \end{aligned} To make it simple, let's ...
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0answers
39 views

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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0answers
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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
61 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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0answers
49 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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121 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 ...
2
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0answers
51 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 ...
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0answers
305 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
136 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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