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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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### My VECM Model Produces The Same Residuals For A Two Asset Portfolio

I have a two asset portfolio with 2 cointegrated ETF's. I would like to see when the ETF's deviates from their equilibrium. Before I show the model, what I expect to happen was that if one ETF's ...
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### Deriving the general form of the best linear predictor $\tilde{X}_{n+m}$ of $X_{n+m}$ for AR(1) process in terms of $X_1, …, X_n$

I'm trying to derive the best linear predictor $\tilde{X}_{n+m}$ for $X_{n+m}$ for a causal, zero-mean AR(1) process $Z_t = X_t - \phi X_{t-1}$. My answer needs to be in terms of $X_1, X_2, ..., X_n$. ...
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### JAGS code for Poisson or negative binomial hurdle (zero-altered) model with autoregressive residual

I am using Bayesian zero-altered Poisson and negative binomial models analyzing time-series data with JAGS. Because the ACF of the Pearson residuals showed autocorrelation, I decided to apply ...
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### How to calculate the autocovariance of a time-series model when the expectation is taken over different lags?

Let $Z_t$ be a weakly stationary stochastic (WSS) process of order $p$ modeled as an autoregressive model. $Z_t = \phi_{1} Z_{t-1} + \phi_{2} Z_{t-2} + \phi_{p} Z_{t-p} + a_t$ . where $a_t$ is ...
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### Kalman EM estimation of observation variance

Let us consider a simple AR(1) process: $$y_{t} = \mu + \beta y_{t-1} + \varepsilon_{t},$$ with $t = 0, \dots, N$. Assume that the parameters $\mu$ and $\beta$ slowly change in time and let's ...
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### Consequences of fitting Unit Root data directly in AR model

I feel it is useful to understand the consequence of violating the assumptions of a model. I check a couple textbooks, but most I can get about the consequence of fitting time series with unit root is ...
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### Converting coefficient of slope to autoregressive factor

I realize this is very fundamental. I apologize. Is there any way to convert the coefficients from a linear model into the decay factor if i want to express it as an autoregressive model? For a ...
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### 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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### Why do these independant variables have significant explanatory power, when 'theoretically' they should have none? (Self contained example inside)

I am putting together a model which involves a simple linear regression, and to aid the development I have put together a process for generating synthetic observations. The idea is that you have ...
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### Comparison of GMM and ML estimators for regression with correlated errors

Consider a linear model with normally distributed, autocorrelated errors \begin{aligned} y&=X\beta+\varepsilon \\ \varepsilon&\sim N(0,\sigma^2_{\varepsilon}) \text{ and autocorrelated.} \end{...
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### 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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### How can I estimate autoregression when non-stationary?

I have a series that I believe has one autoregression characteristic under condition A (example: positive) and another under condition B (example: negative). Is there a way (hopefully in Python) to ...
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### Stationary distribution of AR(1) process with AR(1) shocks

I am trying to find the stationary distribution of an AR(1) process, where the shock terms themselves are an AR(1) process. That is, the process moves subject to the following 2 equations: \begin{...
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### AR(1) process can be estimated using linear regression

Can the $AR(1)$ process represented as $$x_t= ax_{t-1}+\epsilon_t$$ be estimated by regressing $x_t$ on its lagged value $x_{t-1}$.
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### invertibility of $AR(\infty)$?

Here it writes: "Pure AR models are always invertible (since they contain no MA terms)." Is this valid also for the limiting case, that is to say, is $AR(\infty)$ invertible? Why or why not? If ...
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### Using least squares to estimate variance of latent variable

I am having trouble understanding why I can't use least squares to solve an overdetermined system of linear equations using $\bf{x} = (\bf{A}'\bf{A})^{-1} \bf{A}'\bf{b}$. The same model estimated ...
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### 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 ...
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### What is the virtue of loading absolutely-summability in the definition of causality of ARMA model?

An ARMA series $y_t$ is causal function of $\nu_t$ if there exists constants $\psi_j$ such that $\sum_{j=0}^{\infty} |\psi_j|<\infty$ and $y_t=\sum_{j=0}^{\infty} \psi_j\nu_{t-j}<\infty$ for ...
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### Time Series equivalent of the Generalised Linear Model

I have a time series $y_t$ which is measured at regular intervals over a long period of time. The values of $y$ are between $0$ and $1$, it represents a proportion, and these values change slowly over ...
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### the difference between using an AR(1) term (as in GAMM) versus using PM lag variable (in GAM)

I conducted an experiment to predict particulate matter (PM) level using a GAM. To do so I included the lag1 PM (PM value of day before) as well as few meteorological terms. In my second experiment ...
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### Vector Auto Regression handling dummy encoded variables

Firstly, apologies if this question is obvious, I am new to Time Series Forecasting & ML in general. I have an application whereby I collect prices from betting exchanges on an interval. This ...
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### How to read Autocorrelations in GLS models?

I am playing with some marketing data. My response variable is market share and predictors are ...
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### AR model on SMA(k)

If I were to regress Yt+1 on the simple average of Yt, Yt-1, ..., ...
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### Lagrange multiplier test in Mixed Level Model

I want to estimate a mixed level model with AR(1) errors and then conduct a Lagrange Multiplier test. The mixed model allows for rich covariance structures but it does not allow for AR(1) errors. Can ...