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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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27 views

A simple question regarding AR(1) process and CDFs

Somewhat a trivial question, but I struggle to get my head around it. Consider we have an AR(1) process, as follows: $y_t=\rho y_{t-1}+\varepsilon_t,\quad t=1,...,T$. such that $\varepsilon_t$ are $...
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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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13 views

How to calculate auto-regression with missing values for different surnames in one generation?

I do have a dataset consisting of surnames, years and values y. My aim is to analyze whether the value y is dependent on the corresponding value y of the previous generation. Unfortunately, I do not ...
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12 views

Regression for long tailed time series events

I have a set of values which are a time series and follow a long tailed skewed distribution. I would like to understand what the best method might be to predict the next value in the series. Do the ...
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21 views

Z-score from Skewed Student T

I'm implementing the following method. The text is provided for background, but my question is about line (8). Am I understanding this as "a z-score generated from a standardized skewed Student t?" ...
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1answer
19 views

Solving Variance of Time Series AR process [duplicate]

I am trying to solve for the variance of $x[n]$, a time series process. $$x[n] +a_1x[n-1]=w[n]$$where $w[n]$ is white noise with zero mean and variance $\sigma^2_v$. Also $|a_1|<1$. I am aware ...
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29 views

Statistical test for comparing means of two AR(1) time series

Say I have two time series which each follow the AR(1) model: $$ X_{t+1} = X_t + (1 - \theta_X) (\mu_X - X_t) + \epsilon_X(t) $$ $$ Y_{t+1} = Y_t + (1 - \theta_Y) (\mu_Y - Y_t) + \epsilon_Y(t) $$ ...
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1answer
17 views

Showing 0 covariance for special form of AR(1) time series

This is an exercise I have been trying to solve but have not made much progress. Suppose $\{Z_t\}$ is an AR(1) process with $\rho_1 = \phi$. Define the sequence $\{b_t\}$ as $b_t = Z_t - \phi Z_{t+1}$...
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78 views

Superposition of random walk and autoregressive process

Let us consider the following model: $$ y_{t} = c_{t} + \alpha y_{t-1} + v_{t} \\ c_{t+1} = c_{t} + w_{t} $$ where $v_{t} \in \mathcal{N}(0, \sigma^{2}_{v})$ and $w_{t} \in \mathcal{N}(0, \sigma^{2}...
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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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Covariance of prediction errors

I have an exercise to compute the covariance between the prediction errors, but I'm not sure if it is correct, this is the exercise; I have an AR(1) model, $y_t = \phi y_{t-1} + \epsilon_t$, where $\...
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Is it allowed to reduce a dataset of moving averages to run an AR(1) model properly?

I run a simple AR(1) and AR(2) model with the following code: ar.ols(df$y, order.max = 1) ar.ols(df$y, order.max =2) My dataset is as follows: I do have yearly ...
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18 views

Unconditional variance of AR(1) with stochastic volatility

I am trying to find the unconditional variance of an AR(1) process with stochastic volatility: $x_{t+1}=\rho x_{t}+\sigma_{t}\epsilon_{t+1}$ $\sigma_{t+1}^2=(1-\nu)\sigma^2+\nu\sigma_t^2+\sigma_w w_{...
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48 views

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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1answer
31 views

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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28 views

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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35 views

How to change the observation for the first lag in an AR(1) model?

I run a simple AR(1) model in my analysis using ols: ar.ols(df$y, order.max = 1)) However, I work with generations as my unit of analysis. Therefore, the first ...
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40 views

Prediction of $X_{n+1}$ with Yule-Walker estimate

Consider a causal AR(1) process $X_t = \phi X_{t−1} + Z_t$ with $(Z_t)$ iid with mean 0 and finite variance. I am reading in a book, that $\phi X_n$ is the best predictor for $X_{n+1}$ because it ...
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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?
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How to check whether a given ARIMA (p, d, q) process is stationary or not?

I know that a finite MA process $X_t = \Theta(B)Z_t$ is always stationary. Also, whether an AR(p) process is stationary or not can be verified by checking the roots of $\Phi(B)=0$ where the process ...
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43 views

form of the model when using backshift operator

Be $Y_t=X_t + \epsilon_{1,t}$, in which $X_t = X_{t-1} + \epsilon_{2,t}$ and $E[\epsilon_{1,t}\epsilon_{2,s}] = 0 \forall t,s$. How could I say why this process is related with a model on the form $(1-...
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Implementation of 2SLS in regression with AR errors?

Consider the following simple example: $Y_t=\beta X_{t-1}+\varepsilon_t$ $X_t=\gamma Y_t + Z_t +u$, where $\varepsilon_t=\alpha\varepsilon_{t-1}+\eta$, and $E[Z_t\varepsilon_t']=E[uu']=E[\eta\...
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49 views

find the autocovariance function of the process $Y_t$

Consider the processes $X_t = \phi X_{t-1} + v_t$ and $Y_t = \phi Y_{t-1} + X_t + e_t$, in which $|\phi| < 1$ and $v_1$ and $e_t$ are non-correlated random errors with zero mean and variances equal ...
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30 views

Stationarity for an AR(2) process

How can I show that the following AR(2) process is stationary $X_t = X_{t-1} + cX_{t-2}+Z_t$, provided -1 < c < 0 ? I represented the series as $\Phi(B)X_t = Z_t$ and then tried to find out ...
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54 views

Handling serial correlation in time series regression

Suppose that the time series data $(y_1, y_2,..., y_n)$ can be explained through a regression model with $k$ explanatory variables: (1) $y_t = b_0+b_1x_{1t}+b_2x_{2t}...+b_kx_{kt} + \epsilon_t,\ t=1,...
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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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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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1answer
98 views

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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27 views

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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61 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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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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autoregressive time series issue understanding expectation

Consider the following question How is it obvious that $\mathbb{E}(X_t) = 0$? Is it because of the following? Recursively we find that $\mathbb{E}(X_t) = \phi \mathbb{E}(X_{t-1}) = \phi^2 \mathbb{E}(...
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Revert AR process with constant

I have got this task at the Time Series course as a part of Statistical minor. I am math major, and have gone through basic Probability (read:measure theory) course. Let us have $y_t = 0.4y_{t-1}+2+\...
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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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1answer
33 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 ...
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1answer
45 views

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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2answers
52 views

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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1answer
30 views

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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80 views

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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1answer
29 views

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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35 views

AR model on SMA(k)

If I were to regress Yt+1 on the simple average of Yt, Yt-1, ..., ...
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8 views

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 ...
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63 views

When to use different covariance structures?

Log-likelihood could be wrong so comparing covariance pattern models based on log-likelihood and LR tests isn't a perfect way. Intuitively, how do I know when to use Toeplitz versus AR(1) versus ...
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1answer
54 views

Building A Model: Autoregressive?

I've recently read a code about a fisheries productivity model where someone tries to predict the value at time t+1 from its value at time t. There're 23 recorded productivity of tuna from 1967 to ...