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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how to fit ar(1) model with predetermined value of autocorrelation parameter in R? [migrated]

I have the following data: ar <- arima.sim(list(order=c(1,0,0), ar=0.9), n=M1) + 10 How to fit an AR(1) model to simulated data above with ar parameter=0.5? ...
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Time series prediction for a chaotic multivariate data

I am trying to forecast a chaotic multivariate time series, and any architecture I use ( LSTM, MLP or tried to implement Autoregression architectures from few papers like https://arxiv.org/pdf/1704....
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Why the lag number of an AR model doesn't indicate the number of lags in a plotted ACF?

In the below image, there are AR models with differing lags. As far as I know, each autocorrelation function plot has an x-axis that is "number of lags". Can someone help me understand how ...
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ar() in R and Matlab give different results for same dataset

I have a Matlab script that performs some autoregressions and am trying to replicate it in R, but can't get them to match. So as a MWE, I'm trying to get the same results in R as are obtained by this ...
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Negatively correlated estimators for the AR-1 process

I have the following question. Assume we have a stochastic process \begin{equation} y_t = \gamma + \phi y_{t-1} + \epsilon_t, \ \epsilon_t \sim \mathcal{N}(0, \sigma^2), \end{equation} where $|\phi| &...
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How would you convert an ARIMA(0,0,1)(0,1,0)12 model to equation form? [duplicate]

How would you convert an ARIMA(0,1,1)(0,1,1)12 model to equation form?
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Why ACF is used in MA and PACF is used AR models?

I'm reading Analysis of Financial Time Series(Third Edition) RUEY S. TSAY. The author summarizes the model selection of AR(p) and MA(q) as follows: For MA models, ACF is useful in specifying the ...
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Why are AR(p) processes always invertible?

My question is the following: If we have an AR(p) process, then we have the following $$ \Phi(B)X_{t}=Z_{t} $$. I understand that to check for causal/non-causal stationarity, we consider the roots of $...
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Conditional distribution of Ornstein-Uhlenbeck on two fixed points

The conditional distribution of a Ornstein-Uhlenbeck $X(t)$ conditional on $X(0)$ is given by $$ X(t)|X(0) = X(0)e^{-t} + \mu(1 - e^{-t}) $$ This process is usually only defined for $t>0$ (future ...
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Non-Sensible Estimates for MLE of AR Processes

I am taking a course on Time Series Econometrics and I am solving a problem set that requires students to explicitly write maximum likelihood functions for, as an example, AR processes and estimate ...
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Is there a point on performing time series analysis on data that are not gathered with a consistent frequency?

To be more specific, what I have in mind is data gathered from android games. These data wouldn't have any time consistency because a user is free to play as many games as he wants whenever he wants, ...
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Time series - best approach?

Good evening, newbie to the forum but I would appreciate some advice on a new problem I’m working on. We are collecting water samples and measuring a number of things simultaneously in the water, as ...
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Unbiased Estimator of AR(1) Models?

What are the options for unbiased estimators of AR(1) (or AR(p)) models? Bias reduction techniques may also be included (jack knife would be one). I found one paper called "Bias correction using ...
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Equivalence of trend stationary process and stationary process

I am new to Time Series, and I am having some trouble dealing with the constant of an AR(p) process, which of course 'reinforces' itself over time to become a deterministic trend. For simplicity sake, ...
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What does it mean “analyze sample time series data when only a single series is available”?

Since the book says, it will use time series to mean either realization of a process or a process, I have no idea how to interpret the following sentence. "This notion, called weak stationary(i.e....
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Advantages of using PACF

Box-Jenkins approach to time series analyses uses a series of diagnostics, one of which calculating Partial autocorrelation function (PACF). The goal is to determine the order ...
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variance of an autoregressive process

Let $\{x_t\}_{t\in\mathbb{N}}$ be a zero mean strictly stationary sequence of random variables and $c:\mathbb{N}\to\mathbb{R}$ the (auto)covariance function. If the process follows the AR(1) model $$...
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Variance of an AR and ARMA process derived from lag notation

This question concerns the asymptotic variance of an $\text{ARMA}(p,q)$ process. Suppose that an $\text{ARMA}$ process can be rewritten as an $\text{MA}(\infty)$ process, and from this we can in ...
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What kind of statistical test do I need in testing the lag order in an Auto Regression model AR(4) against a restricted AR(3)?

Bit stumped on this one and none of my resources are helping to clarify this one for me. I've been given an estimated Autoregression Model AR(4) model of the sort: Yt = b0 + b1 x Yt-1 + b2 x Yt-2 + b3 ...
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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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Why doesn't PACF cut off for MA processes?

While studying for a time series paper I came across the terms 'partial autocorrelation function' (PACF) and 'autocorrelation function' (ACF) in conjugation to $AR$ and $MA$ processes, why is it such ...
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Causal AR Model?

This questions is about necessary conditions (in form of inequality on coefficients) for the causality of autoregressive models. For instance, $|\phi_1| < 1$ is a necessary condition for an AR(1) ...
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arima (0,1,0). How can I interpret it in this case? [duplicate]

I was predicting time series using auto.arima () in R and I found something I don't know how to interpret. For a certain number of time series, auto.arima () indicated that the best arima model was ...
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Why does my SARIMA model not capture the seasonality?

I have sales data over 100+ days. Every Saturday has 0 sales. For the other days there is also a clear seasonality. Tuesday always has the highest sales, and the order in which the other days follow ...
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Component contributions in Additive Model Time Series

I have trained a model for forecasting time series in a greedy procedure: Fit the Trend component T(t) of the series on the original signal y(t) Fit a Cyclical/Seasonal S(t) component of the series ...
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How to interpret Autocorrelation plots?

I have sales data per day. To create an ARIMA model, they suggest to first look at an autocorrelation plot. How I interpret this is that they look how my sales are correlated to eachother for ...
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Mean of target values at different time points as predictor in multiple regression

I've received regression model that predicts crop yield based on data collected at 3 time points (years). Input data contains multiple attributes and crop yield in the given year for a given location....
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Smoothing autoregressive coefficients

I fit an autoregressive model to a time series with 1837 observations using the R ar() function setting the maximum number of lags to 20. The function selected an AR(19) model using the AIC criterion, ...
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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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ARIMA Model Non-Stationary Time Series

Suppose that the data generated process is the following: Y(t) = 1.2*Y(t-1) + 0.2 The process is clearly non-stationary. My question is why we can't fit an AR(1) model and make predictions?
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AR(p) by iterated vs. lag method. Different results

Reading "Applied Econometrics Time Series" By Walter Enders I am trying to derive the stationary AR(p) model as he does on page 58, fourth edition. This is the AR(P) model \begin{equation} y_t=a_0+\...
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Why do AR(1) times series generated by two methods look similar but have different variance estimate in Python

I come across one question when I use two ways to generate AR(1) sequences. By definition, AR(1) sequence is $x_t = \phi_1 x_{t-1} + \varepsilon_t,\quad \varepsilon_t\sim N(0, \sigma^2)$ I found ...
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Is there a root of AR-polynomial which is the same for any $\phi$?

I am learning timeseries models and got some doubts. Consider an ARIMA(1,1,0) process $Xt$. Let $\phi(z)$ is AR-polynomial. $(1-\phi B )(1-B)=Z_t$. $(1-B)X_t=Y_t$. I read in my study material ...
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Can an AR(1) process with finite past be well-defined?

I am wondering if there is a true need for the infinite past of an AR(1) process to be defined. Usually, an AR(1) is a stationary process defined by the set of equations \begin{equation} X_t = \...
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How fast does a auto-regressive process converge?

Recently I have come across a time series data that happened to fit MA(1) process really well, and I would like to know how fast does this series to mean revert ? I did some google search there seems ...
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Does AR in the TAR model of time series still need to consider stationarity?

For example, first order difference operation? Because I had to implement the TAR algorithm with the MCP penalty function myself, I had to understand the calculation details
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Interpret AR(3) output from `arima` function in R

I have AR(3) like following. I'm not sure whether it is interpreted to $$ Y_t = 5.6923 + 1.0519 Y_{t-1} -0.2292 Y_{t-2} -0.3931 Y_{t-3} + e $$ or other? Thank you. ...
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What is the best way to present the following predictive regression relationship?

If I have a predictive regression with a single regressor of the form \begin{equation} y_t=\beta x_{t-1}+\varepsilon_t \end{equation} where \begin{equation} x_t=\rho x_{t-1}+u_t \end{equation} Then I ...
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Coefficients of the Wold representation of an AR(2) process

So, I am aware that a covariance-stationary AR(p) process can be written as an infinite MA (the Wold representation), taking the form (where there is no constant) $$ y_t=\sum_{i=0}^{\infty}\psi_i\...
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ADF-Test indicates stationarity for a non-stationary time series

I have a minor issue and am not sure what to do. The link below leads to an image of two time series I plotted, the upper being the original, the bottom one obtained by taking the first differences. ...
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How I can simulate autocorrelated data time varying mean in R?

Actually I am working on SQC and I want to fit an AR(1) model to the autocorrelated data with changing mean and use the residuals as charting statistic (shewhart, EWMA, CUSUM) to study the small ...
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How should you determine the order of an AR(p) model using PACF with fluctuating significance?

While plotting the PACF of the sample, the PACF values become insignificant post the second lag, then significant again post the 8th lag and so on. Basically, there's cyclicality in the partial ...
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How to estimate the grandparental influence on the intergenerational transmission of social status?

I do have an educational status for individuals from different families over three generations (y, yparents, ygrandparents). To determine the two-generational social mobility, I run a simple linear ...
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What is the theoretical distribution for this AR(1) model, $(1−0.8B)x_t=ϵ_t,ϵ_t∼N(0,1)$?

The AR(1) model is: $(1−0.8B)x_t=ϵ_t , ϵ_t∼N(0,1)$ and the true mean of the process is $μ≡E(x_t)=0$. Please tell me what is the theoretical distribution under the true AR(1) model. Is it also $N(...
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Masked Autoencoder MADE implementation in TensorFlow vs Pytorch

I am following the course CS294-158 [1] and got stuck with the first exercise that requests to implement the MADE paper (see here [2]). My implementation in TensorFlow [3] achieves results that are ...
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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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Chow test on autoregressive regression

For my master thesis I would like to investigate whether special items (= one-time effects, such as for instance restructuring expenses etc.) do have explanatory power for future operating income. ...
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What is the ACF plot of $x_t = 0.9 x_{t-2} + w_t$

I am just learning time series, and I am wondering about the following AR(2) model: $x_t = 0.9 x_{t-2} + w_t, w_t \sim N(0, \sigma_w^2)$ Please show me the plot of its Autocorrelation Function, or ...
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Parameter estimation for time-varying autoregressive processes in R

I want to estimate the parameters of an autoregressive process with time-dependent coefficients. For example TVAR(1) model with 1 lag: $$ X_t = \phi_t X_{t-1} + \sigma_tW_t $$ where $\phi_t$ and $\...
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Hidden Markov Model with Autoregressive Emissions

So far, all standard HMM implementations I've seen assume some variation of a Gaussian Mixture (GMM) as their emission model. It can of course only have a single mixture component which reduces it to ...

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