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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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The Importance of Initial Conditions in Autoregressive Modeling

I am developing an algorithm to classify time series by using autoregressive modeling. I have used the following two alternative methods, after fitting an AR(p) model to time series: Method 1: ...
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AR(1) Finding $\gamma_l$

I have $\gamma_l = Cov(r_t, r_{t-l})$ as a definition in my notes and now I need to find $gamma_l$ for a series $r_t- m = p(r_{t-1} - m) + a_t$ where $r_t$ is a linear time series with expected value $...
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Stationarity of ADL(p, q) with heteroskedasticity

Suppose I have the model $$y_t = \alpha_0 + \alpha_1 y_{t-1} + ... + \alpha_p y_{t-p} + \beta_0 x_t + ... + \beta_q x_{t-q} + \epsilon_t,$$ where $\{x_t\}$ is a stationary process and $\epsilon_t$ has ...
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Forecast Confidence intervals for for AR(P) [duplicate]

I want to contruct 12-step ahead forecast confidence intervals (CI) for AR(2) models and above. However, the CI calculation seems extremely tedius for forecasts above 2 periods as iteration process ...
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Spatial Temporal Autoregressive Regression and implication on Fixed effects assumption in Difference in Differences

I am currently planning on testing the effects of marginal price change of properties based on an exogenous event using Spatial DID model. The model has a spatio-temporal lagged variable (y) which ...
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(multiple) fractional outcomes & autoregression

Let me start with a broad description of the problem and I will then describe my approach (that might be totally inappropiate). The big goal is to predict the distribution of population of a given age ...
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What is an autoregressive model - terminology with respect to machine learning

In Wikipedia, an autoregressive model is defined in terms of an AR(p) linear process as The autoregressive model specifies that the output variable depends linearly on its own previous values ...
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Tests for predictive models with autoregressive neural networks

I'm working with time series predictions with NNAR autoregressive neural network models (p, P, k) and I'm doubtful for the validation of my models. After making the predictions, I'm selecting those ...
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Approximating AR(1) by finite order MA process - convergence results

I am currently struggling with a result pertaining to the finite order MA approximation of a simple AR$\,(\,1\,)$ process defined on a double sided time-index set $\,T=\mathbb{Z}$. I would be very ...
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VAR Model using Stata

I'm relatively new to the VAR model and have been using Sean Becketti's 'Introduction to Time Series Using State' as reference and wanted to check if I am on the right track. As of now, I have 5 ...
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Is conditional r-squared ever zero?

I am using multivariate auto regressive modeling (MAR) to assess a complex data set (MAR is a form of vector auto regressive modeling, VAR). The output of the MAR method is >1 response variables and >...
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Example: multivariate timeseries model that is uncorrelated at each time step but has higher order interactions

What are examples of timeseries models, say $X_t = [X^{(1)}_t, X^{(2)}_t]^\top, t \in \mathbb{Z}$ (or potentially continuous time), such that they are time-wise uncorrelated, that is $\mathrm{cov}[X^{(...
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Inverting MA(1) to AR(Inf) [duplicate]

While it is $MA(1)$ process there is no dependence between $u(t)$ and $u(t-1)$ i.e $$u(t)=v(t)+Q(1)v(t-1)$$ but when i converted it to AR process i get $u$’s that is dependent on the other $u$’s i.e. $...
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Converting MA(1) to AR(p)

While it is $MA(1)$ process there is no dependence between $u(t)$ and $u(t-1)$ i.e $$u(t)=v(t)+Q(1)v(t-1)$$ but when i converted it to AR process i get $u$’s that is dependent on the other $u$’s i.e. $...
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How to deal with auto-correlation in generalized linear modelling?

I've built a generalized linear model by using glm.nb function (my response is a count type of data) using a single predictor. The model summary is given below. <...
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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 ...
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How to adjust confidence interval

I am reading a journal where it is written "Time series are based on annual-mean translation speeds from 1949-2016. Trends are estimated by linear regression. The P values of the regression are based ...
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mean of autoregressive process [closed]

An autoregressive process of order p, AR(p), is of the form $x_t=\phi_1x_{t-1}+...+\phi_px_{t-p}+w_t$ where $x_t$ is stationary, i.e. $E[x_t]=\mu$ for all $t=0,1,2,...$, and $w_t$ is white noise, i.e....
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ARCH testing and Stationarity

I have a time series y and I need to build the "best" autoregressive model for it i.e. y(q*). What I do: 1. Start with y(1) and test for serial correlation in the errors until I find an AR lag -lets ...
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Recommended tests to determine whether threshold autoregression is suitable

I was wondering if anyone had suggestions about the best sort of test to determine whether a SETAR model is a reasonable specification. I've come across some portmeanteau tests, but I'm not sure ...
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Proof of contemporaneous exogeneity, and its implications for an AR(1) model

It can be shown by contradiction that exogeneity fails to hold for an AR(1) model. Is there any proof that contemporaneous exogeneity does not fail to hold? All I've come across is assuming it does ...
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Autocovariance and autocorrelation function of AR(1) process

I'm preparing the exam about AR models, precisely I have this exercise which I have some issues with points "d" and "e". My try was: Knowing that $W_t=X_t-X_{t-1}$, $h=1$ so: d) $\gamma\left(1\right)...
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Factorised form of Autoregressive Polynomial

I'm new to Time Series Analysis. I've read that when inverting autoregressive characteristic polynomial of arbitrary finite degree, we need to write it in its factorized form: $$\phi_p(x) = \prod_{i=1}...
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Fitting a multivariate AR(1) with covariates?

I have time series data where my main question of interest is making inference on predictive covariates, and accounting for the correlation (one observation each day) is just a nuisance issue. The ...
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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 ...
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1answer
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Interpreting effect sizes in cross-lagged -auto-regressive models

I am running an auto-regressive, cross-lagged panel model between three variables (individual survey responses) to understand the over-time dynamics between them. But I am trying to make sure I ...
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why fit `ARMA` model to residuals when doing residual analysis?

I started my Time Series Analysis not long ago and I am currently at the residual analysis. I found, in the course, the tutor was demonstrating residual analysis by fitting an $AR$, then $ARMA(p,0,q)...
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Projection in AR model

I am currently reading the Brookwell and Davis Book and cuurently read about the PACF. On page 98 they derive the PACF for the AR(1) model $$ X(t)=0.9X(t-1)+Z(t) $$ and say that the orthogonal ...
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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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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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Understanding the outputs of tar (Threshold Autoregressive model estimation, TSA package) command in R

I'm using the TSA::tar command to estimate threshold autoregressive model coefficients. There are a few parameters I would like to know the function of: rss1, rss2, std.res, rms1, rms2 (rms of what?) ...
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How to test for overfitting in a TAR model in R?

I want to fit a threshold autoregressive model, and I'm using the tar package in R. For ARIMA models, I could check if a model was overfit by looking at the values of standard errors as compared to ...
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Reduce the effect of excessive zeros

I am working on an autoregression problem where I use sequential LSTM. My target is well defined, but I think I am facing a problem with the features. As the features were non-stationary, then I ...
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Getting over bid-ask bounce

High-frequency financial data is subject to bid-ask bounce. Description : Unlike traditional data based on just closing prices, tick data carry additional supply-and-demand information in the form of ...
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What is the difference between GARCH, ARGARCH, and DCC-GARCH?

What is the difference between GARCH(1,1), AR(1)GARCH(1,1), and DCC-GARCH?
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Determining whether a model with random walk errors is stationary

If we have a model like an AR(1) except the errors are a random walk (i.e. not iid), then is the model itself stationary? So the model is: $$ x_t=kx_{t-1}+\epsilon_t $$ where $k$ is constant and $0<...
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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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How to compare two variables that have first-order dependencies?

I have the following data, ...
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Time series analysis:How to plot the the following AR(1) graphs?

The equation for AR(1) is : Cases: This is what it looks like: So I came up with this code: ...
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1answer
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Memoryless Property of a Markov Chain of Order 1. Is AR(1) memoryless or of infinite memory?

A stochastic process constitutes a discrete Markov Chain of order 1 if it has the memoryless property, in the sense that the probability that the chain will be in a particular state i, of a finite set ...
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Proof of AR(1) simulation [duplicate]

I am trying to simulate an AR1 process which has a mean of $\mu $. ie. $y_t- \mu = \Phi(y_{t-1}-\mu) + \epsilon_t$ This link here says that we should do :- ...
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1answer
36 views

General Form of Arima(2,1,2)

There is a question in my textbook that asks for the ARIMA(2,1,2) model. I get how to do the AR and the MA parts, but I'm having a little trouble understanding the differencing portion of the model. ...
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Can polynomials, interaction variables, and autoregressive variables trigger exogeneity issues in regression?

I would think the introduction of the mentioned types of variables would introduce exogeneity issues in regression models. However, in such circumstances are these exogeneity issues material, or can ...
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1answer
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What can we predict from the follow ACF and PACF plots?

This is a time series of a wind speed data collected every hour for a month. What can you interpret from the ACF and PACF plots about the trends and seasonal components? Are there any? And which model ...
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Impulse response for general VAR lag-p model: when does it converge?

Consider the VAR lag-p model: $$Bx_t = \Gamma_0 + \sum_{i=1}^p\Gamma_i x_{t-i} + \epsilon_t,\quad x_t\in\Bbb R^n,\,\forall t\in\Bbb Z$$ Setting $B$ to be upper-triangular and $A_0:=B^{-1}\Gamma_0,\,...
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Biasedness of ML estimators for an AR(p) process

Do you know any derivations (or references) which quantify the biasedness of ML estimators of an AR(p) process?
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1answer
150 views

Gaussian distribution of AR(1) model

This is very basic, but I have been stuck here for a while. Consider an AR(1) model $Y_t = c+\phi Y_{t-1} +\epsilon_t$, where $c$ is a constant. If $\epsilon_t \sim i.i.d. N(0, \sigma^2),$ then $...
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Different model result stats::arima and dynlm

I am calculating an autoregressive model with two different libraries (stats and dynlm). Attached you can find the code and the data. I am using in both libraries the same methodology (least squares). ...
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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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1answer
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Solve for inequality of AR model

I was working through my textbook and found this problem that I got stuck at: Consider the AR(2) Model $$X_t = \phi_1X_{t-1}+\phi_2X_{t-2}+\epsilon_t$$ We can assume $\phi_2 > 0$, so the roots of ...