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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193 views

Time Series AR1, coefficient accuracy different from linear regression

I am interested to know why running the R arima function's result has a different accuracy from my "manual" way of linear regression. ...
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800 views

Autoregression vs Sliding Window method

I'm a beginner at machine learning and have a question regarding time series. I have a data set dependent over time, with a single feature and I am trying to predict the future value of this. This far ...
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1answer
508 views

Autoregressive model for time series with structural breaks

I'm using a structural break model (threshold model or regime switching model) to examine the dynamics of a time series. The ADF test shows that the series has a unit root. Right now I'm regressing $y$...
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1answer
276 views

What is the distribution of the difference between two AR(1) processes?

I am reading a paper published in a good economics journal. An econometric model is presented in the paper. A part of the described model is not very clear to me. Please let me state a couple of ...
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1answer
38 views

Mixture of normals, dependent on past

I have the following probability model: $(X_k|\text{PastHistory}_{k-1}, \theta_0,\theta_1,\theta_2) \sim (\pi\cdot N(\theta_1+\theta_0\cdot X_{k-1},1)+(1-\pi)\cdot N(\theta_2+\theta_0\cdot X_{k-1},1))...
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1answer
390 views

Is this explanation of the Box-Jenkins approach correct?

Can someone please tell me what they think of my explanation of the Box-Jenkins approach to modeling time series? Do you have anything to add (in particular to my explanation as to the intuition ...
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2answers
1k views

Need a clear and simple auto-regressive model example

This may be hard to find, but I'd like to read a well-explained auto-regressive model example that: uses minimal math extends the discussion beyond building a model into using that model to forecast ...
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1answer
36 views

Seasonailty in time series: adding seasonal lags versus detrending using Fourier Transform?

There are a number of posts on Cross-Validated about seasonality in time-series and detrending a dataset, in the context of classical time series models like AR, MA, ARIMA, etc. But my question was ...
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1answer
186 views

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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1answer
60 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
453 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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1answer
358 views

How do the forecast intervals from an AR model behave when the time series is inherently stationary?

I'm trying to wrap my head around two contradictory intuitions behind how forecast intervals should behave when we use an AR process to model a stationary time series: (a) On one hand, since the time ...
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1answer
299 views

First difference of AR(1) process

Given AR(1): $$X_t - \mu = \phi(X_{t-1}-\mu) + \epsilon_t$$ where $$ \mu = 0.85 \\ \phi=0.59 $$ and $$ W_t = X_t - X_{t-1} $$ Compute $$ Corr(W_t,W_{t-1})=-0.205 \\ Cov(W_t,W_{t-4})=-0.43 \\ Corr(...
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1answer
165 views

Joint AR(1) posterior distribution explicit under conjugate prior

I have encountered a problem in my textbook 'The Bayesian Choice' by Christian P. Robert. It goes something like this: $"$For a particular case of AR(1) model, $(x_t)_{1\leq t\leq T}$. Where $x_t = \...
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1answer
111 views

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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1answer
168 views

Why is auto.arima modeling an AR(1) process as an MA(1)?

Playing around with auto.arima to see how effective it is at model selection. I first simulated an $AR(1)$ process with $X_{t+1} = 0.9 X_t + \epsilon_t$ ...
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1answer
176 views

Is this an Autoregression or OLS?

Say I have a time series $y_t = \frac{1}{n} \sum_{i=1}^n x_t^{(i)}$. For example, $y_t$ can be the returns of the S&P 500 index at time $t$ and the $x^{(i)}_t$ is return of the $i^{th}$ company in ...
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878 views

An autoregressive process has to be always gaussian?

It is well known that if a generic autoregressive process of $n$ order, $AR(n)$, has a gaussian white noise error term ("innovations"), then it is gaussian too. So I presume that if the error term is ...
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1answer
249 views

How to learn 'end of sequence' for continuous sequence?

Consider Autoregressive model (i.e. RNN Language model) which try to output next token given all previous tokens. When generating sequence with this model, model need to learn when should be end of ...
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1answer
11k views

Fitting model AR(1) with R

I've sampled 100 variables from a Gauss distribution with mean 0 and standard deviation 1. > set.seed(1) > wn=rnorm(100) Then I've fitted an AR(1) model ...
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1answer
182 views

How do I quantify the decay in the initial condition of an AR process?

I'm working on basic code that generates data from AR and VAR processes; the code generates enough observations to dampen the effect of the initial conditions. For example, if I want to generate 30 ...
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1answer
180 views

How to identify relationship between response time series(Yt) & input time series(Xt) only in terms of Yt-1 & Xt?

I have a response time series(Y) & Input time series Xt & Zt. My only objective is to identify functional form Yt=f(Yt-1,Xt,Zt) where f(Yt-1,Xt,Zt) contains only lags of Yt , Xt & Zt as ...
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1answer
4k views

Meaning of output of function “ar” in R

How should I read the output of the function ar in R. For example, take this VAR model: ...
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489 views

Evaluation of Autocorr/Part Autocorr values

I am practicing MA and AR modelling by using autocorrelation and partial autocorrelation values. My data is in the image below; I can see that only at lag 12 there is a value that might be considered ...
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1answer
222 views

Autoregressive model with exponential lags

I have a very highly sampled time series that I would like to fit an autoregressive model (AM) to (~3 million samples). From knowing what they represent, I have believe there should be unique ...
3
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1answer
3k views

Different results of Engle's Lagrange multiplier test for conditional heteroscedasticity from SAS and FinTS

To fit a simple AR(5) model, I use SAS PROC AUTOREG. I called the option ARCHTEST=(QLM) which provides Engle’s Lagrange ...
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2answers
9k views

Stationary ARMA model as infinite AR or MA process

How can a stationary, invertible ARMA(1,1) process be represented as either an infinite order AR or infinite order MA process?
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29 views

Burg's method for estimation of AR-models

I am trying to get an intuitive understanding of Burg's method for estimation the coefficients of an AR-model. Say we have an AR(1)-process with \begin{equation*} X_t = a X_{t-1} + \varepsilon_t \...
3
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1answer
88 views

Day-of-week effects on regression coefficients in autoregressive model?

I have a timeseries (sampled daily, weekdays only) whose volatility clearly shows dependency on day of week. In particular the standard deviation of the differenced series $\Delta y_t$ is smallest on ...
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79 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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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: ...
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417 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 ...
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233 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 ...
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84 views

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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341 views

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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80 views

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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69 views

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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81 views

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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80 views

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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374 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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69 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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1k 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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217 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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69 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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670 views

Derivation of conditional expectation and variance of the AR(1) process [duplicate]

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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191 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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0answers
273 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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309 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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2answers
271 views

Method to remove bad values in time series (bad values known to take on a particular value)

This sounds easy, but I don't know of a good statistical method for it. I have a time series that has (good) data points that range from ~3.5 to 30. The data are collected by an automated sensor. ...
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1answer
74 views

Prove the Variance of an AR(2) Model

Take a stationary AR(2) model, $y_t=\alpha+\phi_2y_{t-2}+\epsilon_t$ We know that $$Var[y_t]=E[y_t-E[y_t]]^2$$ Which is, $$Var[y_t]=E[y_t^2-E^2[y_t]]\\$$ \begin{aligned} Var[Y_t]&=E[y_t(\alpha+\...

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