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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1k views

Modelling auto-correlated binary time series

What are the usual approach to modelling binary time series? Is there a paper or a text book where this is treated? I think of a binary process with strong auto-correlation. Something like the sign of ...
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Why is OLS estimator of AR(1) coefficient biased?

I am trying to understand why OLS gives a biased estimator of an AR(1) process. Consider $$ \begin{aligned} y_{t} &= \alpha + \beta y_{t-1} + \epsilon_{t}, \\ \epsilon_{t} &\stackrel{iid}{\...
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Estimation of unit-root AR(1) model with OLS

Given a random walk $x_t$, $$x_t=x_{t-1}+\varepsilon_t,$$ consider estimating the slope coefficient $\beta$ in $$x_t=\beta x_{t-1}+\varepsilon_t$$ by OLS. This question and the following answer ...
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Auto-regression versus linear regression of x(t)-with-t for modelling time series

What difference precisely does autoregression (for AR(p), p=1,2,...) have when compared to linear regression of that time series random variable w.r.t time axis? Explanation with diagrams clarifying ...
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1answer
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Conceptual questions: Variance of a process

Wikepedia, at Variance of Autoregressive model, mentions an expression of variance for an AR(1) process. I am learning signal processing (beginner level) and facing difficulty in understanding some ...
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How does ACF & PACF identify the order of MA and AR terms?

It's been more than 2 years that I am working on different time series. I have read on many articles that ACF is used to identify order of MA term, and PACF for AR. There is a thumb rule that for MA, ...
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2answers
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Why do we care if an MA process is invertible?

I am having trouble understanding why we care if an MA process is invertible or not. Please correct me if I'm wrong, but I can understand why we care whether or not an AR process is causal, ie if we ...
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How to write an AR(2) stationary process in the Wold representation

I managed to write an AR(1) process in the Wold representation with help from the geometric series. I am having trouble with a stationary AR(2). How could I do?
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AR(1) coefficient is correlation?

Is the ar1 coefficient from an AR(1) model the "first order correlation of the noise" of a time series? I'm using R's aws package and one of the arguments of the <...
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1answer
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How to build a function with the result of auto.arima in R?

I use: fit = auto.arima(Y, xreg=X) in R to get ARIMA(1,0,0), result as follows: ...
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Generate a random variable which follow Gamma distribution and AR(1) process simulatenously

Is it possible to generate numbers from Gamma distribution (with parameters shape=10, scale=15, say) which also follow a AR(1) process, simultaneously? If it's possible, than how to do that?
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How to understand SARIMAX intuitively?

I'm trying to understand a paper about electric load forecasting but I'm struggling with the concepts inside, specially the SARIMAX model. This model is used to the predict the load and uses many ...
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Under what circumstances is an MA process or AR process appropriate?

I understand that if a process depends on previous values of itself, then it is an AR process. If it depends on previous errors, then it is an MA process. When would one of either of these two ...
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1answer
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Simulate AR(1) process in R with specified nonzero mean and AR coefficient [closed]

I need to simulate an AR(1) process with the following equation in R: $$ X_{t} = 5 + 0.5X_{t-1}+Z_t $$ Where $Z_t$ ~ White Noise(0,1) and $T=500$. I know I should be using the ...
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Cross correlation influenced by self auto correlation

I have two stationary time series ts1, ts2, I wanna find the cross correlation ($\textrm{CCF}$) between them. As a result, it ...
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2answers
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What's wrong if I fit the auto-regression with OLS?

I am doing auto-regress by usual linear regression package. e.g. $y_t=φx+ε_t$ with $x =y_{t-1}$ My reason is that, Auto-regression does assumes iid errors, same for linear regression. Linear ...
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1answer
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Auto regressive process, maximum likelihood estimator

A first-order autoregressive process, $X_0,\dots,X_n$, is given through the following conditional distributions: $X_i | X_{i-1},\dots,X_0 \sim \mathcal{N}(\alpha X_{i-1},1)$, for $i = 1,2,\dots,n$ and ...
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Skipping lags in autoregressive modeling?

is it possible to skip immediately preceding time points? So that, if, for example, you are using model order 2, that is, two time points, but not the two immediately previous time points, but rather, ...
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2answers
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What is the difference between AR and VAR?

What is the exact difference between an autoregressive (AR) and vector autoregressive model (VAR)? I always thought that VAR would just be for more than two variables, until I learned that AR can ...
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AR(1) forecasting

So I have a small question about forecasting using an AR(1) model. I have $Y_t=4-0.6Y_{t-1}+e_t$ with {$e_t$} as W.N. with $\sigma^2_e=2$ I am asked to forecast $\hat{Y_t}(1)$ for which I am using ...
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If an auto-regressive time series model is non-linear, does it still require stationarity?

Thinking about using recurrent neural networks for time series forecasting. They basically implement a sort of generalized non-linear auto-regression, compared to ARMA and ARIMA models which use ...
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992 views

Unbiased estimator for AR($p$) model

Consider an AR($p$) model (assuming zero mean for simplicity): $$ x_t = \varphi_1 x_{t-1} + \dotsc + \varphi_p x_{t-p} + \varepsilon_t $$ The OLS estimator (equivalent to the conditional maximum ...
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Random walk estimation with AR(1)

When I estimate a random walk with an AR(1), the coefficient is very close to 1 but always less. What is the math reason that the coefficient is not greater than one?
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1answer
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The distribution of the initial point of an AR process

Consider a stochastic process $\{X_t, t = 1, 2, \ldots\}$ following the model $$X_t = \alpha X_{t-1} + e_t,$$ where $e_t \thicksim f$. Can I say that the distribution of the initial point, $X_1$, ...
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Poisson with an autoregressive term

I want to fit a fairly "standard" Poisson model, but with an autoregressive term. $N_i \sim \mathrm{Pois}( \lambda_i E_i)$ with $\log \lambda_i = X_i \beta + \delta$ $\delta \sim AR(1)$ $X_i$ is a ...
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Steps to perform time series analysis

I'm trying estimate an autoregressive model with an exogenous variable. It's about the impact of changes in oil prices on the economy. I'm planning on regressing gdp growth rate on its own lagged ...
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2answers
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Invertibility of AR(p) model

Notation: $\dot{Z}_t = Z_t - E(Z_t)$, so that it is centered at 0. $a_t$ stands for the residual and we assume the $a_t$ are independent and normally distributed with mean 0 and constant standard ...
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1answer
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Parameters in Autoregressive representation of an ARCH model

Suppose we have a $0$ mean time serie representing stock index returns about a title, $r$. I also know it follows an $ARCH(p)$ model with parameters $\omega$ and $\alpha$, specified in the following ...
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Can I use OLS to estimate an ARX(1) model? [duplicate]

Is a statistically correct to find the parameters of the model $$ Y_t = \beta_1 Y_{t-1} + \beta_2 X + \epsilon $$ by just using OLS? For example, there won't be any problems with bias of the ...
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1answer
444 views

Good Resource For Converting ARIMA output in R to equation form?

I've seen this question asked a few times but I still haven't seen a place where I can get some good examples on how to convert an arima() output in ...
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2answers
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Breusch-Pagan Test for ARIMA Model in R [closed]

I am testing my model using the Breusch-Pagan Test, but have not been able to find anything online regarding how to calculate it for an ARIMA Model. My AR1 Model is: ...
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1answer
956 views

AR.OLS isn't matching to an OLS on the autoregressive lags, Why?

I am using R and running ar.ols() on some data. And trying to compare to a more "manual" method of computing an AR model by doing lm() using the autoregressive lags as my independent variables. ...
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1answer
79 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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1answer
117 views

I understand why stationarity is a requirement for AR(p) models, but why is it necessary for MA(q) models?

I have (or at least I think I have) a good intuition for why stationarity is a requirement for modeling $AR(p)$ models: an $AR(p)$ model with coefficients $a_1,....,a_p$ : $ Y_t = a_1Y_{t-1}+a_2Y_{...
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What is the reason for not including an intercept term in AR and ARMA models?

In econometric literature it is usually argued that in case of estimating an equation, an intercept term must be always included regardless of its statistical importance because removing the constant ...
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1answer
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Breusch-Godfrey Test and the length of the lag, p

I'll use Breusch-Godfrey (BG) test to test correlation of an AR(1) model. In order to perform a BG test, the simple regression model is first fitted by ordinary least squares to obtain a set of sample ...
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How to fit log-linear poisson autoregressive mixed model?

I have time-series count data $N_{i,j}$ (population sizes in site $i$ and year $j$) and I want to correlate year-to-year changes with the environmental conditions $x_{i,j}$. For this, I want to fit ...
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1answer
357 views

Why are exponential smoothing models not considered auto-regressive?

I've seen so far two definitions of the term "auto-regressive" model when it comes to time series modeling: The first definition is just basic AR models and their relatives such as ARMA and ARIMA, ...
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2answers
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Finding variance of AR process

$\newcommand{\E}{\mathbb{E}}$How do I find the variance of an autoregressive AR(1) process $$y_t=\phi y_{t-1}+\varepsilon_{t}$$ where $\lvert {\phi}\rvert<1$ and knowing that $$y_t=\sum_{j=0}^\...
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Why are we not concerned about the distribution of the $x_t$ in an AR(1) model?

I am trying to investigate the reasons why we don't bother about the distribution of the $x_t$ in an autoregressive model. Why do we concern ourselves about the distribution of $e_t$? And why are ...
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1answer
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How to write variance covariance matrix of AR(1) process in R

I'm trying to write autocovariance matrix of AR(1) process in R and I'm having difficulty. The autocovariance matrix that I'm using in my project takes the form as shown in the picture: I also ...
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4answers
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Time Series for each customer

Is it possible to create Time Series Analysis for each customer? Say if have 100 customers and I wanted to predict how much amount they are going to spend next. I have done the Time Series for the ...
3
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1answer
122 views

Are all $AR(p)$ processes for which $|a_1|,…,|a_p| < 1$ stationary?

For an $AR(p)$ process $ Y_t = a_1Y_{t-1}+a_2Y_{t-2}+...+a_qY_{t-q}$ : Is having the coefficients $|a_1|,....,|a_p| < 1$ just a necessary condition for stationarity, or is it sufficient as well?
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170 views

Unable to calculate the density function for AR

The model is an AR(p) process excited by a white Gaussian noise $\epsilon_t$, \begin{align} Y_t = &c+ \phi_1Y_{t-1} + \phi_2 Y_{t-2}+ \ldots+ \phi_p Y_{t-p} + \epsilon_t\\ \epsilon_t = &\...
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1answer
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Constants in determining stationarity of a time series

After reading about AR(p) processes I have one question regarding the characteristic polynomial of AR(p) processes and its roots. Let's say that you want to determine whether the time series $$y_t ...
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1answer
552 views

Variance of AR(1) process using lag operator

Suppose for the AR(1) model, $$Y_t=\phi_1Y_{t-1}+e_t$$ I want to find the variance $Var(Y_t)$ using lag operator: $$Y_t=(1-\phi_1L)^{-1}e_t$$ My way is simply taking the variance, $$Var(Y_t)=(1-\...
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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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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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1answer
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lag in prediction outputs in one-step ahead neural network autoregressive model

I am working on an ARX forecasting problem mostly using feed-forward neural networks in MATLAB. The functional model is of the form $y(t) = f(y(t-1),...,y(t-n),u(t))$. My data is at half hourly ...
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1answer
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Auto correlation function of AR(p) process

I am doing a time series course and in the theory part there are few things I don't understand.In obtaining auto correlation function for AR(p) process it is done as: $$\newcommand{\Var}{{\rm Var}}\...