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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525 views

Model selection and estimation for pseudo out-of-sample forecasting

I have quarterly data on inflation from 1990 Quartal 1 to 2016 Quartal 3. If I want to perform the pseudo out-of-sample forecasting one quarter ahead with an autoregressive function, do I have to ...
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278 views

Describe AR process with additive white noise using ARMA process

Disclaimer: This is a homework problem This is a problem from "Adaptive Filter Theory" by Haykin. Problem 2.10 (2nd edition). Problem A discrete-time stochastic process $\{x(n)\}$ that is real-...
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What is the exact log-likelihood of an AR(2) model?

Let's say we have the following AR(2) model: $y_t=\phi_0+\phi_1y_{t-1}+\phi_2y_{t-2}+e_t, \; e_t\sim N(0,\sigma^2_e)$ with T observations in total. Working out the conditional log-likelihood is ...
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Longitudinal data analysis where meaning and metric of response variable varies over time

Determining what factors predict change over time is a topic of investigation in many fields and there are a variety of readily implemented methods for analysing repeated measures in the same metric. ...
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Determining parameters in AR model for non-stationary time series

I am currently trying to fit an AR model to some financial data. The time series $Y_t$ in levels is clearly non-stationary; however it appears the first differences $dY_t$ are stationary (and this is ...
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487 views

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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446 views

Fit Negbin glm model with autoregressive correlation structure

I am attempting to estimate the effect of various variables on the time-series of counts of reported cattle stillbirths. We investigate the effect of day-of-week, month, holidays etc…and also the ...
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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 ...
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237 views

Variance of sum of AR(2) processes

We have $$y_t = a\mu_t + b\mu_{t-1} + c\mu_{t-2} + d\mu_{t-3}$$ and $\mu$ itself is an AR process, let's assume here an AR(2) $$\mu_t = \phi_1 \mu_{t-1} + \phi_2 \mu_{t-2} + \epsilon_t$$ where $\...
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257 views

Differencing a time series

I am looking to find the ACF of a time series, but after it is differenced. $y_t = a_1y_{t-1} + \epsilon_t , \mid a_1 \mid < 1$ I understand that to find the ACF this process needs to be ...
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Stationarity of AR(1) process, stable filter

This section of the Wikipedia article about the Autoregressive Model reads: An AR(1) process is given by: $$X_t = c + \varphi X_{t-1}+\varepsilon_t$$ where $\varepsilon_t$ is a white noise process ...
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How to show the inconsistency of the OLS estimator for unit-root AR(1) processes by simulation?

From what I understand, OLS gives consistent estimates for stationary AR(1) time series but not for unit-root ones. I am trying to illustrate this phenomenon with a small simulation in R but the OLS ...
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1answer
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How many lags should I include in time series prediction?

I'm wondering: how do I select the number of previous time steps to use to predict the current one? I'm just plotting the autocorrelation plot and picking previous time steps that have statistically ...
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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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714 views

Reproducing sinusoid with autoregressive discrete model

I wonder how to reproduce sinusoid with autogressive discrete model : y = sin(t) with or without additive noise is my target, t ...
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1answer
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Difference between different autoregressive models

I am trying to understand the difference between these three different specifications of an autoregressive model for variable var in Stata: ...
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466 views

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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202 views

Doubts in linear regression

If a linear regression model has a constant term say 1 or 0.2, for example if the original model is $y(t) = 0.2 + ay(t-1) $, then what does this constant term imply? Will it hamper the estimates if ...
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3k views

Backshift operator applied to a constant

This questions is two part: 1) What happens when you apply the backshift operator to a constant? For example, if I have the AR process $$(1-\phi B)(y_t-\mu)=\epsilon_t$$ does that equal $$y_t-\mu-\...
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1answer
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Problem simulating AR(2) process

OP EDIT: There where no problem with this. The problem was with the method I was using for obtaining the PACF. Apparently it doesn't work quite well in this case (I was using the scikits/tsa python ...
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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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Difference between MA and AR

I fail to see the difference between Moving Average (MA): $x_t=\epsilon_t+β_1\epsilon_{t−1}+…+β_q\epsilon_{t−q}$ Autoregressive (AR):   $x_t=\epsilon_t+β_1x_{t−1}+…+β_qx_{t−q}$ $x_t$ is ...
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770 views

Why does a AR(1) model that's mean reverting revert back to B0 as opposed to B0 + B1*B0?

In time series analysis, an AR$(1)$ model takes the form: $$x_t = \beta_0 + \beta_1 \cdot x_{t-1} + w_t,$$ where $w_t$ is the white noise term. In order for the model to be stationary and to ...
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112 views

AR model with vector valued variables (in R)

I want to estimate a vector-valued model $$\mathbf{y}_t = a\mathbf{y}_{t-1}+b\mathbf{y}_{t-2}+\cdots$$ Here, each $\mathbf{y}_t\in\mathbb{R}^n$ and the coefficients $a,b,\dotsc$ are real numbers (...
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954 views

AR(1) working correlation matrix with GEE

I'm attempting to fit a GEE model and I have a question about using the AR(1) working correlation matrix. I've read some conflicting information about this correlation matrix. In some books and ...
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320 views

Residual based bootstrap autoregressive series in MATLAB

I have defined the model as follows. Let $$y_1 = 0$$ and $$ y_i = \alpha + \beta y_{i-1} + \epsilon_i $$ for $i_2\ldots i_T$, where $\alpha$ and $\beta$ are the estimated coefficients and $\...
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1answer
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Fit a moving average (MA) time series model to the data (R:stats::ar equivalent)

I am looking for some tools for automatic fitting of moving average (MA) time series model to my data in R. I know R:stats::ar ...
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1answer
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AR(1) parameter estimation

Given a time series, I'd like to estimate the parameters of an AR(1) model for it. As explained on wikipedia, there are different ways for doing that. What may be called a naive method is to compute ...
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1answer
20 views

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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1answer
117 views

Anomaly detection using vector autoregression

I want to detect anomalies in multivariate time series using statistical approaches. In particular. I want to use a vector autoregression model like VAR, VARMA or VARIMA, to predict a time stamp $x_t$ ...
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27 views

Uniquely defined autoregression

Let $\{w_t\}, t \in \mathbb{Z}$ be a random noise. Given a sequence $\{w_t\}$, does the autoregression $x_t = x_{t-1} - 0.9 x_{t-2} + w_t$, $ t \in \mathbb{Z}$ uniquely define a sequence $\{x_t\}$? ...
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176 views

How can differencing in an ARIMA model be implicit instead of explicit?

In this post - Rob Hyndman explains that: Even with that correction, the two models are not quite equivalent. In the Eviews code, the differencing is done before estimation, whereas in the R code ...
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175 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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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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2answers
749 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
451 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
658 views

What is $\hat{\sigma}^2$ for AR(1) non-causal case?

Assume that $|\phi|>1$ and {$X_k$} is the stationary solution of the non-causal AR(1) equations, $$X_k=\phi X_{k-1}+Z_k$$ where {$Z_k$} is white noise with mean $0$ and variance $\sigma^2$. Show ...
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1answer
233 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
2k views

Practical issues with dynamic panel data modeling

Unfortunately for me, I've got a situation where I need to control for the lag of a dependent variable as a robustness check against an alternative interpretation of my main regression. The baseline ...
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1answer
37 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
373 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
994 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
71 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
46 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
316 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
242 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
217 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
124 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
86 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
137 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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