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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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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146 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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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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55 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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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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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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62 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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289 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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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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806 views

Can autoregressive coefficient values be greater than 1?

I am using multivariate autoregressive (MAR) models to fit my long-term dataset of species abundances and environmental variables but when I use only the data from a specific period of the year (e.g.,...
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266 views

Ljung-Box and Breusch-Godfrey for univariate time series

I am testing autocorrelation of a raw time series, not the residuals with Ljung-Box and Breusch-Godfrey tests. The problem is that I am getting contradicting p-value results for some series: Ljung-...
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803 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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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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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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570 views

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

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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762 views

The inverse of AR correlation matrix

I want to find the inverse of the following matrix: $$ R_{k-1}=\begin{pmatrix} 1 &\rho &\rho^2 &\cdots &\rho^{k-2} \\ \rho &1 &\rho &\cdots &\...
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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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255 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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295 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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4answers
7k views

How to simulate a third order AR model

I'm trying to understand AR models but it's getting pretty difficult for me. Just wanted to ask you some hints on how to simulate an AR(3) model driven by a zero mean WN for 1000 values in Matlab, ...
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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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2answers
51 views

In AR(1), why is $X_i|X_{i-1}=x_{i-1}\sim N(\alpha x_{i-1},\sigma^2)$?

The AR(1) model starting with $X_0=0$ is $$X_i=\alpha X_{i-1}+\epsilon_i, \ i=1,...,n, \ -1<\alpha<1$$ where $\epsilon_i\sim N(0,\sigma^2)$ are independent error terms. Why then is $$X_i|X_{i-...
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Confusion related to the calculation of autocovariance

I have a confusion related to the calculation of autocovariance Suppose $X_t = \phi X_{t-1} + \epsilon_t$ Then how the autocovariance $E(X_{t+n}X_t) - \mu^2 = \frac{\sigma_{\epsilon}^2}{(1-\phi^...
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51 views

Autoregressive process with random walk perturbation (with drift)

Suppose we have an autoregressive process, $$y_t=\phi y_{t-1} +u_t$$ where $|\phi|<1$. If $u_t$ is an i.i.d random variable this process is stationarity. What if $$u_t=u_{t-1}+g+\epsilon_t$$ where $...
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1answer
50 views

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 ...
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1answer
623 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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1answer
6k views

Using `ar` function in R to fit a AR(p) with predetermined exact order p. [closed]

I am playing with R's sunspot data and I have $X$ as the yearly sum of sunspots(hence $X$ is a vector of length 235). I want to fit different AR models with different orders, $p = 1, 2, ..., 20$. I ...
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1answer
64 views

Stationarity of AR(1) process whose autoregressive parameter could change over time

Imagine an AR(1) has an autoregressive parameter which could change in time. $y_t-\mu=\phi_t (y_{t-1}-\mu)+\varepsilon_t\,$, where $\phi_t$ is not always constant but still lies inside the usual ...
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2answers
657 views

what is the intuition behind stationarity condition for AR(p) process?

i get that you have to find the roots of the characteristic polynomial but can someone explain the intuition behind the roots must be outside the unit circle? what is a unit circle? before anyone ...
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605 views

How can I convert a Gauss-Markov process to i.i.d. Gaussian process?

I am wondering is there any straight forward approach to convert a Gauss-Markov process, i.e., a First order autoregressive process with i.i.d. Gaussian input, with the covariance matrix $K=Toeplitz(1,...
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1answer
59 views

time series model with additional, time-independent regressors?

How does one introduce time-independent regressors into a time-series model? Let's say that you want to model house prices based on mortgage valuations from the past 5 years AND based on additional ...
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1answer
94 views

Moments of an AR(1) Process

Definition of an AR(1) process In an Autoregressive Process, a time series can be generated based on a stochastic difference equation. \begin{align} X_t = c + \phi \, X_{t-1} + \epsilon \end{align} ...
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3answers
532 views

predictions for AR(1) model

I don't understand how predictions can trace the actual data so closely (see the code below)? Does that make sense? The model is $Y_t = \theta Y_{t-1} + Z_t$ where $Z_t$ is random noise. Hence the ...
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1answer
567 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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1answer
511 views

Why use Granger causality instead of autoregression?

I'm working on an analysis on GDP growth. I want to test whether regional growth in GDP in a time frame can be explained by the growth of air traffic in a preceding period (and controlling for some ...
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2answers
38 views

Expand a power of the difference operator in terms of time series $z_t$

I am trying to use excel to plot different time series. I have the equation $(1-L)^2 * z_t$ I know that $(1-L)*z_t$ is equal to $z_t-z_{t-1}$ Can I just expand $(1-L)^2$ using basic algebra and ...
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1answer
1k 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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Simple Example of Autoregressive and Moving Average

I am really trying, but struggling, to understand how Autoregressive and Moving Average work. I am pretty terrible with algebra and looking at it doesn't really improve my understanding of something. ...
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2answers
3k views

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
112 views

Stochastic Volatility Model

In Kim et al. (1998) stochastic volatility model is specified as: $y_t=\beta\exp({\frac{h_t}{2}})\varepsilon_t,\quad t\geqslant1$ $h_{t+1}=\mu+\phi(h_t-\mu)+\sigma_\eta\eta_t$ $h_1\sim N(\mu,\frac{...
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225 views

Proving for an AR(2) process that $E[X_t | F_{t-1}]=E[X_t | F_{t-2}]=E[X_t | F_{t-3}]$

An exercise states: Using the law of iterated expectations applied to an AR(2) process, verify that $E_{t−k} . . . E_{t−1} (X_t ) = E(X_t |F_{t−k} ) $ for $ k = 1, 2, 3 $ where $ E_{t−k} (X_t ) = E(...
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2answers
2k views

How to decide the optimal AR-model order?

I'm trying to create AR-model on wheather data and I wondered is there a method or algorithm which can find the optimal order for an AR-model? I'm using Matlab for my data-analysis, is there a ...
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2answers
456 views

Test for independence of random variables

I have a time series of data (about 300-750 elements, depending on the sample) and a model that has some random residues. I used the Kolmogorov–Smirnov test to make sure that the normality hypothesis ...
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1answer
34 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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1answer
43 views

A simple question regarding AR(1) process and CDFs

Somewhat a trivial question, but I struggle to get my head around it. Consider we have an AR(1) process, as follows: $y_t=\rho y_{t-1}+\varepsilon_t,\quad t=1,...,T$. such that $\varepsilon_t$ are $...
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1answer
34 views

Feasibility of running mixed-effects poisson/logistic regression with correlation structure such as AR(1), Toeplitz

I'm not aware of any R package that lets me use specify the covariance pattern model such as in the package nlme and run the mixed effects poisson/logistic ...
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1answer
52 views

What is the virtue of loading absolutely-summability in the definition of causality of ARMA model?

An ARMA series $y_t$ is causal function of $\nu_t$ if there exists constants $\psi_j$ such that $\sum_{j=0}^{\infty} |\psi_j|<\infty$ and $y_t=\sum_{j=0}^{\infty} \psi_j\nu_{t-j}<\infty$ for ...
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1answer
51 views

the difference between using an AR(1) term (as in GAMM) versus using PM lag variable (in GAM)

I conducted an experiment to predict particulate matter (PM) level using a GAM. To do so I included the lag1 PM (PM value of day before) as well as few meteorological terms. In my second experiment ...
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1answer
27 views

Is there any theory on the order of Autoregression model for periodic time series? [closed]

Say M periodic signals, then one can safely say using AR-M model can achieve the perfect prediction. But how about further, in a more general sense, is there any publications on this? Update: Here ...
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291 views

Approximate AR(p) with a product of AR(1) and AR(2)

Literature suggests that any AR(p) ARIMA model can approximated as a combination of AR(1) and AR(2) processes. For example, one book suggests that an AR(3) model with the following coefficients: ...