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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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12 views

Infering time-series autoregression coefficients from time series with different lengths

I have $500$ time-series that represent different occurences of the same class of events. As such they have similar properties but not the same length (lengths vary from 30 to 150). I suspect the ...
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ACF Plot - sinusoid appears after 1st order difference - dissapears after 2nd order difference

I have a dataset of stock prices and wanted to make it stationary. I did a difference using lag 1 and then did the difference again using lag 1. After the first differencing the Augmented Dickey ...
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34 views

Is an autoregressive model considered a model for independent or dependent data?

It seems to me that, in the statistical literature, data, models, and inference seem to conflate the terminology of dependent and independent. Case 1: a cross sectional sample of weight and height ...
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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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265 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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23 views

Automatic process to determine stationarity of AR(p) model

I have read that an AR(p) process is stationary if all of the roots of it's characteristic equation are greater than one in absolute value. Does this mean that I can find out if my data set is ...
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What happens when using Durbin-Watson Test for AR(2)?

In my textbook, it says Durbin-Watson Test can be used only for AR(1) because d-statistic becomes biased if error term isn't follow AR(1) process. I'm curious why d-statistic gets bias when using DW ...
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40 views

Autocorrelation in Elo ratings

FiveThirtyEight uses the following formula for their NFL Elo ratings: $$ R_i^{k+1} = R_i^k + K \cdot M(z) \cdot A(x) \cdot (S_{ij} - \sigma(x)) $$ where $z$ is the game's margin of victory, $x=R_i^k - ...
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34 views

Variance of linear combination of AR(1) process

Let $ \{X_t\}$ ~ AR(1): $$ X_t=2.62-0.84X_{t-1}+\epsilon_t, \ \ \ \epsilon_t\sim WN(0,2.27)$$ Compute the variance of $$ \overline{X}= \frac{1}{3}\sum_{t=1}^{3} X_t $$ The solution is: Var($\...
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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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Autoregression model prediction

I am using autoregression for predicting next 10 steps ahead, but if I am giving more than 8 input values, it predicts negative value, otherwise the prediction is good. What is the reason behind it? ...
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29 views

How does one forecast next point in time series using GAS package in R?

I am using the GAS (Generalised Auto regressive score) package in R in order to forecast a chosen time series. I have read package documentation as well as author published paper and I struggle with ...
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What is the difference between the results using different AR(p) estimation methods?

There are three different ways to do AR(p) estimation. OLS MLE Yule=Walker Equation What are the differences between the results using these three methods? // http://www2.econ.osaka-u.ac.jp/~...
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Calculating AR(2) model variance and mean

AR(2) model is $$ Y_t=2.25+0.75Y_{t-1}+0.45Y_{t-2}+e_t $$ How can we calculate variance and mean?
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What is the difference between the p parameter in ARIMA(p,d,q) and the lag value used by AR?

I understand that p represents the order of the AR model used within the ARIMA model, but does that have anything to do with the lag value that is calculated by the IC (eg. aic, bic, hic) in the AR ...
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1answer
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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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52 views

General formula for AR($p$) auto-regressive time series

I'm trying to find a reference (including the full formula) for the following. If $X_n = a_1 X_{n-1} + \cdots a_p X_{n-p} + e(n)$ where $\{e(n)\}$ is a white noise, then $$ X_n=g(e_0,e_1,\ldots,e_n)+\...
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Distribution of the Sum of an AR(1) Model Time Series

I have the following model for my model $\Delta X_{t} = \mu \Delta t + \rho \Delta X_{t-1} + \sigma \sqrt{\Delta t} Z_t$ with the following initial conditions - $\Delta X_{1} = \mu \Delta t + \...
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29 views

On the stationary density of an autoregressive model of order 2

Consider a stochastic process $\{X_t, t = 1, 2, \ldots\}$ following a stationary AR(2) model $$X_t = \theta_1 X_{t-1} + \theta_2 X_{t-2} +e_t,$$ where $e_t \thicksim N(0, \sigma^2)$. I want to find ...
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Is an AR(p) process with roots inside the unit circle non-stationary? [duplicate]

The title pretty much says it all: is an AR(p) process with roots inside the unit circle non-stationary?
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Derivation of the distribution of $\hat{\phi}=[\hat{\phi}_1, \cdots, \hat{\phi}_p]$ in AR(p) models

Background Consider the following AR($p$) model: $$ \dot{X_t} = \phi_1 \dot X_{t-1} + \phi_2 \dot X_{t-2} + \cdots + \phi_p \dot X_{t-p} + \epsilon $$ where $\dot{X} := X - \mu = X - \mathbb{E}(X)$...
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Model residuals vs test “residuals” correlation

Suppose I have an autoregressive univariate model fitted with a given period, so we obtain residuals produced in that process. We want to know the correlation of that residuals with other variables of ...
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Papers regarding panel data with autoregressive errors

I've been playing around, using the fixed effects and first difference models to estimate regression coefficient for DGP:s that have autoregressive error terms, and I'm wondering if anybody knows of ...
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30 views

Running two MCMC chains in parallel while minimizing Kullback-Leibler divergence between both sample distributions

I want to sample from a distribution $p(X)$ with $X \in R^n$. However, I can only evaluate the likelihoods of $Z = AX$ and $Z = BX$ with $A,B \in R^{m \times n}$ and $m = n-1$. Now my idea is to run ...
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35 views

Eviews- Error correction estimation using BDM's one-step procedure

I am trying to estimate an equation for the average wage using quarterly data. I want to build an ECM which can bes estimated using Banerjee-Dolado-Mestre's approach to cointegration. So far, I haven'...
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How many lags to use in ADF test?

So I've ran a ADF test on my data multiple times with different lags and all up to a lag of 4 have a p-value below .05. So in this case how many lags do you decide to use? Could this also provide a ...
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41 views

What is the difference between an AR process and autocorrelation?

Or is it maybe the same thing? I see that autocorrelation is when Yt is correlated with its lag Yt-1. But isn't that essentially what an AR process (say AR(1)) is? We are assuming that there IS ...
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104 views

Showing the expectation of a lognormal AR(1) process

Suppose I have a lognormal AR(1) process: $$\log(y_{t+1}) = (1-\theta)c + \theta \log (y_t) + \varepsilon_{t+1},$$ $$\varepsilon \sim N(0,\sigma^2)$$ To show $\operatorname{E}(y_{t+1})$, is it ...
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45 views

Calculating bias of ML estimate of AR(1) coefficient

I am trying to develop adjustment factors for maximum-likelihood estimates of the auto-regression coefficient in an AR(1) process. By simulation I have discovered that the estimates are positively ...
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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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39 views

Method of moment through covariance derivation

Given a Bivariate INAR(1) Poisson Process: $Y_t^1 = \rho_1 * Y_{t-1}^1+R_t^1$ $Y_t^2 = \rho_2 * Y_{t-1}^1+R_t^2$ Where $R_t^1$ and $R_t^2$ are the innovation terms and follow the bivariate Poisson ...
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45 views

For autoregressive time series modeling, does the AR(p) regressors have to be in order despite insignificance?

I am trying to fit a time series model using data of auto sales (DAUTONSA from FRED) and noticed that there is evidence of serial correlation. I’ve tried fitting a model with 4 lags but noticed that ...
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49 views

Sufficient Condition Stationarity AR(2) process

Given the following AR(2) process: $y_{t} = \phi_{1}y_{t-1} + \phi_{2}y_{t-2} + u_{t}$ I need to prove that the sufficient condition for this process to be stationary is $\phi_{1} + \phi_{2} < 1$....
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Bayesian spatial autoregressive (SAR) model with heteroskedasticity in R

In socio-economic data, I always found heteroskedasticity that can't be solved using transformation.I had read a paper "Spatial autoregressive models with unknown heteroskedasticity:A comparison of ...
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1answer
63 views

Stationarity of AR(p) process

I'm looking for a proof for the stationarity of an AR(p) process, I know a stationary process $Yt$ must fulfill the following conditions: $(1)$-$E(Y_t)=m$ for all values of $t$. $(2)$-$Var(Y_t)=\...
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52 views

Out-of-sample prediction in conditional autoregressive models

I am interested in fitting a conditional autoregressive model to a dataset where observations are grouped by regions, then performing cross-validation by holding out all observations from a single ...
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1answer
97 views

Why doesn't the Wold's decomposition theorem imply a good AR(p) fit?

I'm trying to fit an AR(p) process to the standardized, 10 years long time series of monthly logreturns of a stock index and get extremely poor fit. I'm not surprised, because if I had a good fit, ...
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Why does an AR(p) process require the largest eigenroot to be <1/ characteristic roots lie outsied the unit circle to be stationary?

For reference, this is the paragraph in Wikipedia I'm struggling with. https://en.wikipedia.org/wiki/Autoregressive_model I do understand the simpler proof for under which conditions an AR(1) ...
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1answer
50 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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Regressing across multiple different time series using exogenous variables?

To make this situation clear, I'll use a somewhat silly, but conceptually simple example. Imagine I record teams of movers carrying furniture down the block. I measure the furniture's position/speed ...
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ARIMA MODEL DEGREE OF FREEDOM PROOF

According to arima(p,0,q) model if we have n data and our total parameter is p+q then it is said that degree of freedom is n-(p+q). Could you mathematically demonstrate it? No sufficient information ...
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How to bootstrap single equation autoregressive models?

I have been reading many papers where confidence intervals for the impulse responses of autoregressive process have been boot-strapped. The question is, is the usual way sufficient? Take as an example ...
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Regressing AR noise with statsmodels

I'm looking at a SAS model running like: ...
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1answer
66 views

How to simulate AR(p) model with trend

Backgrounds I have a time series, and I fitted an AR(p) model with trend of $t^2$, with the help of auto.arima, in R package <...
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Dirichilet Process Mixture with dependent likelihood

For a Dirichlet Process Mixture model, is there a version where the conditional distribution of data is not independent? For example, with an autoregressive likelihood. $\theta_n \sim DP(\alpha, G_0)...
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Complex-valued autoregressive and ARMA processes

I’m working with complex-valued discrete time series, and specifically with complex-valued autoregressive and ARMA processes. Could someone provide me with some suggestions about a good papers or ...
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Second-Order stationarity condition for complex-valued autoregressive process

Let $\{c_n\}$ be a complex-valued discrete autoregressive process of order $p$, $\mathsf{AR}(p)$, such that: \begin{equation} \label{cn} c_n = \sum\nolimits_{i=1}^{p}\rho_i c_{n-i} + w_n, \quad n \in (...
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1answer
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OLS estimation of intercept in AR($p$) in R

I investigate the performance of the OLS estimator of an AR($3$) model given by $$ X_t=\mu+\phi_1X_{t-1}+\phi_2X_{t-2}+\phi_3X_{t-3}+\varepsilon_t $$ for $t\in\mathbb Z$ using the following code: <...
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1answer
37 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 $...