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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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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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econometrics: z-type score [on hold]

I have a series of daily balances...lets call it B(t)) . Series is non stationary and fails normality tests and has mean of 5.167 and sigma of 0.774 Lets call the first differences of B(t)-B(t-1) as ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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Simplest model for simulation of inter-arrival times with long-memory

I'm trying to model the inter-arrival times of stochastic point process that I've observed for the purpose of simulating it. The Autocorrelation Function (ACF) of the inter-arrival times shows ...
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Simulate stationary VAR(p)

I would like to simulate a stationary VAR(p) coefficient matrix. However, I only found the following (inefficient) solution: Simulate a coefficient matrix (n x n*p) drawing each coefficient from a ...
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Efficient way to estimate the order p for autoregressive model AR(p)?

I am writing an algorithms to build AR model to estimate stock price in the future. However, I have 88 stocks to look at and wonder whether there is any efficient way to estimate the order p for all ...
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The Importance of Initial Conditions in Autoregressive Modeling

I am developing an algorithm to classify time series by using autoregressive modeling. I have used the following two alternative methods, after fitting an AR(p) model to time series: Method 1: ...
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AR(1) Finding $\gamma_l$

I have $\gamma_l = Cov(r_t, r_{t-l})$ as a definition in my notes and now I need to find $gamma_l$ for a series $r_t- m = p(r_{t-1} - m) + a_t$ where $r_t$ is a linear time series with expected value $...
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Stationarity of ADL(p, q) with heteroskedasticity

Suppose I have the model $$y_t = \alpha_0 + \alpha_1 y_{t-1} + ... + \alpha_p y_{t-p} + \beta_0 x_t + ... + \beta_q x_{t-q} + \epsilon_t,$$ where $\{x_t\}$ is a stationary process and $\epsilon_t$ has ...
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Forecast Confidence intervals for for AR(P) [duplicate]

I want to contruct 12-step ahead forecast confidence intervals (CI) for AR(2) models and above. However, the CI calculation seems extremely tedius for forecasts above 2 periods as iteration process ...
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Spatial Temporal Autoregressive Regression and implication on Fixed effects assumption in Difference in Differences

I am currently planning on testing the effects of marginal price change of properties based on an exogenous event using Spatial DID model. The model has a spatio-temporal lagged variable (y) which ...
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(multiple) fractional outcomes & autoregression

Let me start with a broad description of the problem and I will then describe my approach (that might be totally inappropiate). The big goal is to predict the distribution of population of a given age ...
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What is an autoregressive model - terminology with respect to machine learning

In Wikipedia, an autoregressive model is defined in terms of an AR(p) linear process as The autoregressive model specifies that the output variable depends linearly on its own previous values ...
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Tests for predictive models with autoregressive neural networks

I'm working with time series predictions with NNAR autoregressive neural network models (p, P, k) and I'm doubtful for the validation of my models. After making the predictions, I'm selecting those ...
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Approximating AR(1) by finite order MA process - convergence results

I am currently struggling with a result pertaining to the finite order MA approximation of a simple AR$\,(\,1\,)$ process defined on a double sided time-index set $\,T=\mathbb{Z}$. I would be very ...
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VAR Model using Stata

I'm relatively new to the VAR model and have been using Sean Becketti's 'Introduction to Time Series Using State' as reference and wanted to check if I am on the right track. As of now, I have 5 ...
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Is conditional r-squared ever zero?

I am using multivariate auto regressive modeling (MAR) to assess a complex data set (MAR is a form of vector auto regressive modeling, VAR). The output of the MAR method is >1 response variables and >...
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Example: multivariate timeseries model that is uncorrelated at each time step but has higher order interactions

What are examples of timeseries models, say $X_t = [X^{(1)}_t, X^{(2)}_t]^\top, t \in \mathbb{Z}$ (or potentially continuous time), such that they are time-wise uncorrelated, that is $\mathrm{cov}[X^{(...
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Converting MA(1) to AR(p)

While it is $MA(1)$ process there is no dependence between $u(t)$ and $u(t-1)$ i.e $$u(t)=v(t)+Q(1)v(t-1)$$ but when i converted it to AR process i get $u$’s that is dependent on the other $u$’s i.e. $...
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How to deal with auto-correlation in generalized linear modelling?

I've built a generalized linear model by using glm.nb function (my response is a count type of data) using a single predictor. The model summary is given below. <...
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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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How to adjust confidence interval

I am reading a journal where it is written "Time series are based on annual-mean translation speeds from 1949-2016. Trends are estimated by linear regression. The P values of the regression are based ...