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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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Dealing with auto correlation 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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Interpreting effect sizes in cross-lagged -auto-regressive models

I am running an auto-regressive, cross-lagged panel model between three variables (individual survey responses) to understand the over-time dynamics between them. But I am trying to make sure I ...
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why fit `ARMA` model to residuals when doing residual analysis?

I started my Time Series Analysis not long ago and I am currently at the residual analysis. I found, in the course, the tutor was demonstrating residual analysis by fitting an $AR$, then $ARMA(p,0,q)...
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Projection in AR model

I am currently reading the Brookwell and Davis Book and cuurently read about the PACF. On page 98 they derive the PACF for the AR(1) model $$ X(t)=0.9X(t-1)+Z(t) $$ and say that the orthogonal ...
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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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What guarantees the existence of a finite representation of the Wold decomposition? Mechanics and Intuition

Every covariance stationary process can be written as a linear, infinite distributed lag of white noise. In other words, every covariance stationary process has a Wold representation. Then we go on to ...
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Understanding the outputs of tar (Threshold Autoregressive model estimation, TSA package) command in R

I'm using the TSA::tar command to estimate threshold autoregressive model coefficients. There are a few parameters I would like to know the function of: rss1, rss2, std.res, rms1, rms2 (rms of what?) ...
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How to test for overfitting in a TAR model in R?

I want to fit a threshold autoregressive model, and I'm using the tar package in R. For ARIMA models, I could check if a model was overfit by looking at the values of standard errors as compared to ...
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Reduce the effect of excessive zeros

I am working on an autoregression problem where I use sequential LSTM. My target is well defined, but I think I am facing a problem with the features. As the features were non-stationary, then I ...
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Getting over bid-ask bounce

High-frequency financial data is subject to bid-ask bounce. Description : Unlike traditional data based on just closing prices, tick data carry additional supply-and-demand information in the form of ...
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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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Determining whether a model with random walk errors is stationary

If we have a model like an AR(1) except the errors are a random walk (i.e. not iid), then is the model itself stationary? So the model is: $$ x_t=kx_{t-1}+\epsilon_t $$ where $k$ is constant and $0<...
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Markov Chain order 1 vs. AR(1) … Difference and Implication for Parameter Estimation

As other posts on this site indicate, the difference between a time-homogeneous Markov Chain of order 1 and an AR(1) model is merely the assumption of i.i.d. errors, an assumption that we make in AR(1)...
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How to compare two variables that have first-order dependencies?

I have the following data, ...
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Time series analysis:How to plot the the following AR(1) graphs?

The equation for AR(1) is : Cases: This is what it looks like: So I came up with this code: ...
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Memoryless Property of a Markov Chain of Order 1. Is AR(1) memoryless or of infinite memory?

A stochastic process constitutes a discrete Markov Chain of order 1 if it has the memoryless property, in the sense that the probability that the chain will be in a particular state i, of a finite set ...
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Proof of AR(1) simulation [duplicate]

I am trying to simulate an AR1 process which has a mean of $\mu $. ie. $y_t- \mu = \Phi(y_{t-1}-\mu) + \epsilon_t$ This link here says that we should do :- ...
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General Form of Arima(2,1,2)

There is a question in my textbook that asks for the ARIMA(2,1,2) model. I get how to do the AR and the MA parts, but I'm having a little trouble understanding the differencing portion of the model. ...
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Can polynomials, interaction variables, and autoregressive variables trigger exogeneity issues in regression?

I would think the introduction of the mentioned types of variables would introduce exogeneity issues in regression models. However, in such circumstances are these exogeneity issues material, or can ...
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What can we predict from the follow ACF and PACF plots?

This is a time series of a wind speed data collected every hour for a month. What can you interpret from the ACF and PACF plots about the trends and seasonal components? Are there any? And which model ...
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Impulse response for general VAR lag-p model: when does it converge?

Consider the VAR lag-p model: $$Bx_t = \Gamma_0 + \sum_{i=1}^p\Gamma_i x_{t-i} + \epsilon_t,\quad x_t\in\Bbb R^n,\,\forall t\in\Bbb Z$$ Setting $B$ to be upper-triangular and $A_0:=B^{-1}\Gamma_0,\,...
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Biasedness of ML estimators for an AR(p) process

Do you know any derivations (or references) which quantify the biasedness of ML estimators of an AR(p) process?
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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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Different model result stats::arima and dynlm

I am calculating an autoregressive model with two different libraries (stats and dynlm). Attached you can find the code and the data. I am using in both libraries the same methodology (least squares). ...
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Cross-correlation, cointegration, autoregressive distributed lag model or…what?

I have done my best to understand how to do it in a proper way but I have still a lot of doubts. I have two time series of counts. My a priori hypothesis is that the second time series depends on the ...
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How to relate roots of AR and MA to unit circle

I'm working on these problems and think I figured out most of the steps, but am stuck near the end as I don't understand how to relate my roots back to the unit circle in order to determine ...
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1answer
32 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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Determine if AR(p) model is causal stationary or invertible

I was going through these problems and think I figured out most of them both, but am having some troubles at one of the last steps. The question is for each of the following models: Express them ...
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1answer
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R: GMM Estimators in a dynamic panel

I put up a fixed effects regression using panel data with a time lag of the dependant variable, so somthing like this: ...
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1answer
25 views

Population autocovariance goes to zero, assuming covariance stationary

In time series context, let $\gamma_j=E[(y_t-\mu)(y_{t-j}-\mu)]$ denote population autocovariance, where $\mu$ is population mean of $y_t$, assuming covariance-stationary. Then, $\gamma_j$ goes to $0$ ...
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1answer
135 views

Sum of autocovariances for AR(p) model

Suppose I have the following $AR(p)$ model. $$X_t = \sum_{i=1}^{p} \phi_i X_{t-i} + \epsilon_t\,, $$ where $\epsilon_t$ has mean 0 variance $\sigma^2$. I am not interested in fitting this model, but ...
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How to fit an autoregressive (AR(1)) model with trend and/or seasonality to a time series?

I want to test a model I have on a time series. The model is that the time series adapts to a trend $f(t)$ with a speed $\alpha$. There is also noise in the model. So, the time series is a function ...
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1answer
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Temperature time series forecasting predictions converging to a certain value

I am trying to forecast the value of the ambient temperature based on given data on Python. The data frequency is 15 minutes. In order to predict future values, I am using a simple autoregressive ...
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Convergence of predictions of an autoregressive model

I have performed a simple autogregressive model with lag 2 on a time series data. After obtaining the coefficients, I have computed the predictions. Since the lag is 2 in model, the first prediction $\...
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How to incorporate AR(1) term in a multiple linear regression model

I was trying to model fish catch (CPUE) using a combination of some categorical and numrical predictors. I have the data for 10 years. The data has been collected only in the period from June to ...
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Forecasting autoregressive model. What's the best linear predictor?

Obviously if $X_t = \phi X_{t-1} + Z_t$, then the best linear predictor of $X_t$ given $X_{t-1}$ is $X_t = \phi X_{t-1}$. But if $\phi$ is unknown, one may attempt to substitute $\phi$ by a Yule-...
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Linear representation of AR(2) process

I am trying to determine the linear representation of the causal AR(2) process $$ \phi(B)Y_t = \epsilon_t $$ where $$ \phi(y) = 1-\phi_1y-\phi_2y^2 $$ For AR(1) I'd use power series $$ \chi(y)=\frac{...
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Expected Value of an AR(1) process

I saw the answer on this post and got confused about a couple things in its explanation. Mainly, I am unsure of How the poster immediately knows the process $X_t = c+\phi_1 Y_{t-1} + \epsilon_t$ is ...
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1answer
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AR(2) Characteristic Equation Equivalence

In a recent question I was given the AR(2) process $$ Y_t = \phi_1Y_{t-1} + \phi_2Y_{t-2} + \epsilon_t $$ And I determined that the characteristic equation should be $$ \phi(z)=1-\phi_1z-\phi_2z^2 $$ ...
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1answer
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Degrees of freedom correction in estimation of AR(p) process

Assume that I have a process $y_t$ such that $$y_t = c + \phi_1 y_{t-1} + \ldots + \phi_p y_{t-p} + u_t$$ where $u_t$ is i.i.d. white noise such that $E[u_t] = 0, \forall t$ and $E[u_t u_s]$ is equal ...
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Is there any theory on the order of Autoregression model for periodic time series?

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?
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How do you model time series data with an autogressive model?

$(1.)$ Given an $n$-value time series $\{c_1, \dots, c_{n}\}$, how would you model this with an autoregressive model of order $M$? $(2.)$ Assuming an AR process of order $M = 2$, how would ...
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How Enforce inevitability and Enforce stationarity works in time series?

statsmodels.tsa.statespace.sarimax.SARIMAX () the function here, have 2 parameters that are "enforce_stationarity" and "enforce_invertibility " How do they enforce these 2 properties after ...
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OLS with AutoRegressive Errors on Non-Stationarised Data

I'm working with some time series data (n=40) and trying to fit an OLS with AutoRegressive Errors to model the relationship between my dependent variable and a couple of predictors over time. I'm ...
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Identifying ARMA model

In "Time series analysis with applications in r Jonathan D.Cryer and Kung-skit Chan" Hannan and Rissanen proposed getting the time series order by 2 steps which are (chapter 6, section 6.5): 1-Fitting ...
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1answer
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Is there a way to estimate the probability that the next number in a time series will be 0?

I have a set of 50,000 time series lasting 31 years (each time series for each of 50,000 cells). I would like to find the probability that the next year will go to 0 given the values in previous years....
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divergence of beta estimates between OLS and regression with ARIMA error

I have physiological time-series data: ~60k observations per channel, ~100 Hz sampling. I will model individual channels with ~20 regressors. Under OLS, given temporal autocorrelation in the data, ...
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1answer
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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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Is AR(1) appropriate for measuring a specific pattern in my data?

I am analyzing time series data in which participants rated their thoughts in real time. I am trying to model the shape of the data. Details on the time series (I have about 2,500 of these time ...
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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 ...