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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Expectation of the realized volatility

I was reading Zhang and Wang 2023 and I have some doubts regarding it. The realized Stochastic Volatility Model is expressed as follows: $$\begin{matrix} y_t = \exp \big( \frac{h_t}{2} \big) \...
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auto.arima (Hyndman R package)

I am running an auto arima on a datase that yields two tries as revealed by using trace=TRUE as: ...
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Help with SAR model interpretation

I am working on a research project that investigates the effect of land subsidence and flood risk on property prices. I have used the Spatial Autoregressive (SAR) model to estimate the direct, ...
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Simulation of a time series using the unconditional moments

$y_{t}$ follows a covariance stationary AR(1) process $ y_t = \phi y_{t-1} + \varepsilon_t \hspace{1cm} \varepsilon \sim \mathcal{N}(0,\sigma^2) $ I want to simulate a time series $y_{1:T}$. Can I use ...
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Question on Cochrane–Orcutt estimation. Can I simply use inverse of the lag operator?

I came across the Cochrane–Orcutt estimation, which is concerned with regression-type estimates in cases where errors are AR(1) correlated. My question is whether I can formally apply the inverse of ...
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Interpreting a Partial Autocorrelation Function Plot

Here's the plot in question. The first two are obviously significant, but I am having trouble determining the next two. The third lag looks too small, but I am borderline on the fourth. Should I be ...
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How should non autocorrelated time series data be modelled?

I'm working on time series predictions for predicting the frequency of 911 calls in a town, based on previous data. But it would seem to me that this time series data isn't autocorrelated. The number ...
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Why are lags of the dependent variable a no-no in traditional random effects models?

This post says: Lagged versions of the dependent variable are a no-no in traditional random effects models. The problem is that they are correlated with the random intercept and produce inconsistent ...
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High autocorrelation in GAMM

I am fitting a GAMM model to understand the evolution of fire elevation in time. I am using the mgcv package from R. My dataset contains 820 fires events with its corresponding elevation, time of ...
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ARIMA flattens out on one order and works perfect on another

I am implementing an ARIMA model on a time series data. I have confirmed that the data is stationary with the adfuller test. I plotted my ACF and PACF graph as below with a lag of 40. Here, I see the ...
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How to Improve forecasting efficacy on an ARIMA model in Python

I am new to Python and pieced some code together to construct a monthly forecast. The data is seasonal but the trend does not seem to be well defined. I've read a lot of posts on the reasons why the ...
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Specifying continuous autoregressive covariance structure in multilevel daily diary model

I have a daily diary dataset with daily ratings of mood (e.g., daily rating of happiness) between two treatment conditions. The complete number of days of ratings vary widely across participants and ...
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Difference ACD and Survival Analysis models

Autoregressive conditional duration (ACD) models are typically used in econometrics for dealing with trade duration (TD) data and it is used to capture the clustering structure. In other areas of ...
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Improving an ARIMA model by eliminating AR/ME terms

I've come over the following two statements to manually improve the fit of an ARIMA model by changing its parameters $p,q$. If the AR coefficients sum to nearly 1 and suggest a unit root in the AR ...
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Simulate an AR(1) process that approximates gamma-like distribution

I’d like to simulate an AR(1) time series that approximates a gamma distribution rather than a normal distribution. I’d like the result to have a specified rate and shape along with a AR(1) process ...
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Analysis of longitudinal data with a change in DV - SEM, MANOVA or not possible?

I have a dataset in which a (latent) construct (e.g., math performance) is measured at different time points using different psychological tests. The first data point is comparable to an assessment ...
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Estimation of AR model with Durbin-Levinson

Let $X_t$ be a stationary time series. We have computed $\rho_1 = 0.8, \rho_2 = 0.5, \rho_3 = 0.4$. If we assume that an AR(3) model without a constant is appropriate, get estimates of $\phi_1$, $\...
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Should the RMSE of an unrestricted VAR model decrease as compared to a restricted Autoregression model when there is Granger Causality

I have 2 time series, say for instance, T1 and T2. T1 granger causes T2 at lag 2. Should this mean that if I make a VAR model with these two time series, and an autoregression model with just T2, the ...
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Should the RMSE of the unrestricted (VAR) model for a time series that is being Granger caused by another be lesser than its restricted counterpart?

I have a couple of time series, say, T1 and T2. I have established (using the grangercausalitytest library of Statsmodels in ...
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Name for the following auto-regressive type data generating model

Suppose $X$ is a $d$-dimensional random vector. The coordinates follow an auto-regressive structure: $$ X_{1} \sim N(\mu_1,\sigma^2_1), \qquad X_{j}|X_{< j} \sim N(a^T_j X_{<j},~ \sigma^2_j), \...
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If $X_t$ is an AR(2) process, what is $Y_t := X_t - X_{t-1}$?

Q: If $X_t$ is an AR(2) process, what is $Y_t := X_t - X_{t-1}$? Attempted solution: $X_t = \phi_1 X_{t-1} + \phi_2 X_{t-2} + W_t$, where $W_t$ is white noise. \begin{equation} \begin{split} Y_t &:...
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interpretation coefficient sign dynamic model

Screenshot of a paper under a section about dynamic interpretations. Source: https://doi.org/10.3200/JECE.36.1.77-92 Does anyone care to explain this to me. Are they stating that the negative ...
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Variance of change after $K$ steps in AR(1) model

I am generating from a standard AR(1) process. Lets assume a decay time lag $\tau=100$ and unit time steps of $\Delta t=1$, so $\phi=\exp(-1 / (\tau/\Delta t))=0.99$. The predicted autovariance is $\...
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How and why center residuals when doing time-series bootstrap?

I am trying to understand and implement Sieve bootstrap (maybe also known s a parametric or model based bootstrap) for time series, where the bootstrap samples are sampled from centered fitted ...
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Is the following true: For the process $y_t =0.879y_{t-1} + e_t$, less than 60% of the value of five periods ago is still reflected in current period?

I know that the process is AR(1) and some of the data is still reflected in this period. But can we just take the power of $5$ of $0.879$?
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Accuracy of probability estimate from generative autoregressive language model

My understanding is that a discriminative classifier such as a CNN that takes an input $x$ and produces a discrete output label $y$ is typically trained to predict the best value of $y$, and would not ...
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What discrete vector timeseries modeling (e.g. autoregression) methods support "continuity" requirements?

This question is motivated by the need to do vector-valued discrete time series forecasting with some guarantees of "continuity" (or rather, a discretized analog of continuity expressed in ...
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When is an AR(1) process strictly stationary?

Suppose I have an AR(1) process $X_t=aX_{t-1}+e_t$, where $e_t$ is a white noise with zero mean and finite variance. Under what conditions do I have $\{X_t\}$ being strictly stationary in the sense ...
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Given an AR(1) process, find the density of the first observation

I'm studying ML estimation and I have a silly question that I'm not able to see its solution. Suppose I have an AR(1) process: $$y_t = c+ \phi y_{t-1}+ u_t, \quad u_t \overset{iid}{\sim}\hbox{Normal}(...
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Does $ARMA(p,q)$ process need to be invertible and have a causal stationary solution to be written in $MA(\infty)$ representation?

Does $ARMA(p,q)$ process need to be invertible and have a causal stationary solution to be written in $MA(\infty)$ representation? And if you write the process in terms of $Z_t$ instead of $X_t$, then ...
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ARDL non-stationary

I have multiple explanatory variables and one dependent variable. Data for all are collected at an annual basis, time series, dependent on their t-1 value. ¨ Will use of ARDL Autoregressive ...
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Writing Yule-Walker Equations for AR(2) with Missing Term

I was wondering how you would find the Yule-Walker equations for an AR(2) (or, really, any AR(P)) stationary time-series model, where there is a "missing" term. For example: $$ X_t - \phi\...
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Partial autocorrelation with categorical data

How do I compute partial autocorrelation values for a time series of categorical (nominal-scale) data? C. H. Weiss (2008, 2018:chapter 6) provides very clear descriptions of how simple autocorrelation ...
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I would like to know how an out-of-sample pseudo prediction is obtained in a mixed autoregressive model

I would like to know how an out-of-sample pseudo prediction is obtained in a mixed autoregressive model from a mathematical point of view. and to understand why it is not possible to make the same ...
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Technical Term for choosing distribution of starting point of a time series in a way such that the time series is stationary

In Time Series Analysis there is this idea of choosing the distribution of the starting point of a time series in a way such that the time series is stationary. Let for example $\{X_t\}_{t=0}^T%$ be a ...
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How to estimate cumulative dynamic multiplier in ARDL model with growth rates

I'm estimating an Panel ARDL model. Generally the dynamic cumulated multiplier is estimated as follows (eq. (1): \begin{equation} \frac{\sum_{i=0}^{m}\beta_{i}}{1-\sum_{j=1}^{n}\phi_{j}} \end{...
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Do I have look-ahead bias?

I have a prediction task at hand, and I'm deciding on how to sample my data and train a model with no look-ahead bias. Given a time series $Z$, my task is to build a simple predictor of size $m$ (...
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Show process is $ARMA(1,1)$

Consider this exercise taken from Brockwell and Davis (1991): I'm a bit confused as to how that implies $Y_t$ is $ARMA(1,1)$. I've tried to show it, but I end up going back in circles, and I'm not ...
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rugarch: Forecast result does not show any AR structure

I am currently working with the rugarch package to forecast the EU-ETS price. While I get reasonable results for the in-sample volatility, the forecast of the of the time series does not look correct ...
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VAR model and AR model

When does a bivariate VAR(2) model equal to a combination of two separate univariate AR(2 ) models?
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A question reagarding stationarity of an AR(2) model

If an AR(2) model is stationary, how to prove that $$\rho_1^2<\frac{\rho_2+1}{2}$$ I know that $$\rho_1=\frac{\phi_1}{1-\phi_2}$$ and $$\rho_2=\frac{\phi_1^2+\phi_2(1-\phi_2)}{1-\phi_2}$$ according ...
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An example of MA($\infty$) process with the property of long-term dependence and strictly stationarity

I am looking for a example of MA($\infty$) process with the property of long-term dependence and stationary on the strong sense. I wolud like you consider the following points to discuss: What ...
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Forecasting the conditional variance of AR(p)-GARCH(1,1) model

How can I derive forecasting formula for the conditional variance $h_{t+k}$, $k\geq1$ for AR(p)-GARCH(1,1)?
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Partial auto-correlation function (PACF)

I wonder how to calculate the partial auto-correlation function of the AR(2) process $Y_t = 0.2 Y_{t−2} + \varepsilon,$ where $\varepsilon_t ∼ \text{N}(0, \sigma^2)$? I found ACF as $\rho_h=0.8\rho_{(...
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Statistically significant AR(1) coefficient, but insignificant coefficients when calculating ARMA(1,1). Why?

I'm using the library statsmodels with a simple financial time series to calculate the coefficients for an AR(1) model, and a ARMA(1,1) model. In the first case the L1 coefficient is statistically ...
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Partial autocorrelation significant at regular lag distance

I am trying to forecast inflation with a simple AR model. I took the natural logarithm of the CPI and subtracted the 12th lag, thus obtaining a measure of inflation. The PACF is significant at the ...
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Do autoregressive coefficients obtained with the Durbin-Levinson algorithm, the Yule-Walker system of equations, and OLS coincide?

I am applying the sieve bootstrap for time series introduced by Kress (1988), which requires the estimation of autoregressive models with the Durbin-Levinson algorithm (generalized to the multivariate ...
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Can any Model be Modified into an "Autoregressive Model"?

In one of Data Science classes, we were given a dataset to explore that contains medical/health information on patients and whether they missed their last doctors appointment or not (e.g. most of the ...
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Marginal distribution of an autoregressive process of order one AR(1)

I'm reading "Econometric Modelling with Time Series" by V. L. Martin, A. S. Hurn and D. Harris ( https://www.researchgate.net/file.PostFileLoader.html?id=56bccdaa6225ff0de28b45a6&...
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Simulating a mixture model in time series

Let $\Phi (\cdot)$ be the cdf of the standard normal distribution. Given $(y_t)_{t \in \mathbb N}$ a time series. Suppose $F(y_t | \mathcal{F}_{t-1})$ is the conditional cumulative distribution ...
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