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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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Unit root stationarity and modelling AR(p) process

I'm reading through Introduction to Econometrics by Gary Koop. I'm a little confused on the process for modelling AR(p) processes. Hopefully someone can help clarify things for me. Let me set out my ...
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Covariance stationarity for an AR(1) with squared terms?

I have a simple, but surprisingly mind-numbing, problem. I am familiar with determining stationarity for an AR(p) process: look at the roots from the characteristic equation. What if we had higher-...
Danny Klinenberg's user avatar
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Autoregressive cross-lagged models

I'm working on a research project with an autoregressive cross-lagged model with two measures three time points. The paths from $t_1$ to $t_2$ were significant, but $t_2$ to $t_3$ were all not ...
Mike's user avatar
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Characteristic Polynomials for AR(p) Processes with intercepts

If we have AR(1) process with no intercept like: $$ x_t=\phi x_{t-1}+w_t, $$ it has a unit root when $|\phi|=1$. If we have an AR(p) process with no constant $$ x_t=\phi_1 x_{t-1}+\phi_2 x_{t-2}+\...
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Manual MLE of AR(1) yields a weird initial value $y_0$

I am playing with a manual implementation of the maximum likelihood estimator (MLE) of the parameters in an AR(1) model $$ y_t = c + \varphi_1 y_{t-1} + \varepsilon_t $$ with $\text{Var}(\varepsilon_t)...
Richard Hardy's user avatar
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Alternative method to deriving autocorrelation function of stationary AR(2) process [duplicate]

I have read this question/answer: Autocorrelation of a stationary AR(2) process How can we derive this using Expectation. Let $Y_t = \phi_0 +\phi_1 Y_{t-1} + \phi_2 Y_{t-2}+\epsilon_t$ I found the ...
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When to include random-effects in zero-inflation model component?

Is it appropriate to specify random-effects (RE) in zero-inflation (ZI) component of the model? My intuition is that whatever RE is appropriate for main component should be appropriate for ZI ...
Suhas Bharadwaj's user avatar
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In repeated measures, how to distinguish regression to the mean from a negative lagged effect?

I have repeated measures for a quantitative variable "cry" for N = 52 participants (how much you cry at a given time), there are 30 repeated measures. The values range from 0 (not at all) to ...
Y45H's user avatar
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Difference between zero-inflated model and zero-altered model

Could someone explain what assumptions I am making (perhaps implicitly) when I specify family = nbinom2() versus ...
Suhas Bharadwaj's user avatar
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Brockwell/Davis seem to say more persistence implies better predictability---do I have a counterexample?

Brockwell/Davis, Introduction to Time Series and Forecasting, p. 40, write (notation slightly adapted; please refer to screenshot below) The best linear predictor $l(Y_{T})=aY_{T}+b$ for a stationary ...
Christoph Hanck's user avatar
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Time series regression on mixed frequency overlapping data

I have an hourly univariate time series. I am trying to see if the next hour, day, week etc changes are forecastable from the past changes. The ACF and PACF of the data both look similar and show some ...
dayum's user avatar
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Gradient flow through sampled tokens when training RNNs (but without teacher forcing)

Suppose we want to train an autoregressive generative language model based on a recurrent neural network (RNN) architecture without teacher forcing: At each timestep, the RNN takes an input token $x_t$...
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Derivation of transition density for AR/ARMA

Suppose $X_{t}$ is given by the following AR(1) model: $$X_{t} = \alpha +\beta X_{t-1} + \epsilon_{t}.$$ As the mean of $X_{t}$ is given by $\mu = \frac{\alpha}{1-\beta}$, we can rewrite the AR(1) ...
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Interpretation and Analysis of a Multivariate Threshold Autoregressive Model

I'm looking to study the asymmetric affect a market rate, like the Fed Funds rate has on an interest rate. In other words, I would like to study the response of interest rate adjustments in different ...
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Different results when fitting ARIMA model for levels vs ARMA model for first differences in R

In the following code I show that I get different forecasts when fitting an ARIMA(2,1,0) for cumulative sums of a generated AR(2) model vs. fitting an ARMA(2,0) for the AR(2) itself. Can anyone point ...
Mr Frog's user avatar
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Using GEE models without clustered data

I am studying a set of longitudinal data with an N-of-1 approach. This means studying each patient in the data separately. I know there is correlation in my response variable and I am looking to use ...
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Autoregressive Process to Moving Average Representation

Suppose that we have an autoregressive process with order 1: $$y_t = a + {\alpha}y_{t-1}+u_t$$ for $t>k$, where $k$ is a positive integer and $\alpha \in (0,1)$. And, $$y_t = b + {\alpha}y_{t-1}+...
user722271's user avatar
2 votes
1 answer
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Difference in coefficients between ar() and lm() using R?

When I use ar(method="ols") it should return the same as lm(), right? It doesn't: ...
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Calculating the fitted values from a gls() object in R

I have created a gls() object to create a linear model with AR(1) errors. By all indications this model is a good fit for the data and the resulting model appears ...
chris202's user avatar
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Estimate of the intercept is off in a simulated AR(1) model

I've been working with a SARIMAX model for forecasting and found myself struggling to accurately interpret its long-term forecasts. To better understand the underlying mechanics and perhaps pinpoint ...
Quant In Spe's user avatar
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2 answers
61 views

Does white noise guarantee that $X_{t-1}$ is uncorrelated with $u_t$?

I have a question about the properties of white noise in a time series context. Specifically, I want to know: If we assume that the error term $u_t$ in a time series model is white noise, does this ...
Newbie's user avatar
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Block length in time series bootstrap of AR(1) model, biased AR coefficient

I'm using block bootstrapping for some simple autoregressive time series models, and I'm running into pretty high bias in the bootstrapped estimates of the autoregressive coefficients, even from large ...
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Covariance matrix of autoregressive process

I am learning about autoregressive processes and there is something that I find unclear about the structure of their covariance matrix. Some sources (e.g. Box and Jenkins, 2016) describe the ...
Lester B. Barnett's user avatar
4 votes
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87 views

Is it legit to estimate an AR(1) model for non-stationary time series?

Suppose ${X_{t}}$ is a non-stationary process. The goal is to estimate the following AR(1) model: $$X_{t}=\alpha +\beta X_{t-1}+\epsilon_t.$$ From classical time series analysis, we know that ...
Sane's user avatar
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2 votes
1 answer
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Which Forecast Evaluation Metric To Use?

It is a forecasting problem. I need an evaluation metric which penalizes under-predictions more than over-predictions. Also I want it's range in certain interval (say 0-100), so that it becomes easier ...
Shardul Pingale's user avatar
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Designing a random factor in a linear mxied effect model to combine repeated and single exposure datasets to prevent p-hacking

I currently have data that can be split into two categories: one is a repeated measures data (collected at multiple time points), whilst another is a single measure dataset (one single time point out ...
Syuma's user avatar
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3 votes
1 answer
164 views

Regarding explosive AR processes and stationarity

I often see this: If we have an $\text{AR}(1)$ process,$x_t=\phi x_{t-1}+w_t, $ $x_t$ is: stationary if $|\phi|<1$ an unit root (nonstationary) if $|\phi|=1$ explosive (and nonstationary) if $|\phi|...
da7666's user avatar
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Multilayer Perceptron vs. Recurrent Neural Network for Time Series Forecasting: Utilizing Multiple Lagged Values

I am currently analyzing daily sales data for a product sold across multiple stores using a Multilayer Perceptron (MLP) model. For simplicity, let's assume it consists of a single layer, structured as ...
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1 answer
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Why are regression models utilizing time anachronistic while autoregressive models are preferred?

I was reading the accepted answer to this question that asked what was the difference between autoregressive models and models that directly utilize time, it states that: Models using time or time-...
Monolite's user avatar
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Autoregession meets multiple regression? - Help with verbiage and approach

Needing some help with verbiage and opinions on how I am approaching this model. I have counts of people over the past 24 months. month count 1 100 2 105 ... ... 24 200 First, I reverse the ...
Tyler Brown's user avatar
2 votes
1 answer
33 views

Can I use an autoregressive (AR1) model to determine if longitudinal data can be treated as individual single time points?

I am currently in a position where I have two datasets, one consisting of longitudinal data collected at 4 different time points, and another consisting of only single time points (e.g. data collected ...
Syuma's user avatar
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1 vote
1 answer
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How to manually evaluate a parameter estimate using the first few values of a time series?

I'm working on a problem (I paraphrase), Consider the $\text{AR}(2)$ model $$y_t=\alpha y_{t-1}-(1-\alpha)y_{t-2}+\epsilon_t, \hspace{1em} \epsilon_t\sim N(0,\sigma^2).$$ The conditional least ...
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Autocovariance function of autoregressive process, generalized to any order

I have calculated the autocovariance function in the AR(1) case, which was quite easy, but I need the autocovariance function for higher order AR(p), assuming it is still stationary. I have been ...
Dylan Way's user avatar
2 votes
1 answer
42 views

What is the natural progression from discrete AR models into continuous time?

Lets say we want to predict a single target variable and we have 10 regressors/features. Assume we would like to predict 30 days ahead (daily predictions up to 30 days ahead) and our data is a daily ...
MilTom's user avatar
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3 votes
1 answer
78 views

How to write the AR polynomial for a model containing a constant?

I'm working on an exercise, Consider the autoregressive time series model $$y_t=c+\frac{1}{3}y_{t-1}-\frac{1}{2}y_{t-2}+\epsilon_t,\tag{$\ast$}$$ where $\epsilon_t$ is white noise with variance $1$. ...
mjc's user avatar
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Covariance matrix of autoregressive process, any order

Consider the model $$x_i=\rho_1x_{i-1}+\rho_2x_{i-2}+\dots+\rho_px_{i-p}+\omega_i,\:\mathbf{\omega}\sim N(\vec0,\mathbf{I}\sigma^2)$$ In the case of order 1 autocorrelation (i.e. where $\rho_2$ and up ...
Dylan Way's user avatar
1 vote
1 answer
49 views

Simulating non-zero mean autoregressive (AR(2)) samples

I asked this question in stackoverflow, with no success. I am hoping that i might get some suggestions here. I am trying to generate non-zero mean AR(2) samples using statsmodels package. But it seems ...
Shew's user avatar
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1 vote
0 answers
21 views

ECM Specification of ARDL model

I have a question regarding the model reparametrization of an ARDL model. Consider the following ARDL$(p,q,q,\ldots,q)$ model: \begin{equation} y_{it} = \alpha_i + \sum_{j=1}^{p} \lambda_{ij} y_{i,t-j}...
Maximilian's user avatar
2 votes
0 answers
23 views

Dirichlet/multinomial dirichlet model with autocorrelation

I need to estimate an inferential statistical model of a variable that is a set of 8 proportions that sum to 1. The data repeat for 25 years and the series is an AR1 process. Is there a statistical ...
Heather Ba's user avatar
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Issue with coefficient estimate in linear trend regression model with autocorrelation of residuals

The question is simple, generally the coefficient estimate is not affected by autocorrelation of residuals when the independent and dependent variable are distinct. I am not sure about the clear ...
Sayooj's user avatar
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1 vote
2 answers
103 views

Risks involved in using Linear Regression model when the residuals are autocorrelated

I am working on building a model which contains time series input variables. I build the data around a Linear Regression model and found that the residuals are autocorrelated. Promptly shifted to ...
Sayooj Balakrishnan's user avatar
1 vote
1 answer
47 views

How does autoregressive training help limit compounding errors at inference?

I'm having a little trouble justifying something in my head and was hoping someone could provide some intuition? I understand for LSTM models or models that maintain some state about a sequence that ...
Kiernan McGuigan's user avatar
2 votes
1 answer
161 views

Compute share moving between deciles of a stationary AR(1) process

I want to compute the probability $P_{ij}$ to move from decile $i$ in one period to decile $j$ in the next period in the distribution of a stationary AR(1) process $$Y_t = \rho Y_{t - 1} + \upsilon_t,$...
Fredrik P's user avatar
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1 vote
0 answers
155 views

Repeated measures of multiple time series processes

I am struggling with a comparison of temporal processes, which are observed in several time series. The problem is as follows: Suppose there are some semi-experimental conditions, with several ...
Sean's user avatar
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0 answers
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Lag operator and stationarity [duplicate]

I just study about time series. I want to ask about in AR(1), why the lag operator, L, need to be bigger than 1 for zt become stationary. And also when |L|>1, it is lie outside of the unit circle ...
Tin's user avatar
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1 vote
0 answers
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At what circumstances will the difficulty for the tasks of density evaluation and sampling be different?

In this tutorial video of normalizing flow, the presenter mentioned that for the original autoregressive flow, the density evaluation is fast and the sampling is slow. In contrast, for the inverse ...
8cold8hot's user avatar
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0 answers
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Is my motivation for ARMA accurate?

I think I finally understand the point of the 'moving average' part of ARMA/ARIMA, but I wanted to confirm here, just in case I am still off. Idea 1: Autoregressive processes are easy to motivate - ...
Terence C's user avatar
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2 votes
0 answers
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Can a covariate also be a random effect in glmmTMB model with ar1 [closed]

I have data consisting of catches of insects at weekly intervals over 2 years, repeated with the same methods at the same location 3 decades later. My main question is, have numbers (total and for ...
IMH's user avatar
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0 votes
1 answer
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AR(n) process error variance in R

I was wondering where does the "sigma^2 estimated as 12.2" in the following example came from. I tried to compute by myself the variance of the residuals, however it seems I probably forgot ...
Beppe's user avatar
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4 votes
1 answer
125 views

Manually compute ARIMAX forecast

I need to forecast a phenomenon using an autoregressive model with an exogenous variable. I've estimated an ARIMA(1,0,0) model, but I can't understand how the forecasts are calculated. Below is the ...
IlRicciardelli's user avatar

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