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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Training ARIMAX model with MLE (Kalman Update)

I refer to this https://www.statsmodels.org/stable/generated/statsmodels.tsa.arima_model.ARIMA.fit.html where it states that the variables of an ARIMAX model is fit by MLE via Kalman Filter. I've used ...
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21 views

Create auto regressive regressors in R (extract from auto.arima)

I have time series with daily data with, this series have 7 frequency. I used auto.arima in order to determine regressors. This function suggest me to use five regressor ar1,ar2,ar3,sar1 and sar2.You ...
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Dealing w/ Collinearity in AR(p) Model

I'm currently looking to study the response of a particular interest rate (Variable A) based on a change in the Federal Funds Rate (Variable B). In other words I would like to know whether or not ...
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31 views

Differencing two AR(1) processes

I am hoping that someone can help me. Taking the AR(1) process $y_t = \alpha y_{t-1}+\epsilon_t$, I am trying to compute the covariance of the differencing process $w_t = y_t - y_{t-1}$. I think ...
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51 views

Why AutoRegression(AR) model in python is giving inaccurate negative prediction

I have time series data with 8 points. I used AR model from statsmodels.tsa.ar_model library. I trained data using 8 points and predicted next 3 points. Though all values in the series are in ...
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23 views

Augmented Dickey-Fuller Test and Lag Length

In R, using the package tseries, one uses the command adf.test for the Augmented Dickey-Fuller Test. However, this assumes a ...
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18 views

The relationship between autoregressive model and distributed-lag model

The autoregressive models (koyck model, adaptive expectation model, potential adjustment model) I have learned so far are all derived from distributed lag models. And intuitively it makes sense since ...
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78 views

crossed random effects with autoregression, glmmTMB

I am working on data that have crossed random effects as well as a autoregressive covariance structure. I would like to check if there is something unlogical about my approach, as the model I would ...
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Transforming AR1 Parameter back to “Non-Stationary” Times Series

I'm kind of stuck on this so any help would be hugely appreciated. My math 'skills' have so far failed me. I have a time series that I am trying to fit into an AR(1) model. Which is expressed as : $...
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25 views

AR process stationarity

For $X[n] =aX[n-1]+W[n]$ When $W[n]$ is iid. One can say that $X[n]$ is the output of $W[n]$ thrown into an LTI system. So how can it be that $X[n]$ is not necessarily WSS, if we know that a WSS ...
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Why does the expected value of the l=lag time series observation multiplied by the white noice equal zero in a stationary autoregressive model?

Suppose we have a stationary AR(1) model. Therefore $y_{t}=b_{0}+b_{1}y_{t-1}+e_{t}$ where e is the white noise. I am just wondering why (A) $E[(y_{t-l}-\mu)e_{t}]=0$. At first, I thought that $y_{t-l}...
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Is Dynamic Factor Model with Gaussian VAR errors equivalent to Linear Gaussian State Space Model?

Linear Gaussian State Space Models (LGSSMs) are generally expressed in the following way: \begin{align} {\bf x}_t &= A {\bf x}_{t-1} + {\bf w}_t, &{\bf w}_t \sim \mathcal N ( 0, Q)\\ {\bf y}...
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55 views

Forecasting an index with google in R

I am trying to predict an index using Google Trend Data. I try to orientate myself by this paper. In this paper the authors use the three variables: Sales, Index and SearchFrequency to forecast the ...
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Best practice for dynamic panel data estimation with multilevel structure in a $T \gg N$ setting

We plan to estimate a dynamic panel model with both, varying intercept and varying slopes. Further, we also want to include group-level predictors for the varying effects in second-stage regressions. ...
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64 views

time series model with additional, time-independent regressors?

How does one introduce time-independent regressors into a time-series model? Let's say that you want to model house prices based on mortgage valuations from the past 5 years AND based on additional ...
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how to interprete this acf and pacf plots

I am trying to interpret ACF and PACF plots correctly, after simulating a 500 long chronological serie following the MA(2) model, I got these ACF and PACF plots , This this mean MA(2) Model is ...
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AR(1) with x-dependent parameter

Let us consider the following model: $$ x_{t} = \alpha + \beta x_{t-1} + v_{t} $$ where $v_{t} \in \mathcal{N}(0, \sigma^{2})$. I'm interested in the steady state distribution of $x$ if $\beta$ can ...
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28 views

Understanding error term in AR model

In AR model, the value at a time $\tau$ is modeled as linear regression of past values and an additional error term ($\epsilon_{\tau}$) at time $\tau$. In this what is the error term?
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Does it make sense to use Autoregression as a test feature in a machine learning algorithm? If yes, then how?

I have a large time-series dataset, which I have splitted in multiple dataframes to make predictions using different machine learning algorithms. This way, I'd like to evaluate my model using the ...
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Understanding AR1 through the glmmTMB package

I've been working through a reproducible example to better understand AR1 covariance matrix using the glmmTMB package. I have a couple of questions, even if only ...
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Cross-lagged model with more than two variables (SEM)

I was wondering if anyone could point me to some literature discussing cross-lagged structural equation models with more than two variables: all the materials I found keep it very simple, and I ...
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What is an autoregressive synchronous panel model in structural equation modeling (SEM)?

I am reading a paper on the use of social networks and political participation that uses SEM for data analysis (Halpern, Valenzuela, & Katz, 2017). The authors employ a two-wave panel survey to ...
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COPAR modeling in R

I was looking for a package supporting the modeling of a COPAR model by Brechmann et. all (see Reference below). However, I was not able to find one. Using the packages VineCopula, Copula.Markov, ...
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Confused about Autoregressive AR(1) process

I create an autoregressive process "from scratch" and I set the stochastic part (noise) equal to 0. In R: ...
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Possible to predict independent variable not among predictors in a vector autoregressive model (VAR)

I have developed a LSTM NN where I predict a variable multiple steps ahead. For this model, the independent variable itself is not among the predictors. I want to compare it with the results of a VAR ...
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Non-linear autoregressions (with financial application)

I stumbled upon a regression that I haven't seen before, if anyone can provide some info that'd be great. Define the dependent variable as: $y_{i,t} = x_{i,t} - x_{i,t-1}$ $y_{i,t} = ax_{i,t} + b$ $...
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Is there a library to fit a Threshold Autoregressive Model (TAR) in Python?

I tried to look into Statsmodel but I couldn't find it. I know that in R there is the TAR package. I would like to find something similar for Python. My entire project is written in Python and I've ...
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Can I estimate base sales in a Marketing Mix Model (MMM) with muddled promotional data?

I am building a marketing mix model (MMM) for an online casino, but they regularly run promotions and marketing campaigns that overlap. Trying to determine a baseline level of sales has been difficult....
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31 views

AR-ARCH conditional variance

Consider a AR(1)+ARCH(1) model: \begin{align*} &x_t=a_0+a_1x_{t-1}+u_t,\\ &u_t=\sigma_t\epsilon_t,\>\>\>\epsilon_t\sim N(0,\sigma^2_{\epsilon}),\\ &\sigma_t=\sqrt{b_0+b_1\sigma^2_{...
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54 views

Autocorrelation of an AR(1) process

I am learning about this AR process. According to the book I'm reading, the autocorrelatio function of a stationary process: $$y_t = c + \phi y_{t-1} + \varepsilon_t, \quad \quad |\phi|< 1$$ is ...
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191 views

Difference between MA and AR

I fail to see the difference between Moving Average (MA): $x_t=\epsilon_t+β_1\epsilon_{t−1}+…+β_q\epsilon_{t−q}$ Autoregressive (AR):   $x_t=\epsilon_t+β_1x_{t−1}+…+β_qx_{t−q}$ $x_t$ is ...
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42 views

Anomaly detection using vector autoregression

I want to detect anomalies in multivariate time series using statistical approaches. In particular. I want to use a vector autoregression model like VAR, VARMA or VARIMA, to predict a time stamp $x_t$ ...
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38 views

Causality between two binary time series

I have the following sample of a big dataframe: ...
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42 views

Autocorrelation function $\rho(s)$ of AR(p), when s goes infinity

Let $\{X_t\}_{t\in\mathbb{Z}}$ is the stacionary autoregressive process of degree p (AR(p)), and autocorrelation function of AR(p) is $$\rho(s)=\phi_1\rho(s-1)+\phi_2\rho(s-2)+\dots+\phi_p\rho(s-p), \...
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59 views

Expected value for an autoregressive process

Cross-post. Consider the following stochastic process: $$Y_i=c\cdot Y_{i-1}+\varepsilon_i,$$ where $$Y_0\sim \mathcal N(0,\sigma^2)$$ is independent of the white noise $$\varepsilon_i \overset{\text{...
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50 views

What is the distribution of the initial point of a stationary VAR(1) process?

I want to generalize the answer here to this case of a VAR(1) model. Suppose that $X_t \in \mathbb{R}^n$ and that $\Lambda \in \mathbb{R}^{n \times n}$. If we have the stochastic process $\left\{ ...
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My VECM Model Produces The Same Residuals For A Two Asset Portfolio

I have a two asset portfolio with 2 cointegrated ETF's. I would like to see when the ETF's deviates from their equilibrium. Before I show the model, what I expect to happen was that if one ETF's ...
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1answer
46 views

A simple question regarding AR(1) process and CDFs

Somewhat a trivial question, but I struggle to get my head around it. Consider we have an AR(1) process, as follows: $y_t=\rho y_{t-1}+\varepsilon_t,\quad t=1,...,T$. such that $\varepsilon_t$ are $...
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when fitting a regression model to a time-series, can I use lagged values of the time-series itself?

I'm fitting a regression model $y_t$ to a time series $x_t$ (not a dynamic model involving ARMA terms!). I saw that useful predictors to put in my model are $t$, seasonality variables and lagged ...
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How to calculate auto-regression with missing values for different surnames in one generation?

I do have a dataset consisting of surnames, years and values y. My aim is to analyze whether the value y is dependent on the corresponding value y of the previous generation. Unfortunately, I do not ...
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Regression for long tailed time series events

I have a set of values which are a time series and follow a long tailed skewed distribution. I would like to understand what the best method might be to predict the next value in the series. Do the ...
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26 views

Z-score from Skewed Student T

I'm implementing the following method. The text is provided for background, but my question is about line (8). Am I understanding this as "a z-score generated from a standardized skewed Student t?" ...
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38 views

Solving Variance of Time Series AR process [duplicate]

I am trying to solve for the variance of $x[n]$, a time series process. $$x[n] +a_1x[n-1]=w[n]$$where $w[n]$ is white noise with zero mean and variance $\sigma^2_v$. Also $|a_1|<1$. I am aware ...
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39 views

Statistical test for comparing means of two AR(1) time series

Say I have two time series which each follow the AR(1) model: $$ X_{t+1} = X_t + (1 - \theta_X) (\mu_X - X_t) + \epsilon_X(t) $$ $$ Y_{t+1} = Y_t + (1 - \theta_Y) (\mu_Y - Y_t) + \epsilon_Y(t) $$ ...
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21 views

Showing 0 covariance for special form of AR(1) time series

This is an exercise I have been trying to solve but have not made much progress. Suppose $\{Z_t\}$ is an AR(1) process with $\rho_1 = \phi$. Define the sequence $\{b_t\}$ as $b_t = Z_t - \phi Z_{t+1}$...
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92 views

Superposition of random walk and autoregressive process

Let us consider the following model: $$ y_{t} = c_{t} + \alpha y_{t-1} + v_{t} \\ c_{t+1} = c_{t} + w_{t} $$ where $v_{t} \in \mathcal{N}(0, \sigma^{2}_{v})$ and $w_{t} \in \mathcal{N}(0, \sigma^{2}...
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37 views

Deriving the general form of the best linear predictor $\tilde{X}_{n+m}$ of $X_{n+m}$ for AR(1) process in terms of $X_1, …, X_n$

I'm trying to derive the best linear predictor $\tilde{X}_{n+m}$ for $X_{n+m}$ for a causal, zero-mean AR(1) process $Z_t = X_t - \phi X_{t-1}$. My answer needs to be in terms of $X_1, X_2, ..., X_n$. ...
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Covariance of prediction errors

I have an exercise to compute the covariance between the prediction errors, but I'm not sure if it is correct, this is the exercise; I have an AR(1) model, $y_t = \phi y_{t-1} + \epsilon_t$, where $\...
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47 views

AR model throws ValueError on a constant time series

Here's my code: ...
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Is it allowed to reduce a dataset of moving averages to run an AR(1) model properly?

I run a simple AR(1) and AR(2) model with the following code: ar.ols(df$y, order.max = 1) ar.ols(df$y, order.max =2) My dataset is as follows: I do have yearly ...

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