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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Mean of target values at different time points as predictor in multiple regression

I've received regression model that predicts crop yield based on data collected at 3 time points (years). Input data contains multiple attributes and crop yield in the given year for a given location....
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Why does my SARIMA model not capture the seasonality?

I have sales data over 100+ days. Every Saturday has 0 sales. For the other days there is also a clear seasonality. Tuesday always has the highest sales, and the order in which the other days follow ...
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When to use AR and when to use MA model?

When to use an AR model and when to use an MA model to model time-series data. What aspects of data are modelled by the AR process which can't be done by MA and vice-versa?
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75 views

ARIMA doesn't include the trend

I have a problem with my ARIMA(1,1,1) predictions. I have a time series with no seasonal component but with an obvious trend. To get rid of it I take the first difference by setting d=1. The model ...
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Component contributions in Additive Model Time Series

I have trained a model for forecasting time series in a greedy procedure: Fit the Trend component T(t) of the series on the original signal y(t) Fit a Cyclical/Seasonal S(t) component of the series ...
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How to interpret Autocorrelation plots?

I have sales data per day. To create an ARIMA model, they suggest to first look at an autocorrelation plot. How I interpret this is that they look how my sales are correlated to eachother for ...
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1answer
126 views

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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16 views

Smoothing autoregressive coefficients

I fit an autoregressive model to a time series with 1837 observations using the R ar() function setting the maximum number of lags to 20. The function selected an AR(19) model using the AIC criterion, ...
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26 views

Cumulative Effect

Let $X_t$ be a causal AR($p$) model. If we have $n$ observations of this time series and fit an AR($m$) model with $m\gt p$ to the data; that is $$X_t = \phi_1 X_{t-1} + \cdots + \phi_m X_{t-m} + W_t,...
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1answer
35 views

ARIMA Model Non-Stationary Time Series

Suppose that the data generated process is the following: Y(t) = 1.2*Y(t-1) + 0.2 The process is clearly non-stationary. My question is why we can't fit an AR(1) model and make predictions?
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AR(p) by iterated vs. lag method. Different results

Reading "Applied Econometrics Time Series" By Walter Enders I am trying to derive the stationary AR(p) model as he does on page 58, fourth edition. This is the AR(P) model \begin{equation} y_t=a_0+\...
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Why do AR(1) times series generated by two methods look similar but have different variance estimate in Python

I come across one question when I use two ways to generate AR(1) sequences. By definition, AR(1) sequence is $x_t = \phi_1 x_{t-1} + \varepsilon_t,\quad \varepsilon_t\sim N(0, \sigma^2)$ I found ...
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Is there a root of AR-polynomial which is the same for any $\phi$?

I am learning timeseries models and got some doubts. Consider an ARIMA(1,1,0) process $Xt$. Let $\phi(z)$ is AR-polynomial. $(1-\phi B )(1-B)=Z_t$. $(1-B)X_t=Y_t$. I read in my study material ...
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Can an AR(1) process with finite past be well-defined?

I am wondering if there is a true need for the infinite past of an AR(1) process to be defined. Usually, an AR(1) is a stationary process defined by the set of equations \begin{equation} X_t = \...
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249 views

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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288 views

How to model auto-correlation with a sinusoidal decay pattern in time series data?

I have ambient temperature data recorded at 90-minute intervals (16 readings per day) over approximately one year. I’m using GAMs to characterise the daily temperature profile in different seasons but ...
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How fast does a auto-regressive process converge?

Recently I have come across a time series data that happened to fit MA(1) process really well, and I would like to know how fast does this series to mean revert ? I did some google search there seems ...
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Interpret AR(3) output from `arima` function in R

I have AR(3) like following. I'm not sure whether it is interpreted to $$ Y_t = 5.6923 + 1.0519 Y_{t-1} -0.2292 Y_{t-2} -0.3931 Y_{t-3} + e $$ or other? Thank you. ...
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Does AR in the TAR model of time series still need to consider stationarity?

For example, first order difference operation? Because I had to implement the TAR algorithm with the MCP penalty function myself, I had to understand the calculation details
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435 views

Nonstationary solutions for stationary ARMA equations

By "stationary" I mean "weakly stationary". Consider a "stationary" AR(1) equation: $$X_t=\varphi X_{t-1}+\varepsilon_t,$$ where $t\in\mathbb{Z}$ are discrete time moments, $\varepsilon_t$ a zero-...
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Determining standard error of the mean from a correlated, stationary time series using known autocorrelation without block averaging

I'd like to determine the SEM of measurements taken from a stationary time series. SEM calculation using all measurements isn't accurate because adjacent measurements may be highly correlated, so the ...
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What is the best way to present the following predictive regression relationship?

If I have a predictive regression with a single regressor of the form \begin{equation} y_t=\beta x_{t-1}+\varepsilon_t \end{equation} where \begin{equation} x_t=\rho x_{t-1}+u_t \end{equation} Then I ...
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518 views

Model selection and estimation for pseudo out-of-sample forecasting

I have quarterly data on inflation from 1990 Quartal 1 to 2016 Quartal 3. If I want to perform the pseudo out-of-sample forecasting one quarter ahead with an autoregressive function, do I have to ...
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53 views

Coefficients of the Wold representation of an AR(2) process

So, I am aware that a covariance-stationary AR(p) process can be written as an infinite MA (the Wold representation), taking the form (where there is no constant) $$ y_t=\sum_{i=0}^{\infty}\psi_i\...
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25 views

ADF-Test indicates stationarity for a non-stationary time series

I have a minor issue and am not sure what to do. The link below leads to an image of two time series I plotted, the upper being the original, the bottom one obtained by taking the first differences. ...
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How I can simulate autocorrelated data time varying mean in R?

Actually I am working on SQC and I want to fit an AR(1) model to the autocorrelated data with changing mean and use the residuals as charting statistic (shewhart, EWMA, CUSUM) to study the small ...
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1answer
16 views

How should you determine the order of an AR(p) model using PACF with fluctuating significance?

While plotting the PACF of the sample, the PACF values become insignificant post the second lag, then significant again post the 8th lag and so on. Basically, there's cyclicality in the partial ...
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39 views

What is the theoretical distribution for this AR(1) model, $(1−0.8B)x_t=ϵ_t,ϵ_t∼N(0,1)$?

The AR(1) model is: $(1−0.8B)x_t=ϵ_t , ϵ_t∼N(0,1)$ and the true mean of the process is $μ≡E(x_t)=0$. Please tell me what is the theoretical distribution under the true AR(1) model. Is it also $N(...
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VAR vs STAR for space-time autoregression in Python

I want to use autoregressive model to build a predictor for some sets of spatio-temporal data. For example, I have historical traffic data (speeds at various segments of freeways). similarly, I have ...
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How to estimate the grandparental influence on the intergenerational transmission of social status?

I do have an educational status for individuals from different families over three generations (y, yparents, ygrandparents). To determine the two-generational social mobility, I run a simple linear ...
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1answer
39 views

Masked Autoencoder MADE implementation in TensorFlow vs Pytorch

I am following the course CS294-158 [1] and got stuck with the first exercise that requests to implement the MADE paper (see here [2]). My implementation in TensorFlow [3] achieves results that are ...
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29 views

Can we express an AR(1) process as follows?

If $X_t$ follows an AR(1) process as follows \begin{equation} X_t=\rho X_{t-1}+\varepsilon_t \end{equation} Would it be correct to express the above as \begin{equation} X_t=\mathbb{E}\{X_t\mid X_{t-...
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Chow test on autoregressive regression

For my master thesis I would like to investigate whether special items (= one-time effects, such as for instance restructuring expenses etc.) do have explanatory power for future operating income. ...
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159 views

First difference of AR(1) process

Given AR(1): $$X_t - \mu = \phi(X_{t-1}-\mu) + \epsilon_t$$ where $$ \mu = 0.85 \\ \phi=0.59 $$ and $$ W_t = X_t - X_{t-1} $$ Compute $$ Corr(W_t,W_{t-1})=-0.205 \\ Cov(W_t,W_{t-4})=-0.43 \\ Corr(...
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1answer
29 views

What is the ACF plot of $x_t = 0.9 x_{t-2} + w_t$

I am just learning time series, and I am wondering about the following AR(2) model: $x_t = 0.9 x_{t-2} + w_t, w_t \sim N(0, \sigma_w^2)$ Please show me the plot of its Autocorrelation Function, or ...
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27 views

$z_t=x_{t+7}/x_t$. Solve back for x. model is $z_t=alpha*z_{t-1}$

I want to create a model of x, Now my issue is that to get this fit I need to transform the original data such that $z=x_{t+7}/x_t$ the absolutely best fit I could get is by regressing $z_t=alpha*...
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Parameter estimation for time-varying autoregressive processes in R

I want to estimate the parameters of an autoregressive process with time-dependent coefficients. For example TVAR(1) model with 1 lag: $$ X_t = \phi_t X_{t-1} + \sigma_tW_t $$ where $\phi_t$ and $\...
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Hidden Markov Model with Autoregressive Emissions

So far, all standard HMM implementations I've seen assume some variation of a Gaussian Mixture (GMM) as their emission model. It can of course only have a single mixture component which reduces it to ...
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1answer
31 views

Does the Transformer decoder query based on the previous token?

Consider the decoder part of the popular Transformer architecture; briefly put, the decoder module consists of a composition of self-attention layers and performs auto-regressive prediction. Because ...
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53 views

How to estimate a single AR model for multiple time series?

I have ACF of a time series data, I want to interpolate the ACF using AR model for which I need to calculate the coefficient of the AR model The problem is I have (6000x6000) matrix of ACF and I cant ...
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1answer
46 views

How AR model parameters are estimated?

I know that AR(p) is defined like this: $$ y_{t} = c + \phi_{1}y_{t-1} + \phi_{2}y_{t-2} + \dots + \phi_{p}y_{t-p} + \varepsilon_{t}, $$ From what I understood, this means $y_{t}$ value is predicted ...
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1answer
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Exogenous variable in both the mean and variance model?

\begin{aligned} y_t &= \beta_0 + \beta_1 y_{t-1} + \beta_2 x_{t-1} + \epsilon_t, \\ \epsilon_t &= \sqrt{h_t}\eta_t, \\ h_t &= \alpha_0 + \alpha_1 \epsilon_{t-1}^2 + \alpha_2 h_{t-1} + \...
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AR model estimated with Yule-Walker equation is poor

Hello :) I am an undergraudate student studying time series analysis as a way to kill time during the Covid-19 self-quarantine. I write this to ask you, what could be the possible reasons behind my ...
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In predictive regressions, can we assume that the predictor follows the Ornstein-Uhlenbeck process?

A predictive regression is a regression of the form \begin{equation} y_t=\beta x_{t-1}+\varepsilon_t \end{equation} where $x_{t-1}$ is generally assumed to be a highly persistent stochastic variable, ...
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How do you find the coefficient values for the unique causal solution of $\text{AR}(p)$?

I am working with a causal $\text{AR}(p)$ model. I know I can express this in causal form as: $$X_t = \psi(B) W_t \quad \quad \quad \psi(z) = \theta(z)/\phi(z).$$ I also know that the parameters ...
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1answer
76 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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Why are the inference results of autoregressive model different according to batch size?

I trained the Transformer network and got inference results from the network using batch-size 100. And, the results were [100 x 256 (max decoding length)] fixed-shape integer matrix. When I used batch-...
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2answers
27 views

Can we assume the initial value of a sample following an AR(1) process is a constant?

Say we have sample from a population that follows an AR(1) process: \begin{equation} x_t=\rho x_{t-1}+\varepsilon_t \end{equation} Is it correct to assume that $x_0$ is a constant? Or does the $x_0$ ...
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1answer
218 views

Proof of contemporaneous exogeneity, and its implications for an AR(1) model

It can be shown by contradiction that exogeneity fails to hold for an AR(1) model. Is there any proof that contemporaneous exogeneity does not fail to hold? All I've come across is assuming it does ...
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1answer
45 views

How to recursively express an AR(p) process

We know that an AR($1$) process \begin{equation} x_t=\rho x_{t-1}+u_t,\quad \lvert\rho\lvert<1 \end{equation} can be recursively expressed as \begin{eqnarray} x_t&=&\rho(\rho x_{t-2}+u_{t-...

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