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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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How to change the observation for the first lag in an AR(1) model?

I run a simple AR(1) model in my analysis using ols: ar.ols(df$y, order.max = 1)) However, I work with generations as my unit of analysis. Therefore, the first ...
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Prediction of $X_{n+1}$ with Yule-Walker estimate

Consider a causal AR(1) process $X_t = \phi X_{t−1} + Z_t$ with $(Z_t)$ iid with mean 0 and finite variance. I am reading in a book, that $\phi X_n$ is the best predictor for $X_{n+1}$ because it ...
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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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Simulate cointegrated prices and VAR model [closed]

I am trying to simulate cointegrated stock prices and use a VAR Model to make forecasts. This is the code I wrote so far: ...
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How to check whether a given ARIMA (p, d, q) process is stationary or not?

I know that a finite MA process $X_t = \Theta(B)Z_t$ is always stationary. Also, whether an AR(p) process is stationary or not can be verified by checking the roots of $\Phi(B)=0$ where the process ...
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form of the model when using backshift operator

Be $Y_t=X_t + \epsilon_{1,t}$, in which $X_t = X_{t-1} + \epsilon_{2,t}$ and $E[\epsilon_{1,t}\epsilon_{2,s}] = 0 \forall t,s$. How could I say why this process is related with a model on the form $(1-...
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Implementation of 2SLS in regression with AR errors?

Consider the following simple example: $Y_t=\beta X_{t-1}+\varepsilon_t$ $X_t=\gamma Y_t + Z_t +u$, where $\varepsilon_t=\alpha\varepsilon_{t-1}+\eta$, and $E[Z_t\varepsilon_t']=E[uu']=E[\eta\...
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find the autocovariance function of the process $Y_t$

Consider the processes $X_t = \phi X_{t-1} + v_t$ and $Y_t = \phi Y_{t-1} + X_t + e_t$, in which $|\phi| < 1$ and $v_1$ and $e_t$ are non-correlated random errors with zero mean and variances equal ...
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Stationarity for an AR(2) process

How can I show that the following AR(2) process is stationary $X_t = X_{t-1} + cX_{t-2}+Z_t$, provided -1 < c < 0 ? I represented the series as $\Phi(B)X_t = Z_t$ and then tried to find out ...
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Handling serial correlation in time series regression

Suppose that the time series data $(y_1, y_2,..., y_n)$ can be explained through a regression model with $k$ explanatory variables: (1) $y_t = b_0+b_1x_{1t}+b_2x_{2t}...+b_kx_{kt} + \epsilon_t,\ t=1,...
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Consequences of fitting Unit Root data directly in AR model

I feel it is useful to understand the consequence of violating the assumptions of a model. I check a couple textbooks, but most I can get about the consequence of fitting time series with unit root is ...
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Converting coefficient of slope to autoregressive factor

I realize this is very fundamental. I apologize. Is there any way to convert the coefficients from a linear model into the decay factor if i want to express it as an autoregressive model? For a ...
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Context in which an AR(1) error term can be considered a random effect?

We have the following situation: \begin{aligned} y_t &= f(x_t)+u_t, \\ u_t &= au_{t-1}+\epsilon_t, \\ \epsilon_t &\sim N(0,\sigma^2). \end{aligned} To make it simple, let's assume $f$ is ...
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Why do these independant variables have significant explanatory power, when 'theoretically' they should have none? (Self contained example inside)

I am putting together a model which involves a simple linear regression, and to aid the development I have put together a process for generating synthetic observations. The idea is that you have ...
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Comparison of GMM and ML estimators for regression with correlated errors

Consider a linear model with normally distributed, autocorrelated errors \begin{aligned} y&=X\beta+\varepsilon \\ \varepsilon&\sim N(0,\sigma^2_{\varepsilon}) \text{ and autocorrelated.} \end{...
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Moments of an AR(1) Process

Definition of an AR(1) process In an Autoregressive Process, a time series can be generated based on a stochastic difference equation. \begin{align} X_t = c + \phi \, X_{t-1} + \epsilon \end{align} ...
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How can I estimate autoregression when non-stationary?

I have a series that I believe has one autoregression characteristic under condition A (example: positive) and another under condition B (example: negative). Is there a way (hopefully in Python) to ...
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Stationary distribution of AR(1) process with AR(1) shocks

I am trying to find the stationary distribution of an AR(1) process, where the shock terms themselves are an AR(1) process. That is, the process moves subject to the following 2 equations: \begin{...
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AR(1) process can be estimated using linear regression

Can the $AR(1)$ process represented as $$ x_t= ax_{t-1}+\epsilon_t$$ be estimated by regressing $x_t$ on its lagged value $x_{t-1}$.
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autoregressive time series issue understanding expectation

Consider the following question How is it obvious that $\mathbb{E}(X_t) = 0$? Is it because of the following? Recursively we find that $\mathbb{E}(X_t) = \phi \mathbb{E}(X_{t-1}) = \phi^2 \mathbb{E}(...
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Revert AR process with constant

I have got this task at the Time Series course as a part of Statistical minor. I am math major, and have gone through basic Probability (read:measure theory) course. Let us have $y_t = 0.4y_{t-1}+2+\...
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invertibility of $AR(\infty)$?

Here it writes: "Pure AR models are always invertible (since they contain no MA terms)." Is this valid also for the limiting case, that is to say, is $AR(\infty)$ invertible? Why or why not? If ...
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Using least squares to estimate variance of latent variable

I am having trouble understanding why I can't use least squares to solve an overdetermined system of linear equations using $\bf{x} = (\bf{A}'\bf{A})^{-1} \bf{A}'\bf{b}$. The same model estimated ...
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Feasibility of running mixed-effects poisson/logistic regression with correlation structure such as AR(1), Toeplitz

I'm not aware of any R package that lets me use specify the covariance pattern model such as in the package nlme and run the mixed effects poisson/logistic ...
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What is the virtue of loading absolutely-summability in the definition of causality of ARMA model?

An ARMA series $y_t$ is causal function of $\nu_t$ if there exists constants $\psi_j$ such that $\sum_{j=0}^{\infty} |\psi_j|<\infty$ and $y_t=\sum_{j=0}^{\infty} \psi_j\nu_{t-j}<\infty$ for ...
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2answers
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Time Series equivalent of the Generalised Linear Model

I have a time series $y_t$ which is measured at regular intervals over a long period of time. The values of $y$ are between $0$ and $1$, it represents a proportion, and these values change slowly over ...
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1answer
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the difference between using an AR(1) term (as in GAMM) versus using PM lag variable (in GAM)

I conducted an experiment to predict particulate matter (PM) level using a GAM. To do so I included the lag1 PM (PM value of day before) as well as few meteorological terms. In my second experiment ...
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Vector Auto Regression handling dummy encoded variables

Firstly, apologies if this question is obvious, I am new to Time Series Forecasting & ML in general. I have an application whereby I collect prices from betting exchanges on an interval. This ...
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How to read Autocorrelations in GLS models?

I am playing with some marketing data. My response variable is market share and predictors are ...
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AR model on SMA(k)

If I were to regress Yt+1 on the simple average of Yt, Yt-1, ..., ...
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Lagrange multiplier test in Mixed Level Model

I want to estimate a mixed level model with AR(1) errors and then conduct a Lagrange Multiplier test. The mixed model allows for rich covariance structures but it does not allow for AR(1) errors. Can ...
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62 views

When to use different covariance structures?

Log-likelihood could be wrong so comparing covariance pattern models based on log-likelihood and LR tests isn't a perfect way. Intuitively, how do I know when to use Toeplitz versus AR(1) versus ...
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1answer
54 views

Building A Model: Autoregressive?

I've recently read a code about a fisheries productivity model where someone tries to predict the value at time t+1 from its value at time t. There're 23 recorded productivity of tuna from 1967 to ...
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1answer
57 views

Linear Regression : Can I use both levels and changes in the same model?

I have a linear model with 1 predictor variable in the form of: $Y = a + b_{1}*X$ Both $X$ and $Y$ are stationary variables and the fit of the model is good. I have also created 2 other models based ...
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Simulate AR(1) with a matrix of returns in R

I downloaded the closing prices of stocks. Then I computed the returns and saved in a matrix. Now i want to simulate an AR(1) with the value of the returns. How can i do it? P.S - in R studio
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Infering time-series autoregression coefficients from time series with different lengths

I have $500$ time-series that represent different occurences of the same class of events. As such they have similar properties but not the same length (lengths vary from 30 to 150). I suspect the ...
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ACF Plot - sinusoid appears after 1st order difference - dissapears after 2nd order difference

I have a dataset of stock prices and wanted to make it stationary. I did a difference using lag 1 and then did the difference again using lag 1. After the first differencing the Augmented Dickey ...
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Is an autoregressive model considered a model for independent or dependent data?

It seems to me that, in the statistical literature, data, models, and inference seem to conflate the terminology of dependent and independent. Case 1: a cross sectional sample of weight and height ...
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How do the forecast intervals from an AR model behave when the time series is inherently stationary?

I'm trying to wrap my head around two contradictory intuitions behind how forecast intervals should behave when we use an AR process to model a stationary time series: (a) On one hand, since the time ...
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3answers
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predictions for AR(1) model

I don't understand how predictions can trace the actual data so closely (see the code below)? Does that make sense? The model is $Y_t = \theta Y_{t-1} + Z_t$ where $Z_t$ is random noise. Hence the ...
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Automatic process to determine stationarity of AR(p) model

I have read that an AR(p) process is stationary if all of the roots of it's characteristic equation are greater than one in absolute value. Does this mean that I can find out if my data set is ...
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What happens when using Durbin-Watson Test for AR(2)?

In my textbook, it says Durbin-Watson Test can be used only for AR(1) because d-statistic becomes biased if error term isn't follow AR(1) process. I'm curious why d-statistic gets bias when using DW ...
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Autocorrelation in Elo ratings

FiveThirtyEight uses the following formula for their NFL Elo ratings: $$ R_i^{k+1} = R_i^k + K \cdot M(z) \cdot A(x) \cdot (S_{ij} - \sigma(x)) $$ where $z$ is the game's margin of victory, $x=R_i^k - ...
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Variance of linear combination of AR(1) process

Let $ \{X_t\}$ ~ AR(1): $$ X_t=2.62-0.84X_{t-1}+\epsilon_t, \ \ \ \epsilon_t\sim WN(0,2.27)$$ Compute the variance of $$ \overline{X}= \frac{1}{3}\sum_{t=1}^{3} X_t $$ The solution is: Var($\...
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1answer
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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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Autoregression model prediction

I am using autoregression for predicting next 10 steps ahead, but if I am giving more than 8 input values, it predicts negative value, otherwise the prediction is good. What is the reason behind it? ...
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How does one forecast next point in time series using GAS package in R?

I am using the GAS (Generalised Auto regressive score) package in R in order to forecast a chosen time series. I have read package documentation as well as author published paper and I struggle with ...
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What is the difference between the results using different AR(p) estimation methods?

There are three different ways to do AR(p) estimation. OLS MLE Yule=Walker Equation What are the differences between the results using these three methods? // http://www2.econ.osaka-u.ac.jp/~...
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Calculating AR(2) model variance and mean

AR(2) model is $$ Y_t=2.25+0.75Y_{t-1}+0.45Y_{t-2}+e_t $$ How can we calculate variance and mean?
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What is the difference between the p parameter in ARIMA(p,d,q) and the lag value used by AR?

I understand that p represents the order of the AR model used within the ARIMA model, but does that have anything to do with the lag value that is calculated by the IC (eg. aic, bic, hic) in the AR ...