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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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### 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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### 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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### Out-of-sample prediction in conditional autoregressive models

I am interested in fitting a conditional autoregressive model to a dataset where observations are grouped by regions, then performing cross-validation by holding out all observations from a single ...
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### Why doesn't the Wold's decomposition theorem imply a good AR(p) fit?

I'm trying to fit an AR(p) process to the standardized, 10 years long time series of monthly logreturns of a stock index and get extremely poor fit. I'm not surprised, because if I had a good fit, ...
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### Why does an AR(p) process require the largest eigenroot to be <1/ characteristic roots lie outsied the unit circle to be stationary?

For reference, this is the paragraph in Wikipedia I'm struggling with. https://en.wikipedia.org/wiki/Autoregressive_model I do understand the simpler proof for under which conditions an AR(1) ...
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### Complex-valued autoregressive and ARMA processes

I’m working with complex-valued discrete time series, and specifically with complex-valued autoregressive and ARMA processes. Could someone provide me with some suggestions about a good papers or ...
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### Second-Order stationarity condition for complex-valued autoregressive process

Let $\{c_n\}$ be a complex-valued discrete autoregressive process of order $p$, $\mathsf{AR}(p)$, such that: \begin{equation} \label{cn} c_n = \sum\nolimits_{i=1}^{p}\rho_i c_{n-i} + w_n, \quad n \in (...
### OLS estimation of intercept in AR($p$) in R
I investigate the performance of the OLS estimator of an AR($3$) model given by $$X_t=\mu+\phi_1X_{t-1}+\phi_2X_{t-2}+\phi_3X_{t-3}+\varepsilon_t$$ for $t\in\mathbb Z$ using the following code: <...
Suppose we have an autoregressive process, $$y_t=\phi y_{t-1} +u_t$$ where $|\phi|<1$. If $u_t$ is an i.i.d random variable this process is stationarity. What if $$u_t=u_{t-1}+g+\epsilon_t$$ where \$...