# 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 understand SARIMAX intuitively?

I'm trying to understand a paper about electric load forecasting but I'm struggling with the concepts inside, specially the SARIMAX model. This model is used to the predict the load and uses many ...
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### Under what circumstances is an MA process or AR process appropriate?

I understand that if a process depends on previous values of itself, then it is an AR process. If it depends on previous errors, then it is an MA process. When would one of either of these two ...
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### How does ACF & PACF identify the order of MA and AR terms?

It's been more than 2 years that I am working on different time series. I have read on many articles that ACF is used to identify order of MA term, and PACF for AR. There is a thumb rule that for MA, ...
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### Why do we care if an MA process is invertible?

I am having trouble understanding why we care if an MA process is invertible or not. Please correct me if I'm wrong, but I can understand why we care whether or not an AR process is causal, ie if we ...
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### If an auto-regressive time series model is non-linear, does it still require stationarity?

Thinking about using recurrent neural networks for time series forecasting. They basically implement a sort of generalized non-linear auto-regression, compared to ARMA and ARIMA models which use ...
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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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### Boosted AR for time series forecasting?

I have time series data recorded at multiple locations, stored in a matrix $Y$. I have fit a Vector Autoregressive Model to it which forecasts the data pretty well on a test set. However, if I plot ...
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### 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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### Unbiased estimator for AR($p$) model

Consider an AR($p$) model (assuming zero mean for simplicity): $$x_t = \varphi_1 x_{t-1} + \dotsc + \varphi_p x_{t-p} + \varepsilon_t$$ The OLS estimator (equivalent to the conditional maximum ...
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### How to write an AR(2) stationary process in the Wold representation

I managed to write an AR(1) process in the Wold representation with help from the geometric series. I am having trouble with a stationary AR(2). How could I do?
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### Random walk estimation with AR(1)

When I estimate a random walk with an AR(1), the coefficient is very close to 1 but always less. What is the math reason that the coefficient is not greater than one?
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### Intuition behind the characteristic equation of an AR or MA process

Ok, so I've just started learning Time Series Analysis. We can write an $MA(q)$ process as $Y_t = \theta(L) \varepsilon_t$ and an $AR(p)$ process as $\varepsilon_t = \phi(L) Y_t$ in terms of 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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### R and EViews differences in AR(1) estimates

The main problem is: I cannot obtain similar parameter estimates with EViews and R. For reasons I do not know myself, I need to estimate parameters for certain data using EViews. This is done by ...
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### What's a stationary VAR?

What is a stationary VAR (vector autoregression)? Can a VAR with non-stationary variables be stationary? How do you test whether a VAR is stationary or non-stationary? (Example in ...
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### The explosive AR(1) process with $\varphi>1$, where was this first represented as a stationary, but non-causal, time-series?

According to this question and answer Explosive AR(MA) processes are stationary? the AR(1) process (with $e_t$ white noise): $$X_{t}=\varphi X_{t-1}+e_{t} \qquad , e_t \sim WN(0,\sigma)$$ is a ...
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### How to create a markov chain with gamma marginal distribution and AR(1) coefficient of $\rho$

I want to generate a synthetic time series. The time series needs to be a markov chain with a gamma marginal distribution and an AR(1) parameter of $\rho$. Can I do this by simply using a gamma ...
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### Is VAR a MANOVA with auto regression?

What are the differences between VAR (vector auto regression) and MANOVA?
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### What's wrong if I fit the auto-regression with OLS?

I am doing auto-regress by usual linear regression package. e.g. $y_t=φx+ε_t$ with $x =y_{t-1}$ My reason is that, Auto-regression does assumes iid errors, same for linear regression. Linear ...
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### Simulate AR(1) process in R with specified nonzero mean and AR coefficient

I need to simulate an AR(1) process with the following equation in R: $$X_{t} = 5 + 0.5X_{t-1}+Z_t$$ Where $Z_t$ ~ White Noise(0,1) and $T=500$. I know I should be using the ...
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### Variance of a stationary AR(2) model

I have two questions: 1) When one says an ARMA process is 'stationary,' do they mean strongly stationary or weakly stationary? 2) Is there a quick way to find the variance of a stationary AR(2) ...
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### Time series regression with lagged dependent and independent variables

I have monthly data for air passengers, oil price and unemployment. I'm trying to create a model to forecast air travel demand using oil price and unemployment as explanatory variables but are facing ...
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### What is an autoregressive decoder?

I saw that this was part of a deep belief network I was looking at. I'm not sure what it means. Is it a layer that transforms few inputs into many outputs and has a connection to itself? What is an ...
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### What is the expected value of the sample variance under a linear regression with omitted variables of an AR(2) process?

Lately, I have been interested in phenomenons related to omission of variables. For example, it can be shown that the expected value of the sample variance under the inclusion of one variable $x_1$ ...
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### The distribution of the initial point of an AR process

Consider a stochastic process $\{X_t, t = 1, 2, \ldots\}$ following the model $$X_t = \alpha X_{t-1} + e_t,$$ where $e_t \thicksim f$. Can I say that the distribution of the initial point, $X_1$, is ...
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### What is the reason for not including an intercept term in AR and ARMA models?

In econometric literature it is usually argued that in case of estimating an equation, an intercept term must be always included regardless of its statistical importance because removing the constant ...
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### Independent Bernoulli trials vs markov chain

Original Question Suppose we have a sequence of Bernoulli trials $X_1, X_2, \cdots X_T$ which are ordered in time and may or may not be independent. I am interested in understanding the probability of ...
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### Poisson with an autoregressive term

I want to fit a fairly "standard" Poisson model, but with an autoregressive term. $N_i \sim \mathrm{Pois}( \lambda_i E_i)$ with $\log \lambda_i = X_i \beta + \delta$ $\delta \sim AR(1)$ $X_i$ is a ...
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### Causality and stationarity of AR models

Studying AR models, I found that there are two properties that these models can have stationarity and causality. For what concerns stationarity, I have studied that this condition is satisfied if ...
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### How to interpret coefficients in a vector autoregressive model?

Can I interpret the coefficients in a VAR model in the same way as I do in a normal OLS regression?
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### Estimation of unit-root AR(1) model with OLS

Given a random walk $x_t$, $$x_t=x_{t-1}+\varepsilon_t,$$ consider estimating the slope coefficient $\beta$ in $$x_t=\beta x_{t-1}+\varepsilon_t$$ by OLS. This question and the following answer ...
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### Joint distribution of AR(1) model

Let $$y_1 \sim \mathcal{N}(\phi_0,\,\sigma^2),$$ $$y_t|y_{t-1},\, \phi_0,\,\phi_1,\,\sigma^2 \sim \mathcal{N}(\phi_0+\phi_1(y_{t-1} - \phi_0),\,\sigma^2),$$ for $t=2,3,\cdots,T$. I want to ...
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### What is better for time series prediction: AR or ARIMA?

I am trying to make a prediction in a time series with window 512 and horizon 2. I want to know if it's worth using ARIMA, that seems to be hard to understand, instead of the simple Autoregressive ...
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### How to compute the standard error of the mean of an AR(1) process?

I try to compute the standard error of the mean for a demeaned AR(1) process $x_{t+1} = \rho x_t + \varepsilon_{t+1} =\sum\limits_{i=0}^{\infty} \rho^i \varepsilon_{t+1-i}$ Here is what I did:  \...
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### Why this OLS fitting will converge to (0,-0.5)?

In this AR1 model, if we fit y with diff(y), regardless the true coefficient, when N is large, it seems the model will converge ...
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Notation: $\dot{Z}_t = Z_t - E(Z_t)$, so that it is centered at 0. $a_t$ stands for the residual and we assume the $a_t$ are independent and normally distributed with mean 0 and constant standard ...