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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521 views

Good Resource For Converting ARIMA output in R to equation form?

I've seen this question asked a few times but I still haven't seen a place where I can get some good examples on how to convert an arima() output in ...
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2answers
576 views

When is ARMA(p,q) strictly (strongly) stationary and what is its unconditional distribution?

It seems that the discussions on ARMA are always focused on weak (second-order) stationary, but what about strong stationary? What are the conditions on the coefficients for it to be strictly ...
2
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1answer
199 views

Does stationarity of AR(p) imply innovations are i.i.d.?

My lecture notes give the following definition: A stochastic process $(X_t)_{t\in\mathbb{Z}}$ is called autoregressive of order $p$ if it satisfies: $$X_t=\phi_1X_{t-1}+...+\phi_{t-p}X_{t-p}+W_t.$$ ...
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1answer
425 views

Can I use a spatially lagged Dependent Variable while using Spatial Error Model?

I have following issue: I run spatial diagnostics on dependencies for my Log-Log Transformed regression model. LM Tests (including Robust) are highly significant. Since I am using GeoDa, I cannot ...
2
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2answers
853 views

Generate AR(1) process with different $y_0$ values in R [closed]

I need to plot an AR(1) graph for the process $$ y_k = 0.75y_{k-1} + \varepsilon_k $$ for $y_0=1$ and another graph for $y_0=10$. Assume $\varepsilon_k$ is uniformly distributed on interval $[-0.5,0....
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1answer
496 views

What's the meaning of the expansion coefficient of the AR model?

I am trying to understand the meaning of the phi parameter of the AR modeling. A bit of background: I am digging into statistical parametric mapping (SPM) and the prewhitening method, used to get rid ...
2
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1answer
115 views

AR(2) process: are leptokurtic residuals OK?

I have a time series of logarithmic returns. After inspection of the ACF and PACF plots, I tried to fit AR(2), MA(2) and ARMA(1,1) models and eventually found out that the AR(2) version can possibly ...
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1answer
288 views

what R function fits a smoothing spline regression model with correlated errors?

I want to fit a smoothing spline regression model with correlated errors (it's a time series) using R. All I could find is function ssr, from library ...
2
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1answer
388 views

What does “AR(p) filtered series” mean?

I guess this means that omitting some variables in a certain interval, say, $(x_1, x_2, x_3, x_4, x_5) \to (x_1, x_5)$ in AR(4) model. Is it right? Or does this means eliminating autocorrelations ...
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1answer
653 views

Vector autoregression - number of lags

I am constructing a Vector autoregression model and I have used AIC to find how many lags I should use. Does 7 lags seem unreasonable? I am trying to find the impact the property market has had on the ...
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1answer
20 views

AR(1) Finding $\gamma_l$

I have $\gamma_l = Cov(r_t, r_{t-l})$ as a definition in my notes and now I need to find $gamma_l$ for a series $r_t- m = p(r_{t-1} - m) + a_t$ where $r_t$ is a linear time series with expected value $...
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1answer
120 views

Population autocovariance goes to zero, assuming covariance stationary

In time series context, let $\gamma_j=E[(y_t-\mu)(y_{t-j}-\mu)]$ denote population autocovariance, where $\mu$ is population mean of $y_t$, assuming covariance-stationary. Then, $\gamma_j$ goes to $0$ ...
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1answer
551 views

Temperature time series forecasting predictions converging to a certain value

I am trying to forecast the value of the ambient temperature based on given data on Python. The data frequency is 15 minutes. In order to predict future values, I am using a simple autoregressive ...
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1answer
1k views

Expected Value of an AR(1) process

I saw the answer on this post and got confused about a couple things in its explanation. Mainly, I am unsure of How the poster immediately knows the process $X_t = c+\phi_1 Y_{t-1} + \epsilon_t$ is ...
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1answer
67 views

A first order autoregressive process with lag-1 auto-correlation 0.5

I was reading my professor's notes by myself, it says that: ''A first-order autoregressive process with lag-1 auto-correlation 0.5 can be generated from $\theta^{(t+1)}|\theta^{(t)} \sim N(0.5\theta^{...
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1answer
581 views

Interpreting Regression Coefficients in time series model with seasonal & trend component AND a lagged variable

I'm working on a time series model which predicts daily sales. The model is based on seasonal-trend decomposition by Loess, basically. (I'm using an R package similar to Prophet by facebook.) Since ...
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1answer
39 views

AR(1) - autocorrelation calculation

I am practicing deriving proofs and I cant seem to yield the correct answer for the covariance of an AR(1) model: $$X_t=pX_{t−1}+e_t.$$ Would greatly appreciate if someone could tell me where I am ...
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1answer
383 views

Autocorrelation at lag 1 but scatterplot shows no linear relationship

I have a time series consisting of 490 days and for each day I have the residual of a forecasting model. I wanted to check if the residuals somehow correlate and calculated the ACF at lag 1 which is <...
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1answer
52 views

Proving stationairty of AR(1)

Let me set this up. We have an AR(1) process: $x_1 = w_1$ and $x_t = \frac{1}{2}x_{t-1} + w_t$ for $t \geq 2$ and where the $w_t \sim N(0, \sigma^2)$. I have read that this process is stationary - ...
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1answer
497 views

ACF and PACF of AR(p)

Why does the PACF of AR(p) model cut off past the order of the series? Why does the ACF tail off to zero? What is the intuitive reason behind this?
2
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1answer
617 views

Conditional maximum likelihood of AR(1) UNIFORM PROCESS

Let $Z_t = \phi Z_{t-1} + u_t$ where $u_t \sim uniform[-1,1]$ and $|\phi|<1$ I I am facing problems coming up with conditional maximum likelihood estimate of an AR(1) process with uniform errors. ...
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1answer
93 views

Constants in determining stationarity of a time series

After reading about AR(p) processes I have one question regarding the characteristic polynomial of AR(p) processes and its roots. Let's say that you want to determine whether the time series $$y_t ...
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1answer
62 views

length of time series to estimate $\phi_2$ for an AR(2)

So I have this question that I am kinda stumped on... I have an AR(2) model with $\phi_1=0.4$ $\phi_2=0.5$, I need to find how long of a time series I would need to estimate $\phi_2$ with 95% ...
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1answer
405 views

Posterior distributions of parameters in a AR(1) model

Consider a AR(1) model with states given by $x_t=\phi x_{t-1}+a_{t}$, $a_{t}\sim\mathcal{N}(0,\tau^2)$ and the observations given by $y_t=x_{t}+e_{t}$, $e_{t}\sim\mathcal{N}(0,\sigma^2)$ for $t=1,...
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1answer
4k views

conditional vs unconditional forecast variance in AR(1)

I have trouble showing that conditional forecast error of AR(1) has smaller variance than the unconditional one. I can show that cond. forecast error is: $$ Y_{T+1}=aY_{T}+\epsilon_{T+1} $$ $$ \...
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1answer
78 views

AR(4) - show $\mathbb{E}(Y_t)$ = 0

$\newcommand{\E}{\mathbb{E}}$I was wondering if anyone could help me show that $\E(Y_t) = 0$? I have had a go myself but have got stuck. Could someone please point me in the right direction? ...
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1answer
174 views

Unable to calculate the density function for AR

The model is an AR(p) process excited by a white Gaussian noise $\epsilon_t$, \begin{align} Y_t = &c+ \phi_1Y_{t-1} + \phi_2 Y_{t-2}+ \ldots+ \phi_p Y_{t-p} + \epsilon_t\\ \epsilon_t = &\...
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1answer
179 views

What is the “scale” parameter in “continuous autoregressive model” in cts package?

I am trying to use the "car" command in "cts package" in R program and I see the "scale" parameter there. I wonder whether this can be assumed to be equivalent to time intervals for time series ...
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1answer
4k views

calculating the expected value and variance of a log AR(1) process

I have an AR(1) process that looks like this: $$ \ln(g_t) = (1 - \rho_g)(\ln(\mu_g) - c) + \rho_g\ln(g_{t-1}) + \epsilon^g_t $$ where $|\rho_g| < 1$, $\epsilon^g_t \sim N(0, \sigma^2_g)$, and ...
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1answer
118 views

Is a Stationary VAR Process with Zero Mean Gaussian Innovations a Gaussian Stationary Process?

Consider the stationary VAR process $${\bf X}_t = \sum_{\tau = 1}^{L} A_\tau {\bf X}_{t-\tau} +{\bf \epsilon}_t$$ If the innovations $\epsilon_t \sim MVN({\bf 0},\Sigma)$ then is ${\bf X}_t$ a ...
2
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1answer
218 views

Autoregressive model with exponential lags

I have a very highly sampled time series that I would like to fit an autoregressive model (AM) to (~3 million samples). From knowing what they represent, I have believe there should be unique ...
2
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1answer
2k views

Do data and residuals of a VAR model have to be of normal distribution?

Does vector autoregression (VAR) model require data to be of normal distribution? What are the pitfalls if the residuals are not of normal distribution?
2
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1answer
681 views

Long term and short term effect of estimators on AR model

How do we find out the long term and short term effect of estimators on AR model?
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19 views

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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34 views

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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38 views

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 &\overset{iid}{\sim} N(0,\sigma^2). \end{aligned} To make it simple, let's ...
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29 views

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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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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1answer
56 views

General formula for AR($p$) auto-regressive time series

I'm trying to find a reference (including the full formula) for the following. If $X_n = a_1 X_{n-1} + \cdots a_p X_{n-p} + e(n)$ where $\{e(n)\}$ is a white noise, then $$ X_n=g(e_0,e_1,\ldots,e_n)+\...
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33 views

On the stationary density of an autoregressive model of order 2

Consider a stochastic process $\{X_t, t = 1, 2, \ldots\}$ following a stationary AR(2) model $$X_t = \theta_1 X_{t-1} + \theta_2 X_{t-2} +e_t,$$ where $e_t \thicksim N(0, \sigma^2)$. I want to find ...
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100 views

How many lags to use in ADF test?

So I've ran a ADF test on my data multiple times with different lags and all up to a lag of 4 have a p-value below .05. So in this case how many lags do you decide to use? Could this also provide a ...
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48 views

Regressing across multiple different time series using exogenous variables?

To make this situation clear, I'll use a somewhat silly, but conceptually simple example. Imagine I record teams of movers carrying furniture down the block. I measure the furniture's position/speed ...
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195 views

What's the variance of an AR(1)/ARCH(1)

The main question is: an AR(1)/ARCH(1) process has the variance of the ARCH(1)? I've tried to compute the unconditional variance of an AR(1)/ARCH(1) model, so an AR(1) in which the noise is modelled ...
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0answers
119 views

Markov Chain order 1 vs. AR(1) … Difference and Implication for Parameter Estimation

As other posts on this site indicate, the difference between a time-homogeneous Markov Chain of order 1 and an AR(1) model is merely the assumption of i.i.d. errors, an assumption that we make in AR(1)...
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256 views

How to relate roots of AR and MA to unit circle

I'm working on these problems and think I figured out most of the steps, but am stuck near the end as I don't understand how to relate my roots back to the unit circle in order to determine ...
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0answers
146 views

Convergence of predictions of an autoregressive model

I have performed a simple autogregressive model with lag 2 on a time series data. After obtaining the coefficients, I have computed the predictions. Since the lag is 2 in model, the first prediction $\...
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64 views

Forecasting autoregressive model. What's the best linear predictor?

Obviously if $X_t = \phi X_{t-1} + Z_t$, then the best linear predictor of $X_t$ given $X_{t-1}$ is $X_t = \phi X_{t-1}$. But if $\phi$ is unknown, one may attempt to substitute $\phi$ by a Yule-...
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77 views

divergence of beta estimates between OLS and regression with ARIMA error

I have physiological time-series data: ~60k observations per channel, ~100 Hz sampling. I will model individual channels with ~20 regressors. Under OLS, given temporal autocorrelation in the data, ...
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0answers
458 views

LASSO in AR-Models

I couldnt find such a post here. I am highly interested in applying the lasso to different situations. However, I am actually dealing with time series models of high order. I have found some research ...
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65 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 ...