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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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1answer
194 views

Approximate AR(p) with a product of AR(1) and AR(2)

Literature suggests that any AR(p) ARIMA model can approximated as a combination of AR(1) and AR(2) processes. For example, one book suggests that an AR(3) model with the following coefficients: ...
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0answers
60 views

Fit ARMA model to ACF

If I have the autocovariance function $\gamma_\tau$ (numerically over a given set of lags $\tau = 0 \ldots n - 1$) of a stationary linear stochastic process, is there an efficient way to determine the ...
4
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0answers
127 views

Stationary Distribution of Multiplicative Autoregressive Model

I know for the additive autoregressive model the stationary distribution of $\{X_t\}$ can be found, if it exists, in the following way: \begin{align} X_t &= \alpha X_{t-1} + \epsilon_t\\ \...
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1answer
36 views

How to calculate the average age of observations in forecasting models of various types?

Suppose you have a time series $\ Z_t$ that is used as a forecasting model for $\tau$ steps ahead from origin $T$. $\ Z_t$ is defined as: $\ Z_t = 0.05 Z_T + 0.10 Z_{T-1} + 0.15 Z_{T-2} + 0.20 Z_{T-...
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0answers
258 views

Assumptions on Neural Networks (NNETAR)

Are there any assumptions that must be covered when fitting an NNETAR model? non-correlation, normality, or something? I've already saw Rob Hyndman post where he says NNETAR doesn´t need stationarity, ...
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0answers
518 views

AR(2) process- covariance stationary- complex roots

I am trying to check if this process is covariance stationary. I have an AR(2) process given by: $Y_t(1-1.1L+0.8L^{2})=\epsilon_t$ I saw that to check if the process is stationary, instead of ...
2
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1answer
441 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 ...
16
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2answers
962 views

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 ...
3
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1answer
553 views

Why does a AR(1) model that's mean reverting revert back to B0 as opposed to B0 + B1*B0?

In time series analysis, an AR$(1)$ model takes the form: $$x_t = \beta_0 + \beta_1 \cdot x_{t-1} + w_t,$$ where $w_t$ is the white noise term. In order for the model to be stationary and to ...
2
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1answer
48 views

Effect of strong auto-correlation on forecasting?

Suppose a wise-sense stationary univariate time series has relatively strong auto-correlation of lag-length of 1, say, around -0.7 Then how would it affect the forecast? Conversely, if a ...
2
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1answer
489 views

Why use Granger causality instead of autoregression?

I'm working on an analysis on GDP growth. I want to test whether regional growth in GDP in a time frame can be explained by the growth of air traffic in a preceding period (and controlling for some ...
2
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1answer
536 views

Variance of AR(1) process using lag operator

Suppose for the AR(1) model, $$Y_t=\phi_1Y_{t-1}+e_t$$ I want to find the variance $Var(Y_t)$ using lag operator: $$Y_t=(1-\phi_1L)^{-1}e_t$$ My way is simply taking the variance, $$Var(Y_t)=(1-\...
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892 views

Fitting GLM (family = inverse.gaussian) on simulated AR(1)-data

I am encountering quite an annoying and to me incomprehensible problem, and I hope some of you can help me. I am trying to estimate the autoregression (influence of previous measurements of variable X ...
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1answer
69 views

neural network AR opinion and math background

I'm trying to predict a time series using the neural network approach. I saw this function "nnetar", in the "forecast" package, what do you think? I can not find a mathematical explanation about it. ...
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0answers
47 views

AR(p) selection

I am facing difficulty in determining what AR(p) model to specify for my regression. By trial and error depicted below, it seems to me that an AR(4) model should be specified. ...
2
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0answers
43 views

Learning the raw mathematics behind VAR modelling to implement it myself

I have recently been using R, along with the very handy vars package, to model time series and generate forecasting based on the results it produces. I have found a ...
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1answer
1k views

What is the variance for time series data? How can it be computed?

I have learnt time series courses but may have forgotten some important and basic pieces. For typical times series data say AR data, there is only one observation at a time. How do you define the ...
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2answers
1k views

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 ...
2
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1answer
345 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
359 views

Reference that AR(p) are strictly stationary if and only if it is causal

I am interested in conditions that an autoregressive model AR(p) is strictly stationary. It turns out that these lecture notes https://www.bauer.uh.edu/rsusmel/phd/ec2-3.pdf (see bottom of page 13) ...
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64 views

Question regarding derivation of variance for AR(1) when rho is less than 1

In class my professor derived the variance for the AR(1) model in the case of $|\rho| < 1$. I am having some issues with derivation. $$Var[X_k] = E[X_k^2] - E[X_k]^2 = E[X_k^2] - 0 = E[X_k^2]$$ $$...
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1answer
397 views

random walk and covariance stationary

I was preparing for CFA and encountered this question, which is quite puzzling. To use autoregressive model, it has to be covariance stationary (same mean, covariance). If a model's residual is not ...
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0answers
48 views

How to quantify correlation between two non-Gaussian AR(1) time-series?

I am trying to understand the need for lateral transshipment within the same echelons in our supply-chain. Lateral transshipment would be more helpful if the stock-levels at two stocking-points are ...
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0answers
57 views

Discrete choice model with autoregressive error term

I consider consumer choice over time $t$ over a set of alternatives $J$, whereby a consumer's perceived utility of option $j$ is $u_{jt}=x_{j}+\mu_{jt}$ with $\mu_{j0}=\epsilon_{j0}\sim\mathcal{F}(\...
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0answers
149 views

Transforming a multivariate binary time series to be stationary

I have a multivariate (multi-response) dataset with, for example, 10 different binary responses. I'm interested in an AR(p) model, determining how the responses at previous time steps relate to the ...
1
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1answer
77 views

Bayesian autoregressive model with second peak at 1 in posterior distirbution of AR parameter

I am trying to run a Bayesian hierarchical AR1 model for a set of fairly short time series. In some of the series I get a second peak around 1 in the posterior distribution of the AR1 parameter. ...
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0answers
328 views

Solve For ACF/ACVF of An AR(3) Process

I am currently doing an online course on Time Series and this is a self-assessment question from the homework, I won't see the answer until I submit, so I would appreciate hints/leads. I have made ...
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1answer
1k views

Why is the dickey fuller test different from a simple t-test

I am trying to understand why should there be different distribution for t-statistic, in case of AR model, Dickey-Fuller test For e.g. Say, the model is $Y_t = \beta_lY_{t-1} + \varepsilon_{t}$. ...
1
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1answer
415 views

Deriving method of moments estimator for AR(1) process

The method of moments estimator for AR processes can be had with the Yule-Walker equations. But how is it derived? The equation for AR(1): $$Y_t =aY_{t-1}+\epsilon_t$$ Where $\epsilon $ ~ $N(0,\...
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0answers
34 views

Handling error with spatially lagged variable that is related to but distinct from outcome variable. SAR, CAR, Durbin, or lagged X?

I am estimating a diffusion model in which I am using spatial lags to predict an outcome variable, but I'm having trouble specifying the distribution of the error term because the lagged variable is ...
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1answer
3k views

Autocorrelation of a stationary AR(2) process

Consider the stationary AR$(2)$ process of the form: $y_{t} = \alpha + \phi_{1} \ y_{t-1} + \phi_{2} \ y_{t-2} + u_{t}$ where $u_{t}$ is i.i.d. white noise. Just as a head's up, we have not covered ...
3
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1answer
112 views

How can differencing in an ARIMA model be implicit instead of explicit?

In this post - Rob Hyndman explains that: Even with that correction, the two models are not quite equivalent. In the Eviews code, the differencing is done before estimation, whereas in the R code ...
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0answers
202 views

Derive MA (Moving average) representation of a first-difference-process

I have a non-stationary AR(1)-process. After taking the first difference, how can I derive the MA representation of the resulting „difference process“ Delta_xt? As an example, consider xt = 1.5xt-1 ...
3
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1answer
122 views

Are all $AR(p)$ processes for which $|a_1|,…,|a_p| < 1$ stationary?

For an $AR(p)$ process $ Y_t = a_1Y_{t-1}+a_2Y_{t-2}+...+a_qY_{t-q}$ : Is having the coefficients $|a_1|,....,|a_p| < 1$ just a necessary condition for stationarity, or is it sufficient as well?
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1answer
117 views

I understand why stationarity is a requirement for AR(p) models, but why is it necessary for MA(q) models?

I have (or at least I think I have) a good intuition for why stationarity is a requirement for modeling $AR(p)$ models: an $AR(p)$ model with coefficients $a_1,....,a_p$ : $ Y_t = a_1Y_{t-1}+a_2Y_{...
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0answers
1k views

Stationarity of AR(1) model

Okay this might be a stupid question but, I understand that a (weakly)Stationary Time Series is one where 1) $E[X_t]$ = constant 2) $Var(x_t)$ = constant 3) $cov(x_t,x_{t-h})$ = constant, at any ...
2
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1answer
50 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
39 views

Why do the fundamental time series model explain many dynamical phenomena

What is so special about the way Moving average models, Autoregressive models and their combinations (ARMA, ARIMA) are defined that they seem to fit many of the univariate time series we observe in ...
4
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2answers
768 views

How do I Estimate Joint Entropy Using a Histogram?

I am trying to estimate the entropy for two time series, defined by random variables $X$ and $Y$, each distributed according to an unknown PDF which is to be estimated empirically (using a histogram ...
2
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1answer
478 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
600 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. ...
2
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1answer
69 views

Would multiple-regression give the same results as auto-regression?

From looking at textbooks, I see that the equation used for estimating auto-regression is different from the equation used to estimate multiple-regression. But, if I used successive values from a ...
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0answers
156 views

Convert Spatial Weight Matrix File?

I want to create a spatially lagged version of my response variable to estimate a spatial autoregressive model in R. To quantify the spatial relationship that exists among the features in my data I ...
0
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1answer
586 views

How to fit an Autoregression model in Spark? [closed]

I'm having a look at the implementation of Autoregression model in Scala https://github.com/sryza/spark-timeseries/blob/master/src/main/scala/com/cloudera/sparkts/models/Autoregression.scala Now if I ...
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1answer
57 views

Could somebody help me read these ACF and PACF plots?

So, I have this time series that I have already forecasted using an ARMA model, but I am new to this and am therefore not at all sure whether or not I did this (somewhat) correctly. I got the best ...
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0answers
72 views

How to simulate a time-dependent AR(2) model?

I'm working with time series and I need to simulate a time-dependent AR(2) model like this: $Y_t = a_tY_{t−1} − 0.5Y_{t−2} + \epsilon_t ,t =1, 2, ..., 1024,$ where $a_t=0.8*cos(t)$ How can I ...
0
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1answer
905 views

External regressors in mean or variance equation of AR(1)-GARCH(1,1)?

What is the difference between entering my external regressors in the mean equation and entering them in the variance equation in an AR(1)-GARCH(1,1) model? I get more explanatory results with the ...
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1answer
57 views

How to randomly generate a data with an error which is not normally distributed but follows an empirical distribution $(0, \sigma^2)$

I have this model AR model for multiple time series $ Y_{it} = \phi y_{it-1} + \delta_1y_{i-1 ,t-1} + \delta_2 y_{i+1 , t-1} + \lambda_i + \epsilon_{it} $ where $\epsilon_{it}$ is a function of ...
0
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1answer
481 views

Estimate AR(1) using Log returns

I am studying share price log returns and AR(1) model. I downloaded data from FTSE100 and used the Adj.close column to find the Ln returns: Now I am trying to understand how can I estimate an AR(1) ...
0
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1answer
641 views

When to use deep learning instead of AR models?

Autoregressive models & deep learning(rnn-lstm) models both are used for time series prediction. As we choose the 'look back' for lstm's, provision to choose optimal lag by viewing acf-pacf plot ...