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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Using for loop to find a string in a varying coefficient AR [on hold]

I am trying to output the following text: s(a[i,j,k], by=data[,c(2)])+s(a[i,j,k], by=data[,c(3)]) + s(a[i,j,k], by=data[,c(4)])+s(a[i,j,k], by=data[,c(5)]) My ...
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47 views

Mean reversion in AR(1) process

Mean reverting level in following AR(1) process is $b/(1-a)$. $$x(t) = a + bx(t-1).$$ I understand this. I read that the mean reverting level for AR(1) process given below with finite differencing ...
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1answer
37 views

Practical issues with dynamic panel data modeling

Unfortunately for me, I've got a situation where I need to control for the lag of a dependent variable as a robustness check against an alternative interpretation of my main regression. The baseline ...
3
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3answers
118 views

Under what circumstances is an MA process or AR process appropriate?

I have a very basic question. Please let me know if this has been asked before, but in my defence I haven't seen it on Cross Validated. I understand that if a process depends on previous values of ...
2
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8 views

Methods for measuring snowball effects in a “complete” longitudinal dataset

I'm looking for ways to test for "cumulative advantage" effects in a longitudinal dataset (see image) I guess the data set is principally similar to this: http://www.caldercenter.org/whatis.cfm , ...
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1answer
38 views

Is the Moving Average of ARMA the same of Moving Average of Stock Market?

I'm studying time series prediction and I have some questions. Is the Moving Averages movel studied the methods of the ARMA family has the same concept as the methods studied in Moving Averages ...
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13 views

How do you calculate standard error in a Dickey-Fuller test?

So in everything I've found, they tell you have to calculate $\rho$, or how to test for confidence interval for it. What I am trying to figure out is how to calculate the SE which would get us our ...
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1answer
32 views

Predicting time series with OpenBUGS

I have a number of fairly short time series (about 4–100 observations) which I need to forecast into the future. I decided to use Bayesian inference, because there is external information about each ...
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24 views

How can I correct for residual autocorrelation in a fixed effect panel model?

The residuals have an AR(2) structure. Is it appropriate to add AR terms to a fixed-effects panel model?
2
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1answer
36 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
64 views

R fit restricted AR(p) model

I have a question about using R to fit an AR model. If we want fit a AR(p) model, the equation will be $Y_t = φ_1Y_{t-1} + φ_2Y_{t-2} + ... + φ_pY_{t-p} + Z_t$. What about I only want to fit the model ...
2
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1answer
52 views

Fit a moving average (MA) time series model to the data (R:stats::ar equivalent)

I am looking for some tools for automatic fitting of moving average (MA) time series model to my data in R. I know R:stats::ar ...
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12 views

Why must the solutions of the characteristic equation be greater than 1 for an Auto-regressive model to be stationary?

According to my notes, stationarity occurs if: All the solutions of the characteristic equation of the $AR(P)$, $1-\phi_1X-\phi_2X^2-...-\phi_PX^P=0$, are greater than 1 in modulus. No ...
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1answer
60 views

Conceptual issues in AR model representation

Q1 : I am looking for a $k$ lagged (k order) AR model where k >2 conveniently k >20. I am unable to find any large AR model other than the popular second order AR model, $x_t = 0.195x_{t-1} - ...
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1answer
55 views

How to interpret the expression of MA(1) as AR($\infty$)

When AR(1) is expressed as MA($\infty$), I can interpret it as: let's say my wage this year depends only on last year's wage and a random shock (my boss' mood). But last year's wage also depends on ...
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1answer
80 views

Determining order of AR model

Suppose that we have the following model $$ y[t] = A_1\sin(\omega_1 t+\phi_1)+A_2\sin(\omega_2 t + \phi_2)+ \cdots + A_p \sin(\omega_pt + \phi_p) + z(t) . $$ Let us call this signal as B. Then in ...
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1answer
51 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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155 views

Differencing a time series

I am looking to find the ACF of a time series, but after it is differenced. $y_t = a_1y_{t-1} + \epsilon_t , \mid a_1 \mid < 1$ I understand that to find the ACF this process needs to be ...
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23 views

AR(2) simulation problem

Take covariances $Cov[X_{t-2},X_{t}]$, $Cov[X_{t-1},X_{t}]$ and $Cov[X_t,X_t]=Var[X_t]$ and calculate the parameters for the AR(2) process ($a_1$, $a_2$ and $\sigma^2$ (the variance of the error ...
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41 views

How to interpret the characteristic roots of moment equation of a AR(2) model?

I am learning the financial time series using the book 'Analysis of financial time series' by Ruey Tsay. In chapter 2, they introduced AR(2) models. The moment equation (which is the function between ...
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35 views

Autoregressive Markov chain simulation and the likelihood ratio test for Markov property

I am trying to estimate a Markov chain of second order (Markov chain that fulfills $P[X_t|X_{t-1},X_{t-2}]=P[X_t|X_{t-1},X_{t-2},...,X_{t-p}]$) using an AR(2) process. Once I have simulated the ...
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127 views

How to fit log-linear poisson autoregressive mixed model?

I have time-series count data $N_{i,j}$ (population sizes in site $i$ and year $j$) and I want to correlate year-to-year changes with the environmental conditions $x_{i,j}$. For this, I want to fit ...
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72 views

ARDL, Lag Terms and Singularity

I am interested in fitting an ARDL model that has 4 lags for each explanatory variable. However, when I fitting the model in R. R says that coefficients are not defined because of singularities. Is ...
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29 views

Computing Standard Errors in EM algorithm

I'm applying the EM to a hidden markov chain (the $\mathbf{Z}=\{Z_1,...,Z_n\}$ variable), with observations(the $\mathbf{Y}=\{Y_0,...,Y_n\}$ variable) dependent not only on the hidden markov chain, ...
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1answer
31 views

Mixture of normals, dependent on past

I have the following probability model: $(X_k|\text{PastHistory}_{k-1}, \theta_0,\theta_1,\theta_2) \sim (\pi\cdot N(\theta_1+\theta_0\cdot X_{k-1},1)+(1-\pi)\cdot N(\theta_2+\theta_0\cdot ...
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1answer
97 views

Conditional expectation in AR(1) process

Suppose we have a stationary AR(1) process: $Y_{t+1}=a+ \rho Y_{t} + \epsilon_{t+1}$ where $\epsilon_{t+1}$ is white noise with probability density function $\phi(.)$. Now say we have a equation ...
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56 views

Fit Negbin glm model with autoregressive correlation structure

I am attempting to estimate the effect of various variables on the time-series of counts of reported cattle stillbirths. We investigate the effect of day-of-week, month, holidays etc…and also the ...
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34 views

Vector autoregression with gradient descent

I am no expert in statistics, but I have been asked to implement a VAR model using gradient descent in R. I have written a code that, from what I have been told, it makes sense. However, the estimated ...
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28 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 ...
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188 views

Is non-stationary AR(p) process constant in mean?

A non-stationary $AR(1)$ process, which is a random walk, is constant in mean, but not constant in variance. How about the other $AR(p)$ processes with the order $p>1$? Are they constant in mean?
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67 views

Estimating a VAR model with variable coefficients

I want to estimate a VAR model based on the Dufour and Engle paper "Time and the Price Impact of a Trade" (2000). There, the parameter $ b_{i} $ of the endogenous variable $ x_{i} $ is dependent on ...
2
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1answer
110 views

Determining the amount of lag in an autoregressive model

I have done a lot of work in regression (time-invariant) but I am just now studying forecasting. My question is about determining the amount of lag to use in an autoregressive model. I assume that ...
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1answer
163 views

Auto-Regressional & Moving Average Model Formula Properties

I seeking help in understanding specific values underlying the formula's for the MA(p) model & the AR(q) model. I am attempting to implement the models (building up to the combined ARIMA model) in ...
2
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1answer
195 views

How to plot spectra of an AR(2) process

I am stuggling with this problem and was hoping to find some guidance to answer it. Let $y_t=\phi_1y_{t-1}+\phi_2y_{t-2}+\epsilon_t$, with $\epsilon_t\sim N(0,1)$. Now, I want to plot the spectra ...
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30 views

Determining the posterior distribution for an Autoregressive or order 1 model

Question: For this question, note that the notation $y_{1:T} = (y_1, y_2, \cdots, y_T)$, ie, a vector of random variables. Consider the following AR(1) model: \begin{align*} y_{t+1} = \phi y_t + ...
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1answer
164 views

Backshift operator applied to a constant

This questions is two part: 1) What happens when you apply the backshift operator to a constant? For example, if I have the AR process $$(1-\phi B)(y_t-\mu)=\epsilon_t$$ does that equal ...
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1answer
94 views

How to do Univariate Heteroscedasticity Test

I just wanted to know how to do Heteroscedasticity Test on a Univariate Model? ex: an univariate autoregressive model ex: an univariate ARCH/GARCH model If it is possible, how does one do that in ...
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1answer
92 views

Auto regressive process, maximum likelihood estimator

A first-order autoregressive process, $X_0,\dots,X_n$, is given through the following conditional distributions: $X_i | X_{i-1},\dots,X_0 \sim \mathcal{N}(\alpha X_{i-1},1)$, for $i = 1,2,\dots,n$ and ...
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45 views

Tips when regressing growth rates

I have 20 months of Year over Year growth rates for a X independent variable and Y dependent variable. The correlation between these two variables is 0.72. I would like to predict Y using X for ...
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75 views

Can I get an univariate ARMA(2,1) representation from a bivariate VAR process?

Suppose the VAR is on (x,y) and I want to get an ARMA(2,1) expresion for x, how can i do that? For example, $\left[ \begin{array}{l} x_t\\ y_t \end{array} \right] = \left[ \begin{array}{l} ...
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1answer
393 views

Meaning of output of function “ar” in R

How should I read the output of the function ar in R. For example, take this VAR model: ...
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1answer
163 views

Is this explanation of the Box-Jenkins approach correct?

Can someone please tell me what they think of my explanation of the Box-Jenkins approach to forecasting time series? Do you have anything to add (in particular to my explanation as to the intuition ...
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213 views

Variance of a smoothed AR(1) process

The query I have relates to calculating the variance of AR(1) processes that are smoothed with a simple moving average. So: In an AR(1) process of the form: $$ X_t=c+\varphi X_{t-1}+\varepsilon_t, ...
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1answer
185 views

Time series: correcting the standard errors for autocorrelation

I have performed a number of tests to detect any presence of autocorrelation in my monthly return series. The test results confirm that the standard errors are not independent. A Durbin-Watson test ...
2
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1answer
428 views

AR(1) coefficient is correlation?

Is the ar1 coefficient from an AR(1) model the "first order correlation of the noise" of a time series? I'm using R's aws package and one of the arguments of the ...
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307 views

AR(1) process with heteroscedastic measurement errors

1. The problem I have some measurements of a variable $y_t$, where $t=1,2,..,n$, for which I have a distribution $f_{y_t}(y_t)$ obtained via MCMC, which for simplicity I'll assume is a gaussian of ...
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1answer
238 views

How to estimate a model with fixed and random effects for a long panel dataset?

NOTE: I am using Stata for doing this. I have a long panel dataset, meaning my N is much smaller than my T. I have N = 5, T = 61. I tried to estimate my model, but I get an error related to the ...
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1answer
176 views

Residuals in double seasonal exponential smoothing

I have a time series with muliple seasonal cycles, which are 24 and 168 hours for my case. I would like to use Double Seasonal Exponential Smoothing method to forecast, which was published by James W. ...
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67 views

What is geometric autoregressive process?

Can anyone give a definition for Geometric Autoregressive Process? Any specific properties? And, in what fields is this mostly applied? To add some context to the question, here is a section of the ...
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23 views

Number of areas in conditional autoregressive models

This is a simple question on Bayesian spatial modelling via conditional autoregressive modelling. What is, according to your judgement (and possibly some methodological source), the minimum number ...