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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Autoregressive model with input variables in proc arima procedure

I am currently working on the time series analysis for series Y but I have to use other two variable A and B as an input variable in SAS proc arima procedure. But I am unable to interpret the cross ...
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14 views

Problem simulating AR and MA models using filters

I do not know how to use filter to simulate AR and MA models. To me it looks the same way for both MA and AR Estimate AR so then how do I know that the model is AR or MA ? For example, Problem1: For ...
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30 views

Whitening a regression with an AR process

I was reading a research paper: $Y_{t}\text{=}\beta_{0}+\beta_{1}X_{1t}+\beta_{2}X_{2t}$ (where $Y_{t}$ is stock returns and not the change in stock returns) ($X{}_{1t}$ is the return of a stock ...
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Is there a convenient form for this large covariance matrix?

Consider the following bivariate vector autoregression: $$X_t=\mu +X_{t-1}A+\varepsilon_t,\ \varepsilon_t \overset{iid}{\sim} MVN(0, V),\ X_t=(X_{1,t},X_{2,t})',$$ where the assumptions on the ...
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1answer
13 views

Reference requested for Moving Average model

I am not from econometrics background and hence not familiar with text books which may contain a large moving average and an auto regressive model. I have found AR model from Simon Haykin's Adaptive ...
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14 views

Multivariate model and large regression

I am not familiar with the concept of multivariate model and just learning about regression model. I am familiar with Autoregressive model and Moving Average. Multivariate regression model provided ...
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10 views

Do AR models with GARCH errors have a positive spectral density?

Can someone please help me to verify that AR models with GARCH errors have a positive spectral density and are bounded?
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33 views

Determining parameters in AR model for non-stationary time series

I am currently trying to fit an AR model to some financial data. The time series $Y_t$ in levels is clearly non-stationary; however it appears the first differences $dY_t$ are stationary (and this is ...
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2answers
120 views

Steps to perform time series analysis

I'm trying estimate an autoregressive model with an exogenous variable. It's about the impact of changes in oil prices on the economy. I'm planning on regressing gdp growth rate on its own lagged ...
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24 views

Geographic regression

I'm working on a project to estimate real estate and started with some classique techniques, such as linear regression etc. The obtained results are already going in the good direction, but to get ...
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27 views

R: Package 'sphet' on spatial autoregressive models

I'm using the function gstslshet from the package sphet in R, but I'm wondering what the outcome is. Is it a model which can be ...
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1answer
48 views

Spatially auto-regressive two-stage model

I'm working on a project in which I use a 'Generalized Spatial Two-Stage Least Squares' model, mostly known as $y= X \beta + \lambda W y + u$ and $u = \rho M u + \epsilon_n$ where $y$ and $u$ are ...
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53 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
57 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 ...
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3answers
129 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 ...
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14 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
47 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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15 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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33 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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37 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?
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42 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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84 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 ...
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67 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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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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58 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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89 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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53 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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158 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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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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44 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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37 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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130 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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80 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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38 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
108 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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61 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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38 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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2answers
197 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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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 ...
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1answer
119 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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173 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 ...
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1answer
208 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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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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177 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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98 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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93 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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46 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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78 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} ...