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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conditional/unconditional expectation and variance for an AR(1) process

We have an AR(1) process, $X_t=\alpha X_{t-1}+\varepsilon_t$ with $\varepsilon\sim(0,\sigma^2)$, $X_0=0$ and $|\phi|<1$. We have the conditional expected a value with respect to $X_{t-1}$: ...
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The Impact Of Education On Economic in Mozambique [closed]

I am trying to study The Impact Of Education On Economic Growth : The Case Of Mozambique using MRW’s (1992) augmented neoclassical model. The variables are GDP, Human Capital (HC) and Physical ...
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15 views

Multivariate linear regression with prior information on variances

I have a slight variation to a classic problem, which might have a simple answer - but if it does, I cannot find it. My problem is a multiple linear regression, of the type that is common in ...
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9 views

How to include seasonality in the data ARMAX model that has multiple periodicities?

I am doing regression with ARIMA errors. The residuals are as shown in the figure below. Clearly, the scatter plot shows that this time series has some sort of periodicity or seasonality, but its very ...
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42 views

Relation between AR(1) and Vasicek model

The discrete time version of a Vasicek model is equivalent to an AR(1) model with opportunely chosen parameters, as showed in this paper: http://www.damianobrigo.it/toolboxweb.pdf. Following this ...
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AR(1) - Zero or One coefficient

Suppose I have the following model for GDP growth: $g_{t+1} = \phi_0 + \phi_1 g_t +\epsilon_{t+1 }$ What are the implications of having $\phi_1=0$? What about $\phi_1=1$? In which case is GDP ...
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34 views

Random effects: Can location be nested within time period?

This is a bit philosophical: I have mutiple responses at multiple sites in multiple years. Can I legitimately nest the random effect Site within Year, or must Site and Year be crossed effects? Given ...
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24 views

Causal form coefficients from AR(2) model

I'm following Shumway & Stoffer's "Time SEries Analysis and Its Applications With R Examples: EZ -- Third Edition". At the top of page 86, an AR(2) forecasting model $ x^n_{n+m} = 6.80 + 1.35 ...
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A closed form formula for the normalizing constant in standard normal auto-regressive series?

Let $Z_t = c_1Z_{t-1} + c_2Z_{t-2} + ... + c_nZ_{t-n} + c\epsilon_t$ where $Z_t, \epsilon_t \sim \mathtt{N}(0,1)$ are iid variables and $Z_s \sim \mathtt{N}(0,1)$ for all $s$. Given the values of ...
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AR(2) model is causal

AR(2) model is : $$X_t=\phi_1X_{t-1}+\phi_2X_{t-2}+W_t$$ Where $W_t\sim N(o,\sigma^2)$ I want to prove AR(2) model is causal . So , I tried as : $$X_t-\phi_1X_{t-1}-\phi_2X_{t-2}=W_t$$ ...
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Getting VAR parameters from research paper

Many econometrics papers provide the parameters used in their VAR model. If I notate my VAR model as $$z_{t+1} = c + B z_{t} + \Sigma \epsilon_{t+1}$$ where $\epsilon \sim N(0, I)$, then I need to ...
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29 views

Autoregression for 20 numbers

If I had 20 numbers over a certain time period say 20 days and would like to figure out the next 5 numbers in the series, I am assuming I use an AR(20) Autoregression. However, I don't know how to do ...
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22 views

Selecting optimal sample rate for time series prediction

Is there a procedure to choose the optimal sample rate (every second, minute, hour, ...) for time series prediction (say fitting an ARMA model)? I guess it depends on how many steps I want to predict ...
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33 views

On the Beta-t-EGARCH and the score

I am going to define the Beta-t-EGARCH model utilizing the more familiar GARCH model as does Harvey in Dynamic models for volatility and heavy tails (2008), I hope this will serve the purpose to ...
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10 views

estimate the PSD using Yule walker

Please can anyone simply explain about the yule walker AR method ?. because i have a discrete noise signal data to estimate the Power Spectrum Density using yule walker method in C++. So I'm trying ...
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1answer
49 views

How to identify relationship between response time series(Yt) & input time series(Xt) only in terms of Yt-1 & Xt?

I have a response time series(Y) & Input time series Xt & Zt. My only objective is to identify functional form Yt=f(Yt-1,Xt,Zt) where f(Yt-1,Xt,Zt) contains only lags of Yt , Xt & Zt as ...
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53 views

AR(1) working correlation matrix with GEE

I'm attempting to fit a GEE model and I have a question about using the AR(1) working correlation matrix. I've read some conflicting information about this correlation matrix. In some books and ...
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17 views

Effects of covariance structures on mixed effects models

What are generally the effects of using a covariance structure on a mixed effect model ? More specifically, in a mixed model, what should be the expected effect of using an AR(1) covariance ...
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1answer
41 views

Diferencing an autoregressive model

By differencing an AR(1) Model could we get an MA(1)? I mean $Y_t = U + Y_{t-1} + e_t$ $\Delta Y_t = Y_t - Y_{t-1} = U + E_t = U + E_t + 0 * E_{t-1}$ >> Meaning MA(1) ?
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Mean of Empirical Auto-Covariance

Chris Chatfield's "The Analysis of Time Series: An Introduction" 6th ed, gives the mean of the empirical auto-covariance as: E(r_k)=-1/N where E is the expectation r_k is the empirical ...
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41 views

Using ARIMA to Create a Model in R

I'm trying to get understand why the values for my model are different when using two different functions. The first one is from Example 9.2 (International Visitors to Australia), using the ...
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1answer
72 views

Breusch-Pagan Test for ARIMA Model in R

I am testing my model using the Breusch-Pagan Test, but have not been able to find anything online regarding how to calculate it for an ARIMA Model. My AR1 Model is: ...
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Analysis of Multiple Time Series Data with Exogenous Shocks

Real Life ProblemThis one is a tough one and some crowd sourcing seems like a good way to get some feedback. I am trying to determine the effect of Non-Farm Payroll surprises on a subsector of the ...
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50 views

AR(2) model interpretation

If I have a negative sign in my AR(2) model equation, (for example, $y = 100 - 50x_{t-1} + 25x_{t-2}$) and if my AR(1) and AR(2) has same r-square value, is it okay to interpret it as model overfit? ...
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How do I interpret weak exogeneity in an ADL model?

First year econ graduate student here; looking at an ADL (Autoregressive Distributed Lag) model for the first time. Consider $Y_t = \omega Z_t + \alpha Y_{t-1} + \beta Z_{t-1} + \mu + \epsilon_t$, ...
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48 views

How do I interpret lagsarlm output from R's spdep?

I've run lagsarlm on my dataset, using a mixed model and using a row-standardized adjacency matrix. I have results that I think are good, but would am not sure how ...
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Longitudinal data analysis where meaning and metric of response variable varies over time

Determining what factors predict change over time is a topic of investigation in many fields and there are a variety of readily implemented methods for analysing repeated measures in the same metric. ...
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19 views

How to fit an logistical autoregression in R?

I modeled the relationship of $X$ and $Y$ by the logistic function. The residual plot displays autocorrelation which I'd like to rid. I want to try adding trend component to $X$, thus the model ...
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How can I compute cross-correlation and auto-correlation in R using pooled data?

I'm trying to perform a lagged linear regression on time series data sourced from ~10,000 hospital patients, for the purpose of estimating causal relationships between administration of a drug and a ...
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ARX model selection

I have an autoregressive model with exogenous variables: $S_{t} = \sum_{i=1}^{p} a_i S_{t-i} + \sum_q \sum_{i=1}^{r} b^q_i X^{q}_{t-i}$ where $S_t$ is the signal I want to predict and $X^q_t$ the ...
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171 views

How to interpret the Durbin-Watson test output in R [closed]

Just for "train" with linear regression in R I'm doing a Durbin-Watson test over the residuals of a regression (over stock ...
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1answer
118 views

Difference between different autoregressive models

I am trying to understand the difference between these three different specifications of an autoregressive model for variable var in Stata: ...
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Modeling Non-Stationary Time Series Data

Data set: response and predictors are all non-stationary, time series variables After performing Box-Cox transformations and testing a variety of power transformations on each variable, the ...
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3answers
178 views

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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366 views

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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Is a random walk + white noise modal an ARIMA(0,1,1)? [closed]

Let $Y_t=Y_{t-1}+\epsilon_t$ be a random walk and $Y_0=0$ Why is it true that the process $X_t=Y_t+\eta_t$, where $\eta_t$ is a white noise, so that $cov(\epsilon_t,\eta_s)=0$ for all $t,s$?
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Simulating a dynamical system

Basically I need to replicate Hartley's 'A User's Guide to Solving Real Business Cycle Models' . Specifically (to make question relevant to stats.stackexchange), I want to simulate the dynamical ...
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109 views

autoregressive coefficient and white noise

I have one variable of Monthly rainfall data. I have developed a forecasting model using Back-propagation Neural Network Model (using Matlab), with 3 input units and one output unit, the three inputs ...
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What is the difference between autoregressive, auto-regression and multiple linear regression?

I have developed three neural network models for monthly rainfall forecasting the first has three input, which is the values of the previous three months. the second has six inputs and the third has ...
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Regressor vs. AR/MA terms

I'm running a regression to model Hong Kong's office rentals series: Source:Private Office - Average Rents by Grade by District (from 1982) Source:All HK Property Market Statistics ...
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2answers
281 views

Autocorrelation and Partial Correlation plots in ARMA models

Consider the following input and its Autocorrelation and Partial Autocorrelation plots (source). What are the shaded blue areas in these plots? I often see them when studying ARMA models. What do ...
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69 views

Autoregressive model in R

Citation: As the data represent repeated measurements from individual plots, within-plot correlation may result in inefficient estimates and underestimation of standard error. Therefore the ...
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1answer
45 views

Stationarity of AR(1) process whose autoregressive parameter could change over time

Imagine an AR(1) has an autoregressive parameter which could change in time. $y_t-\mu=\phi_t (y_{t-1}-\mu)+\varepsilon_t\,$, where $\phi_t$ is not always constant but still lies inside the usual ...
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Autoregressive Model

I am currently attempting to build a regression model explaining Current Inflation as measure by monthly CPI. I am considering the following model; CPI = B0 + B1(LAG_CPI) + B2(Lag_Oil_Price) + ...
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146 views

Autocorrelation *across* random effects in nlme:lme?

I have response data measured at the site and month level. I wish to fit a year trend to the data and month to remove the seasonal trend. However, to avoid pseudoreplication, I have fitted year also ...
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Time series causation and probability

I have a series for an individual that looks like this: There are observations of the individual at random times. At each time they may experience an event and the outcome fo that event is binary ...
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Is is possible to determine conditional conjugacy in this case?

I'm working on a problem where I have to extract sufficient statistics for parameter estimation in a state-space model. Usually these come from the quantities used for conjugate updates. I'm OK with a ...
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122 views

Nonlinear Autoregressive model parameter estimation from time series

I'm working on a nonlinear multivariate autoregressive model of order 1 (markovian). It is a discrete-time dynamical system which models exchange of mass between compartments in a compartmental model ...
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1answer
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Residual based bootstrap autoregressive series in MATLAB

I have defined the model as follows. Let $$y_1 = 0$$ and $$ y_i = \alpha + \beta y_{i-1} + \epsilon_i $$ for $i_2\ldots i_T$, where $\alpha$ and $\beta$ are the estimated coefficients and ...
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55 views

What is the auto-covarriance of a stationary AR1 process?

Say a stationary AR(1) process is given by: $$ X_t = c + \phi X_{t-1} + \epsilon_t $$ where $ \epsilon_t $ is a white noise process with zero mean and constant variance $ \sigma^2 $. Wikipedia ...