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For this I think you could just specify a variable that contains the time ordering, rather than year and nest that within site. So you want a variable time (say) that is the integers 1, 2, 3, ..., T in site 1, and again in site 2 and so on. This would be pretty straight forward to code in R if you have no missing time points in any of the sites. It needs to ...

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It is the ARMA(1,1) with parameters $c$, $d$ and $\sigma_\xi^2$ that is given. The question is then what are the parameters of the AR(1)$+$white noise representation (if such a representation exist). This is found by solving (5.3) w.r.t. $a$, $\sigma_\eta^2$ and $\sigma_\varepsilon^2$ which leads to \begin{align} a&=c \\ \sigma_\varepsilon^2&=\...

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AR(n) is not a stable class of models under aggregation. In your example, this assumption "Assuming 𝑦𝑡 follows an AR(1) as well" is wrong and yearly data is not AR(1). You can get an intuition for why it is wrong by writing down an example for aggregating monthly AR(1) into bimonthly AR(1). $$x_t = ax_{t-1} + e_t$$ $$y_t = x_t + x_{t-1}$$ $$y_{t-... 1 You can use ARMA.autocov in the ts.extend package The ts.extend package contains a number of functions that compute theoretical aspects of stationary ARMA models, including the auto-correlation and auto-covariance functions, the auto-covariance matrix, and the standard probability functions for the stationary Gaussian ARMA model (i.e., density, ... 0 I think you have two options with option one using a SAR(IMA) model. If you have a reoccurring effect every weekday (dummy), then you can already improve the model. Option 2: As you stated - I would use an AR and then fit a GARCH on the errors. Or even use the SAR(IMA) and then a GARCH. The combination would result in 3 candidate models. SAR, AR->Garch, ... 1 Use the rGARMA function in the ts.extend package If you don't mind using R instead of MATLAB, you can generate random vectors from any stationary Gaussian ARMA model (including AR models) using the ts.extend package. This package generates random vectors directly form the multivariate normal distribution using the computed autocorrelation matrix for the ... 7 The question suggests some basic confusion between the equation and the solution The Equation Let {\varphi} > 1. Consider the following (infinite) system of equations---one equation for each t\in \mathbb{Z}:$$ X_{t}=\varphi X_{t-1}+e_{t}, \mbox{ where } e_t \sim WN(0,\sigma), \;\; t \in \mathbb{Z}. \quad (*) $$Definition Given e_t \sim WN(0,\sigma)... 1 ... the AR(1) process (with e_t white noise):$$X_{t}=\varphi X_{t-1}+e_{t} \qquad , e_t \sim WN(0,\sigma) is a stationary process if $\varphi>1$ because ... It seems me not possible as showed there: https://en.wikipedia.org/wiki/Autoregressive_model#Example:_An_AR(1)_process for wide sense stationarity $-1 < \varphi < 1$ must hold. Moreover, ...

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