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A VAR(p) model has multiple dependent variables: $$y_t=A_1 y_{t-1}+\dots+A_p y_{t-p}+\varepsilon_t$$ where $A_1,\dots,A_p$ are coefficient matrices. The current values of the dependent variables are $y_t$; this is a vector of length $k$: $(y_{1,t},\dots,y_{k,t})$. The current values depend on past values (vectors) $y_{t-1}, \dots, y_{t-p}$. In addition, ...

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It's perfectly normal when you have only 100 samples. Take it n=1000 and you'll see an exponentially dropping autocorrelation curve. Small samples have less tendency to obey theoretical results.

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A key difference which I failed to appreciate: the MA model predictions of $x_t$ include $\epsilon_{t-1}$ in its computation whereas the AR model only predicts based on $x_{t-1}$ without (explicit) regard as to whether the prediction at $t-1$ was under- or over-estimating $x_{t-1}$. Fleshing out what @user1587692 highlighted, the MA model averages across ...

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This is the simplest example that I could come up with to help visualize AR, MA and ARMA processes. Note that this is just a visual aid for an intro into the subject and by no means rigorous enough to account for all possible cases. Assume the following: We have two agents in a competition who are tasked with performing a certain kind of action (jump ...

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A finite AR model can be expressed as an MA model and vice-versa , If one has an ar(1) model with coefficient .333333333 then the models are (nearly) identical . Consider the case for an ar(1) with coefficient of .3 x(t)=ϵ(t)+.3*x(t−1) since x(t-1)=e(t-1)+.3*x(t-2) we can substitute for x(t-1) and get x(t)=ϵ(t)+.3* [ e(t-1) + .3*x(t-2) ] x(t)=ϵ(t)+.3*...

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You should consider the $\epsilon_t$ innovations rather than residuals. In the MA case, you average across the recent innovations, whereas in the AR case you average across the recent observations. Even if the models are stationary and have no deterministic terms, the innovations and the observations are different. For example, suppose the $\epsilon_t$ are ...

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A principled approach is to make probabilistic predictions. This gives a probability distribution over possible values at each timepoint, rather than a simple point prediction. The methods you described can provide this. The model should capture the ordinary behavior of the system, so it should ideally be fit to data containing no anomalies. Then, a point ...

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For stationarity we require $\text{var} (X_2) = \text{var} (X_1) \equiv B$. This leads to the "fixed point" equation $$B = \Lambda B \Lambda^{\top} + \Sigma.$$ Using the work from @JarleTufto here the "fixed point" equation leads to: \begin{align} \text{vec} (B) &= \text{vec} (\Lambda B \Lambda^{\top} + \Sigma) \\ &= \text{vec} (\Lambda B \Lambda^{\...

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For binary data correlation does not suit well, but there are many similarity indexes that you can use like the Jaccard index (https://en.wikipedia.org/wiki/Jaccard_index).

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