# ARMA - var(delta wt = var(wt E(delta wt = delta B*Xt = Xt-1...

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With a moving average model the dependency in the MA model has a cutoff effect, MA(1) cuts off after 1 lag any theta will give you a stationary process autoregressive model it exponentially decays...the acf gradually goes to 0 and then stays there higher constant, decays slower if phi is greater than 1, the process is no longer stationary For MA: ACVF E(Xt) = E(Wt + theta Wt-1) = 0 Var(Xt) = Var(Wt + theta Wt-1) = Var(Wt) + Var(theta Wt-1) = (1+theta^2)*sigma^2 cov(xt, t+h) = 0 when abs(h) > 1 cov(wt +theta Wt-1 + Wt+h + theta Wt+h-1) = 0 AR: tail off MA: cut off ----- Second ARMA notes: When you have trend, things are no longer stationary...it depends on time Delta Xt = Xt - Xt-1 = Delta + wt

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Unformatted text preview: var(delta + wt) = var(wt) E(delta+wt) = delta B*Xt = Xt-1 B^2 * Xt = B(B Xt) = Xt-2 delta = 1 - B delta(delta(xt)) = (1-B)[(1-B)(xt)) = (1-2B+B^2)*Xt = Xt - 2Xt-1 + Xt-2 In R, use diff for differencing delta(constant B0) = 0------------gamma(h) = cov(xt, t+h) gamma(0) = variance(xt) p(h) = autocorrelation btwn xt and xt+h p(h) = cov(xt, xt+h) / stdev(xt)*stdev(xt+h) but the 2 stdev are the same so it’s the variance on the bottom ph() = gamma(h) / gamma(0)[email protected] differencing might introduce additional dependencies that makes the model useless AR and MA models capture some type of dependency---------------...
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