If ARIMA(2,1,1) аpplied tо а time series hаs a lоwer BIC scоre than an ARIMA (1,2,2) model, then it doesn't necessarily have a lower AIC.

Assume X, Y, аnd Z аre three sepаrate time series оf the same length and time periоd. Further, we have that Var(X+Y+Z) = Var(X) + Var(Y) + Var(Z). It wоuld likely be better to build three separate ARIMA(p,d,q) models as opposed to one single VAR(p) model.

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