WHAT AN ARIMA COMPONENT IS

• Consider a regression : Y(t) = v0*X(t) + e(t) (1) where e is an i.i.d. ( gaussian ) error distribution
  where an auto-projective model for X exists such that
  X(t) = w0*X(t-12) + a(t)
  where a is an i.i.d. ( gaussian ) error distribution
  or X(t) = ( [1 - w0*B**12 ] **-1 ) a(t) (2)
  Substituting (2) into (1) we get
  Y(t) = v0 ( [1 - w0*B**12] **-1 ) a(t) + e(t)
  Thus since both a and e are normal random variables we can get
  Y(t) = [é(B)/í(B)] A(t)
  {é(b)/í(b)} = ARMA model for unobserved series A(t)
  or restated as a Rational Expectations Model
  Y(t) = VO + V1*Y(t-1) + V2*Y(t-2) + ...
  
  Thus an ARIMA model is simply a regression model in sheep's clothing.
  The effect of X has been captured in the history of Y, thus while the ARIMA is explicit in the history of Y it is implicit in the history of X.
  
  What this means in practice is that for whatever reason , if you ignore or simply can't capture an X variable you can incorporate its effect throught the memory of Y. The important thing to remember is that this structure may reflect either omitted stochastic (probabalistic) or omitted deterministic series. If the omitted variable is deterministic (intervention) and arose randomly then intervention detection (pulses) will deal with that effect while if the omitted deterministic variable arose every S periods this will be dealt with using a seasonal pulse intervention.
  
  Examining the residuals in order to assess whether or not they can be predicted from the history of residuals is exactly equivalent to extracting information from the history of the Y series.
  
  

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