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Univariate Transfer Function Estimation

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V.II.2 Univariate Transfer Function Estimation

Generally, the same remarks should be made w.r.t. the estimation procedure of TF(r,s,b) models as with ARIMA models (see section V.I.2).

An additional point however, should be made. The TF(r,s,b) model involves (at least one) exogenous variable. The exogenous (c.q. input) variable could be correlated with the noise component N(t). In fact, N(t) represents an infinite amount of (exogenous) influences which have played a role in determining the (stationary) output series, whereas the effect of one individual input series is measured explicitly. This is a causality test by means of a stochastic experiment: the input series is said to have a Granger-causal effect on the output series if it enables to significantly improve the one-step-ahead forecast performance. In case of correlation of the input and the noise however, this interpretation will be distorted.

Therefore it is imperative to estimate (if possible) TF(r,s,b) models that have been identified by means of a PCCF, since in the PCCF the input is uncorrelated with the noise. In any other case, a strong a-priori causality knowledge (c.q. assumption) must be made in order to estimate a TF(r,s,b) model.

Furthermore, it must be checked whether the relationship between the input and output series is a unidirectional one, because otherwise a simultaneity bias might occur. In case of simultaneity (which might be tested by TF(r,s,b) models in both directions), a multivariate time series analysis should be employed.
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