# Univariate Transfer Function Identification

#### V.II.1 Univariate Transfer Function Identification

A relationship between two variables under investigation may be linear or nonlinear. Sometimes it is possible to transform a nonlinear relationship into a linear one by use of power transformations.

We may identify nonlinearities by using half-slopes. First we partition the observations of a scatter plot into three equal parts (each part consisting of approximately the same amount of observations). Then we compute the medians of each partition yielding a set of three pairs of coordinates:

The half-slopes are computed by

(V.II.1-1)

whereas the half-slope ratio is

(V.II.1-2)

If this ratio is equal to 1, no transformation is required. On the other hand, if

(V.II.1-3)

Remark that if the ratio is negative, no power transformation can be used to linearize the relationship.

The dependence between the input variable (c.q. the exogenous variable), and the output variable (c.q. the endogenous variable) must not be restricted to linear models. Let us therefore have a closer look at the so-called impulse response function (IRF). A simple example of an IRF is

(V.II.1-4)

In order to have the transfer function allow only finite incremental changes in the output series (if the input series undergoes a finite change), the model must be checked for stability.

(V.II.1-5)

is called the steady gain of the model. Note that this gain converges in stable models.

On using the more general (and parsimonious) formulation of the pure (c.q. without noise) transfer function model

(V.II.1-6)

(V.II.1-7)

This result can be used to obtain

(V.II.1-8)

Obviously a transfer function model depends on the parameters r, s, and b (c.q. TF(r,s,b)). Hence, on using (V.II.1-8), for any TF(r,s,b) model the theoretical impulse-response function can be obtained. A likewise procedure can be used to obtain the step impulse functions for any TF(r,s,b) model.

If one is concerned with identifying a transfer function relationship between the input and output variable, and if the input variable is not a white noise series, a prewhitening step should precede the computation of a cross correlation function (CCF). In order to see this, consider the identification of the impulse response function without prewhitening (c.q. when the input series is not white noise).

If W(t), X(t), and N(t) are stationary series then the impulse response function

(V.II.1-9)

(V.II.1-10)

On assuming that the input variable, and N(t) are uncorrelated; and on taking expectations, the covariance function is obtained

(V.II.1-11)

from which it can be seen that a relationship exists between the impulse-response function and the covariance function. If however the input X(t) is not white noise, the cross covariance function (and hence also the CCF) is distorted due to autocorrelation, since (V.II.1-10) becomes

(V.II.1-12)

where it is implicitly assumed that X(t) is uncorrelated with N(t); W(t), and X(t) are stationary (though autocorrelated) time series with zero mean; and k = 0, 1, 2, ...

In order to compute the prewhitened cross correlation function (PCCF), we first filter the stationary input variable X(t) through its univariate stochastic (ARIMA) model

(V.II.1-13)

such that the white noise series a(t) is obtained. Second, the stationary output series Y(t) is transformed through the same filter

(V.II.1-14)

from which it can be deduced that

(V.II.1-15)

(V.II.1-16)

where in practice the estimated standard errors and correlation coefficients are used.

Thus, the PCCF can be used for transfer function identification purposes:

- the residual ACF and PACF (of the transfer function model) can be used for checking purposes, and provides a mean to adapt inadequately identified models.

This is a graphical summary of various types of Transfer Function-Noise models that can be identified by the use of the PCCF:

This is a graphical summary of various types of Impulse Response Functions that can be investigated:

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