QUESTION:

I understand regression, but I don't understand what a Transfer Function is. Can you explain the difference?

ANSWER:

A regression model, like any other model, is only as good as its form. What I mean by form is; a. Correct lag/lead specification for the user specified x's b. Correct specification for the error process. Economists refer to model specification bias as a major problem, which unfortunately can never be remedied by estimation. The fix for MSB is a redefined model. Let us first examine (a). The exact form of the relationship between the series must be specified either initially or via diagnostic checking ultimately. Consider the case where Y(t) = W X(t-3) + N(t). If you incorrectly specify Y(t) = W X(t) + N(t) then you conclude that there is no relationship. The problem is that the relationship is not-contemporary but a lagged one of order 3. Temporal relationships like leads are prevalent in the consumer products industry where sales arise in advance of the holiday and often are reduced the week of the holiday. The correct lag structure Y(t) = W(B) X(t) + N(t) might require a polynomial lag, not necessarily describable by a smooth curve. Sometimes the relationship is optimally modeled by the ratio of two poynomials W(B) and D(B).

                       W(B)                 T(B)

           Y(t)   =  --------  X(t)  +  ----------  A(t)

                      D(B)                  P(B)

The problem or opportunity of incorporating lead, contemporaneous and lag structures does not exist in cross-sectional data. Thus the teaching of regression is much simpler for those problems where time is non-existent in the problem specification. The issue with point b is that if you don't have the correct form of the noise process then the estimates that you get for W(B) and D(B), albeit the correct form are effected by the incorrect model specification. Some researchers assume that this can be done independently but not so as the sample space between W(B)/D(B) and T(B)/P(B) is not orthogonal or independent. Identification of all 4 components requires the approach of Transfer Function Identification, a subject that will be treated elsewhere. There is another assumption of the form of the error process that requires special mention. The concept of Robust Regression deals with the identification and the handling of anomalies. The analog of this in time series is called Intervention Detection or Outlier Detection. In time series these can take on four different but relatable forms;
  1. Pulse
  2. Step ( a consecutive sequence of pulses all with similar magnitude and direction)
  3. Seasonal Pulse ( a sequence of pulses all with similar magnitude and direction spaced s periods apart )
  4. Trend ( a sequence of values that fall approximately on a linear equation in time beginning at point i and ending at point j )