1. WHY WE FILTER TO IDENTIFY THE RELATIONSHIP
Title Slide

2. Flawed When Applied To Time Series Data
The notion of using cross-correlations to identify regression structure requires that the data be jointly bi-variate normal. Among other things this requires independent observations.
More generally the misleading inference comes about through applying the regression theory for stationary series to series that have auto-regressive structure.
Recognizing this , early researchers attempted to extract the within relationship (autoregressive structure) and then proceed to examine cross-correlative (among) relationships.

3. Flawed When Applied To Time Series Data
The notion of using cross-correlations to identify regression structure requires that the data be jointly bi-variate normal. Among other things this requires independent observations.
More generally the misleading inference comes about through applying the regression theory for stationary series to series that have auto-regressive structure.
Recognizing this , early researchers attempted to extract the within relationship (autoregressive structure) and then proceed to examine cross-correlative (among) relationships.

4. The Correlation Coefficient
This amounts to finding least squares estimate of , regressing y on x, with error having modulus.

5. THE PROBLEM IS THE X’S ARE CROSS-CORRELATED THUS WE ARE UNABLE TO USE STANDARD IDENTICATION TO DETERMINE WHAT LAGS IN X ARE NECESSARY

6. EARLY ATTEMPTS TO ADJUST FOR THE TIME SERIES COMPLICATION
Initial attempts to adjust for within relationships included “de-trending” and/or differencing. Both are usually presumptive and often lead to “Model Specification Bias”.

7. CONTRIBUTION BY BOX AND JENKINS
Box and Jenkins codified this process by recognizing that an ARIMA filter is the optimum transform to extract the “within structure” prior to identifying the “among structure”. They pointed out that both “de- trending” and “differencing” are particular cases of a filter, whose optimized form is an ARMAX model potentially containing both ARIMA and Dummy Variables such as Trends.

8. How to Identify the Relationship
The first step to this process is to develop an ARIMA model for each of the user-specified input time series in the equation.
Each series must then be made stationary by applying the appropriate differencing and transformation parameters from its ARIMA model.

9. How to Identify the Relationship
Each input series is pre-whitened by its own ARIMA model AR (autoregressive) and MA (moving average) factors. The output series is filtered by the input series AR and MA factors, t(B) and p(B) respectively.
The cross correlations between the pre- whitened input and output reveal the nature of this interrelationship.

10. Why We Filter to Identify
Y(t ) = W(B)X(t) + V(t ) (equation 1)
Where
Y(t ) is the vector of future values at time t of the output
W(B) is the matrix of cross correlations operating on the vector of inputs, X(t)
and V(t ) is the vector of interventions/errors
Let [ t(B)/p(B)] represent the successive application of first the MA filters, p(B), and then the AR filters, t(B) to the white noise processes x(t) that result in the vector of inputs X(t),
Now if X(t) =[t(B)/p(B)]x(t) then x(t)= [p(B)/t(B)] X(t)
Using [p(B)/t(B)] on equation (1) we get
[p(B)/t(B)] Y(t ) = W(B) [p(B)/t(B)] X(t) + [p(B)/t(B)] V(t)
  y(t ) = W(B) x(t) + [p(B)/t(B)] V(t) which allows a clear identification of W(b)

11. Why We Filter to Identify
Using [p(B)/t(B)] on equation (1) we get
[p(B)/t(B)] Y(t ) = W(B) [p(B)/t(B)] X(t) + [p(B)/t(B)] V(t)
Note that we filter the output Y(t ) by the AR and MA factors of the causals X(t) to generate the process y(t )
  y(t ) = W(B) x(t) + W(t ) (equation 2)
This enables identification of W(B) since x(t) is a white noise process and cross-correlations between y(t) and x(t) are meaningful as
compared to the useless cross-correlations between Y(t ) and X(t)
 Note that [p(B)/t(B)] plays no role in W(B)

12. Accounts for omitted stochastic cause variables
Response Function
Accounts for the timing and form of the impact of the known cause variable (s)
Error Component
Accounts for omitted stochastic cause variables

13. EARLY RESEARCHERS RECOGNIZED THAT IDENTIFYING RELATIONSHIP USING CROSS-CORRELATIONS WAS FLAWED BECAUSE OF THE INTERDEPENDENCE within THE X’S

14. IDENTIFICATION THEN PROCEEDED WITH THE SURROGATE VARIABLES
ONE APPROACH SUGGESTED WAS TO DETREND THE X AND TO DETREND THE Y AND CREATE x and y WHICH WOULD (hopefully) BE FREE OF THE DISEASE CALLED TIME DEPENDENCY (AUTOREGRESSIVE STRUCTURE ) . NO CONSIDERATION WAS GIVEN TO THE IDEA OF MULTIPLE TRENDS AND/OR LEVEL SHIFTS.
IDENTIFICATION THEN PROCEEDED WITH THE SURROGATE VARIABLES

15. IDENTIFICATION THEN PROCEEDED WITH THE SURROGATE VARIABLES
OTHER RESEARCHERS SUGGESTED AN ASSUMED ARIMA FILTER OF FIRST DIFFERENCES OF BOTH X AND Y TO CREATE x and y WHICH WOULD (hopefully) BE FREE OF THE DISEASE CALLED TIME DEPENDENCY (AUTOREGRESSIVE STRUCTURE). NO CONSIDERATION WAS GIVEN TO THE IDEA OF OTHER FILTERS .
IDENTIFICATION THEN PROCEEDED WITH THE SURROGATE VARIABLES

16. Accounts for omitted stochastic cause variables
Response Function
Accounts for the timing and form of the impact of the known cause variable (s)
Error Component
Accounts for omitted stochastic cause variables