Transfer Function Clarification I'm seeking a little clarification on the specific application of transfer functions for time series. 
I've followed the Box-Jenkins approach for selecting potential exogenous predictors... using R's prewhiten function to "identify" the independent time series and then filter the Y and check for significant lags in the CCF plot. Let's assume that in doing so, a statistically significant lag was found for all predictors.  yay! 
However, when I plug all variables into the full transfer function model, I find that I have to make adjustments to the individual Transfer Function parameters of each IV in order to 1) lower the BIC and 2) obtain statistically significant p-values.  I'm mostly adjusting the delay parameter of the transfer functions - the Numerator and Denominator terms are mostly the same based on their ARIMA identity. 
Am I doing something wrong?  Is this the "subjectivity" Pankratz mentioned... 
as an example, let's say one IV has the form: (1,0,1), and significant CCF with y at lag 6 - found using the prewhitening operation. However, once in the model with other variables, I may have to change that delay from 6 to 4 or to 0 in order for the variable to be statistically significant or see a drop in the BIC. 
maybe this is not really allowed?  I'm not sure. 
However... using this "subjective" approach, I was able to fit a nearly perfect model with a MAPE (yes I know about the problems) of less than 1% and all predictors being statistically significant and an 3-point reduction in BIC.  believe me, I'm not that good at this.
Thank you, as always.
EDIT 1: 
data can be found here: Data
image of the final multivariate transfer function 

image of the model summary statistics 

model residuals

 A: Following the broad guidelines laid out in The theory behind fitting an ARIMAX model I introduced your 36 monthly values and 12 time series into AUTOBOX. After the AUTOBOX modelling process http://www.autobox.com/pdfs/A.pdf is followed I will briefly critique the differences between your over-parameterized SAS model and the  AUTOBOX result.
To begin with I scaled the output series (your DV) by 1000 to make the data more commensurate, probably unnecessary but as a safety measure. Following is the graph of the DV  and a curious V6 showing that early values were assumed to be constant rather than reverse forecasted .
Transfer Function Model Identification is not done by "identifying significant cross correlations between each IV and the DV" but rather by employing pre-whitening filters to compute pre-whitened cross-correlations for identification purposes. I believe that this important/critical feature was not available to you in the software you used BUT I am not a SAS expert. 
For completeness purposes I present the pre-whitened results here for V6 . Here are the two filters )one for V6 the other for Y  and the pre-whitening results . Notice that the differecing opertaor for V6 is a 1 reflecting the systematic effect of a set of constants in the history of v6.
AUTOBOX developed the following model  and here  and  . It used 6 predictors and 4 identified pulse anomalies (curious that the three periods 25,24 and 26 were on the list !) along with a constant (11 parameters in total based upon 33 estimable equations) to obtain 
The base 36 values (Y) were filtered to obtain the following residual plot  with an acf here  .
The Actual/Fit and Forecast is here  with Actual/Cleansed here 
CRITIQUE (GENTLE !) OF YOUR MODEL AND MODELLING APPROACH:
1) The data should be scaled to make it more commensurate to avoid possible numerical estimation issues.
2) Missing early value should be reverse forecasted rather than using a constant
3) Identification is done with pre-whitened data
4) You had 36 observations . You used a model for V6 (delay 9 + first differences)  which effectively dropped the number of estimable equations to 26 (36-9-1) which you then used to estimate 17 parameters yielding 9 degrees of freedom. This in my opinion is wildly over-fitting and is responsible for the r-sq of .994 ( as compared to AUTOBOX's .769 ) that you reported .... which you were questioning anyway.
I believe your r sq of .994  (26 estimable equations ...17 parameters) lead you to say  "I was able to fit a nearly perfect model with a MAPE (yes I know about the problems) of less than 1% " . This is an unfortunate result of excessive parameterization using a targeted approach as you sought out cross-correlations to be fit.
5)The p values of 0.0000 for your 17 estimated parameters should have been a red flag for you ( and probably were !)
6) Modelling lags of 7,8,9,10 etc periods with 36 observations for monthly data is a bit much and probably spurious unless supported by prior domain knowledge.
All of my comments here are made in good faith to help you and others better understand the transfer function modelling approach and the impact of available software on that.
