QUESTION: 
 
 I am trying to estimate the following time series model:
 
 y(t) = c + beta*x(t) + rho*y(t-1) + e(t) - theta*e(t-1)
 
 I thought that the following SAS statements would do the trick:
 
 proc arima data=testdata;
 identify var=y crosscorr=x;
 estimate p=1 q=1 input=x method=ml;
 run;
 
 
Unfortunately, these statements seem to estimate the following 
different time
 series model:
 
 y(t) = c + beta*x(t) - RHO*BETA*X(t-1) + rho*y(t-1) + e(t) - theta*e(t-1)
 
Can I estimate the original time series model using proc arima? 
If not, is there another procedure that I should use?
 
 Thanks for your help,
 
 Max
 
 
 
ANSWER: 
 
MAYBE YOU SHOULDN'T
 
 
The model that you wish to estimate is
 
y(t) = c + beta*x(t) + rho*y(t-1) + e(t) - theta*e(t-1)
 
or
 
y(t t) - rho*y(t-1) = c + beta*x(t) + e(t)- theta*e(t-1)
 
or
 
[ 1 - rho*B ]y(t) = c + beta*x(t) + [1-theta*B]e(t)
 
where B is the BACKSHIFT VARIABLE ( also known as L to econometricians )
 
again
 
[ 1 - rho*B ]y(t) = c + beta*x(t) + [1-theta*B]e(t)
 
or dividing through by [1-rho*B ]
 
y(t)=[c]/[1-rho*B]+[beta*x(t)]/[1-rho*B]+{[1-theta*B]/[1-rho*B]}e(t)
 
or more generally
 
EQUATION (1)
 
y(t)={[c]/[1-a1*B]}+{[beta]/[1-a2*B]}*x(t)+{[1-theta*B]/[1-a3*B]}*e(t)
 
The problem with your equation is that it constrains a1=a2=a3 and
 Furthermore may be an incorrect specification ( Model Specification Bias ) and
consequentially may be totally inappropriate for the data at hand 
( y and x ).
Specifically constraining both X and e to have identical denominator
structures in order to satisfy some theoretical , rather than
 identified , model can have nasty side effects. I am not saying that 
one should not introduce prior models without seeing or listening to
 the data but care should be taken.
 
Estimation , even Non-Linear estimation or Maximum Likelihood , can go
 Astray if
 
either y or x have unusual values ( PULSES ,SEASONAL PULSES , LEVEL
 SHIFTS
and/or LOCAL TIME TRENDS )
 
2. the variance of the errors is not constant
 
3. the parameters themselves change over time .
 
or
 
4. an unsuitable model is in place ..as estimation does not help you if
 you have an inappropriate model.
 
AUTOBOX can estimate such models. I believe, but I could be wrong ,
 that SAS can estimate a general Transfer Function as above ( EQUATION
BUT it can't constrain parameters to be a certain value or to be
 equal as your model
requires. I suggest that might use the data to suggest an appropriate
Transfer Function rather than trying to shoe-horn your data into some
arbitrary textbook model. AUTOBOX has an automatic Transfer Function
heuristic that can be quite useful in building these kinds of models.
The user can if they wish specify their own model and proceed to do
 diagnostics.
 
 
 
SIDE BAR: A WOLF IN SHEEP'S CLOTHING
 
 
Sometimes ARIMA models are scoffed at or spoken about in a pejorative 
Way as not being "CAUSAL" thus not being useful. The truth is that an
OMITTED X will have caused previous value(s) of Y .... thus if X has
some sort of autoregressive process then previous Y's can be used to
proxy the omitted X.
 
y(t)={[c]/[1-a1*B]}+{[beta]/[1-a2*B]}*x(t)+{[1-theta*B]/[1-a3*B]}*e(t)
 
will lead to some ARIMA ,if you omit x(t) from the equation.
 
Consider if P(B)x(t)= T(B)a(t)
 
or x(t)= {T(B)/P(B)]*a(t)
 
substituting for x(t) we get
 
y(t)={[c]/[1-a1*B]}+{[beta]/[1-a2*B]}*{T(B)/P(B)]*a(t)+{[1-theta*B]/[1-a3*B]}*e(t)
 
which upon clearing fractions
 
{(1-a1*B)}{(1-a2*B)}{(1-a3*B)}*{P(B)/T(B)}*y(t)=
 
 constant + {beta}*a(t)*(1-a3*B) + {[1-theta*B]*(1-a2*B)*e(t)
 
and since both e(t) and a(t) are gaussian I.I.D. we can combine them and
get
 
 PP(B)y(t)= TT(B)u(t)
 
which explains why an ARIMA model really contains the effect of the
omitted X's and will be useful if x(t)= {T(B)/P(B)]*a(t) continues into
the future.
 
In September 1977 , in the American Economic Review an article by
Fernandez " An Empirical Inquiry on the Short-Run Dynamics of Output
 And Prices" .... a footnote on page 598 treats the subject of ARIMA
 And Transfer Functions rather well. I have never seen this
 discussed as well in any text.
 
 
 
Consider this more concrete example:
 
Suppose that y(t) follows the following time series process:
 
 y(t) - rho*y(t-1) = mu + beta*x(t) + e(t) - theta*e(t-1)
 
Suppose further that x(t) is an AR(1) process.
 
 x(t) - gamma*x(t-1) = v(t)
 
Equivalently, if abs(gamma) < 1, then x(t) has an MA(finite) representation
 
 x(t) = GAMMA(B)v(t),
 where GAMMA(B) is a finite order lag polynomial
 
Using this result, the original process for y(t) can be expressed as
 
 y(t) - rho*y(t-1) = mu + beta*GAMMA(B)v(t) + e(t) - theta*e(t-1)
 
Or, equivalently,
 
 y(t) - rho*y(t-1) = mu + LAMBDA(B)*z(t),
 where LAMBDA(B) is an finite order lag polynomial
 
The point is that y(t) can be expressed as some ARMA process without 
x(t) entering the equation at all. Furthermore the order of the
 "original" process will change after this transformation. In this
 case, it goes from an ARMA(1,1) with an input series x(t) to an
 ARMA(1,finite) without the input series x(t).