Test for Equivalence of two time series
Hi: Testing for H0: $\phi_2 = \phi_1$ in an  AR(1) where $\phi_1$ can be done using the offset function in R.  Do you use R and, if so, are you familiar with the offset function ? If so, I can explain how to do this in more detail. 

Note that what I will do here is definitely an approximation ( econometricians might cringe. I'm not sure)  because $\phi_1$ is an estimate from the original model, rather than a true constant. But, for "government" work, this approximation might be okay. I don't know how precise you want to be.

Okay. So, suppose you're using R and the estimate of the AR(1) for the first series is $\phi_1$. Then, you want to use R to test whether the coefficient of the AR(1) for the second series is also $\phi_{1}$. You should look at the help for offset by doing ?offset but what it does in a nutshell is allow one to put constants in the call to lm where these constants are viewed as coefficients not to be estimated. They are assumed given.

So, suppose you wanted to simply estimate an AR(1). Since, we don't have to worry about stationarity conditions ( by definition, $\phi$ from an AR(1) has to be between -1.0 and 1.0 ), let's just use lm() rather than arima(). 

The first step is to lag the original series so if the first series is $y1$, let's assume that y1lag is the same series but lagged by 1 element. ( just do y1lag &lt;- c(NA, y1[-1] ) ).

So, the standard call to lm ( assuming zero intercept ) would be 

ARlm1 &lt;- lm(y1 ~ -1 + y1lag, na.ction = na.omit, data = whatever) and then you could do a summary(AR1lm) to get whatever you want including  the AR1 coefficient estimate, call it $\hat{\phi}_1$. 

Now, suppose you have the second series, along with its lag so $y2$ and $y2lag$.

So, to run the same model you would do

ARlm2 &lt;- lm(y2 ~ -1 + y2lag , na.action = na.omit, data = whatever). 

But we don't want to use the above call in this case. We want to test something. The way to do this is to  force the $\phi_1$ estimate on to the model and then test whether another coefficient is zero. The call which will allow  for such a test is:

AR1lm2test&lt;- lm(y2 ~ -1 + offset( $\hat{\phi}_1$ * y2lag) + y2lag, na.action = na.omit, data = whatever).

Do you see what I'm doing with offset ? I force $\phi_1$ to be the coefficient of the lagged term but then I add the ylag term on the end which is not offsetted. You can think of what above is doing as testing H0: $\phi_1 + \phi_2 = \phi_1$ versus $\phi_1 + \phi_2 \neq \phi_1$

In this manner, we end up  pseudo estimating an AR1, but the coefficient estimate outputted ( $\phi_2$, the coefficient of $y2lag$ ) represents how far away from the null we are. This coefficient can be tested to see whether it's value is zero. If the null hypothesis that it's zero cannot be rejected, then this is evidence that $\phi_2$ does equal $\phi_1$. If the test indicates that there is enough evidence to reject the null, then this is evidence that the $\phi_2$ does not equal $\phi_1$. The tstat from the output of summary can be used to do the test. 

I hope this made sense. Again,  even if it does makes sense, I still should emphasize that this procedure is approximate. Another similar ( maybe slightly better )  approach would be to concatenate the two series and then do some test for structural change but then you have to start using other packages along with the arima function and things get messier. Sticking with lm allows for the use of offset. I'm not sure if the base arima() function in R allows for offset. It might but the approach described will be more straightforward while sacrificing some rigor.

                                                        Mark 


ADDENDUM:

Coming back to this after some time away so it's an addendum.

I just realized that the call to AR1lm2test should be such that the sum of the coefficients $\hat{\phi_1}$ + $\hat{\phi_2}$ needs to be restricted to be between -1.0 and +1.0 and the lm() function doesn't allow for this type of constraint. There are two ways to deal with this problem.

1) Run AR1lm2test and, if the constraint holds, then everything is fine
and the output can be trusted. If the restriction doesn't hold, then the output has no useful meaning.

2) Check out if there is a version of arima() that allows for the use of offset. If there is such a function, then the restriction will be automatically enforced by that arima() function. There are many packages that have various arima() functions.  The CRAN Time Series task site is useful for finding packages that do things in time-series.