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This question has been asked before with very good (but incomplete) answers. This and this are the two best answers that I found. But following is my doubt:

Top answers from both (by IrishStat) the links suggest to fit a common model to both the time series and then globally. I understand that if the same series was split in two parts, chow test can be used comparing the estimated models of the two splits with the estimated global model.

However, if the two time series are different (say urban and rural inflation), and the comparison is to be done for the same time window, IrishStat suggests "..( putting the second series behind the first ) . Make sure that your software recognizes the beginning of the scond series.."

Question: how to do this? Do we need to add a dummy (LS outlier type?). If so, would it not disturb the estimate of the global model. In fact there should be no global model in the strict sense. Joining two time series assumes a dependence structure between last point of first series and first point of second series, where there may not be any.

Dayne
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    *How* do you want to compare these series, for what purpose, and what assumptions can you make concerning any common properties they might have? – whuber Jun 11 '19 at 10:53
  • The question is pretty straight forward. You have two series. You want to check whether they are statistically same or not, i.e. same data generation process. This is akin to mean comparison for two IID samples using t-test. The same question was very well articulated [here](https://stats.stackexchange.com/questions/19103/how-to-statistically-compare-two-time-series). – Dayne Jun 11 '19 at 11:09

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I used AUTOBOX and took 21 values representing three distinct subjects ( 7 values each) enter image description here and specified an ar(1) model here

enter image description here

The idea is that the last point of first series and first point of second series should be restricted from affecting the model coefficients.

I directed that there were 3 groups here enter image description here

The results shown here detail how the 1st , 8th and 15th residual were restricted to 0.0

enter image description here

with final model here reflecting a pooled estimate enter image description here and here enter image description here

to test the hypothesis of a common parameter across the 3 groups , simply estimate each group separately and sum the 3 error sum of squares and perform an F test. The CHOW test can be used comparing the estimated models of the two splits with the estimated global model.

IrishStat
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  • Thanks for answering. What's the rationale behind restricting the residuals to be zero at break points? And wouldn't doing so immediately calculate the coefficients? For example, $x_t=\alpha x_{t-1}+e_t$. If we know that for a given value of $x_t$, $e_t$ is zero, then $\alpha=\frac{x_t}{x_{t-1}}$? – Dayne Jun 12 '19 at 04:07
  • the model used was an ar(1) . there is no forecast possible for period 1 . – IrishStat Jun 12 '19 at 05:02
  • I meant, using this at, say, 8th residual. Since the three series are put together, the last point of series 1 is $x_7$ and the first point of series 2 is $x_8$. If by construction, we are forcing $e_8=0$, then shouldn't $\alpha=x_8/x_7$ (assuming no constant, in this example)? – Dayne Jun 12 '19 at 05:34
  • a is the "best" coefficient for the model y(i)=a*y(i-1)+e(i) for i=1,21 and where e(1),e(8) and e(15) are constrained to be 0.0 and y(0)=0.0 – IrishStat Jun 12 '19 at 06:38
  • I have already upvoted but want to understand how it is correct. So you said that $\alpha$ is the best estimate with such restrictions on residuals. Two doubts: (1) How to get this best $\alpha$? (2) If we just fit a common ARIMA specification model separately on both the series. Then also we can compare the estimated coefficients, isn't it? What is the need of having a global model at all and do this joining of time series? – Dayne Jun 13 '19 at 10:48
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    perhap we can hash this out on SKYPE as we are having a communications problem with one-sided material. Please SKYPE me or call my land line so I can help. You seem to want a parameter by parameter test , rather than a comprehensive test , a portmanteau test , which is easily conducted as there could be many parameters in the model not just one.. – IrishStat Jun 13 '19 at 11:17