[R] Estimate an initial value for the level series from a return series

[Moohwan Kim]

Dear R family,

Today I have a tough question related to the model specification issue.
I obtain two log return series by first differencing the associated
price series each.
Then I produce two filtered log return series.
My objective is to check if there is cointegration relation between the two.
In the process, the level series should be used.
As you guess, I got far away from the original level series because I
twisted (i.e., filtered) the original log returns.
That is, with the filtered returns, I am not able to use the original
level series because its attributes or structures may change in the
middle of the data transformation.
My idea of overcoming the designated problem above is that I have to
recover the level series based on the filtered log returns.
In order to accomplish this task, I must estimate an initial value for
the level series. Actually I have to have two initial values because
there are two series.

My question is as follow:
Given the filtered log return series, is there a reasonable way to
estimate an initial value for the price series?

Best regards
moohwan kim