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I am regressing time series on time series.

I have tested for cointegration on the entire time sample (3 years) and the series are cointegrated.

I need to make a rolling window of regressions (to get a time series of betas) over this entire time sample. My rolling window will probably be 50 or 100 days.

What if my time series on this short period will not be cointegrated (even if I know it's cointegrated over the entire period)? Can I use a specific estimator (GMM or other) to estimate this regression? What is the bias on my estimated beta if I would regress using OLS?

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Co-integration is a theoretical, and unconditional relationship between stochastic processes, not a relationship between two finite series of specific realizations of these processes. Therefore when samples of different sizes give different co-integration tests results, the result from the largest sample size prevails.

Using OLS to estimate the co-integrating vector, has the following issues: to apply OLS we have to normalize the co-integrating vector and treat one variable as the "dependent variable" of the specification. If the theory behind the model does not specify which should this variable be, then the choice will be arbitrary - and the OLS estimates are affected by the normalization -and worse, the tests for the existence of co-integration are also affected (see for example Hamilton p. 589).

Second, the usual OLS may have large bias, as mentioned by Hayashi p. 651 . Hayashi calls this estimator "SOLS" (Static OLS) and advocates the "DOLS" estimator (Dynamic OLS), where in the RHS of the specification we include in addition a number of first-differences of the regressors (past and future), which result in the the regressors being strictly exogenous. Studies indicate that the DOLS estimator dominates in root mean squared error both the SOLS estimator, but also the Full-Information Maximum Likelihood estimator of Johansen (1988 and 1995).

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