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I have two time series that I am investigating, acc and amb, the time frequency is daily data. They are both non stationary, as evidenced by the follows:

adf.test(df$acc)

    Augmented Dickey-Fuller Test

data:  df$acc
Dickey-Fuller = -2.7741, Lag order = 5, p-value = 0.2519
alternative hypothesis: stationary

> adf.test(df$amb)

    Augmented Dickey-Fuller Test

data:  df$amb
Dickey-Fuller = -1.9339, Lag order = 5, p-value = 0.6038
alternative hypothesis: stationary

I am looking to test for cointegration between the two time series but the problem I am running into is that the cointegrating vector seems to change in time.

1) First 200 points

Johansen-Procedure

Test type: maximal eigenvalue statistic (lambda max) , with linear trend

Eigenvalues (lambda):  
[1] 0.0501585398 0.0003129906

Values of teststatistic and critical values of test:

          test 10pct  5pct  1pct
r <= 1 |  0.06  6.50  8.18 11.65
r = 0  | 10.19 12.91 14.90 19.19

Eigenvectors, normalised to first column: (These are the cointegration relations)

        acc.l2    amb.l2
acc.l2  1.0000000  1.000000
amb.l2 -0.9610573 -2.237141

Weights W: (This is the loading matrix)

       acc.l2       amb.l2
acc.d -0.03332428 -0.002576070
amb.d  0.03986111 -0.001591227

2) First 1000 points

Johansen-Procedure

Test type: maximal eigenvalue statistic (lambda max) , with linear trend

Eigenvalues (lambda):
[1] 0.019211132 0.001959403

Values of teststatistic and critical values of test:

          test 10pct  5pct  1pct
r <= 1 |  1.96  6.50  8.18 11.65
r = 0  | 19.36 12.91 14.90 19.19

Eigenvectors, normalised to first column: (These are the cointegration relations)

        acc.l2   amb.l2
acc.l2  1.0000000  1.00000
amb.l2 -0.8611314 15.76683

Weights W: (This is the loading matrix)

        acc.l2        amb.l2
acc.d -0.008993595 -0.0002419353
amb.d  0.027935684 -0.0002067523

3) Whole History

Johansen-Procedure

Test type: maximal eigenvalue statistic (lambda max) , with linear trend

Eigenvalues (lambda):
[1] 0.0144066813 0.0008146258

Values of teststatistic and critical values of test:

          test 10pct  5pct  1pct
r <= 1 |  1.16  6.50  8.18 11.65
r = 0  | 20.64 12.91 14.90 19.19

Eigenvectors, normalised to first column: (These are the cointegration relations)

        acc.l2    amb.l2
acc.l2  1.0000000   1.00000
amb.l2 -0.8051537 -25.42806

Weights W: (This is the loading matrix)

       acc.l2       amb.l2
acc.d -0.01003068 7.009487e-05
amb.d  0.02128464 6.980209e-05

You can see the marginal change the coefficient values, from -0.96 to -0.86 to -0.80.

My question is how to interpret this, what is the optimal look back period, what is the true relationship I should use for future prediction?

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Two cointegrated time series share a common trend on the long run but there can exist large discrepancies between them on the short term.

The optimal look back period would be the very beginning of the data generating process at work in your phenomena. Unfortunately, you are limited to a finite data sample to estimate this potential cointegration relationship and nothing can guarantee that those data points were not sampled during a period with high or small discrepancies.

So I would definitely go for the largest data set available as it is the less likely to be mislead by a "one-time" perturbation.

NB: all of this is true provided that the data generating process remain the same during the whole time of observation. Now, you might want to consider applying a Chow test on your data in order to check for any structural break in your series.

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