# Cointegration structure

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


       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


        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


       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?