Hi all, I am a newbie and looking to seek some help.

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?

Many thanks.

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