## #StackBounty: #time-series #least-squares #econometrics #cointegration #unit-root Time series regression with stationary and integrated…

### Bounty: 100

I am estimating structural impulse response functions of a five-variable model (say $$x_1$$, … , $$x_5$$) using
Jorda’s local projection method and an external shock series.

The local projections are an alternative to the VAR(p).
The method basically boils down to estimating for every variable $$y in {x_1, dots x_5 }$$ and for every horizon $$h in {0, dots, H}$$ the following linear regression.

$$y_{t+h} = alpha_0^h + sum_{i = 1}^palpha_1^h x_{1,t-i} + sum_{i = 1}^palpha_2^h x_{2,t-i} + sum_{i = 1}^palpha_3^h x_{3,t-i} + sum_{i = 1}^palpha_4^h x_{4,t-i} + sum_{i = 1}^palpha_5^h x_{5,t-i} + beta^h shock_t+ u_t.$$

And the impulse response function is $${beta^h}_{h=0}^H$$.

Note: I am only interested in the consistent estimation of $$beta^h$$. I don’t care about the other parameters.

The problem is that on a couple of the data sets I have one or more $$I(1)$$ variables.

I think that I have figured out what to do in the case where there is only one $$I(1)$$ variable:

When this variable -say $$x_1$$ is on the left hand side of the equation –
$$x_{1, t+h} = alpha_0^h + sum_{i = 1}^palpha_1^h x_{1,t-i} + sum_{i = 1}^palpha_2^h x_{2,t-i} + sum_{i = 1}^palpha_3^h x_{3,t-i} + sum_{i = 1}^palpha_4^h x_{4,t-i} + sum_{i = 1}^palpha_5^h x_{5,t-i} + beta^h shock_t+ u_t.$$

the regression is balanced as $$E(x_{1, t+h} – x_{1, t-i}) = 0$$.

I also think that adding a time trend when a stationary variable is on the right hand side solves the problem with the stochastic trend of $$x_1$$ although I have a harder time showing this.

I however don’t know what to do when there are more I(1) variables in the system. Both in the case with cointegration and in the case without cointegration among the I(1)-variables.

I don’t feel like differencing some of the trending variables. For example – the interest rate, as I would have a hard time interpreting the results.

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## #StackBounty: #time-series #cointegration #augmented-dickey-fuller #granger-causality Trouble interpreting cointegration test results

### Bounty: 50

I’m struggling with testing the cointegration of 2 time series (or rather interpreting the test results properly).

So I got 2 time series `x` and `y` each containing 36 monthly data points (oil prices).

From looking at those time series, I’d say they are cointegrated.

However when applying different cointegration tests, they don’t seem to be:

1) Augmented Dickey-Fuller

`````` from statsmodels.tsa.stattools import adfuller
from statsmodels.api import OLS

ols_result = OLS(y, x).fit()
``````

returns

`````` (0.6614451366946532,
0.9890361840444819,
10,
25,
{'1%': -3.7238633119999998, '5%': -2.98648896, '10%': -2.6328004},
84.12263429255607)
``````

i.e. a p-value of 0.98; null hypothesis cannot be rejected, time series are not cointegrated.

2) Engle-Granger

`````` coint_t, p_value, _ = coint(y, x)
p_value
0.06910078732250052
``````

returns a p-value of 0.069 i.e. not cointegrated.

What am I doing wrong here?

PS: there seems to be Granger-Causality between the 2 time series (tested using `statsmodels.tsa.stattools.grangercausalitytests`)

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## #StackBounty: #time-series #cointegration Understanding coefficients from Johansen Cointegration

### Bounty: 50

I understand that the main purpose of the Johansen test is to check the number of cointegrating relationships, if there are any. However, Im trying to get a better understanding of the eigenvectors and eigenvalues produced by it. I have built an ECM model with 1 dependent and 3 explanatory variables, and determined that the eigenvectors produced, after sorting, by doing a Johansen cointegration test on it are as follows:
``` [[ -4.82 -0.78 0.71 1.35] [ 3.08 -3.96 12.45 -2.32] [ 29.47 -34.32 52.14 -9.70 ] [ -5.48 9.78 -11.23 6.32]] ```
and the eigenvalues are `[0.321 0.093 0.061 0.006]`.

Based on this page, the eigenvector associated with largest eigenvalue would be “the most mean-reverting” and hence I assume it could serve as the coefficients of the time-series. However, I find that the first column of my eigenvectors above, which should be the coefficients, based on the aforementioned logic, are quite different from the coefficients initially obtained through OLS. I divide the first column in the eigenvector matrix by -5.48 as this coefficient is associated with the dependent, to give `[0.88 -0.56 -5.38 1]`. This is rather different from the OLS coefficients which are `[-0.05 0.71 4.12 1]`. I was wondering if what I’ve done is wrong and if so, why it is, or if there is a reason why I shouldnt expect them to be the same?

Thanks

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