#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.

Many thanks in advance!


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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.

36-month time series

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()
 result = adfuller(ols_result.resid)

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?

Thanks in advance!

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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