#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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