# #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.