## #StackBounty: #interaction #panel-data #stationarity #unit-root what should I do about a non-stationary variable in a panel-data intera…

### Bounty: 50

We have panel data on immigration stocks, immigration flows, and immigration policy for 30 countries and 10-30 years. We would like to test the theory that the effect of immigration flows (i.e., annual numbers of incoming immigrants as % of pop) on immigration policy depends on immigrant stocks (i.e., non-citizens as % of pop). In other words, immigration flows affect policy, but only when there are few existing immigrants to begin with.

It seems to me that an interaction between immigration stocks and flows will allow a test of this theory. However, while our dependent variable (immigration policy) and our main independent variable (immigration flows) appear to be stationary, immigrant stocks is not. Standard solutions like first-differencing immigrant stocks won’t help because that would transform stocks into another measure of annual flows, which will not allow us to test the theory.

Another way of putting this is to ask: does stationarity matter only for the dependent variable? Or also for all independent variables?

Advice on how to proceed will be greatly appreciated!

Get this bounty!!!

## #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.

Get this bounty!!!

## #StackBounty: #time-series #stationarity #sas #unit-root #kpss-test Evaluating the importance of a unit-root

### Bounty: 100

I have a monthly time series and I’m trying to determine if such set of data is stationary or not; the dataset is about composed by 160 record.

Specifically, I’m running 2 test found in literature:

1. KPSS: if $$H_0$$ has been rejected then one cannot assume the time series is stationary;
2. Phillips-Perron test: if $$H_0$$ has been rejected then one cannot assume that the time series has a unit-root (then it is stationary);

I preferred to implement the Phillips-Perron test in place of the most common Augmented Dickey-Fuller test since the Phillips-Perron test adjusts for the heteroschedasticity and serial correlation.

Here below, one can find the output of such analysis.

The KPSS test returns not significant p-values both for single-mean, implying that you cannot infer that the time series is not stationary; likewise, the Phillips-Perron test returns significant p-values for the single-mean and trend component, but not for the zero-mean case.

How should I consider or interpret such result?

I wonder if one can evaluate the importance and the strength of such unit-root; for instance, in the question the user @ferdi deals with the variance ratio test to argue the framework to evaluate the importance of a unit root in a time series.

Could you suggest some reference about?

I’m currently running the analysis in SAS, but any programming language would be nice.