Hell No! The regressand (y) must be nonstatioary or I(1) and your regress is a vector X with 0

Actually stationarity is required in order to obtain consinstent estimates with OLS. Therefore, the best thing to do is to make all independent time series stationary. On the other hand, what you can do by using the actual data you have is to perform a causality test following the procedure proposed by Toda & Yanamoto (1995), which allows for possible different order of integration (also I(0) and I(1) can be the case).

Hence you'll be able to assess if "inflation threshold" can cause (in the Granger sense) economic growth.

I do not know what Raffaele Mattera

is talking about

I said that inference is not possible without making series stationary. However I suggested another applicaton he can try with the data he has, based on Toda & Yanamoto paper ("Statistical inference in vector autoregressions with possibly integrated processes") published on Journal of Econometrics in 1995. Actually you can find several paper using this approach in testing (no-)causality.

The I(0) for the dependent variable and I(1) for all the independent variables is the results of DFuller and PPerron gives almost the same result as DF: I(0) for dependent variable and a mix of I(0) and I(1) for independent variables. Should I use additional unit root test like KPSS or the ones Raffaele Mattera is prescribing before taking decision?

Well, I advice you to perform also more advanced tests for unit root as you said, even if it really depends on the time series characteristics. However what I suggested to you is a strategy for testing Granger sense of causality and it is a good idea in my opinion do that for supporting the identifying assumption that you propose.

Therefore, I can suggest the following. If after a more accurate unit root test you end up with a situation where you can estimate ARDL (let's say, also the independent you need are stationary), let's do the same a causality test. However if the estimation is not possibile, you could at last propose a causality test by following the procedure I suggested before.

Yes, you can have a mixture of I(1) and I(0) variables in the ARDL model. Pesaran, Shin, and Smith (J. App. Econometrics, 2001), who originated the ARDL technique, state this to be an advantage of the approach. Let the variables be z, partitioned into y and some x’s. Then one of many possibilities is for y to be I(0) and all the x’s to be I(1), which is the case mentioned by the question poser, Mr. Koroma. In fact, in a detailed blog entry, Dave Giles presents a 2-variable example where y might be I(0) according to the ADF test. https://davegiles.blogspot.com/2013/06/ardl-models-part-ii-bounds-tests.html

But what kind of ARDL results might one get if y is truly I(0) and all x’s are truly I(1)? My take is that the results are not likely to be very useful.

I now construct two models to illustrate some possibilities (but with no necessary economic rationale!). Let “d” be the difference operator, let “*” be the multiplication symbol, and assume a 3-variable model for simplicity. I.i.d. mean 0 errors are u and v. Variables x1 and x2 are random walks and not cointegrated. There could be relationships among the three variables that maintain the stationarity of y. Possibility 1:

y(t) = a1*dx1(t) + a2*dx2(t) + v(t).

Here, the level of y (GDP growth rate for Mr. Koroma) is related to the changes in or growth rates of x1 and x2. Is there an economic model that suggests that, say, the GDP growth rate is related to CHANGES in the growth rates of population and govt. expenditure, as apparently suggested for Mr. Koroma’s variables? If so, why not just estimate that model, where all variables are stationary? But if ARDL is applied, the corresponding ARDL equation is

dy(t) = -y(t-1) + a1*dx1(t) + a2*dx2(t) + v(t).

The ARDL t and F tests will likely both be significant because of the error correction coefficient on y(t-1); this just measures the tendency of y to return to its mean (0 here, for simplicity). The equation correctly shows no long-run relationship of y to x1 or x2 (whose coefficients on their lagged values are implicitly 0 in the equation). That is, x1 and x2 wander around as random walks, but y stays near 0. (I am assuming the model is dynamically stable.)

Now suppose x1 and x2 are cointegrated, giving possibility 2:

x1 = b*x2 + u(t). (x1 and x2 are again random walks.)

And suppose (for some strange reason) y is related to the lagged cointegration errors, u(t-1):

y(t) = a*u(t-1) + v(t) = a*x1(t-1) – a*b*x2(t-1) + v(t).

In a 3-variable version of Mr. Koroma’s case, the model says that, say, the population and government spending growth rates tend to stick together in an equilibrium (cointegration) relationship (this seems plausible), but temporary deviations from the equilibrium cause temporary fluctuations in the GDP growth rate. Is there a theory that suggests this possibility?? Anyway, now the ARDL equation is:

dy(t) = -y(t) + a*x1(t-1) – a*b*x2(t-1) + v(t).

This shows that ARDL estimation will likely give significant nonzero coefficients for lagged x1 and x2, but the coefficients do not lead to what we would call cointegration vectors (when divided by the negative of the error correction coefficient, the coefficients become cointegration coefficients in the pure I(1) case). In the present model they do not represent a long-run relationship between y and the x’s in the sense meant by cointegration. Instead, each coefficient shows the response of y to the stationary cointegration errors when decomposed into separate x1 and x2 movements: e.g., x1’s coefficient is the effect on dy of changing x1 holding x2 constant, which is identically the effect of a change in u, the cointegration error.

Thus, in both my examples, the ARDL results reflect relationships among stationary fluctuations, not involving the overall movements in the righthand I(1) variables themselves.