You could probably try an ARDL framework; it allows for a mix between variables that are stationary in level and first difference at the same time. Then you could model the long-term relation using VECM, for example. The framework developed by Pesaran is very intuitive in doing this, I think. You could also consider a causality test using the Toda and Yamamoto approach.

Best of luck,

Fayçal

You have a mix of integration orders: Inflation and GDP are stationary at level I(0). Balance of payments is stationary at first difference (I(1)). This mixed order of integration makes standard VAR or VECM inappropriate. I would suggest use a Vector Autoregressive Distributed Lag (VARDL) model (also known as the ARDL bounds testing approach) because it handles mixed I(0) and I(1) variables and it allows testing for both short-run dynamics and long-run relationships.

If you have a mix of I(0) and I(1) variables, the VAR model in levels is not suitable. Instead, you can use a VECM if all variables are I(1) and cointegrated. If not, consider transforming the I(1) series to first differences or use ARDL models which can handle mixed stationarity.

Use ARDL or separate modeling for I(0) and I(1) variables if no cointegration exists

When faced with a stationary mix that is, some variables are integrated of order zero, I(0), and others are integrated of order one, I(1) a traditional VAR or VECM model becomes problematic. VECM specifically requires that all variables be I(1) and cointegrated. If you proceed with a mix, you risk model misspecification. In this context, the Autoregressive Distributed Lag (ARDL) model developed by Pesaran and Shin (1999) becomes a robust alternative. The ARDL bounds testing approach can handle a combination of I(0) and I(1) variables, provided none is integrated of order two, I(2). It allows for the simultaneous estimation of both short-run dynamics and long-run equilibrium relationships. Alternatively, if the I(1) variables are cointegrated and the I(0) variable has theoretical relevance, you may include it in the VECM model as an exogenous or control variable. However, this should be grounded in solid economic theory. Prior to model estimation, it is critical to conduct thorough unit root testing using ADF, PP, or KPSS tests, followed by cointegration tests such as Johansen’s or the ARDL bounds test, to guide model selection and specification.

If you wish to use ARDL, you are estimating one equation. One of your variables is the dependent variable. The others are explanatory variables. One of the conditions underlying the validity of the ARDL process is that there is no contemporaneous feedback from the dependent variable to the explanatory variables (weak exogeneity requirement). Thus, ARDL can not be used to measure the simultaneous interaction between a group of variables.

It is generally found that real GDP is non-stationary but that log(GDP) is I(1) - log(real GDP) is non-stationary but its first difference is stationary. You will need to justify your conclusion that real GDP is stationary.

You need to reconsider your model. For example, there is no reason to suppose that inflation and real GDP are related. Inflation may be due to monetary or fiscal policy, an output gap, or demand or supply shocks. Any particular level of inflation may occur at any level of GDP. If you do find any correlations between your variables, they will not be stable and will change if you estimate them over a different sample period.

I should have added

The relationship between the GDP and the balance of payments is the accounting identity.

Y = C + I + G + X -M.

X-M is the balance of payments. A one unit increase in the balance of payments will give rise to a one unit increase of one unit in GDP. Any estimated model must give this result

It is a bad (but unfortunately far-spread) habit of econometricians to apply econometric (i.e. statistical) methods without economics. They take a set of data and simply "throw" it into a program package.

Devinta, you have obviously not yet specified the model you want to estimate (test) and all answers except John's do not care about the specification and the different nature (dimensions) of the variables. I would e.g. never correlate a percentage change (like inflation) to an "absolute" variable (like GDP). By the way, (for a time series analysis) I would never take real GDP as such (or its log), but always take its (relative) changes. Another problem is the relation between real GDP and the BoP, which remains a financial (i.e. nominal) variable, although one could construct a "real BoP" via the GDP identity. This identity may be considerably different from the number shown in National Banks statistics (because of income transfers, timing of payments etc).

I cannot imagine an effect of BoP on inflation. If you are unlucky to get a significant relation, you would have troubles to explain me and others how this functions.