Table 2 shows the result of the ADF test. The variable lnoilinfillwell shows a t-value of -4.016, which is higher than the 5% critical value of -2.920, leading to the rejection of the null hypothesis. In contrast, variables like lnoilpr, lngaspr, lnmaturity, lnrigs, lnvoloil, and lnvolgas have t-values below the critical value, indicating they are non-stationary. 

Autoregressive Distributed lag (ARDL) 

The ARDL cointegration approach, developed by Pesaran and Shin (1999), offers three main advantages over traditional methods:

1. It allows variables to be integrated at different orders (I(0), I(1), or fractional).
2. It is more effective with small sample sizes.
3. It provides unbiased long-run estimates.

Key points of the ARDL model:

• Includes lagged values of the dependent variable and both current and lagged values of regressors.
• Uses a mix of endogenous and exogenous variables, unlike the VAR model.
• A unit root test ensures no variables are I(2).
• Can handle variables with different integration orders or all variables as I(1).
• If cointegrated, both short-run (ARDL) and long-run (VECM) models are specified; if not, only the short-run model is used.
• The ARDL (p, q) model is defined as follows: 

Where    is a vector and the variables in are allowed to be purely I(0) or I(1) or cointegrated; β and  are coefficients; is the constant; i=1,2,3, …k; p, q are optimal lag orders; is a vector of the error terms-unobservable zero mean white noise vector process (serially uncorrelated or independent). 

1. The dependent variable is a function of lagged dependent variable, current and lagged value of the other explanatory variable(s) in the model. 

1. The lag lengths for p, q may not necessarily be the same. 

1. p lags: used for the dependent variable 

1. q lags: utilized for the exogenous variables