ARDL model: lag order selection for valid inference

Question

I estimated an ARDL model of 6 variables. After several attempts (using different lags) to find a better estimate, I got a selected ARDL model using AIC as (1,1,0,0,1,2) while using SIC is ARDL (1,0,0,0,1,2).

My questions are:

1. Can I still use this model given these lag selections?
   ARDL (1,1,0,0,1,2) = AIC
   ARDL (1,0,0,0,1,2) = SIC
2. Will the inference from these models be valid?

Answer 1

1. You can use the model for forecasting, for example, but how well it will work depends on the situation. You might have selected the best model (which is a relative term) from the candidate set, but that model may or may not be good on absolute terms.
2. Proper inference after model selection is terribly difficult. However, most practitioners ignore the model selection step and proceed (incorrectly) as if the model was fixed from the beginning. Such inference (which is conditional on the selected model) is straightforward as long as the basic assumptions for the particular inference method are satisfied; e.g. the assumption that errors are independent of regressors when you are doing a $t$-test in a linear regression model. So that is what most of the people do even though they are mistaken to ignore the model selection stage and thus their results are unreliable (typically too optimistic).