You shouldn't choose the standard errors based on whether this delivers "improved" statistical significance. In general, I would not expect the standard errors to become smaller - i.e., coefficient estimates to become more significant - when using Newey-West standard errors.
You are right that a p-value of 0.066 is not very comfortable. This could be used as a justification to use Newey-West standard errors. Alternatively, you might want to address the serial correlation in the first place by allowing for higher-order lags in the ARDL specification. If your sample size allows, you could increase the maximum lag order with the ardl option maxlag(), and/or use the AIC instead of the BIC model selection criterion.
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