# Maximum likelihood and OLS estimation of ARDL model under nonstationarity

Consider the simple ARDL(1,1) model $$y_t=\beta_0+\beta_1y_{t-1}+\beta_2x_{t}+\beta_3x_{t-1}+\epsilon_t$$

If $$y_t$$ and $$x_t$$ are non-stationary can I fit the model with OLS? If not, is assuming that $$\epsilon_t$$ is normally distributed enough to use MLE? Or do I have to determine the distribution of $$x_t$$ too?

• For an interesting exposition on the ARDL, see hendry's "dynamic econometrics". He characterizes each type of ARDL quite clearly and that's one of them. But, in general, the safest way to fit an ARDL is to write out the likelihood and maximize it numerically. Andrew Harvey's text Economic Time Series Analysis is beautiful for explanations-details involved with doing that in terms of the gradients etc. The other ways MLE, OLS, might work in special cases but it's safer to just maximize the empirical likelihood. May 10, 2019 at 10:21
• I wonder if the problem of choosing between MLE and OLS might be tangential to the problem of nonstationarity. In some other time series models, neither MLE nor OLS will be appropriate under nonstationarity, so a better choice would be to reformulate the model to achieve a stationary formulation. ARDL might be special in that it could handle nonstationary series seamlessly, though; I do not know enough about ARDL. May 10, 2019 at 10:31
• @RichardHardy so could MLE not be consistent under non-stationarity? May 10, 2019 at 10:43
• I do not know about MLE for ARDL models, but in general under neglected nonstationarity, I think, yes. Think about regressing two independent random walks on one another and using MLE derived for two stationary time series instead, it would probably be inconsistent. Also, even in the cases where MLE is consistent, you might be getting wrong estimates of precision (standard errors and confidence intervals). May 10, 2019 at 11:22