Difference between long run coefficient and non stochastic steady state coefficient ARDL model I am a little bit confused on the definition of long run equilibrium coefficient. Suppose I have an ARDL model as:
$y_t = \rho_1 y_{t-1} + \rho_2 y_{t-2} + \beta_1x_{t-1} + \beta_2x_{t-2} $
The steady state non stochastic solution is:
$y=\frac{ \beta_1  +  \beta_2 }{1- \rho_1 -\rho_2}x$
What is the definition of long run equilibrium coefficient for $x$ here? Is it defined as the coefficient of the steady state non stochastic solution $\frac{ \beta_1  +  \beta_2 }{1- \rho_1 -\rho_2}$? What is the difference between the two concepts?
yes, the term that you showed for the ALDR non-stochastic steady state:
$$\frac{ \beta_1  +  \beta_2 }{1- \rho_1 -\rho_2}$$
is long-run multiplier or sometimes also called long run equilibrium coefficient (see Verbeek 2008 Guide to Modern Econometrics 4th ed.  pp 340). As  far as I can understand there is not much difference between the two concepts in the context of ARDL model and they are used interchangeably.
The term non-stochastic steady state has wider application especially in macroeconomics (e.g. in growth theory), but often steady state results are just called long-run equilibrium because in most models steady state is attained only in long-run equilibrium.