I have a dataset with T=11(2010-2020) and N=63(provinces). I want to estimate the causality relationship in short run and long run between 2 variables. Which model can be suitable?
Within model without time effects
Always consistent exept if you use lag variable
Maybe your data is too sparse to yield meaningful results (from any model or method), but the nature of this problem looks like it should be amenable to all 4 of the methods described in Bretherton et al. (1992), J. Climate: "An intercomparison of methods for finding coupled patterns in climate data". (https://doi.org/10.1175/1520-0442(1992)005%3C0541:AIOMFF%3E2.0.CO;2 ).
These should all identify "coupled modes of variability between time series of two fields" (a "field" being a map of one of your variables).
As I remember, all those methods (but certainly the singular value decomposition, or SVD) allow you to make statements like "X% of the variance of variable 1 is explained by the variability of variable 2", which is probably about as much as any model can do to establish a causality relationship. You may be able to apply some common sense analysis too: e.g., if your variables are, say, rainfall rate and size of rice harvest, then rainfall may plausibly affect the rice harvest, but not vice versa.
In https://doi.org/10.1175/1520-0469(1994)051%3C1244:TIOITS%3E2.0.CO;2, we were able to use those methods to estimate that ~10% of extratropical 250mb height variability could be attributed to the Madden-Julian wave in the tropics - of which ~5% was due to physical processes and ~5% to chance. Seemed about right to me, given all the other sources of extratropical variability. Reducing those other sources of extratropical variability (i.e., baroclinic waves and instabilities) by reducing the heating rate allowed that 10% due to the M-J wave to increase to ~40%.
See too this J. Climate paper by Cherry (1996) on SVD and canonical correlation analysis (CCA): https://doi.org/10.1175/1520-0442(1996)009%3C2003:SVDAAC%3E2.0.CO;2