My model is: $$Y=\beta_0+\beta_1X_1+\beta_2X_2+\beta_3X_3+\beta_4X_4.$$
I want to check if 'weather' (not included in the above model) is an appropriate instrument. $X_1$ represents price. If I determine that 'weather' is an appropriate instrument I will use my new model (which includes this instrument) to obtain the own price elasticity.
In order to check if an instrument is appropriate we run a first stage least squares regression. If we deem that the instrument is strong, we then run a two stage least squares regression (TSLS regression).
When running the first stage least squares regression, do we run $Y$ against 'weather' and look at if 'weather' is significant (or check the F-statistic to see if it is below/above $10$ as a rule of thumb), or do we run $X_1$ (price) against 'weather'?
Would I then do the same with the other variables? So, run $X_2$ against 'weather', etc.
Or should I include all the other variables in the regression? And then evaluate if the weather variable is significant? (So run, $Y$ against 'weather', $X_1$, $X_2$, $X_3$ and $X_4$, or run $X_1$ against 'weather', $X_2$, $X_3$ and $X_4$).
When running a TSLS regression, we test for $H_0:$ exogeneity and $H_1:$ endogeneity. When doing this, do we look at the J-statistic and multiply this value by the number of independent variables (here would be $4$)? And then compare to the chi-squared table of values with $4-1=3$ degrees of freedom?
Or, would it be much better to run a Durbin-Wu-Hausman (DWH) test? When I run this test on EViews, I obtain an output with J-statistics and p-values for 'restricted test equation', 'unrestricted test equation' and the difference in J-statistics. What do the 'restricted test equation' and 'unrestricted test equation' represent? If my p-values are not significant, would it be correct to conclude that we do not reject the null hypothesis and so the variables are exogenous and the instrument is not appropriate?
Lastly, does a Cragg-Donald F-statistic of value $5.0323$ signify that the instrument (weather) is weak? And which test provides more conclusive results? (Cragg-Donald, DWH or multiplying the J-statistic by number of instruments and comparing to the chi-squared tables)?