How to choose the best estimator? [Econometrics - panel data] I'm quite new to econometrics and Stata. I'm trying to learn it by my own but I'm having some problems. Here's my problem: I have a balanced panel data and I need to choose the best estimator for my regression model.
I have regressed my panel data in Stata using five different estimators and I don't know which one is the best. I need to choose one between these five.


*

*Pooled OLS estimator

*Between estimator

*Fixed Effects estimator

*First-differences estimator 

*Random effects estimator


I also had calculated the Hausman test but it only tests the Fixed X Random Effects Model. The result of Hausman test was Prob>chi2 = 0.9699.
So I need to use the FE estimator. Is that? And about the others estimators?
 A: Here are the steps that I use to estimate an panel data with Stata:
1, Peform an RE estimation with xtreg,re
2, Peform the Lagrange multiplier test for random effects with xttest0.The LM test helps you decide between a random effects regression and a simple OLS regression. The null hypothesis in the LM test is that variances across entities is zero. This is, no significant difference across units (i.e. no panel effect), and Pooled OLS is applied. If the null hypothesis is rejected, to step 3.
3, Perform an FE estimation with xtreg,fe, then perform an hausman test to compare with the RE estimation.
Differentiating POLS, FE, and RE is the first step to panel analysis for me.
A: Since you labeled this as econometrics, ostensibly you care about consistently estimating the coefficients. In that case, you should think about whether RE or Pooled OLS are justified on economic terms, meaning whether you have an identification problem in terms of omitted variation. 
If the time-invariant heterogeneity is likely correlated with the error term, then RE and pooled OLS will be inconsistent and FE or D-D should be used. 
While the tests that the other answer provides can help with determining whether you should not use RE or pooled OLS, if the tests fails to restrict your estimation method selection, you cannot have too much confidence that RE will be consistent as foolproof testing for endogeneity in this context cannot be done. 
A: Model selection
Based on the information you provided, I assume that you have estimated all the models. If so, you are basically asking how to choose the best model. There are a couple of model selection criteria. The most usual methods are as follows:


*

*The $R^2$. A high $R^2$ indicates a good fit. If you use different independent variables in each model, it is better to use the $\bar{R}^2$ (adjusted $R^2$). Note: don't use this metric to pick the best model if you want to forecast, there is a lot of risk of overfitting.

*Information criteria such as the AIC and the BIC. These metrics are defined as a loss function plus a penalty term (AIC and BIC differ in the penalty term). A loss function should be minimized, so choose the model with the lowest AIC or BIC.

*These slides also mention a Mallows $C_p$, but I have never heard of it during my Bachelor nor during my Master in econometrics.
There are many more methods to decide the best model, but are probably too advanced for beginners in econometrics.
To (hopefully) give you more intuition about the different estimators I suggest you read this guide. It may be a bit technical, but it lists all the advantages and disadvantages of the different estimators.
The Hausman test
The high P-value indicates that you should reject the $H_0$. The $H_0$ in the Hausman test is that the RE is preferred over the FE due to higher efficiency. So yes, you should use the FE estimator.
A: "The high P-value indicates that you should reject the H0."
It should be otherwise; low p-values indicate that one should reject H0, so with a high p-Value should random effects. 
A good introduction to the theory behind the different estimators and the Breusch-Pagan as well as Hausman test is provided here: https://sites.google.com/site/econometricsacademy/econometrics-models/panel-data-models
