As usual, we can estimate by OLS the model (in matrix form) $Y=\alpha+\beta*X+u$ with a sample of $n+m$ observations. The OLS estimator is $\hat{\beta}=(X^{T}X)^{-1}X^{T}Y$. Now, if we partition our sample in two subsamples A and B of sizes n and m, respectively, we could get the OLS estimators for each sumbsample: $\hat{\beta_A}=(X_A^{T}X_A)^{-1}X_A^{T}Y_A$ and $\hat{\beta_B}=(X_B^{T}X_B)^{-1}X_B^{T}Y_B$. We know both of these estimators are unbiased, and their variance is (assuming homoscedasticity) equal to $\sigma^2(X_i^{T}X_i)^{-1}$. Now let's define $\tilde{\beta}=\frac{\hat{\beta_A}+\hat{\beta_B}}{2}$, which is also unbiased, and, assuming no serial correlation, its variance is $V(\tilde{\beta})=\frac{V(\hat{\beta_A})+V(\hat{\beta_B})}{2}$. 
  
How could we prove this estimator $\tilde{\beta}$ is less efficient/worse than $\hat{\beta}$?