# How to show that the estimator is consistent?

$$Y_i=\beta_0+\beta_1X_i+U_i$$ is my regression model for an I.I.D. sample with N=1000 observations. Suppose $$U_i\sim I.I.D.(0,\sigma^2)$$ and Xi are also I.I.D for i=1,2,3......1000. Xi is independent of Ui. How to show that the estimator $$\beta_1={{Y_3-Y_2}\over {X_3-X_2}}$$ is a consistent estimator of the OLS estimator? Simplication gives $$\tilde {\beta}={{\beta_0+\beta_1X_3+U_3-\beta_0-\beta_1X_2-U_2}\over {X_3-X_2}}={{\beta_1(X_3-X_2)+U_3-U_2}\over {X_3-X_2}}=\beta_1+{{U_3-U_2}\over {X_3-X_2}}$$ What is the next step?

• The next step is to infer that what you have is not a consistent estimator. The reason is $\frac{U_3-U_2}{X_3-X_2}$ does not vary with the sample size and will always have the same variance. – Amit Nov 19 '19 at 2:56

Remember that consistency describes how the estimator behaves in the limit as N asymptotically approaches infinity. Assuming no errors in your math up to this point, you need to consider how your error terms $$U_i$$ behave asymptotically as well.
• Well, what does $U_i\sim I.I.D.(0,\sigma^2)$ mean in words? – heh Nov 18 '19 at 16:17