<p>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 <span class="math-container">$U_i$</span> behave asymptotically as well.</p>

CC BY-SA 4.0
2019-11-18T16:16:39.240Z
I don't understand what happens to the error term as n goes to infinity.
Alex Wang
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2019-11-18T16:17:52.917Z
Well, what does $U_i\sim I.I.D.(0,\sigma^2)$ mean in words?
heh
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2019-11-18T16:20:14.500Z
Oh, it should converge to 0. Are we using any weak/strong laws of large numbers?
Alex Wang
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2019-11-18T16:40:09.750Z
Sort of but not really. In this case it follows directly from the specification you've given - by writing the error term out that way, you are supposing that this is true.  The reason you suppose this is so that you can use that assumption later on, such as to resolve your initial question.  :)
heh
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2019-11-18T16:42:20.010Z
Given what you're asking, I would just recommend that you ensure you understand the whole process for showing consistency - at some point, you should be taking limits in probability (so-called "plim" operations) if you're being precise.  But you haven't shown your full working of the problem here, so I'm just assuming you did those steps properly.
heh
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2019-11-18T16:13:50.200Z
2019-11-18T16:13:50.200Z
heh
<p><span class="math-container">$Y_i=\beta_0+\beta_1X_i+U_i$</span> is my regression model for an I.I.D. sample with N=1000 observations. Suppose <span class="math-container">$U_i\sim I.I.D.(0,\sigma^2)$</span> and Xi are also I.I.D for i=1,2,3......1000. Xi is independent of Ui. How to show that the estimator <span class="math-container">$\beta_1={{Y_3-Y_2}\over {X_3-X_2}}$</span> is a consistent estimator of the OLS estimator?
Simplication gives 
<span class="math-container">$\tilde {\beta}={{\beta_0+\beta_1X_3+U_3-\beta_0-\beta_1X_2-U_2}\over {X_3-X_2}}={{\beta_1(X_3-X_2)+U_3-U_2}\over {X_3-X_2}}=\beta_1+{{U_3-U_2}\over {X_3-X_2}}$</span>
What is the next step?</p>

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2019-11-19T02:56:55.320Z
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
<microeconomics><econometrics><mathematical-economics>
How to show that the estimator is consistent?
Alex Wang
# How to show that the estimator is consistent?

(Asked by: username_0 on 2019-11-18)

$Y\_i=\beta\_0+\beta\_1X\_i+U\_i$ 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?

**Tags:** microeconomics, econometrics, mathematical-economics
> **Comments:**
> **username_1**: 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.  

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## 1 answers

### Answer by username_2 on 2019-11-18. Score: 2 | Accepted answer

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.

> **Comments:**
> **username_0**: I don't understand what happens to the error term as n goes to infinity.  
> **username_2**: Well, what does $U_i\sim I.I.D.(0,\sigma^2)$ mean in words?  
> **username_0**: Oh, it should converge to 0. Are we using any weak/strong laws of large numbers?  
> **username_2**: Sort of but not really. In this case it follows directly from the specification you've given - by writing the error term out that way, you are supposing that this is true.  The reason you suppose this is so that you can use that assumption later on, such as to resolve your initial question.  :)  
> **username_2**: Given what you're asking, I would just recommend that you ensure you understand the whole process for showing consistency - at some point, you should be taking limits in probability (so-called "plim" operations) if you're being precise.  But you haven't shown your full working of the problem here, so I'm just assuming you did those steps properly.  

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