<p>In simplified and intuitive way:</p>
<p><strong>Consistency</strong> is ability of the estimator to on average uncover true value of the coefficient. For example, if the true value of some coefficient <span class="math-container">$\beta=2$</span> then estimator <span class="math-container">$E[\hat{\beta}]=\beta=2$</span> as well. An estimator that in expectations would not give you the true beta coefficient would not be consistent.</p>
<p><strong>Efficiency</strong> is the ability of estimator to estimate the value of <span class="math-container">$\beta$</span> as precisely as possible. Due to probabilistic nature of the problem every time you estimate something in statistics there will be some confidence interval. Efficiency, in many standard models that assume normality, means that the estimator has the smaller confidence interval for the true estimate than other possible estimators.</p>
<p>For example, estimator that tells you that <span class="math-container">$E[\hat{\beta}] = 3 \pm 0.5$</span> is more efficient than estimator that tells <span class="math-container">$E[\hat{\beta}] = 3 \pm 1$</span>.</p>
<p>Moreover, note that estimators can be:</p>
<ol>
<li>Consistent and efficient</li>
<li>Consistent but not efficient</li>
<li>Inconsistent but efficient</li>
<li>Inconsistent and not efficient</li>
</ol>
<p>All of the above is possible, it all depends on the asymptotic properties of mode you are looking about.</p>
<blockquote>
<p>I am wondering whether efficiency is AIC and consistency is BIC mentioned in this post?</p>
</blockquote>
<p>No AIC and BIC are information criteria which are estimators in their own right that post there just discusses the properties of AIC and BIC mentioning that the advantage of one is that it is consistent and the other that it is efficient</p>

CC BY-SA 4.0
2021-09-23T10:27:24.533Z
an amazing explanation, respect, 1muflon1, thank you!
Nguyen Lis
CC BY-SA 4.0
2021-09-23T10:56:24.290Z
ah, can I ask, so, asymptotic properties of an estimator is efficiency and consistency right?
Nguyen Lis
CC BY-SA 4.0
2021-09-23T12:30:02.967Z
@NguyenLis yes, although asymptotic property is umbrella term for more than that, any behavior of an random variable or estimator that emerges when the number of observations goes to infinity is some asymptotic behavior or property
1muflon1
CC BY-SA 4.0
2021-09-25T11:48:05.257Z
Your description of consistency looks more like a description of unbiasedness. Consistency is not about averaging but about convergence. Defining efficiency via confidence intervals seems unorthodox and questionable. Also, what do you mean by $\hat{\beta}= E[\beta]$ where $\beta$ (being the true parameter value) is a constant? And then regarding case 3. *inconsistent but efficient*, this probably needs some qualification. Would you have an example? (Cc @NguyenLis)
Richard Hardy
CC BY-SA 4.0
2021-09-25T19:23:16.483Z
@RichardHardy in response to 1, yes but convergence implies that as $n \to \infty$ we have that $E[\beta]= \beta$. Maybe this is incorrect but this is also how it is explained in standard textbooks for example Newbold et al (pp 345) state “[estimator] is said to be a consistent estimator of the parameter $\theta$ if the difference between expected value of the estimator and the parameter decreases as the sample size increases” and later in examples when they talk about asymptotic consistency is always in cases where asymptotically $E[\beta]=\beta$, this being said maybe the book is not correct
1muflon1
CC BY-SA 4.0
2021-09-25T19:29:07.923Z
2. regarding $\hat{\beta}= E[\beta]$ in models like linear reg or t test and so on the estimated coefficient is the expected value of beta, I know this might not be true for any arbitrary estimator but I mentioned at the beging of my answer I am bit oversimplifying but I also edited my q and added extra caveat about that now
1muflon1
CC BY-SA 4.0
2021-09-25T19:34:57.847Z
What does $E(\beta)=\beta$ even mean? It is trivially correct but irrelevant. Did you mean $E(\hat\beta)=\beta$ instead? Newbold's quote is neither sufficient nor even necessary (though the latter depends on precise wording) for consistency. I do not think the simplification is justified, as it seems to have introduced some fundamentally wrong ideas about what consistency is.
Richard Hardy
CC BY-SA 4.0
2021-09-25T19:35:04.237Z
3. regarding the efficiency of estimator again in textbooks like newbold et al (pp  311) you will find stated that "$\hat{\theta}_1$ is said to be more efficient than $\theta_2$ if $VAR(\theta_1)<\var(\theta_2)$" <- this also implies that, ceteris paribus, $\theta_2$ will have wider confidence bound because var{\theta} directly affects standard errors and all else equal higher standard errors the wider confidence interval
1muflon1
CC BY-SA 4.0
2021-09-25T19:35:54.077Z
@RichardHardy oh yea I actually meant $E(\hat{\beta})$
1muflon1
CC BY-SA 4.0
2021-09-25T19:37:11.570Z
$\text{Var}(\hat\beta)<\text{Var}(\tilde\beta)$ does not generally imply the confidence interval for $\hat\beta$ is narrowed than for $\tilde\beta$, though under an additional assumption of e.g. normality it would.
Richard Hardy
CC BY-SA 4.0
2021-09-25T19:38:14.990Z
@RichardHardy thanks for pointing that out, I will add that to my answer if you dont mind
1muflon1
CC BY-SA 4.0
2021-09-25T19:40:52.720Z
I would suggest going back to the definitions and thinking hard how much you can paraphrase them without compromising their meaning. And again, what you present as consistency is unbiasedness, a rather different concept. You may also ask a question on Cross Validated if you have any doubts. The existing questions there already include some fascinating discussions about the subtleties of the notion of consistency. Perhaps also efficiency (not sure about it).
Richard Hardy
CC BY-SA 4.0
2021-09-25T19:43:36.513Z
@RichardHardy, thanks I will first do my own research as one should before posting on SE when I get home since I have some hard copies of also PhD level statistics books that are more rigorous. Also, I tried to simplify the concepts in the above rather than present them in fully rigorous but more complex way, but clearly I did a poor job, thanks for all the feedback as always appreciated, maybe if you have a recommendation for some good text you could leave it another comment
1muflon1
CC BY-SA 4.0
2021-09-26T18:34:15.357Z
@1mouflon1: unbiasedness and consistency are two independent notions, in the sense that an estimator of $\beta$ can be both biased and consistent for $\beta$, while another can be both unbiased and inconsistent.
Bertrand
CC BY-SA 4.0
2021-09-23T09:49:42.820Z
2021-09-25T19:37:27.143Z
2021-09-25T19:37:27.143Z
1muflon1
<p>Normally, when we talk about the asymptotic estimator, we talk about the efficiency and consistency of this estimator. I am wondering whether efficiency is AIC and consistency is BIC mentioned in <a href="https://stats.stackexchange.com/questions/408267/justification-for-and-optimality-of-r2-adj-as-a-model-selection-criterion">this post</a>?
Is there any intuitive way to understand asymptotic properties?</p>

CC BY-SA 4.0
2021-09-25T11:52:25.853Z
You may access a larger pool of experts asking the question on Cross Validated Stack Exchange.
Richard Hardy
<econometrics><nonexpert>
What are the asymptotic properties of an estimator?
Nguyen Lis
# What are the asymptotic properties of an estimator?

(Asked by: username_0 on 2021-09-23)

Normally, when we talk about the asymptotic estimator, we talk about the efficiency and consistency of this estimator. I am wondering whether efficiency is AIC and consistency is BIC mentioned in [this post](https://stats.stackexchange.com/questions/408267/justification-for-and-optimality-of-r2-adj-as-a-model-selection-criterion)?
Is there any intuitive way to understand asymptotic properties?

**Tags:** econometrics, nonexpert
> **Comments:**
> **username_1**: You may access a larger pool of experts asking the question on Cross Validated Stack Exchange.  

---

## 1 answers

### Answer by username_2 on 2021-09-23. Score: 1 | Accepted answer

In simplified and intuitive way:

**Consistency** is ability of the estimator to on average uncover true value of the coefficient. For example, if the true value of some coefficient $\beta=2$ then estimator $E[\hat{\beta}]=\beta=2$ as well. An estimator that in expectations would not give you the true beta coefficient would not be consistent.

**Efficiency** is the ability of estimator to estimate the value of $\beta$ as precisely as possible. Due to probabilistic nature of the problem every time you estimate something in statistics there will be some confidence interval. Efficiency, in many standard models that assume normality, means that the estimator has the smaller confidence interval for the true estimate than other possible estimators.

For example, estimator that tells you that $E[\hat{\beta}] = 3 \pm 0.5$ is more efficient than estimator that tells $E[\hat{\beta}] = 3 \pm 1$.

Moreover, note that estimators can be:

1. Consistent and efficient
2. Consistent but not efficient
3. Inconsistent but efficient
4. Inconsistent and not efficient

All of the above is possible, it all depends on the asymptotic properties of mode you are looking about.

> I am wondering whether efficiency is AIC and consistency is BIC mentioned in this post?

No AIC and BIC are information criteria which are estimators in their own right that post there just discusses the properties of AIC and BIC mentioning that the advantage of one is that it is consistent and the other that it is efficient

> **Comments:**
> **username_0**: an amazing explanation, respect, 1muflon1, thank you!  
> **username_0**: ah, can I ask, so, asymptotic properties of an estimator is efficiency and consistency right?  
> **username_2**: @NguyenLis yes, although asymptotic property is umbrella term for more than that, any behavior of an random variable or estimator that emerges when the number of observations goes to infinity is some asymptotic behavior or property  
> **username_1**: Your description of consistency looks more like a description of unbiasedness. Consistency is not about averaging but about convergence. Defining efficiency via confidence intervals seems unorthodox and questionable. Also, what do you mean by $\hat{\beta}= E[\beta]$ where $\beta$ (being the true parameter value) is a constant? And then regarding case 3. *inconsistent but efficient*, this probably needs some qualification. Would you have an example? (Cc @NguyenLis)  
> **username_2**: @RichardHardy in response to 1, yes but convergence implies that as $n \to \infty$ we have that $E[\beta]= \beta$. Maybe this is incorrect but this is also how it is explained in standard textbooks for example Newbold et al (pp 345) state “[estimator] is said to be a consistent estimator of the parameter $\theta$ if the difference between expected value of the estimator and the parameter decreases as the sample size increases” and later in examples when they talk about asymptotic consistency is always in cases where asymptotically $E[\beta]=\beta$, this being said maybe the book is not correct  
> **username_2**: 2. regarding $\hat{\beta}= E[\beta]$ in models like linear reg or t test and so on the estimated coefficient is the expected value of beta, I know this might not be true for any arbitrary estimator but I mentioned at the beging of my answer I am bit oversimplifying but I also edited my q and added extra caveat about that now  
> **username_1**: What does $E(\beta)=\beta$ even mean? It is trivially correct but irrelevant. Did you mean $E(\hat\beta)=\beta$ instead? Newbold's quote is neither sufficient nor even necessary (though the latter depends on precise wording) for consistency. I do not think the simplification is justified, as it seems to have introduced some fundamentally wrong ideas about what consistency is.  
> **username_2**: 3. regarding the efficiency of estimator again in textbooks like newbold et al (pp  311) you will find stated that "$\hat{\theta}_1$ is said to be more efficient than $\theta_2$ if $VAR(\theta_1)<\var(\theta_2)$" <- this also implies that, ceteris paribus, $\theta_2$ will have wider confidence bound because var{\theta} directly affects standard errors and all else equal higher standard errors the wider confidence interval  
> **username_2**: @RichardHardy oh yea I actually meant $E(\hat{\beta})$  
> **username_1**: $\text{Var}(\hat\beta)<\text{Var}(\tilde\beta)$ does not generally imply the confidence interval for $\hat\beta$ is narrowed than for $\tilde\beta$, though under an additional assumption of e.g. normality it would.  
> **username_2**: @RichardHardy thanks for pointing that out, I will add that to my answer if you dont mind  
> **username_1**: I would suggest going back to the definitions and thinking hard how much you can paraphrase them without compromising their meaning. And again, what you present as consistency is unbiasedness, a rather different concept. You may also ask a question on Cross Validated if you have any doubts. The existing questions there already include some fascinating discussions about the subtleties of the notion of consistency. Perhaps also efficiency (not sure about it).  
> **username_2**: @RichardHardy, thanks I will first do my own research as one should before posting on SE when I get home since I have some hard copies of also PhD level statistics books that are more rigorous. Also, I tried to simplify the concepts in the above rather than present them in fully rigorous but more complex way, but clearly I did a poor job, thanks for all the feedback as always appreciated, maybe if you have a recommendation for some good text you could leave it another comment  
> **username_3**: @1mouflon1: unbiasedness and consistency are two independent notions, in the sense that an estimator of $\beta$ can be both biased and consistent for $\beta$, while another can be both unbiased and inconsistent.  

---

