<p>One example that may occur is with fixed effects. Sometimes, we do run regressions like:
<span class="math-container">$$
y_{i,t} = \alpha_i + X_{i,t} \gamma + \varepsilon_{i,t}.
$$</span>
Here, <span class="math-container">$i$</span> is for example a firm identifier and <span class="math-container">$t$</span> represents time and <span class="math-container">$X_{i,t}$</span> are firm and time varying covariates. If the number of time observations is fixed, but the number of firms goes to infinity, then although <span class="math-container">$\gamma$</span> is consistent, the estimator of <span class="math-container">$\alpha_i$</span> will not be consistent, but it will (usually) be unbiased. See also <a href="https://economics.stackexchange.com/q/48426/28130">here</a> for a related discussion.</p>

CC BY-SA 4.0
2023-09-01T12:57:09.237Z
That is surprising to me. I have always been under the impression that the fixed effects $\alpha_i$ are used as controls to ensure the consistency of $\hat \lambda$. Obviously, it comes down to the definition of 'using an estimator' because simply running a regression as mentioned does not amount to using the estimator of $\alpha_i$ because usually one does not base one's conclusions on the estimates of the fixed effect but instead on $\hat \lambda$.
Jesper Hybel
CC BY-SA 4.0
2023-09-01T15:43:24.217Z
Yeah, okay, I'm familiar with this, it just hadn't occured to me. As Jesper wrote, I think of this as an estimator of $\gamma$.
You are right though if you assume $N\rightarrow \infty$ with $T$ fixed. If $T\rightarrow \infty$, then the example doesn't work.

Thanks!
Michael Gmeiner
CC BY-SA 4.0
2023-09-01T11:38:57.533Z
2023-09-01T11:38:57.533Z
tdm
<p>It is well-known that IV estimators (among others) are biased but consistent. I am just curious if any commonly used estimator is unbiased but inconsistent. I realize there may be a distributional assumption that needs to be imposed to get such an example.</p>
<p>More context: I've seen examples of unbiased but inconsistent estimators of the form &quot;only estimate with the first <span class="math-container">$b$</span> observations in a sample of <span class="math-container">$n$</span>, and hold <span class="math-container">$b$</span> fixed as <span class="math-container">$n$</span> goes to infinity&quot;. I can't imagine anybody actually doing this in real life, and I'm hoping to find a more realistic example.</p>

CC BY-SA 4.0
2023-09-01T20:20:36.167Z
Looking at the most recent observation as the estimate of future values would be inconsistent for iid random variables but is commonly done.
Henry
<econometrics>
Are there any unbiased but inconsistent estimators that are commonly used?
Michael Gmeiner
# Are there any unbiased but inconsistent estimators that are commonly used?

(Asked by: username_0 on 2023-09-01)

It is well-known that IV estimators (among others) are biased but consistent. I am just curious if any commonly used estimator is unbiased but inconsistent. I realize there may be a distributional assumption that needs to be imposed to get such an example.

More context: I've seen examples of unbiased but inconsistent estimators of the form "only estimate with the first $b$ observations in a sample of $n$, and hold $b$ fixed as $n$ goes to infinity". I can't imagine anybody actually doing this in real life, and I'm hoping to find a more realistic example.

**Tags:** econometrics
> **Comments:**
> **username_1**: Looking at the most recent observation as the estimate of future values would be inconsistent for iid random variables but is commonly done.  

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## 1 answers

### Answer by username_2 on 2023-09-01. Score: 2

One example that may occur is with fixed effects. Sometimes, we do run regressions like:
$$
y\_{i,t} = \alpha\_i + X\_{i,t} \gamma + \varepsilon\_{i,t}.
$$
Here, $i$ is for example a firm identifier and $t$ represents time and $X\_{i,t}$ are firm and time varying covariates. If the number of time observations is fixed, but the number of firms goes to infinity, then although $\gamma$ is consistent, the estimator of $\alpha\_i$ will not be consistent, but it will (usually) be unbiased. See also [here](https://economics.stackexchange.com/q/48426/28130) for a related discussion.

> **Comments:**
> **username_3**: That is surprising to me. I have always been under the impression that the fixed effects $\alpha_i$ are used as controls to ensure the consistency of $\hat \lambda$. Obviously, it comes down to the definition of 'using an estimator' because simply running a regression as mentioned does not amount to using the estimator of $\alpha_i$ because usually one does not base one's conclusions on the estimates of the fixed effect but instead on $\hat \lambda$.  
> **username_0**: Yeah, okay, I'm familiar with this, it just hadn't occured to me. As Jesper wrote, I think of this as an estimator of $\gamma$. You are right though if you assume $N\rightarrow \infty$ with $T$ fixed. If $T\rightarrow \infty$, then the example doesn't work.  Thanks!  

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