<p>Typically under the model that looks like this:</p>
<p><span class="math-container">$$ y_{it} = \rho y_{it-1} + u_i + e_{it} = \rho y_{it-1} + v_{it} $$</span></p>
<p>Define the OLS estimator as the estimator for <span class="math-container">$\rho$</span> that estimates the following equation.</p>
<p><span class="math-container">$$ y_{it} = \rho y_{it-1} + u_{it} $$</span></p>
<p>Define the FE estimator as the estimator for <span class="math-container">$\rho$</span> that includes fixed effects:</p>
<p><span class="math-container">$$ y_{it} = \rho y_{it-1} + u_i + e_{it} $$</span></p>
<p>We know that if <span class="math-container">$\rho&gt;0$</span>, then the OLS estimator for <span class="math-container">$\rho$</span> is biased upwards and the fixed effects estimator for <span class="math-container">$\rho$</span> is biased downwards, according to the Nickell bias, and therefore we can bound the coefficient by using OLS and FE estimators.</p>
<p>However, I'm wondering if the case where we have this model:</p>
<p><span class="math-container">$$ y_{it} = \rho y_{it-1} + \beta x_{it} + u_i + e_{it} $$</span></p>
<p>Is the OLS estimator for <span class="math-container">$\beta$</span> (not for <span class="math-container">$\rho$</span>) still biased upwards and the fixed effects estimator biased downwards?
What if we use a nonparametric estimator for <span class="math-container">$\rho$</span>? aka instead of estimating the original equation, we now have:</p>
<p><span class="math-container">$$ y_{it} = f(y_{it-1}) + \beta x_{it} + u_i + e_{it} $$</span></p>
<p>What if we don't know whether the <span class="math-container">$t-2,...$</span> variables factor in?</p>
<p><span class="math-container">$$ y_{it} = f(y_{it-1}, y_{it-2}, y_{it-3}, ...) + \beta x_{it} + u_i + e_{it} $$</span></p>
<p>Thanks for your time in advance!</p>

CC BY-SA 4.0
2024-06-11T08:36:57.523Z
My first thought is that, even if you know the direction of the expected bias, in any finite sample the estimate could be very different. Thus, there's no guarantee for performance.

Having said that, no I don't know, but I'm guessing you can create fake data and simulate this situation, then get an idea. Good luck!
Michael Gmeiner
CC BY-SA 4.0
2024-06-13T18:57:39.273Z
ok, fair, but that could be said of any biased estimator.
Daycent
<econometrics><statistics><time-series><paneldata>
General conditions in which dynamic panel effects are bounded by OLS and FE models?
Daycent
# General conditions in which dynamic panel effects are bounded by OLS and FE models?

(Asked by: username_0 on 2024-06-10)

Typically under the model that looks like this:

$$ y\_{it} = \rho y\_{it-1} + u\_i + e\_{it} = \rho y\_{it-1} + v\_{it} $$

Define the OLS estimator as the estimator for $\rho$ that estimates the following equation.

$$ y\_{it} = \rho y\_{it-1} + u\_{it} $$

Define the FE estimator as the estimator for $\rho$ that includes fixed effects:

$$ y\_{it} = \rho y\_{it-1} + u\_i + e\_{it} $$

We know that if $\rho>0$, then the OLS estimator for $\rho$ is biased upwards and the fixed effects estimator for $\rho$ is biased downwards, according to the Nickell bias, and therefore we can bound the coefficient by using OLS and FE estimators.

However, I'm wondering if the case where we have this model:

$$ y\_{it} = \rho y\_{it-1} + \beta x\_{it} + u\_i + e\_{it} $$

Is the OLS estimator for $\beta$ (not for $\rho$) still biased upwards and the fixed effects estimator biased downwards?
What if we use a nonparametric estimator for $\rho$? aka instead of estimating the original equation, we now have:

$$ y\_{it} = f(y\_{it-1}) + \beta x\_{it} + u\_i + e\_{it} $$

What if we don't know whether the $t-2,...$ variables factor in?

$$ y\_{it} = f(y\_{it-1}, y\_{it-2}, y\_{it-3}, ...) + \beta x\_{it} + u\_i + e\_{it} $$

Thanks for your time in advance!

**Tags:** econometrics, statistics, time-series, paneldata

> **Comments:**
> **username_1**: My first thought is that, even if you know the direction of the expected bias, in any finite sample the estimate could be very different. Thus, there's no guarantee for performance.  Having said that, no I don't know, but I'm guessing you can create fake data and simulate this situation, then get an idea. Good luck!  
> **username_0**: ok, fair, but that could be said of any biased estimator.  

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## 0 answers

