$\sup_{\widetilde{\xi},\alpha}\|\frac{\partial}{\partial\xi^{\prime}}\big{|}_{\xi=\widetilde{\xi}}\mathbb{E}_{\xi_{i0}=\xi}(\mathop{\mathrm{v}ec}\frac{\partial^{2}\ell_{it}(\alpha,\theta_{0})}{\partial\alpha\partial{\alpha}^{\prime}})\|$;
and
$\sup_{\widetilde{\xi},\theta}\|\frac{\partial}{\partial\xi^{\prime}}\big{|}_{\xi=\widetilde{\xi}}\mathbb{E}_{\xi_{i0}=\xi}(\frac{\partial\ell_{it}(\overline{\alpha}(\theta,\widetilde{\xi}),\theta)}{\partial\alpha})\|$
are $O(1)$.

  5. (v)

$\mathbb{E}_{h_{i}=h,\xi_{i0}=\xi}(\frac{\partial\ell_{it}(\overline{\alpha}(\theta,\xi),\theta)}{\partial\alpha})$
is twice differentiable with respect to $h$ and $\xi$, with first and second
derivatives that are uniformly bounded in $\xi$, $h\in{\cal{H}}$, and
$\theta\in\Theta$; and
$\|{\mathop{\mathrm{V}ar}}_{h_{i}=h,\xi_{i0}=\xi}(\frac{\partial\ell_{i}(\overline{\alpha}(\theta,\xi),\theta)}{\partial\alpha})\|=O(1)$,
uniformly in $\xi$, $h$ and $\theta$.

######  Corollary S2.

Let the conditions of Lemmas 1 and 2 hold. Let Assumptions 2, S3, and S4 hold.
Let $K$ be given by (11), with $\gamma=O(1)$. Then, as $N,T,K$ tend to
infinity such that $T^{1+\frac{d}{2}}=O(N)$ we have:

$\displaystyle\widehat{\theta}$ $\displaystyle=$
$\displaystyle\theta_{0}+O_{p}\left(\frac{1}{T}\right)+O_{p}\left(\frac{1}{\sqrt{NT}}\right).$
(S23)

###### Proof.

Let $\delta=\frac{1}{T}+\frac{K}{N}+K^{-\frac{2}{d}}$.444Note that if
$K=\widehat{K}$ is given by (11) with $\gamma{=}O(1)$, then
$K{=}O(T^{\frac{d}{2}})$ and
$\delta{=}O(\frac{1}{T}{+}\frac{T^{\frac{d}{2}}}{N})$, so if
$T^{1+\frac{d}{2}}=O(N)$ then $\delta{=}O(\frac{1}{T})$. To show consistency,
the key step is to show:

$\displaystyle\frac{1}{N}\sum_{i}\left\|\overline{v}(\widehat{k}_{i},\theta)\right\|^{2}=O_{p}\left(\delta\right),\quad\forall\theta\in\Theta.$
(S24)

Let, for all $\theta,h,\xi$:
$\rho(h,\xi,\theta)=\mathbb{E}_{h_{i}=h,\xi_{i0}=\xi}(v_{i}(\overline{\alpha}(\theta,\xi),\theta))$,
and let, for all $i,\theta$:
$\zeta_{i}(\theta)=v_{i}(\overline{\alpha}(\theta,\xi_{i0}),\theta)-\rho(h_{i},\xi_{i0},\theta)$.
One can show, using similar techniques to the proof of Lemma A1, that:
$\frac{1}{N}\sum_{i}\|\overline{\zeta}(\widehat{k}_{i},\theta)\|^{2}=O_{p}(\frac{K}{N})$,
and that this implies (S24).555Note that, in the case of Theorem 1 (i.e., in
the absence of additional heterogeneity $\nu_{i0}$), the left-hand side in
(S24) is $O_{p}(\frac{1}{T})$.

We then show:
$\frac{1}{N}\sum_{i}\frac{\partial\ell_{i}(\widehat{\alpha}(\widehat{k}_{i},\theta_{0}),\theta_{0})}{\partial\theta}=O_{p}(\delta)$,
which will follow from:

$\frac{1}{N}\sum_{i}v_{i}^{\theta}\left(\widehat{\alpha}(\widehat{k}_{i})-\alpha_{i0}\right)=O_{p}\left(\delta\right),$
(S25)

where from now on we omit references to $\theta_{0}$ and $\alpha_{i0}$. We
have:

$\displaystyle\frac{1}{N}\sum_{i}v_{i}^{\theta}\left(\widehat{\alpha}(\widehat{k}_{i})-\alpha_{i0}\right)=\frac{1}{N}\sum_{i}v_{i}^{\theta}\left(\widetilde{\alpha}(\widehat{k}_{i})-\alpha_{i0}+\widetilde{v}(\widehat{k}_{i})\right)+O_{p}\left(\delta\right),$

where $\widetilde{\alpha}(k)$ and $\widetilde{v}(k)$ are as in the proof of
Lemma A2; that is, denoting $w_{i}=(-v_{i}^{\alpha})$, we have
$\widetilde{\alpha}(k)=\overline{w}(k)^{-1}\overline{w\alpha}(k)$ and
$\widetilde{v}(k)=\overline{w}(k)^{-1}\overline{v}(k)$.

Let $\gamma_{v}(h_{i})=\mathbb{E}_{h_{i}}(v_{i})$,
$\zeta_{i}^{v}=v_{i}-\gamma_{v}(h_{i})$,
$\gamma_{w}(h_{i})=\mathbb{E}_{h_{i}}(w_{i})$,
$\zeta_{i}^{w}=w_{i}-\gamma_{w}(h_{i})$,
$\gamma_{v^{\theta}}(h_{i})=\mathbb{E}_{h_{i}}(v_{i}^{\theta})$, and
$\zeta_{i}^{v^{\theta}}=v_{i}^{\theta}-\gamma_{v^{\theta}}(h_{i})$. First, we
have:

$\displaystyle\frac{1}{N}\sum_{i}v_{i}^{\theta}\widetilde{v}(\widehat{k}_{i})=\frac{1}{N}\sum_{i}v_{i}^{\theta}\overline{w}(\widehat{k}_{i})^{-1}\overline{v}(\widehat{k}_{i})=\frac{1}{N}\sum_{i}\overline{v}^{\theta}(\widehat{k}_{i})\overline{w}(\widehat{k}_{i})^{-1}v_{i}$
$\displaystyle{=}\frac{1}{N}\sum_{i}(\overline{\gamma}_{v^{\theta}}(\widehat{k}_{i})+\overline{\zeta}^{v^{\theta}}(\widehat{k}_{i}))(\overline{\gamma}_{w}(\widehat{k}_{i})+\overline{\zeta}^{w}(\widehat{k}_{i}))^{-1}v_{i}{=}\frac{1}{N}\sum_{i}\overline{\gamma}_{v^{\theta}}(\widehat{k}_{i})\overline{\gamma}_{w}(\widehat{k}_{i})^{-1}v_{i}+O_{p}(\delta),$

where for example $\overline{\gamma}_{w}(k)$ is the mean of
$\gamma_{w}(h_{i})$ in group $\widehat{k}_{i}=k$, and we have used that
$\frac{1}{N}\sum_{i}\|\overline{\zeta}^{v^{\theta}}(\widehat{k}_{i})\|^{2}=O_{p}(K/N)$,
$\frac{1}{N}\sum_{i}\|\overline{\zeta}^{w}(\widehat{k}_{i})\|^{2}=O_{p}(K/N)$,
and $\frac{1}{N}\sum_{i}\|v_{i}\|^{2}=O_{p}(1/T)$. Moreover:

$\displaystyle\frac{1}{N}\sum_{i}\overline{\gamma}_{v^{\theta}}(\widehat{k}_{i})\overline{\gamma}_{w}(\widehat{k}_{i})^{-1}v_{i}$
$\displaystyle=\frac{1}{N}\sum_{i}\overline{\gamma}_{v^{\theta}}(\widehat{k}_{i})\overline{\gamma}_{w}(\widehat{k}_{i})^{-1}\gamma_{v}(h_{i})+O_{p}(\delta),$

where we have used that
$\frac{1}{N}\sum_{i}\|\overline{\zeta}^{v}(\widehat{k}_{i})\|^{2}=O_{p}(K/(NT))$.
Lastly, we have:

$\displaystyle\frac{1}{N}\sum_{i}\overline{\gamma}_{v^{\theta}}(\widehat{k}_{i})\overline{\gamma}_{w}(\widehat{k}_{i})^{-1}\gamma_{v}(h_{i})$
$\displaystyle=\frac{1}{N}\sum_{i}{\gamma}_{v^{\theta}}(h_{i}){\gamma}_{w}(h_{i})^{-1}\gamma_{v}(h_{i})$
$\displaystyle+\frac{1}{N}\sum_{i}\left[\overline{\gamma}_{v^{\theta}}(\widehat{k}_{i})\overline{\gamma}_{w}(\widehat{k}_{i})^{-1}-{\gamma}_{v^{\theta}}(h_{i}){\gamma}_{w}(h_{i})^{-1}\right]\gamma_{v}(h_{i}),$

where the first term is $O_{p}(\delta)$ since it is a mean of i.i.d. terms
with mean $O(1/T)$ and variance $O(1/T)$, and the second term is
$O_{p}(\delta)$ since
$\frac{1}{N}\sum_{i}\|h_{i}-\overline{h}(\widehat{k}_{i})\|^{2}=O_{p}(\delta)$
and the $\gamma$ functions are Lipschitz-continuous.

Second, let $v_{i}^{\theta}w_{i}^{-1}=\eta(h_{i},\xi_{i0})+e_{i}$, where
$\mathbb{E}_{h_{i}=h,\xi_{i0}=\xi}(e_{i}w_{i})=0$. We have:

$\displaystyle\frac{1}{N}{\sum_{i}}v_{i}^{\theta}\left(\widetilde{\alpha}(\widehat{k}_{i}){-}\alpha_{i0}\right){=}\frac{1}{N}{\sum_{i}}\eta(h_{i},\xi_{i0})w_{i}\left(\widetilde{\alpha}(\widehat{k}_{i}){-}\alpha_{i0}\right){+}\frac{1}{N}{\sum_{i}}e_{i}w_{i}\left(\widetilde{\alpha}(\widehat{k}_{i}){-}\alpha_{i0}\right),$

where the first term is $O_{p}(\delta)$ since
$\frac{1}{N}\sum_{i}\|h_{i}-\overline{h}(\widehat{k}_{i})\|^{2}=O_{p}(\delta)$,
$\frac{1}{N}\sum_{i}\|\xi_{i0}-\overline{\xi}(\widehat{k}_{i})\|^{2}=O_{p}(\delta)$,
$\frac{1}{N}\sum_{i}\|\widetilde{\alpha}(\widehat{k}_{i})-\alpha_{i0}\|^{2}=O_{p}(\delta)$,
$\eta$ is Lipschitz-continuous, and $w_{i}$ is uniformly bounded (as in the
proof of Lemma A2), and the second term is:

$\displaystyle\frac{1}{N}\sum_{i}e_{i}w_{i}\left(\widetilde{\alpha}(\widehat{k}_{i})-\alpha_{i0}\right)$
$\displaystyle{=}\frac{1}{N}\sum_{i}e_{i}w_{i}\left(\widetilde{\alpha}(\widehat{k}_{i}){-}\overline{\alpha}(\widehat{k}_{i})\right)+\frac{1}{N}\sum_{i}e_{i}w_{i}\left(\overline{\alpha}(\widehat{k}_{i}){-}\alpha_{i0}\right)$
$\displaystyle{=}\frac{1}{N}\sum_{i}\overline{ew}(\widehat{k}_{i})\left(\widetilde{\alpha}(\widehat{k}_{i})-\overline{\alpha}(\widehat{k}_{i})\right)+O_{p}(\delta)=O_{p}(\delta),$

where we have used that the $(e_{i}w_{i})$’s have zero mean given
$h_{1},...,h_{N},\xi_{10},...,\xi_{N0}$ with bounded conditional variance, and
$\frac{1}{N}\sum_{i}\|\overline{ew}(\widehat{k}_{i})\|^{2}=O_{p}(K/N)=O_{p}(\delta)$.

Finally, to show:
$\frac{1}{N}\sum_{i}\frac{\partial^{2}}{\partial\theta\partial\theta^{\prime}}\big{|}_{\theta_{0}}\,(\ell_{i}(\widehat{\alpha}(\widehat{k}_{i},\theta),\theta)-\ell_{i}(\overline{\alpha}(\theta,\xi_{i0}),\theta))=o_{p}(1)$,
we use similar arguments to the proof of Lemma A3.666Although the arguments
are as in the proof of Lemma A3, the target log-likelihood is different since
here $\overline{\alpha}(\theta,\xi_{i0})$ only depends on $\xi_{i0}$, not on
$(\xi_{i0}^{\prime},\nu_{i0}^{\prime})^{\prime}$. In particular, the matrix
$H^{\mathrm{c}ond}$ in Assumption S4 differs from the matrix $H$ in Assumption
3; see (S26) for an example.

#### Example: a linear homoskedastic model.

Consider the model $Y_{it}=X_{it}\theta_{0}+\alpha_{i0}+U_{it}$, where
$X_{it}$ are scalar and $U_{it}$ are i.i.d. with mean zero and variance
$\sigma^{2}$ given $X_{i1},...,X_{iT},\alpha_{i0}$. Let $\widehat{\theta}$ be
the GFE estimator based on a moment
$h_{i}=\varphi(\alpha_{i0})+\varepsilon_{i}$ that satisfies Assumptions 1 and
2 for $\xi_{i0}=\alpha_{i0}$; that is, $h_{i}$ is only informative about
$\alpha_{i0}$, but not about the heterogeneity in $X_{it}$. Let
$\zeta_{i}^{X}=\overline{X}_{i}-\mathbb{E}_{h_{i}}(\overline{X}_{i})$,
$\zeta_{i}^{\alpha}={\alpha}_{i0}-\mathbb{E}_{h_{i}}(\alpha_{i0})$, and
$\zeta_{i}^{U}=\overline{U}_{i}-\mathbb{E}_{h_{i}}(\overline{U}_{i})$. We
assume that $K$ is large enough for the approximation error to be of smaller
order, and that $K/N$ tends to zero, as in Corollary 2. Under appropriate
conditions in the regression model, using similar arguments to the proof of
Corollary S2 (though with no need for any restriction on the relative rates of
$N$ and $T$), one can show that $\widehat{\theta}$ admits the following
expansion:

$\displaystyle\widehat{\theta}{=}\theta_{0}{+}\frac{\frac{1}{N}\sum_{i}\zeta^{X}_{i}(\zeta^{\alpha}_{i}+\zeta^{U}_{i})+\frac{1}{NT}\sum_{i,t}(X_{it}-\overline{X}_{i})(U_{it}-\overline{U}_{i})}{\mathbb{E}[(X_{it}-\overline{X}_{i})^{2}]+\mathop{\mathrm{V}ar}(\zeta^{X}_{i})}{+}o_{p}\left(\frac{1}{T}\right){+}o_{p}\left(\frac{1}{\sqrt{NT}}\right).$
(S26)

Notice two differences between (S26) and the expansion of the FE estimator:
the presence of $\mathop{\mathrm{V}ar}(\zeta^{X}_{i})$ in the denominator, and
the presence of
$\frac{1}{N}\sum_{i}\zeta^{X}_{i}(\zeta^{\alpha}_{i}+\zeta^{U}_{i})$ in the
numerator. In addition, notice that (S26) simplifies to the expression in
Corollary 2 in the absence of additional heterogeneity $\nu_{i0}$.

##  Appendix S3 Simulations

#### Model of wages and participation (see (5)).

We model the initial condition as:
$Y_{i0}=\boldsymbol{1}\left\\{u(\alpha_{i0})\geq
c(1;\theta_{0})+U_{i0}\right\\}$, with $U_{i0}$ standard normal, independent
of $\alpha_{i0}$. We set $c(0;\theta_{0})=0$ and $c(1;\theta_{0})=-1$. We set
$\alpha_{i0}$ and $V_{it}$ to be independent standard normals. In the
simulations based on models (5) and (8) we weight the moments by the share of
between-i variance to total variance.777Specifically, we demean and rescale
$h_{i}$ so that all its components $h_{i\ell}$ have zero mean and unit
variance, and multiply each component $h_{i\ell}$ by:
$\mathop{\mathrm{m}ax}\left(\frac{\sum_{i}h_{i\ell}^{2}-\frac{1}{T^{2}}\sum_{i,t}(h_{it\ell}-h_{i\ell})^{2}}{\sum_{i}h_{i\ell}^{2}},0\right)$.
Using equal weights instead has small effects in these simulations, however we
observed that this particular weighting can improve performance when some
moments are substantially less informative about the heterogeneity than
others.  To compute the variance $\widehat{V}_{h}$ to set the number of groups
in this dynamic model, we use a Newey-West expression with one lag. Lastly,
for kmeans computation we use Lloyd’s algorithm with 100 random starting
values. Table S1 shows additional simulation results for this model.

#### Probit model with time-varying heterogeneity (see (8)).

The $U_{it}$’s are standard normal independent of the $X_{it}$’s and the
$\alpha_{it0}$’s. The data generating process (DGP) for the scalar covariate
is: $X_{it}=\mu_{it0}+V_{it}$, where $V_{it}$ are i.i.d. standard normal
independent of the $U_{it}$’s, $\alpha_{it0}$’s, and $\mu_{it0}$’s, and
$\mu_{it0}=\alpha_{it0}$. We set $\theta_{0}=1$, and set $\xi_{i0}$ and
$\lambda_{t0}$ to be i.i.d. Gamma(1,1) draws, independent of each other. Table
S2 shows additional simulation results for this model, including for the two-
way GFE estimator based on both the cross-sectional moments
$(\frac{1}{N}\sum_{i}Y_{it},\frac{1}{N}\sum_{i}X_{it})^{\prime}$, and the
individual-specific moments $(\overline{Y}_{i},\overline{X}_{i})^{\prime}$.

#### Conditional moments: an example.

Consider the following probit model:
$Y_{it}=\boldsymbol{1}\\{X_{it}^{\prime}\theta_{0}+\alpha_{i0}+U_{it}\geq
0\\}$, where the $U_{it}$ are i.i.d. standard normal independent of the
$X_{it}$’s and $\alpha_{i0}$, and $\theta_{0}$ is a vector of ones. The DGP
for the $k$-th covariate is:
$X_{itk}=\boldsymbol{1}\\{\mu_{i0k}+V_{itk}>0\\}$, where $V_{itk}$ are i.i.d.
standard normal independent of the $U_{it}$’s, $\alpha_{i0}$, and the
$\mu_{i0k}$’s, and $\alpha_{i0}$ and the $\mu_{i0k}$’s follow independent
standard normals. We vary the number of covariates between $1$ and $3$, so the
total dimension of heterogeneity varies between $2$ and $4$. In this model, we
expect the bias of FE to be moderate given the time horizon we consider
($T=20$), since $\alpha_{i0}$ is scalar and FE is a conditional approach. The
question we ask here is how much the use of conditional moments can help
reduce the bias of GFE due to the presence of additional heterogeneity in the
covariates and the increased dimensionality of heterogeneity (see Subsection
4.2).

Consider first using
$h_{i}=(\overline{Y}_{i},\overline{X}_{i}^{\prime})^{\prime}$ as moments. In
Table S3 we show the biases, standard deviations, and root mean squared errors
of FE and GFE among 1000 simulations, for N=1000 and T=20. In the top panel we
report GFE estimates as a function of the number of groups $K$. We see that,
while the bias of GFE remains moderate with one covariate, the bias increases
substantially with the dimension of heterogeneity, in agreement with our
theory. By comparison, the bias of FE in the bottom panel is indeed quite
small, and it only increases moderately with the number of covariates.

The situation is rather different when using conditional moments in GFE. In
the middle panel in Table S3 we show simulation results for GFE based on
covariates-specific conditional means
$\overline{Y}_{i}(x)=\sum_{t=1}^{T}\boldsymbol{1}\\{X_{it}=x\\}Y_{it}/\sum_{t=1}^{T}\boldsymbol{1}\\{X_{it}=x\\}$.
Importantly, in large samples these moments are only informative about
$\alpha_{i0}$, not $\mu_{i0}$. We see that the bias of GFE with conditional
moments increases only moderately with the number of covariates, and that FE
and GFE with conditional moments have comparable — and quite small — biases.

Regarding implementation, note that, for a given $i$, all moments
$\overline{Y}_{i}(x)$ may not be available since $i$’s covariates may never
take the value $x$ in the sample. In Table S3, whenever $\overline{Y}_{i}(x)$
is not available, we set the moment to an imputed value, the overall
conditional mean
$\overline{Y}(x)=\sum_{i,t}\boldsymbol{1}\\{X_{it}=x\\}Y_{it}/\sum_{i,t}\boldsymbol{1}\\{X_{it}=x\\}$.
The imputation does not affect the theory, provided the event that any of the
$\overline{Y}_{i}(x)$’s is not available tends to zero with probability
approaching one in large samples.888To provide intuition in a simple case,
suppose that $X_{it}$ are binary, i.i.d. over time given $\mu_{i0}$, with
$\Pr(X_{it}=1\,|\,\mu_{i0}=\mu)\in(\epsilon,1-\epsilon)$ for all $\mu$, for
some $\epsilon>0$. Then $\Pr(\exists i\,:\,X_{i1}=...=X_{iT}=0)\leq
N(1-\epsilon)^{T}$, which tends to zero whenever $(\ln N)/T\rightarrow 0$.
Moreover, we have obtained similar results using an alternative conditional
first step implementation that does not rely on imputations.999This
implementation is as follows. Let $I_{i}(x)$ be the indicator that there
exists a $t$ such that $X_{it}=x$, and let $x_{1},...,x_{M}$ denote the points
of support of $X_{it}$. In the first step, we use a Lloyd’s-like algorithm to
minimize the function
$\sum_{i=1}^{N}\sum_{m=1}^{M}I_{i}(x_{m})\left(\overline{Y}_{i}(x_{m})-g(x_{m},k_{i})\right)^{2}$,
with respect to $k_{1},...,k_{N}$ and $g(x_{1},1)$, …, $g(x_{M},K)$.

Table S1: Model (5) of wages and participation

T | Bias | std | RMSE | se/std | Bias | std | RMSE | se/std
---|---|---|---|---|---|---|---|---
| GFE, $\eta=1$ | FE, $\eta=1$
5 | -0.570 | 0.058 | 0.573 | 1.082 | -0.835 | 0.064 | 0.837 | 1.066
10 | -0.207 | 0.040 | 0.211 | 1.003 | -0.418 | 0.040 | 0.420 | 1.041
20 | -0.088 | 0.027 | 0.092 | 0.993 | -0.209 | 0.026 | 0.211 | 1.064
30 | -0.055 | 0.023 | 0.060 | 0.960 | -0.140 | 0.023 | 0.142 | 0.991
40 | -0.040 | 0.019 | 0.044 | 1.000 | -0.105 | 0.019 | 0.106 | 1.034
50 | -0.031 | 0.017 | 0.036 | 0.982 | -0.084 | 0.017 | 0.086 | 1.022
| GFE, $\eta=2$ | FE, $\eta=2$
5 | -0.519 | 0.063 | 0.523 | 1.052 | -0.876 | 0.068 | 0.879 | 1.063
10 | -0.163 | 0.043 | 0.169 | 0.985 | -0.442 | 0.041 | 0.444 | 1.070
20 | -0.049 | 0.031 | 0.058 | 0.929 | -0.225 | 0.028 | 0.227 | 1.042
30 | -0.032 | 0.024 | 0.040 | 0.964 | -0.153 | 0.022 | 0.154 | 1.068
40 | -0.019 | 0.020 | 0.028 | 0.981 | -0.113 | 0.019 | 0.115 | 1.045
50 | -0.015 | 0.019 | 0.024 | 0.944 | -0.091 | 0.018 | 0.093 | 1.000

Notes: $1000$ simulations, $N=1000$. “RMSE” is root mean squared error, “se”
is the average of standard error estimates across simulations, “std” is the
standard deviation of the estimator across simulations. $\eta$ is the risk
aversion parameter.

Table S2: Probit model (8) with time-varying heterogeneity

T | Bias | std | RMSE | se/std | Bias | std | RMSE | se/std | Bias | std | RMSE | se/std | Bias | std | RMSE | se/std
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| 2-way GFE, $\sigma{=}-1$0 | GFE, $\sigma{=}-1$0 | FE, $\sigma{=}-1$0 | IFE, $\sigma{=}-1$0
5 | -0.045 | 0.035 | 0.057 | 0.927 | -0.044 | 0.035 | 0.056 | 0.926 | 0.442 | 0.071 | 0.448 | 0.706 | 0.116 | 0.064 | 0.133 | 0.473
10 | -0.016 | 0.024 | 0.028 | 0.939 | -0.014 | 0.024 | 0.028 | 0.939 | 0.198 | 0.036 | 0.201 | 0.762 | 0.100 | 0.036 | 0.107 | 0.488
20 | -0.003 | 0.016 | 0.016 | 1.014 | -0.000 | 0.016 | 0.016 | 1.014 | 0.098 | 0.019 | 0.100 | 0.911 | 0.087 | 0.020 | 0.089 | 0.596
30 | -0.000 | 0.013 | 0.013 | 1.009 | 0.003 | 0.013 | 0.013 | 1.013 | 0.069 | 0.014 | 0.070 | 0.966 | 0.059 | 0.014 | 0.061 | 0.675
40 | 0.001 | 0.011 | 0.011 | 1.021 | 0.005 | 0.011 | 0.012 | 1.016 | 0.055 | 0.012 | 0.057 | 0.949 | 0.044 | 0.012 | 0.046 | 0.676
50 | 0.001 | 0.010 | 0.010 | 0.995 | 0.006 | 0.010 | 0.012 | 0.994 | 0.048 | 0.011 | 0.049 | 0.947 | 0.036 | 0.011 | 0.037 | 0.677
| 2-way GFE, $\sigma{=}0$ | GFE, $\sigma{=}0$ | FE, $\sigma{=}0$ | IFE, $\sigma{=}0$
5 | -0.044 | 0.042 | 0.060 | 0.883 | -0.043 | 0.042 | 0.060 | 0.882 | 0.488 | 0.091 | 0.497 | 0.654 | 0.152 | 0.089 | 0.176 | 0.398
10 | -0.022 | 0.026 | 0.035 | 0.951 | -0.021 | 0.026 | 0.034 | 0.949 | 0.226 | 0.045 | 0.231 | 0.710 | 0.118 | 0.040 | 0.125 | 0.494
20 | -0.009 | 0.018 | 0.020 | 0.969 | -0.006 | 0.018 | 0.019 | 0.964 | 0.108 | 0.023 | 0.110 | 0.843 | 0.099 | 0.023 | 0.101 | 0.571
30 | -0.004 | 0.014 | 0.015 | 1.001 | 0.001 | 0.014 | 0.014 | 1.000 | 0.072 | 0.017 | 0.074 | 0.918 | 0.068 | 0.017 | 0.070 | 0.615
40 | -0.002 | 0.013 | 0.013 | 0.989 | 0.004 | 0.013 | 0.013 | 0.985 | 0.056 | 0.014 | 0.058 | 0.922 | 0.051 | 0.014 | 0.052 | 0.639
50 | -0.001 | 0.011 | 0.012 | 0.965 | 0.005 | 0.012 | 0.013 | 0.961 | 0.046 | 0.012 | 0.047 | 0.926 | 0.040 | 0.012 | 0.042 | 0.643
| 2-way GFE, $\sigma{=}1$ | GFE, $\sigma{=}1$ | FE, $\sigma{=}1$ | IFE, $\sigma{=}1$
5 | -0.049 | 0.056 | 0.074 | 0.754 | -0.048 | 0.056 | 0.074 | 0.754 | 0.565 | 0.146 | 0.583 | 0.506 | 0.207 | 0.117 | 0.238 | 0.359
10 | -0.032 | 0.029 | 0.043 | 0.981 | -0.030 | 0.029 | 0.042 | 0.979 | 0.251 | 0.062 | 0.258 | 0.603 | 0.141 | 0.043 | 0.147 | 0.513
20 | -0.014 | 0.020 | 0.024 | 0.986 | -0.010 | 0.020 | 0.022 | 0.983 | 0.114 | 0.027 | 0.117 | 0.825 | 0.125 | 0.029 | 0.128 | 0.514
30 | -0.007 | 0.016 | 0.017 | 0.996 | -0.001 | 0.016 | 0.016 | 0.992 | 0.074 | 0.019 | 0.077 | 0.889 | 0.085 | 0.021 | 0.088 | 0.561
40 | -0.005 | 0.014 | 0.015 | 0.985 | 0.001 | 0.014 | 0.014 | 0.980 | 0.055 | 0.016 | 0.057 | 0.915 | 0.063 | 0.016 | 0.065 | 0.611
50 | -0.003 | 0.012 | 0.012 | 1.027 | 0.004 | 0.012 | 0.013 | 1.025 | 0.044 | 0.014 | 0.046 | 0.951 | 0.050 | 0.014 | 0.052 | 0.632
| 2-way GFE, $\sigma{=}1$0 | GFE, $\sigma{=}1$0 | FE, $\sigma{=}1$0 | IFE, $\sigma{=}1$0
5 | -0.016 | 0.075 | 0.076 | 0.691 | -0.015 | 0.075 | 0.076 | 0.692 | 0.706 | 0.262 | 0.753 | 0.386 | 0.300 | 0.255 | 0.394 | 0.218
10 | -0.013 | 0.035 | 0.037 | 0.946 | -0.010 | 0.035 | 0.036 | 0.947 | 0.323 | 0.097 | 0.337 | 0.486 | 0.183 | 0.060 | 0.192 | 0.458
20 | -0.002 | 0.024 | 0.024 | 0.975 | 0.003 | 0.024 | 0.024 | 0.967 | 0.150 | 0.037 | 0.154 | 0.719 | 0.168 | 0.036 | 0.172 | 0.474
30 | 0.002 | 0.019 | 0.019 | 0.991 | 0.008 | 0.019 | 0.021 | 0.983 | 0.100 | 0.025 | 0.104 | 0.814 | 0.121 | 0.030 | 0.125 | 0.456
40 | 0.003 | 0.016 | 0.016 | 0.989 | 0.010 | 0.016 | 0.019 | 0.985 | 0.076 | 0.020 | 0.079 | 0.851 | 0.091 | 0.021 | 0.093 | 0.536
50 | 0.002 | 0.014 | 0.015 | 0.995 | 0.010 | 0.014 | 0.018 | 0.996 | 0.061 | 0.017 | 0.063 | 0.911 | 0.073 | 0.017 | 0.075 | 0.592

Notes: See notes to Table S1. IFE is interacted fixed-effects with one factor.
$\sigma$ is the substitution parameter.

Table S3: Probit model with binary covariates

K | Bias | std | RMSE | Bias | std | RMSE | Bias | std | RMSE
---|---|---|---|---|---|---|---|---|---
| GFE, 1 covariate | GFE, 2 covariates | GFE, 3 covariates
5 | -0.189 | 0.029 | 0.191 | -0.293 | 0.031 | 0.295 | -0.362 | 0.042 | 0.365
10 | -0.083 | 0.027 | 0.088 | -0.205 | 0.032 | 0.207 | -0.275 | 0.035 | 0.278
20 | -0.017 | 0.029 | 0.033 | -0.118 | 0.030 | 0.122 | -0.206 | 0.033 | 0.209
30 | 0.006 | 0.029 | 0.030 | -0.081 | 0.030 | 0.086 | -0.166 | 0.032 | 0.169
40 | 0.018 | 0.029 | 0.035 | -0.056 | 0.030 | 0.064 | -0.136 | 0.033 | 0.140
50 | 0.026 | 0.030 | 0.039 | -0.040 | 0.031 | 0.051 | -0.116 | 0.033 | 0.120
| Cond. GFE, 1 covariate | Cond. GFE, 2 covariates | Cond. GFE, 3 covariates
5 | -0.060 | 0.035 | 0.069 | -0.085 | 0.037 | 0.093 | -0.111 | 0.039 | 0.117
10 | -0.045 | 0.033 | 0.056 | -0.073 | 0.043 | 0.085 | -0.100 | 0.044 | 0.109
20 | -0.015 | 0.034 | 0.038 | -0.046 | 0.037 | 0.059 | -0.075 | 0.045 | 0.087
30 | 0.008 | 0.036 | 0.036 | -0.031 | 0.036 | 0.047 | -0.061 | 0.043 | 0.075
40 | 0.025 | 0.035 | 0.043 | -0.020 | 0.037 | 0.041 | -0.050 | 0.042 | 0.065
50 | 0.034 | 0.035 | 0.049 | -0.012 | 0.036 | 0.038 | -0.040 | 0.041 | 0.057
| FE, 1 covariate | FE, 2 covariates | FE, 3 covariates
- | 0.062 | 0.031 | 0.069 | 0.074 | 0.034 | 0.081 | 0.088 | 0.039 | 0.097

Notes: $1000$ simulations, $N=1000$, $T=20$. In the top panel we show GFE
estimates based on unconditional moments for different $K$ values, in the
middle panel we show GFE estimates based on conditional moments for different
$K$ values, in the bottom row we show FE estimates.