3
$\begingroup$

I want to do a linear regression: $y_{i}=\psi_{j(i)}+X_{i}^{\prime} \xi+\varepsilon_{i}$, where $\psi_{j(i)}$ is a high dimensional fixed effect on group $j$, $X_i$ are the covariables of each observation $i$, and $\varepsilon_{i}$ is error assumed to be exogenous.

This linear regression can be done by many statistic packages through LSMR method avoids inverting the regressor matrix.

However after this regression I want to calculate the size weighted variance of the fixed effects $$\theta_{\psi}=\sum_{j=1}^{J} s_{j}\left(\psi_{j}-\bar{\psi}\right)^{2}$$, where $s_j$ is the share of the group. However the simple "plug-in" estimates $$\begin{aligned} \hat{\theta}_{\psi} =\sum_{j=1}^{J} s_{j}\left(\hat{\psi}_{j}-\hat{\bar{\psi}}\right)^{2} =\sum_{j=1}^{J} s_{j}\left(\hat{\psi}_{j}\right)^{2}-(\hat{\bar{\psi}})^{2} \end{aligned}$$ is upward biased because $$\begin{aligned} \mathbb{E}\left[\hat{\theta}_{\psi}\right] &=\sum_{j=1}^{J} s_{j} \mathbb{E}\left[\left(\hat{\psi}_{j}\right)^{2}\right]-\mathbb{E}\left[(\hat{\psi})^{2}\right] \\ &=\sum_{j=1}^{J} s_{j}\left\{\psi_{j}^{2}+\mathbb{V}\left[\hat{\psi}_{j}\right]\right\}-(\bar{\psi})^{2}-\mathbb{V}[\hat{\bar{\psi}}] \\ &=\theta_{\psi}+\underbrace{\sum_{j=1}^{J} s_{j} \mathbb{V}\left[\hat{\psi}_{j}\right]-\mathbb{V}[\hat{\bar{\psi}}]}_{\text {bias }} \end{aligned}$$.

A simple correction is to use the conventional HC standard error: $$\theta_{\psi} \approx \hat{\theta}_{\psi} - \sum_{j=1}^{J} s_{j} \hat{\mathbb{V}}_{H C}\left[\hat{\psi}_{j}\right]$$.

However the high dimensional effects estimated from LSMR method does not provide std errors. And thus I have no idea how to do this correction.

Another way to do correction is to assume $\mathbb{V}[\varepsilon]=I \sigma^{2}$ and then direct calculate $$\mathbb{V}[\hat{\psi}]=\left(\tilde{F}^{\prime} \tilde{F}\right)^{-1} \sigma^{2}$$, where $\tilde{F}$ is residualized version of $F$, and $F$ is the dummy matrix of ${\psi}_{j(i)}$.

However I have no idea how to regress $F$ on $X$ to get the residual $\tilde{F}$ (how can I regress a dummy matrix on 𝑋?) and also how to inverse the $\left(\tilde{F}^{\prime} \tilde{F}\right)$.

$\endgroup$
3
$\begingroup$

I guess this follows from the Frish-Waugh Lovell theorem.

If you have $K$ dummies and say $n$ observations, so $F$ is $n \times K$, you need to regress every dummy in $F$ on all variables in $X$ separately ($K$ regressions). The residuals make up the matrix $\bar F$ which is again of dimension $n \times K$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.