In the early days of recording, the microphone could only cleanly capture a narrow frequency range. So engineers were like: we’ll just record what the microphone handles. Fair enough.

Then the microphone got better. Then digital audio arrived. Then we could capture everything.

But by then, scholars had already proven theoretically that music only contains those frequencies. The proof was very elegant. It won a Gramophone award. And to this day, grad students learn that a symphony is just a low-frequency hum with robust standard errors.

Well, but nevertheless you can still do a lot with double bass, bassons and tubas.

Consider the following panel data regression:

$$ y_{it} = x_{it}'\beta + \eta_i + \delta_t + v_{it}, \forall t.\tag{1} $$

Assume strict exogeneity

$$ \text{Strict Exogeneity:}\quad \mathbb E[u_{it}\mid x_{i1}, \ldots, x_{iT}] = 0, \forall s, t. $$

We observe $i\in [N]$ and multiple periods $t\in [T]$ of $\lbrace y_{it}, x_{it}\rbrace$. We don’t observe $\eta_i, \delta_t$. We want to estimate $\beta$.

Within Group (EG) estimator

Demean: let $\ddot y_{it} = y_{it} - \bar y_i - \bar y_t$ ($y_i = \sum_t y_{it}/T$ and vice versa for $\bar y_t$). Let $\ddot x_{it} = x_{it} - \bar x_i - \bar x_t$. Then according to $(1)$ we have

$$ \ddot y_{it} = \ddot x_{it}'\beta + \tilde v_{it}, \forall t.\tag{2} $$

Where, $\tilde v_{it} = v_{it} - \bar v_i - \bar v_t$. Stack $(2)$ up in matrix form, let:

$$ \ddot y_i := \begin{bmatrix} y_{i1}\\ \cdots\\ y_{iT}\\ \end{bmatrix}, \quad \ddot X_i := \begin{bmatrix} x_{i1}'\\ \cdots\\ x_{iT}'\\ \end{bmatrix}\text{ ($T\times K$ matrix)}, \quad \ddot v_i := \begin{bmatrix} \tilde v_{i1}\\ \cdots\\ \tilde v_{iT}\\ \end{bmatrix} $$

$(2)$ becomes $\ddot y_i = \ddot X_i\beta + \ddot v_i.$ When we got to observe a lot of individual’s panel data $i$, the within-group GLS estimator of $\beta$ is

$$ \hat \beta^{WG} = \frac{\frac1N\sum_{i\in [N]}\ddot X_i^T \ddot y_i}{(\frac 1N\sum_{i\in [N]} \ddot X_i^T\ddot X_i)} = \frac{\frac1N\sum_{i\in [N]}\ddot X_i^T (\ddot X_i\beta + \ddot v_i)}{(\frac 1N\sum_{i\in [N]} \ddot X_i^T\ddot X_i)} = \beta + \frac{\frac1N\sum_{i\in [N]}\ddot X_i^T \ddot v_i}{(\frac 1N\sum_{i\in [N]} \ddot X_i^T\ddot X_i)} $$

(note: matrix on denominator means inversion). From strict exogeneity we know that $\mathbb E[\hat \beta^{WG} ] = \beta$. By a standard CLT (independence across $i$, finite moments):

$$ \sqrt{N}(\hat{\beta}_{WG} - \beta) \xrightarrow{d} \mathcal{N}(0, \Sigma_{XX}^{-1} \Sigma_{m} \Sigma_{XX}^{-1})\tag{3} $$$$ \begin{aligned} & \Sigma_{XX} = \text{plim }\frac{1}{N}\sum_{i=1}^N \ddot{X}_i'\ddot{X}_i\cr & \Sigma_m = \text{plim } \frac{1}{N}\sum_{i=1}^N \ddot{X}_i' \ddot{v}_i \ddot{v}_i' \ddot{X}_i \end{aligned} $$

A note on correlation. Expand the “meat” variance for a single unit:

$$ \ddot{X}_i'\ddot{v}_i\ddot{v}_i'\ddot{X}_i = \left(\sum_{t=1}^T \ddot{x}_{it}\ddot{v}_{it}\right)\left(\sum_{s=1}^T \ddot{v}_{is}\ddot{x}_{is}'\right) = \sum_{t=1}^T\sum_{s=1}^T \ddot{x}_{it}\,\ddot{v}_{it}\ddot{v}_{is}\,\ddot{x}_{is}' $$

This is a $K \times K$ matrix that contains all cross-products $\ddot{v}_{it}\ddot{v}_{is}$ for every pair $(t,s)$. The term $E[\ddot{v}_{it}\ddot{v}_{is} \mid X_i]$ are nonzero whenever $v_{it}$ and $v_{is}$ are correlated. In other words, when taking the expectation to obtain $\Sigma_m$, the outer product $\ddot{v}_i\ddot{v}_i'$ automatically “picks up” whatever dependence structure exists.

Analgous analysis for the DiD estimator

For the TWFE model $y_{it} = x_{it}'\beta + \eta_i + \lambda_t + v_{it}$, the DiD estimator uses “double-demeaned” data: subtract both the unit mean and the time mean, then add back the grand mean:

$$ \tilde{x}_{it} = x_{it} - \bar{x}_i - \bar{x}_t + \bar{x}. $$

The formula is identical in structure — just replace $\ddot{x}$ with $\tilde{x}$ everywhere:

$$ \widehat{V}_{cluster}(\hat{\beta}_{DID}) = \left(\sum_{i=1}^N \tilde{X}_i'\tilde{X}_i\right)^{-1} \left(\sum_{i=1}^N \tilde{X}_i'\hat{\tilde{v}}_i\hat{\tilde{v}}_i'\tilde{X}_i\right) \left(\sum_{i=1}^N \tilde{X}_i'\tilde{X}_i\right)^{-1} $$

The logic is similar: independence across $i$ allow us to treat $\tilde{X}_i'\tilde{v}_i$ as i.i.d. draws, and the outer product of each unit’s score vector absorbs all within-unit serial correlation.