# Standard Error of Mean in Panel Data

Consider a panel with data $$(y_{i,t})$$ for $$i\in\{1,...,N\}$$ and $$t\in\{1,...,T\}$$.

Problem: I want to know how the cross-sectional average of $$y$$ behaves in an average time period.

My approach:

• For each $$t$$, I calculate the cross-sectional average of $$y_{i,t}$$: $$\bar{y}_t=\frac{1}{N}\sum\limits_i y_{i,t}$$.
• I take the time series mean of that series: $$\bar{\bar{y}}=\frac{1}{T}\sum\limits_t \bar{y}_t$$.
• The standard error of $$\bar{\bar{y}}$$ is $$\frac{\sigma}{\sqrt{T}}$$, where $$\sigma=\sqrt{\frac{1}{T-1}\sum\limits_t (\bar{y}_t-\bar{\bar{y}})^2}$$.

Question:

• The standard error of $$\bar{\bar{y}}$$ takes the variation across time periods into account, but not the cross-sectional variation. If there is large variation across entities but little variation across time, wouldn't my standard error be misleading? Indeed, the standard errors of the cross-sectional means (depending on the sample size $$N$$) are irrelevant for the standard error of the time series average.

Consider the extreme case: every time period is identical: $$y_{i,t}=y_{i,t'}$$ for all $$t,t'\in\{1,...,T\}$$. Then, the standard error of $$\bar{\bar{y}}$$ would be zero, regardless of how dispersed $$y_{i,t}$$ is in the cross-section.

Instead, I would almost like to calculate the time series mean of the standard errors of $$\bar{y}_t$$ to get a feeling for how much $$y_{i,t}$$ varies cross-sectionally in the average time period?