# Notation for the summation in unbalanced panel data

Unbalanced panel data, where $$i$$ stands for sections and $$t$$ stands for time points, have different numbers of $$t$$ for every $$i$$ (or different numbers of $$i$$ for every $$t$$, but let me use the first framing).

Then, if I want to compute the mean of some variable (let me use $$x_{it}$$), then what is a clear but simple notation? If I tried to be meticulous, it might be like:

$$\frac{1}{n}\sum_{i=1}^{n_{i}}(\sum_{t=1}^{n_{t}(i)}x_{it})$$

where I let $$n_i$$ mean the number of sections, and $$n_t(i)$$ mean the number of time points for section $$i$$ (because it is unbalanced panel data, $$n_t$$ must be difference for different $$i$$). But this looks unusual and unnecessarily complicated to me.

One simplification might be: $$\frac{1}{n}\sum_{i,t}^{n}x_{it}$$. Or, $$\frac{1}{n}\sum_{i,t}x_{it}$$, as $$n$$ speaks only to the total number of observations. Do either or both of these simplified forms still keep the meaning clear, if it is already clarified that data are panel data (regardless of being balanced or unbalanced)?

I would appreciate your help!

## 1 Answer

I remember that there are some notations used. It reminds me of writing my PhD thesis which uses unbalanced panel data for solving and coding some statistical models. Among others, Baltagi and Wu (1999) and Baltagi and Liu (2020) are useful, which are available online.

• Thanks for the papers; indeed they have relevant full notations! Just out of curiosity, do you think my simplified notation, 1/n \sum_{i,t} x_{i,t}, is also clear enough to mean it is the mean of x in the data, if it is already clarified that x_{i,t} is a variable in unbalanced panel data? Dec 8, 2021 at 9:53
• One that I can currently think of is \frac{1}{n} \sum_{i=1}^{N} \sum_{t=1}^{T_i} x_{i,t} where n = \sum_{i=1}^{N} T_i and N is the number of sections.
– dkim
Dec 8, 2021 at 19:43
• Thanks! Indeed, it may be better to spell it out fully like what you suggest Dec 10, 2021 at 14:26