Panel data: why the general model can not be estimated?

Question

I am reading the book

Baum, C. F. (2006). An Introduction to Modern Econometrics Using Stata (Stata Press, ed.).

In Chapter 9 there is written that

Given panel data, we can define several models that arise from the most general linear representation: $$ y_{it} = \sum\limits_{k=1}^K x_{kit} \beta_{kit} + \epsilon_{it} \quad i=1,\dots,n \;\; t=1,\dots,T $$ [...] Assume a balanced panel in which there are $T$ observations for each of the $N$ individuals. Since this model contains $k \times N \times T$ regression coefficients, it cannot be estimated from $N \times T$ observations.

I had two questions:

1. Why the book says I have $N \times T$ observations? I have one observation for each individual, for each time period AND for each regressor. So I should have $k \times N \times T$ observations, right?

2. Why this general model can not be estimated?

Answer 1

ad 1, I assume he refers to the number of units (e.g., persons) times time periods to count the sample size. You multiply by the number of regressors $k$ (you might also say $k+1$, since there is also the dependent variable...).

Even so, fitting $knT$ coefficients will not be possible. Forget about the panel aspect for a second, and assume you want to fit a standard regression model with $k$ regressors, but only observe one person (i.e., $n=1$). If $k>1$, you cannot tell apart these $k$ influences from just one person.

Answer 2

It is not so much that the model cannot be estimated, but rather that if the estimates are completely unrestricted, multiple estimates could fit the data equally well. Once you start to assume that observations for the same subject are related (the typical solution for panel data) or start to assume prior distributions for the coefficients, this issue goes away.

Answer 3

The problem Baum is referring to is that, with $N \times T$ obesrvations, the set of $k \times N \times T$ coefficients in the model is not identifiable. What this means is that there are different coefficient arrays that will lead to the exact same sampling probabilities for the $N \times T$ observations, and so there is no basis to distinguish these different coefficient arrays. Since this is a form of linear model, you need at least as many observations as coefficients in order for the coefficients to be identifiable (plus additional observations if you want to be able to estimate the error variance).

Answer 4

As to your question #1:

I have one observation for each individual, for each time period AND for each regressor. So I should have $k \times N \times T$ observations, right?

No, you have $N \times T$ observations. Each observation for unit $i$ at a given point in time $t$ contains information on $y_{it}$ and each of the regressors $x_{kit}$. Think of an observation as a row in an Excel file. (As a side, this is the so-called long format of storing panel data. There is also a wide format where information for each time period on a given unit $i$ is stored in a single row. If Baum had this in mind, then he would say that we have $N$, not $N \times T$ observations.)

As to question #2, this model cannot be estimated because it contains $K \times N \times T$ coefficients and we only have $N \times T$ observations. Intuitively, we are allowing each regressor to have an effect that is potentially different for each unit-time (i.e. $it$) combination: this is too much to ask to the data we have.