Regression with 'stacked' data I've occasionally seen people do something like the following. Let's say we have a survey battery with k questions, for instance an evaluation of red jelly beans, an evaluation of green jelly beans, etc. Then we have another battery of related questions, such as an evaluation of the color red, green, and so forth. The data might look like this:




id
y1
y2 
y3 
x1
x2
x3




1
10
9
5
6
10
5


2 
9
 5
10
0
10
5




The data is then 'stacked', so that for each respondent there are k rows:




id
y
x




1
10
6


1
9
10


1
5
5


2
9
0


2
5
10


2
10
5




Finally, a model such as OLS is fitted, with y as the dependent variable and x as the independent variable. Most likely there are several independent variables, and they may have been constructed in more complicated ways.
I'm trying to understand whether this is a sensible thing to do, and what issues might arise. It feels a bit fishy: observations are not independent (there are k observations for each respondent, and n (the number of respondents) observations for each type of jelly bean). A model with mixed effects would seem reasonable, as suggested in answers to similar questions here and here.
If, however, the analysis was performed as described, what should I be cautious about when interpreting the results?

Indeed, as you already point out:

observations are not independent (there are k observations for each respondent, and n (the number of respondents)

The OLS parameter estimates will be unbiased, as long as there is exogeneity ($\mathbb{E}[\epsilon_i|x_i] = 0$). The covariance matrix of the parameters, however, in OLS given by:
$\text{Var}[\hat{\beta}∣X] = \sigma^2(X^\top X)^{−1}$
will over- or underestimate the covariance of the parameter estimates (to estimate the covariance, the unknown quantity $\sigma^2$ is replaced with its estimate $s^2$). Standard errors, which are used for inference (e.g., hypothesis tests), are given by the diagonal of this matrix. For valid inference, the estimated covariance matrix needs to account for the correlated nature of the observations. Zeileis et al (2020) give an extensive overview of such corrections, a.k.a. 'clustered covariances' or 'robust standard errors':
Zeileis, A., Köll, S., & Graham, N. (2020). Various Versatile Variances: An Object-Oriented Implementation of Clustered Covariances in R. Journal of Statistical Software, 95(1), 1–36. https://doi.org/10.18637/jss.v095.i01
