Regression in 3 dimensions

[Frank Giaquinto]

I'm reading an article in which the following regression model is estimated:
Y_ict= b₀ + b₁ X_ict + PI_it +RHO_ci + ZETA_ct + u_ict ,
where i,c,t denote the industry sector, the country, and time, respectively. The authors claim that this panel approach includes industry-year (PI_it), country-sector (RHO_ci), and country-year(ZETA_ct) fixed effects.
What kind of regression model is this? Could you suggest any reference about it? Why is the intercept b₀ included if we have fixed effects?
Is it feasible to estimate a regression model like this:
Y_ict= b₁ X_ic,t-1 + b₂ X_ict + b3 H_ct + b4 K_ct + PI_it +RHO_ci + ZETA_ct+ u_ict
where I added the lagged value of the continuous explanatory variable X, and I added two new predictors. These predictors consist of a categorical variable (H_ct ) and a continuous variable (K_ct) both of which remain constant across county-industries.
Any references about a suitable STATA command?

[Andrew Musau]

Clearly, the authors have data at the firm level. As a referee, I would expect them to clarify why their model lacks firm effects to accommodate firm-level heterogeneity. It's possible that the authors do not have panel data at the firm level but instead have data on multiple firms spanning various industries across several years (pooled cross-sections). In such a scenario, I would classify their model as Pooled Ordinary Least Squares (POLS) with industry-year, country-sector, and country-year dummies. However, they would need to introduce an additional index on their equation for firms. Since the authors have firm-level data, the intercept can be estimated. Generally, including an intercept in a fixed effects model is feasible, but estimating such an intercept separately from the fixed effects requires imposing certain constraints (see https://www.stata.com/support/faqs/s...effects-model/). Therefore, my guess is that the authors are stressing the term "fixed effects" throughout their analysis due to its association with robustness in panel data, to sell their paper.

To estimate the authors' model, one could use the reghdfe command from from https://github.com/sergiocorreia/reghdfe, absorbing the various dummies as follows:

reghdfe y x1 ... xn, absorb(industry#year country#sector country#year)

However, if the dataset consists of firm-level panel data, the authors would need to justify why their model is better than the standard two-way Fixed Effects model:

reghdfe y x1 ... xn, absorb(firm year) vce(cluster(firm))

[Frank Giaquinto]

Dear Professor, I greatly appreciate your prompt reply.
To provide further clarity (and my apologies for the oversight), the authors employ input-output tables to track the flow of outputs utilized as intermediate inputs by manufacturing industries across various countries. Consequently, the lowest level of aggregation observed is at the industry-sector level (denoted by 'i'), and different industry-sectors are observed at various time points for multiple countries ('c'). Specifically, we have t=1,2,...,30; i=1,2,...,17; c=1,2,...,76.
In such a scenario, is it problematic to utilize predictors that vary across countries over time (denoted by 'c' and 't') but remain constant across industries ('i'), namely, using variables indexed by 'ct' rather than 'ict'?

[Andrew Musau]

Consequently, the lowest level of aggregation observed is at the industry-sector level (denoted by 'i')

If that were the case, then an observation is an industry-year combination, and having industry-year dummies in the model would mean that they are including a dummy for each observation, which is impossible. As I said, my suspicion is that an observation is at a lower level of aggregation.

[Frank Giaquinto]

I see. If I would do something similar, I might use single dummies at industry, country, and time levels, that is, delta_i , gamma_c, tau_t, right?
In such a scenario, is it problematic to utilize a predictor that varies across countries over time (denoted by 'c' and 't') but remains constant across industries ('i'), namely, using a predictor indexed by 'ct' rather than 'ict'?
Could you suggest any reference dealing with such panels and considering the fact that t=1,2,...,30; i=1,2,...,17; c=1,2,...,76
Thanks for the help

[Andrew Musau]

It's not an issue because you do not have fixed effects at the unit level. As I stated in #2, your model is POLS and both within and between variation are used. Unless you can demonstrate the absence of heterogeneity at the unit level, any results you obtain are descriptive rather than causal. Standard panel data econometrics textbooks discuss POLS and generally do not recommend it, except if the RHS variables capture the heterogeneity.