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I am trying to expand my knowledge about the different interpretations of combinations of fixed effects.

I am using a pooled cross section dataset with observations at the firm level. The dataset spans multiple countries over 2 years (2005-2010).

EDIT: Please see sample data below

My question is very simple. In this scenario, what is the difference of interpretation between including country and year fixed effects, country-year fixed effects, or both?

Is there a case to be made for each option when taking into account my dataset?

I read the following on another site:

When you interact state and year dummies (i.e. when you include state, year, and state*year in the regression, which by the way is the same as creating state-year dummies and including them in the regression), you are assuming that the unobserved state-level heterogeneity varies over time. Also, you are assuming the time effect to vary by state. If you include state and year separately and no interaction, you are assuming that the unobserved state-level heterogeneity is constant over time.

If I read this, I get the feeling that it is always better to include the interactions. But in statistics, nothing seems to come for free. So what is the downside here?

Please see my thought process below (and please correct me if I am wrong):

  1. I separately add a country dummy, I account for static differences per country. In other words, I control for time constant omitted variable bias. And I add a year dummy to account for change (I would say world trends in this case).

  2. If I add a country-year dummy, I am controlling for trends that are country specific.

  3. If I add a country dummy, a year dummy and a country-year dummy, I am doing all of this at once?

For 1 and 2 I am pretty much okay.

By point 3 I begin to wonder: do I need to? Should I always include this? At what cost do I include this country-year dummy?

If I have a country-year dummy without a country and year dummy, does that make sense?

Or should I therefore put them all in? Coming to point 4..

Data

panelID= c(1:50)
year= c(2005, 2010)
country = c("A", "B", "C", "D", "E", "F", "G", "H", "I", "J")
urban = c("A", "B", "C")
indust = c("D", "E", "F")
sizes = c(1,2,3,4,5)
n <- 2
library(data.table)
library(dplyr)
library(AER)
set.seed(123)
DT <- data.table(   country = rep(sample(country, length(panelID), replace = T), each = n),
                    year = c(replicate(length(panelID), sample(year, n))),
                    sales= round(rnorm(10,10,10),2),
                    industry = rep(sample(indust, length(panelID), replace = T), each = n),
                    urbanisation = rep(sample(urban, length(panelID), replace = T), each = n),
                    size = rep(sample(sizes, length(panelID), replace = T), each = n))
DT <- DT %>%
group_by(country) %>%
mutate(base_rate = as.integer(runif(1, 12.5, 37.5))) %>%
group_by(country, year) %>%
mutate(taxrate = base_rate + as.integer(runif(1,-2.5,+2.5)))
DT <- DT %>%
group_by(country, year) %>%
mutate(vote = sample(c(0,1),1), 
votewon = ifelse(vote==1, sample(c(0,1),1),0))

# No interaction

summary(ivreg(sales ~ taxrate + as.factor(size) + as.factor(urbanisation) + country + as.factor(year) | 
as.factor(votewon) + as.factor(size) + as.factor(urbanisation) + country + as.factor(year), data=DT))

summary(ivreg(sales ~ taxrate + as.factor(size) + as.factor(urbanisation) + as.factor(vote) + country + as.factor(year) 
| as.factor(votewon) + as.factor(size) + as.factor(urbanisation) + as.factor(vote) + country + as.factor(year), data=DT))

# Interaction

summary(ivreg(sales ~ taxrate + as.factor(size) + as.factor(urbanisation) + country:as.factor(year) | 
as.factor(votewon) + as.factor(size) + as.factor(urbanisation) + country:as.factor(year), data=DT))

summary(ivreg(sales ~ taxrate + as.factor(size) + as.factor(urbanisation) + as.factor(vote) + country:as.factor(year) 
| as.factor(votewon) + as.factor(size) + as.factor(urbanisation) + as.factor(vote) + country:as.factor(year), data=DT))

# Both

summary(ivreg(sales ~ taxrate + as.factor(size) + as.factor(urbanisation) + country*as.factor(year) | 
as.factor(votewon) + as.factor(size) + as.factor(urbanisation) + country*as.factor(year), data=DT))

summary(ivreg(sales ~ taxrate + as.factor(size) + as.factor(urbanisation) + as.factor(vote) + country*as.factor(year) 
| as.factor(votewon) + as.factor(size) + as.factor(urbanisation) + as.factor(vote) + country*as.factor(year), data=DT))
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  • $\begingroup$ Hi Tom. Could you provide a reference to the site where you received the information/quote above? $\endgroup$ – Thomas Bilach May 29 at 14:02
  • $\begingroup$ I have added it, but it was just a forum with more or less the same discussion. But not a very constructive one regretfuly. $\endgroup$ – Tom Kisters May 29 at 14:04
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    $\begingroup$ Adding a dummy for time isn't quite the same thing as accounting for trends. It's a binary, categorical or qualitative dummy which is accounting for change. A trend would be better defined by a continuously linear or polynomial variable in the presence of 3 or more periods. Also, pooled time series model interpretation is dependent on the presence of an intercept where the absence of one eliminates the need for k-1 country dummies. Regardless, specifying model 4. would result in linear combinations of the main effects wrt the interaction, making the main effect parameters redundant. $\endgroup$ – user332577 May 29 at 14:53
  • $\begingroup$ Okay, so point 4 does not exist, we can remove that from the discussion. $\endgroup$ – Tom Kisters May 29 at 15:02
  • $\begingroup$ I guess that the main downside of including all these fixed effects is that you remove all between variation meaning that the estimates become more sensitive to a limited number of influential observations (I read this somewhere but can't recall where exactly). In addition, it could lead to overfitting. So the model will describe the data quite well but possibly has little predictive power for new data. Given the popularity of this modeling technique I guess that inference takes precedence over generalisability, in some academic quarters at least. $\endgroup$ – horseoftheyear May 29 at 15:30
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Include country and year when you include the country and year interaction. Then one interpretation of your model is that you have estimated country effects at 2005 and country effects at 2010, with the difference = the interaction estimate. When you leave out the interaction, effects of each variable are estimated independently of the other. If you only put in the interaction, the estimate is rarely meaningful as it confounds the individual effects of country and year with the interaction estimate. Best, B.

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  • $\begingroup$ @ BIJones, thank you for your commentanswer. Could you perhaps elaborate at bit more (maybe with some references), especially since your answer appears to contrast what has been said in the comments. $\endgroup$ – Tom Kisters Jul 6 at 8:15

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