Difference-in-Difference-in-Difference Intepretation

[Matt Tucker]

Dear Statalisters,

I am conducting a triple difference regression to determine the impact of a new Policy on stock returns. I am interested in the impact of the Policy on treated stock returns, as well as the impact on treated financials stocks, which is a sub-group within the sample.

Below are my baseline DIDID results, where the dependent variable is Log of stock returns, FTSE is a dummy which equals 1 for treated stocks, Post is a time dummy which equals 1 for dates after the Policy was introduced, and Banks is a dummy equalling 1 for financial stocks.

All Stocks
Dependent Variable: Log of Stock Returns
15 days
FTSE	0.0971***
	(0.0228)
Post	0.1703***
	(0.0134)
Banks	-0.0360
	(0.0296)
FTSE x Post	-0.1237***
	(0.0172)
Banks x Post	-0.0212
	(0.0535)
FTSE x Banks	-0.0210
	(0.0416)
FTSE x Post x Banks	0.0752
	(0.0735)
Size	0.0101***
	(0.0037)
Income	-0.0125***
	(0.0037)
Headquarters	-0.0017
	(0.0151)
Currency	-0.0561***
	(0.0168)
Constant	-0.0107
	(0.0331)
R2	0.015
Root MSE	1.4696
Observations	152,945

I believe that by exponentiating the triple interaction term FTSE x Post x Banks, it suggests treated financials stock returns increase by 10.60% over 15 days, due to the introduction of the policy.
Is this how to interpret the triple interaction term FTSE x Post x Banks?

Thank you very much for your help!

[Clyde Schechter]

No, it's not. That term represents the difference in treatment effect between financial stocks and other treated stocks. It is not the treatment effect for any particular group of stocks.

Interpreting these models gets complicated. Fortunately, Stata has the -margins- command to simplify the task considerably. But to avail yourself of -margins-, you need to run your regression using factor-variable notation. Read -help fvvarlist- to learn about the latter. For an outstanding introduction to the -margins- command, I recommend the excellent Richard Williams' https://www3.nd.edu/~rwilliam/stats/Margins01.pdf. It does not have any examples of 3-level interactions, but it will introduce you with great clarity to the overall workings of that command.

If you have questions about reading the -margins- output, or structuring the -margins- command to get what you want, post back showing the outputs. (Note: When showing regression outputs here it is best to show the immediate, direct results of the regression command itself, not the version that has been laundered through -estout- or other pretty-print commands.