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From a paper of Gao (2021), I saw a paparagraph

We deviate from this traditional specification in three ways: we examine only money held by corporations, we include real GDP growth in the regression to control for economic fluctuations over the business cycle, and we use real instead of nominal interest rates. This last choice is for internal consistency with the theoretical framework that follows

I understand the internal consistency from this description

For example, if a respondent expressed agreement with the statements "I like to ride bicycles" and "I've enjoyed riding bicycles in the past", and disagreement with the statement "I hate bicycles", this would be indicative of good internal consistency of the test.

But I am wondering when we should use "internal consistency" and why the author need to perform the robustness test for it afterwards?

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But I am wondering when we should use "internal consistency" and why the author need to perform the robustness test for it afterwards?

You do not use internal consistency that is something that your model either has or does not have. Internal consistency is property not a method.

For example:

$$a+b=1 \tag{1}$$

$$5a+5b=15 \tag{2}$$

Is an inconsistent system of equations. The equation $(1)$ tells us that $a+b=1$, but the equation $(2)$ tells different story:

$$ 5a+5b=15 \implies 5(a+b) =15 \implies a+b=3 \tag{3} $$

Clearly, $a+b$ cannot be at the same time both 1 and 3, so here we have some internally inconsistent system. You can see it further by substituting 3 into 1. You will get:

$$a+1-a=3 \implies 1 = 3$$

which is obvious contradiction.

People always strive to have internally consistent models, if you have internally inconsistent model that is tantamount of making a modeling mistake somewhere.

why the author need to perform the robustness test for it afterwards?

They do not do robustness test to check for internal consistency, they use robustness test, because in order to make their model internally consistent, they were forced to use real interest rate, but they were also interested in knowing if the effect holds for nominal interest rate as well even though that was not their preferred specification.

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