Meaning of term "Linear" in Linear Regression

One of the most frequently asked questions in analytics interviews that interviewers use to confuse people :
What do you mean by term "Linear" in Linear Regression ? 

Few interviewers phrase the question in another way  :
If you regress a Y variable with X-square or Log(X) as one of the independent variables, would the equation Y = Beta(0) + Beta(1) X-Square + Beta(2) &%$#*&^ .....  be considered as a equation of linear regression ?

For complete explanation :


Well, the answer of the question is :


Term "Linear" refers to Linearity of Beta Coefficients of model,
not the linearity in Y or X variables. 

Interviewer may counter (to confuse you) :

How come equation Y = Beta(0) + Beta(1) X-Square + Beta(2) .... be an equation of Linear regression?

You need remind him that :

To fulfill the basic assumption of linearity, we do transformations with X variables (Square, Cube, Roots of various orders, Log etc.) and make the variables linear with Y. Post transformation, we need not write the variable as such, we can give them new name.

E.g.  If we transform X to X-square, we can call it "P" , so the new equation would be :

     Y = Beta(0) + Beta(1)+ Beta(2) .........., and it starts looking like a linear equation now.


One of the key assumptions of linear regression is assumption of linearity. People often mix this assumption of "linearity" with the term "Linear" in linear regression and interviewers helps them confuse further. Please be aware  !

So what actually Linearity of Beta Coefficients means ?

Linearity in the B coefficients means that they are not raised to any power, or log of something or in a form of a/b ... something like Pie (22/7). Beta coefficients are plain and vanilla, a rational number.



Reference :

Damodar Gujrati, Econometrics by example
Chapter 1, Page 3




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