No, Indian Institute of Technology (IIT), Madras does not offer Integrated M.A. in Economics, instead it offers:
Integrated M.A. in Development Studies
Integrated M.A. in English Studies

To apply for any of the above mentioned courses, you need to clear its Entrance Exam - Humanities and Social Sciences Entrance Examination (HSEE). Its details are as follows:

It is for admission to the five-year integrated Master of Arts (M.A.) Program at IIT Madras for admission in Integrated MA

Eligibility Criteria

One applying for HSEE should have either completed or should be appearing in any one of the following qualifying examinations (QE):

(a) The final exam of the 10+2 system, conducted by a Central or State Board recognized by the Association of Indian Universities

(b) Intermediate or 2 year Pre-University Exam conducted by a Board or University recognized by the Association of Indian Universities

(c) Final exam of 2 year course of the Joint Services Wing of the National Defence Academy; senior secondary school exam conducted by the National Institute of Open Schooling with a minimum of five subjects

(d) Any Public School, Board or University Exam in India or in a foreign country recognized as equivalent to the 10+2 system by the Association of Indian Universities (AIU)

(e) HSC vocational exam

(f) General Certificate Education (GCE) exam (London, Cambridge or Sri Lanka) at the Advanced (A) level

(g) High School Certificate Exam of the Cambridge University or International Baccalaureate Diploma of the International Baccalaureate Office, Geneva

Candidates who have completed Class XII (or equivalent) exam outside India or from a Board not specified above need to produce a certificate from the Association of Indian

If Class XII exam is not a public exam, one must have passed at least one public (Board or Pre-University) exam earlier

Nationality:
One who is not citizen of India at the time of registering for HSEE (by birth or naturalized) are treated as foreign nationals

Percentage of Marks in QE:
One belonging to the GEN Category & OBC Category must secure a minimum of 60% marks in aggregate in the QE.
Candidates belonging to SC/ST & PwD Categories must secure a minimum of 55% in aggregate in the QE

Physical Fitness:
Every qualified candidate need to submit a Physical Fitness certificate from a registered medical practitioner in the prescribed format that is made available along with the letter of offer of admission