EX-99.906

This certification is provided pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, 18 U.S.C. § 1350, and accompanies the report on Form N-CSR for the period ended June 30, 2015 of The Endowment Institutional TEI Fund W, L.P. (the “registrant”).

I, John A. Blaisdell, the Co-Principal Executive Officer of the registrant, certify that, to the best of my knowledge:

1. The Form N-CSR fully complies with the requirements of Section 13(a) or 15(d) of the Securities Exchange Act of 1934 (15 U.S.C. 78m(a) or 78o(d)); and

2. The information contained in the Form N-CSR fairly presents, in all material respects, the financial condition and results of operations of the registrant.

August 27, 2015

Date

/s/ John A. Blaisdell

John A. Blaisdell

Co-Principal Executive Officer

This certification is being furnished solely pursuant to Section 906 of the Sarbanes-Oxley Act of 2002 and is not being filed as part of Form N-CSR or as a separate disclosure document. A signed original of this written statement required by Section 906 has been provided to the registrant and will be retained by the registrant and furnished to the Securities and Exchange Commission or its staff upon request.

This certification is provided pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, 18 U.S.C. § 1350, and accompanies the report on Form N-CSR for the period ended June 30, 2015 of The Endowment Institutional TEI Fund W, L.P. (the “registrant”).

I, Andrew B. Linbeck, the Co-Principal Executive Officer of the registrant, certify that, to the best of my knowledge:

1. The Form N-CSR fully complies with the requirements of Section 13(a) or 15(d) of the Securities Exchange Act of 1934 (15 U.S.C. 78m(a) or 78o(d)); and

2. The information contained in the Form N-CSR fairly presents, in all material respects, the financial condition and results of operations of the registrant.

August 27, 2015

Date

/s/ Andrew B. Linbeck

Andrew B. Linbeck

Co-Principal Executive Officer

This certification is being furnished solely pursuant to Section 906 of the Sarbanes-Oxley Act of 2002 and is not being filed as part of Form N-CSR or as a separate disclosure document. A signed original of this written statement required by Section 906 has been provided to the registrant and will be retained by the registrant and furnished to the Securities and Exchange Commission or its staff upon request.

This certification is provided pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, 18 U.S.C. § 1350, and accompanies the report on Form N-CSR for the period ended June 30, 2015 of The Endowment Institutional TEI Fund W, L.P. (the “registrant”).

I, John E. Price, the Principal Financial Officer of the registrant, certify that, to the best of my knowledge:

1. The Form N-CSR fully complies with the requirements of Section 13(a) or 15(d) of the Securities Exchange Act of 1934 (15 U.S.C. 78m(a) or 78o(d)); and

2. The information contained in the Form N-CSR fairly presents, in all material respects, the financial condition and results of operations of the registrant.

August 27, 2015

Date

/s/ John E. Price

John E. Price

Principal Financial Officer

This certification is being furnished solely pursuant to Section 906 of the Sarbanes-Oxley Act of 2002 and is not being filed as part of Form N-CSR or as a separate disclosure document. A signed original of this written statement required by Section 906 has been provided to the registrant and will be retained by the registrant and furnished to the Securities and Exchange Commission or its staff upon request.