Foundations and Universities typically have endowments. These types of organizations are typically formal, nonprofit 501(c)3 designated organizations. Endowed foundations in the US would be researched in nonprofit studies, philanthropy studies, and public policy studies, as well as by the topic, e.g. public health or leadership

Thanks Richard, but in UK, nonprofit organization especially which are operated by private institution are not under Public sector study. 

In my view this a gray area.  Public institutions, defined as government is most common in my experience.  I generally use the term 'non-governmental organization' when I'm referring to non-profits that have social outcome goals, although depending on how you look at it that could include lobbying groups.  

In the US, the corporate status 'non-profit' is decided by state governments.  Tax exemption (i.e. you can deduct money donated when calculating federal income tax) is determined by the Treasury Department Internal Revenue Service pursuant to the Internal Revenue Code section 501(c)(3).  This in NOT automatic for all organizations created under state law as 'non-profit'.  To be tax-exempt under section 501(c)(3) of the Internal Revenue Code, an organization must be organized and operated exclusively for exempt purposes.  Among the activities prohibited is lobbying.  There are many other IRS determined tax classifications that may be assigned to organizations created under state law as 'non-profits'.  

All things considered, I'd not classify them in the 'public sector'.

Thanks for the info William.

In case endowment institution which is owned by the government, Can we put this as public sector?

In addition, endowment institution which is owned by the private sector, but serving exactly the same like endowment institution which is owned by the government. Can we put this also under public sector study?