Filtering is the process of housing units changing price over time and therefore tending to be inhabited by people of different income/wealth than previously. Originally and most commonly, it refers to "filtering-down," whereby housing that was built for or occupied by a higher-income group becomes less expensive and occupied by households of lower income. However, Lowry  analyzed it as change in value which could be up or down, and later the concept of gentrification has been described as "filtering up."
At least back to the 1940s in the United States, the concept of filtering has been used to support the argument that private, or market-rate, housing development can meet the housing needs of much of the population, even if it is often originally built for or occupied by higher-income residents. While often made, the argument mostly lacked good empirical proof until Rosenthal  demonstrated that filtering occurs widely, although significantly less so in recent decades in high-cost, supply-constrained areas such as the Bay Area and Boston.
Filtering is not the only way in which new market-rate housing may address affordability for other income groups, however. Two other reasons are 1) fungibility, and 2) adaptation.
The degree to which different housing units, or any good, may equivalently serve someone's needs is sometimes called fungibility, or substitutability. See Wikipedia: fungibility. In the case of market-rate housing, it is likely that often there is enough fungibility between available housing units that the buyer/renter, if that MR housing had not been developed, would have sought out another unit in the area, excluding or potentially displacing another, probably lower-income person, from that unit.
Second, housing may be used differently or converted from its original intended use, possibly serving other and lower-income residents. Houses and apartments become shared between multiple tenants, either informally as house- or apartment-shares, or possibly by being divided into multiple legal residences. This can occur even with the first tenants of new housing, for example in San Francisco where vacant new "luxury" apartments are sometimes divided and shared between many tenants.
"the changing of occupancy as the housing that is occupied by one income group becomes available to the next lower income group as a result of decline in market price."
"I propose to define 'filtering' simply as a change in the real value (price in constant dollars) of an existing dwelling unit. ... To analyze filtering as a market process, its causes and consequences, four basic constructs should be kept in mind: (1) An array of all dwelling units according to their real values ... (2) An array of all dwelling units according to their quality (by some quantifiable measure other than price). (3) An array of all households according to their real incomes ... (4) An array of supply prices of new dwelling units in each quality class...."
- Hertz, Daniel. "What filtering can and can’t do." City Observatory, 10.11.2015. http://cityobservatory.org/what-filtering-can-and-cant-do/.
- Lowry, Ira S. "Filtering and Housing Standards: A Conceptual Analysis." Land Economics, Vol. 36, No. 4 (Nov., 1960), pp. 362-370
- Ratcliff, Richard Updegraff. "The Economics of the Housing Problem: Preliminary Draft." (1945)
- Ratcliff, Richard U. Urban Land Economics (1949).
- Rosenthal, Stuart S. . "Are Private Markets and Filtering a Viable Source of Low-Income Housing? Estimates from a 'Repeat Income' Model." American Economic Review, 104(2): 687-706 (February 2014). DOI: 10.1257/aer.104.2.687. Preprint (2013): http://faculty.maxwell.syr.edu/rosenthal/recent%20papers/Is_Filtering_a_Viable_Source_of_Low-Income_Housing_%206_18_13.pdf.
- Zuk, Miriam, and Karen Chapple. "Housing Production, Filtering and Displacement: Untangling the Relationships." Berkeley Institute of Government Studies - Research Brief, May 2016.