There is not a clear answer to that question. DNA is not specific to a region, but to inter-related groups of individuals. The labels we give to regions are somewhat arbitrary, driven by political boundaries and history. Ancestry DNA has sampled some groups whose ancestors have lived in the same region for centuries. This is from their "white paper", which you can read by going to your ethnicity results and clicking on the little tool icon on the upper right.:"\
"The "Examples of Native People" chart shows individual ethnicity estimates for five sample individuals native to each region. Individuals whose family trees suggest deep ancestry in a particular region are considered "native" to that region. AncestryDNA has gathered thousands of these "native" DNA samples from across the globe. Members of our native sample collection are unique because their family has lived in their region for centuries."
They have also included in their analysis people whose families have been in the same area for several generations, usually individuals who have four grandparents from the same area.
So, the answer to your question is that it can vary from a few generations to many centuries. There is some good background material on how they came up with the sample populations. It's a bit technical but worth studying to put your ethnicity estimate in perspective.