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I am confused if it is possible for to randomly select a sample such that SRF and PRF do not intersect. I have thought of the concept of parallel lines, which allows for such a case but I am unsure if concepts of probability and statistics allow for it.

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  • $\begingroup$ (PRF: Population Regression Function, SRF: Sample Regression Function) $\endgroup$ – keepAlive Oct 9 '18 at 9:48
  • $\begingroup$ Welcome to the site. I have posted an answer below, but am puzzled as to what practical economic problem might prompt this question. $\endgroup$ – Adam Bailey Oct 9 '18 at 21:17
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It is theoretically possible, but extremely unlikely if the random sample is large.

To see that it is possible, consider the very simple case shown below in which the population consists of just four points in the X-Y plane which happen to lie at the corners of a parallelogram.
enter image description here Assuming linear regression lines, the population regression function (PRF) will be as represented by the continuous line. If a sample regression function (SRF) is fitted to a sample of two points and the points happen to be A and B, then the function will be as represented by the dashed line which is parallel to the PRF.

Note however that even in this case, the probability that a random sample of two points will produce a SRF parallel to the PRF is only one third. There are six possible such samples, two of which (AB,CD) yield SRF's parallel to the PRF, and four of which (AC,AD,BC,BD) do not. In a realistic case in which both the population and the sample are much larger, and the points do not lie in such a simple geometrical pattern, the probability of a SRF parallel to the PRF will be very small indeed.

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