Let the time series of 1-year estimates be y1, y2, y3, …, yt, …. Let and be 3-year and 5-year period estimates, respectively, where t denotes the last 1-year estimate in the multiyear estimate. Under the first assumption, the multiyear estimates are simply 3- or 5-year moving averages of the 1-year estimates. Thus, for example, for 3-year estimates,
and, for 5-year estimates,
With the 3-year period estimates, pairs of estimates that are only 1 year apart have 2 years in common, and those that are 2 years apart have 1 year in common. There is no overlap when the pair of years is more than 2 years apart. With the 5-year period estimates, estimates that are only 1 year apart have 4 years in common, those that are 2 years apart have 3 years in common, those that are 3 years apart have 2 years in common, and those that are 4 years apart have 1 year in common. It is only when two 5-year period estimates are 5 or more years apart that there is no overlap.
The extent of overlap between two multiyear estimates determines the estimand that the difference between them is estimating. Consider the difference between two 3-year estimates, with one being and the other being with t > 3. With t = 4, because of the 2-year overlap, the difference between and is
that is, one-third of the change between year 1 and year 4. With t = 5, with a one-year overlap, the difference between and reduces to [(y4 – y1)