I am a little bit confused by the distinction. What are the formal definition of paired and unpaired? More specifically, if I have 2 time series measuring the same quantity (e.g. vacancy rate) with slightly different underlying data, are the data paired or unpaired?
Paired refers to the method of comparison. If you are comparing two data sets the comparison is paired if you take differences but matching data points from the two data sets. So this would be done if the two sample sizes are equal. So with the time series you are pairing if you match the two series at each time series. It is the difference between comparing the mean paired difference versus the difference of the two sample means. When data are not paired you can test for difference between means with unequal sample sizes.