Can we do sampling with different periods for time series data? Briefly, I want to estimate the effects of average annual advertising spending on sales of $n$ listed companies (say $C_1, C_2, \cdots, C_n$) on a particular stock market. My question is that can I select different periods for different stocks? For example can I use a dataset with a stock $C_1$ in $2010-2017$ period but for stock $C_2$ in $2013-2019$, both are annual data?
I guess that I don't need to choose all stocks with THE SAME periods, right? If possible, can you tell me what results in statistics or probability (e.g theorems) might explain why I am correct/incorrect?
Any comments or suggestions are highly appreciated! Thanks!
 A: While you could use different periods for different stocks, results that are extrapolated from this data could be misleading, since the data does not account for one-off events in different time periods. For example, if one of your time periods is $2006-2012$ while another time period is $2011-2015$, the latter time period may account less of the $2008$ financial crisis than the former time period. Another drawback of this approach is that average annual advertising spending could also be correlated with the length of the time period, so comparing two time periods of different lengths would not be reasonable.
One thing you could do is estimate average annual advertising spending on sales several times using different time periods and then average your results. However, I suspect you will need to average over a very large number of different time periods to minimize their effect, which could be intractable. This is an example of experimental design. Have a look at this and this for details.
If you are interested to know why averaging works, this is just the law of total probability.
