# Bootstrapping time series data: Circular block bootstrap

I have some very basic questions on circular block bootstrap applied to time series (dependent data).

Let's suppose, I have a time series data like the one below. I know it's non stationary, but for demonstration purpose let's assume this to be an index of time series data:

x <- c(1,2,3,4,5,6,7,8,9,10)


Applying a Moving block bootstrap I get the following results with the block length of 3 with replacement of size 10:

   [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,]    3    4    5    1    2    3    2    3    4     2
[2,]    7    8    9    2    3    4    4    5    6     8
[3,]    3    4    5    2    3    4    7    8    9     5
[4,]    7    8    9    6    7    8    4    5    6     8
[5,]    4    5    6    8    9   10    3    4    5     8
[6,]    7    8    9    3    4    5    1    2    3     7
[7,]    8    9   10    3    4    5    2    3    4     6
[8,]    6    7    8    4    5    6    2    3    4     6
[9,]    8    9   10    6    7    8    8    9   10     6
[10,]    8    9   10    5    6    7    7    8    9     2


In order for us to avoid edge effects (both 1 and 10 will always start or end respectively) a variant of the moving block bootstrap is circular block boot strap:

Below are my questions:

1. Should I just create a new vector by just appending the x with x as below and then apply block bootstrapping? Is this circular block bootstrapping ?

x.circle <- c(x,x)

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

2. If I apply block bootstrap with same size block size 3 and 10 replicates I get the following results, How should I trim the data to reduce it to length 10? should I pick the first 10 elements or should I start some where in the middle and pick 10 elements :

1. What are the benefits of Circular block bootstrapping ? I intuitively understand what an edge effect is, can you someone please explain why it is important ?
• If your data is circular, it makes sense to use it. Asymptotics will be more accurate. – Chan-Ho Suh Jun 19 '15 at 18:31
• @Chan-HoSuh thanks, my question is around how to make the data circular. – forecaster Jun 19 '15 at 20:33
• Let's say your data was the 24 hours if the day. It would make sense to be circular. – RegressForward Jun 20 '15 at 13:32
• @RegressForward, thank you very much for very clear explanation. So for instance, if I have data from Jan 2013 thru Mar 2014, I would not be able to apply circular bootstrap? In other words, the start and end calendar month should be same in case of monthly data? Thanks – forecaster Jun 21 '15 at 18:02