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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 :

enter image description here

  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 ?
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  • $\begingroup$ If your data is circular, it makes sense to use it. Asymptotics will be more accurate. $\endgroup$ – Chan-Ho Suh Jun 19 '15 at 18:31
  • $\begingroup$ @Chan-HoSuh thanks, my question is around how to make the data circular. $\endgroup$ – forecaster Jun 19 '15 at 20:33
  • $\begingroup$ Let's say your data was the 24 hours if the day. It would make sense to be circular. $\endgroup$ – RegressForward Jun 20 '15 at 13:32
  • $\begingroup$ @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 $\endgroup$ – forecaster Jun 21 '15 at 18:02

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