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I am a PhD student and would like to get a mixture real data that may exhibit upper and lower tails dependencies along with other dependencies structure. I am working on a pair-copula models.

These models mean that, if we have a multivariate data, then we can build an acyclic tree where only two variables are modelled at a time. These models help to model non-normal dependence structures.

I would like to have a multivariate data that exhibit mixture dependencies between each variables and hope these dependencies including tail dependencies. Most of this data can be financial.

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According to the wikipedia article Tail Dependence:

stock returns ... commonly exhibit tail dependence.

To look at generic stock data, one of the easiest sources of historic data is Quandl. Here's a blog post (June 2017) from them giving an overview.

Quandl has some paid products, but many/most data sources are free. They also have an API. Blog about API, and example of simple CSV export:

https://www.quandl.com/api/v3/datasets/WIKI/AAPL.csv

and endpoints for many codes.

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