Theory and Application of PKU（北大的理论和应用）.pdf

Theory and Application of Copula CCERCCER 卯光宇 mgy0410@ 1010.3030 Econometrics （计量经济学） • What is Econometrics？ CombinationCombination ofof economics,economics, mathematicsmathematics andand  statistics. • WhWhat iis thhe ffocus off econometriics?? Estimatingg Various Conditional Moments E(Y|X), Var(Y|X), P(Y=q|X) Estimating Various Structural Models from  Economic Theory. MyMy FocusesFocuses • Modelling Time Series NonstationaryNonstationary DataData andand NonlinearNonlinear ModelsModels • Monitoring Structural Change • Applying Copula Functions AA HeuristicHeuristic ExampleExample How to construct a two‐variate joint  distribution function whose margginals are  respectively but not jointly normally  distributed?distributed? X ~ F (x ), Y ~ G (y ), (X ,Y ) ~ H (x , y ) H (x, y ) H (F −1(F (x )),G −1(G (y ))) C(u, v) ( F( x), G(y )) where C (u, v) H (F −11(u),G−11(v)) SklarSklar TheoremTheorem (1959)(1959) Let H be a n-dimensional distbution function with margins F ,...,F . Then there exists an n-copula C 1 n such that for all x ∈R n , HH ((xx ,...,,..., xx )) CC ((FF ((xx ),...,),...,FF ((xx )))) 11 n 11 11 n n C is unique if F ,...,F are all continuous. 1 n Conversely, if is C a n-copula and F ,...,F are 1 n distributiondistribution functions,functions, thenthen HH defineddefined ababoveove isis anan n-dimensional distribution function with margins F ,...,F . 1 n CopulaCopula • Defi