lppl_estimate_rob_3s: 3rd step LPPL estimation procedure by Geraskin and Fantazzini...
In deanfantazzini/bubble: Tests for financial bubbles and explosive behavior

Description Usage Arguments Details Value Examples

This function performs the 3rd step of the LPPL estimation procedure by Geraskin and Fantazzini (2013) and Fantazzini (2016)

1	lppl_estimate_rob_3s(x, par1, par2)

`x`	is a T x 1 data vector
`par1`	is a 6 x 1 vector containing the parameters estimated in the 1st step [beta, omega, A, B, C1, C2]
`par2`	is a 1 x 1 scalar containing the parameter estimated in the 2nd step [tc]

This function performs the 3rd step of the LPPL estimation procedure by Geraskin and Fantazzini (2013) and Fantazzini (2016) using the LPPL formula by Filimonov and Sornette (2013): Using the estimated parameters in the first and second stages as starting values, it estimates all the 7 LPPL parameters [beta, omega, A, B, C1, C2, tc] by using a quasi-Newton method algorithm. Moreover, it computes the KPSS test statistic with the LPPL residuals to check their stationarity (a new condition introduced by Lin et al. (2014)).

par_est is a 8 x 1 vector containing the estimated 7 LPPL parameters [beta, omega, A, B, C1, C2, tc], togother with the KPSS test statistic computed with the LPPL residuals to check their stationarity (a new condition introduced by Lin et al. (2014))

 ## Not run: 
 tparm=c(0.35, 4.15, 2.07, 7.16,-0.43, 0.035, 0.00007, 530)
 aa=lppl_simulate(500,tparm)
 # 1st estimation step
 bb1=lppl_estimate_rob_1s(aa)
 bb1
 # 2nd estimation step
 bb2=lppl_estimate_rob_2s(aa,bb1)
 bb2
 # 3rd estimation step
 bb3=lppl_estimate_rob_3s(aa,bb1,bb2)
 bb3
 # The original C parameter can be retrieved using standard trigonometric
 # functions
 C.param=bb3[5]/cos(atan(bb3[6]/bb3[5]))

 # The first major condition for a bubble to occur within the LPPL framework
 # is that 0 < beta < 1, which guarantees that the crash hazard rate accelerates.
 # The second major condition is that the crash rate should be non-negative,
 # as  highlighted by  Bothmer and Meister (2003), which imposes that
 # b = [- B x beta - |C| x sqrt(beta^2  + omega^2)] >=0.
 # For the original parameters we have:
 -tparm[5]*tparm[1]-abs(tparm[6])*sqrt(tparm[1]^2+tparm[2]^2)
 # while for the estimated paramaters we have:
 -bb3[4]*bb3[1]-abs(C.param)*sqrt(bb3[1]^2+bb3[2]^2)
   
## End(Not run)