'=========================================================
'the performance of wold decomposition in nonlinear series
'=========================================================








create u 1 1000

'========================================================================================
' the ture DGP is:  y(t) =epsilon(t) + a1*epsilon(t-1) + a2*epsilon(t-1)*epsilon(t-2)
' which is nonlinear if a2 is not 0, and weakly stationary
'========================================================================================

scalar sumvar1=0 
scalar sumvar2=0  
scalar sumvar3=0 
 
'=======parameter values=======                              
scalar a1=0.2                                    
scalar a2=0.8
'==============================


for !t=1 to 1000                                  


'======generate epsilon_t, which is N(0,1)=======
series eps=nrnd  
'================================================


                             
'======generate y_t=======

smpl @first @first
series y=eps

smpl @first+1  @first+1
series y=eps+a1*eps(-1)

smpl @first+2 @last
series y=eps+a1*eps(-1)+a2*eps(-1)*eps(-2)

smpl @first @last

'======================


equation appro1.ls y eps(-1) eps(-1)*eps(-2)        ' (1  approximate the model assuming we know the DGP
equation appro2.ls y y(-1 to -10)                   ' (2  approximate y_t with a AR(10)
series u=resid                                      ' (3  get the residuals u(t) from the AR(10) regression
equation appro3.ls y u(-1 to -10)                   ' (4  approximate y(t) with the lags of u(t)
 

sumvar1=sumvar1+(appro1.@ssr)/990             
sumvar2=sumvar2+(appro2.@ssr)/990 
sumvar3=sumvar3+(appro2.@ssr)/990 


next

scalar estimatevar1=sumvar1/1000      '"estimatevar1" is the estimation of the variance in the first regression, which should be 1
scalar estimatevar2=sumvar2/1000      '"estimatevar2" is the estimation of the variance in the second regression, which should be larger than 1
scalar estimatevar3=sumvar3/1000      '"estimatevar1" is the estimation of the variance in the third regression, which should be lager than 1.

'=======================================================================
'if the DGP is linear, (a2=0), then all the three estimates should be 1
'=======================================================================