' ==============================================================================================
'                           Ergodic Theorem for the Mean
'
'     1.-  Generate an ergodic process (AR(1) causal) and a non-ergodic process (Random Walk)  yt = my + phi1* yt-1 + eyt  and xt = mx +  xt-1 + ext
'     2.-  Estimate the mean of both processes, for instance by OLS: 
'                   yt  = c + et and xt= c + et 
'            and see where do they converge asymptotically
'     3.-  Compute the t-statistics, the DW test and the R2 and save them in the matrices "t", "DW", and "R2", respectively
'     4.-  Iterate 1000 times
'     5.- Calculate the rejection frequency of the significance test of mean x=0 and also mean y=0
'     
' Other possible things to do:
' HAC (Newey-West) : equation eq1.ls(cov=hac) yt c  or 
' equation eq1.ls(cov=hac, bw=neweywest) yt c t
' Regression in forst difference: equation eq1.ls(cov=hac) d(yt) c 
'==============================================================================================

create u 1 1000

rndseed 12345678

' Generate Data and Statistics

scalar rep=1000
vector(rep)  meany=0
vector(rep)  meanx=0
vector(rep) tmeanx = 0
vector(rep) tmeany = 0

scalar phi1=0.3
scalar my = 0
scalar mx = 0

for !i = 1 to 1000

     smpl @first @first
     series yt = nrnd
     smpl @first+1 1000
     series yt =  my + phi1*yt(-1)+ nrnd 

     smpl @first @first
     series xt = nrnd
     smpl @first+1 1000
     series xt =  mx + xt(-1)+ nrnd 
     
     equation eq1.ls yt c
 	equation eq2.ls xt c

     smpl @first @last
	
	meany(!i)=eq1.@coefs(1)
     tmeany(!i)=eq1.@tstats(1)
	meanx(!i)=eq2.@coefs(1)
     tmeanx(!i)=eq2.@tstats(1)

   

next

smpl @first @last

' Rejection frequency Ho: mean (y) = 0

scalar freqmeany=0

for !i = 1 to 1000
     if  abs(tmeany(!i)) > 1.96 then
     freqmeany=freqmeany+(1/1000)
     endif
next

smpl @first @last

' Rejection frequency Ho: mean (x) = 0

scalar freqmeanx=0

for !i = 1 to 1000
     if  abs(tmeanx(!i)) > 1.96 then
     freqmeanx=freqmeanx+(1/1000)
     endif
next