# R/quasi_sym.R In cquad: Conditional Maximum Likelihood for Quadratic Exponential Models for Binary Panel Data

#### Documented in quasi_sym

```quasi_sym <- function(eta,s,dyn=FALSE,y0=NULL){

# Compute quasi-symmetric function and its derivatives for given parameters eta
# and total equal to s
# if dyn then the last element of is gamma, in this case y0 must be precised
# preliminaries

TT = length(eta)
if(dyn){
ga = eta[TT]
eta = eta[-TT]
TT = TT-1
uT1 = c(rep(0,TT),1)
}
# initialization
if(dyn){
g0 = c(1,0)
g1 = c(0,exp(eta[1]+y0*ga))
E0 = matrix(0,TT+1,2)
E1 = matrix(0,TT+1,2)
E1[1,2] = exp(eta[1]+y0*ga)
E1[TT+1,2] = y0*exp(eta[1]+y0*ga)
F0 = array(0,c(TT+1,TT+1,2))
F1 = array(0,c(TT+1,TT+1,2))
F1[1,1,2] = exp(eta[1]+y0*ga)
F1[1,TT+1,2] = F1[TT+1,1,2] = F1[TT+1,TT+1,2] = y0*exp(eta[1]+y0*ga)
}else{
f = c(1,exp(eta[1]))
D1 = t(c(0,exp(eta[1])))
D2 = array(D1,c(1,1,2))
}
if(TT>1) for(t in 2:TT){
if(dyn){
g00 = g0; g10 = g1
E00 = E0; E10 = E1
F00 = F0; F10 = F1
# function g
g0 = c(g00+g10,0)
g1 = c(0,g00*exp(eta[t])+g10*exp(eta[t]+ga))
# first derivative
E0 = cbind(E00+E10,0)
E1 = cbind(0,E00*exp(eta[t])+E10*exp(eta[t]+ga))
E1[t,-1] = E1[t,-1] + g00*exp(eta[t])+g10*exp(eta[t]+ga)
E1[TT+1,-1] = E1[TT+1,-1] + g10*exp(eta[t]+ga)
# second derivative
F0 = array(0,c(TT+1,TT+1,t+1))
F0[,,1:t] = F00+F10
F1 = array(0,c(TT+1,TT+1,t+1))
for(h in 1:t){
ut = rep(0,TT+1); ut[t] = 1
ut2 = ut+uT1
Tmp1 = E00[,h]%o%ut
Tmp2 = E10[,h]%o%ut2
F1[,,h+1] = F1[,,h+1]+(F00[,,h]+Tmp1+t(Tmp1)+g00[h]*(ut%o%ut))*exp(eta[t])+
(F10[,,h]+Tmp2+t(Tmp2)+g10[h]*(ut2%o%ut2))*exp(eta[t]+ga)
}
}else{
f0 = f; D10 = D1; D20 = D2
# function f
f = c(f,0)+c(0,f)*exp(eta[t])
# first derivative
st = min(s,t)
s1 = 1:(t-1); s1 = s1[s>=s1]
ee = exp(eta[t])
D1 = matrix(0,t,st+1)
if(length(s1)>0){
D1[1:(t-1),s1+1] = D10[,s1+1]+D10[,s1]*ee
D1[t,s1+1] = f0[s1]*ee
}
if(s>=t){
D1[1:(t-1),t+1] = D10[,t]*ee
D1[t,t+1] = f0[t]*ee
}
# second derivative
D2 = array(0,c(t,t,st+1))
if(length(s1)>0){
D2[1:(t-1),1:(t-1),s1+1] = D20[,,s1+1]+D20[,,s1]*ee
D2[1:(t-1),t,s1+1] = D2[t,1:(t-1),s1+1] = D10[,s1]*ee
D2[t,t,s1+1] = f0[s1]*ee
}
if(s>=t){
D2[1:(t-1),1:(t-1),t+1] = D20[,,t]*ee
D2[1:(t-1),t,t+1] = D2[t,1:(t-1),t+1] = D10[,t]*ee
D2[t,t,t+1] = f0[t]*ee
}
}
}
# output
if(dyn){
f = g0+g1; f = f[s+1]
D1 = E0+E1; d1 = D1[,s+1]
D2 = F0+F1; D2 = D2[,,s+1]
lf = log(f); dl1 = d1/f; Dl2 = D2/f-dl1%o%dl1
}else{
f = f[s+1]; d1 = D1[,s+1]; D2 = D2[,,s+1]
lf = log(f); dl1 = d1/f; Dl2 = D2/f-dl1%o%dl1
}
out = list(f=as.vector(f),d1=as.vector(d1),D2=D2,lf=as.vector(lf),dl1=as.vector(dl1),Dl2=Dl2)
return(out)

}
```