Non linear regression for exponential decay model in R - r

Question

I have the following problem:
I asked 5 people (i=1, ..., 5) to forecast next period's return of 3 different stocks. This gives me the following data:
S_11_i_c <-read.table(text = "
i c_1 c_2 c_3
1 0.150 0.70 0.190
2 0.155 0.70 0.200
3 0.150 0.75 0.195
4 0.160 0.80 0.190
5 0.150 0.75 0.180
",header = T)
In words, in period t=10 participant i=1 expects the return of stock c_1 to be 0.15 in period t=11.
The forecasts are based on past returns of the stocks. These are the following:
S_t_c <-read.table(text = "
time S_c_1 S_c_2 S_c_3
1 0.020 0.015 0.040
2 0.045 0.030 0.050
3 0.060 0.045 0.060
4 0.075 0.060 0.060
5 0.090 0.070 0.060
6 0.105 0.070 0.090
7 0.120 0.070 0.120
8 0.125 0.070 0.140
9 0.130 0.070 0.160
10 0.145 0.070 0.180
",header = T)
In words, stock c=1 had a return of 0.145 in period 10.
So, the variables in table S_11_i_c are the dependent variables.
The variables in table S_t_c are the independet variables.
The model I want to estimate is the following:
My problem with coding this is as follows:
I do only know how to express
with the help of a loop. As in:
Sum_S_t_c <- data.frame(
s = seq(1:9),
c_1 = rnorm(9)
c_2 = rnorm(9)
c_3 = rnorm(9)
)
Sum_S_t_c = 0
for (c in 2:4) {
for (s in 0:9) {
Sum_S_t_c[s,c] <- Sum_S_t_c + S_t_c[10-s, c]
Sum_S_t_c = Sum_S_t_c[s,c]
}
}
However, loops within a regression are not possible. So, my other solution would be to rewrite the sum to
However, as my actual problem has a much larger n, this isn*t realy working for me.
Any ideas?