Fitted incorrect with small values

Modeling
1

I have noticed that my model for linear regression gives very accurate fitted values, except when they are really small. Is this loss of precision normal? How can I get an appropriate fit with small values?

I use the following model:

data {
    int<lower=0> N;
    vector[N] x;
    vector[N] y;
}
parameters {
    real intercept;
    real first_degree;
    real<lower=0> sigma;
}
model {
   intercept ~ normal(7.7e-07,7.7e-08)
   first_degree ~ normal(6.9e-11,6.9e-12)
   sigma ~ normal(1.9e-12,1.9e-13)
    y ~ normal(first_degree*x+intercept, sigma);
}

with the following data:

intercept=7.661272e-07
first_degree=6.862693e-11
sigma=1.895320e-12
x = sample(0:500,500,replace=T) 
y = first_degree * x + intercept + rnorm(N, sd=sigma)
df = data.frame(x=x,y=y)
2

Have you tried the non-centered parameterisation? It might make it easier for the sampler:

data {
    int<lower=0> N;
    vector[N] x;
    vector[N] y;
}
parameters {
    real intercept_raw;
    real first_degree_raw;
    real<lower=0> sigma_raw;
}
transformed parameters {
    real intercept = 7.7e-07 + intercept_raw * 7.7e-08;
    real first_degree = 6.9e-11 + first_degree_raw * 6.9e-12;
    real sigma = 1.9e-12 + sigma_raw * 1.9e-13;
}
model {
   intercept_raw ~ normal(0,1);
   first_degree_raw ~ normal(0,1);
   sigma_raw ~ normal(0,1)
    y ~ normal(first_degree*x+intercept, sigma);
}
3

@andrjohns Thank you for your reply, the non-centered parameterisation gave a better result, closer to the real values but the chains still tend to diverge.