Maximum likelihood estimator for classical simple linear regression

Question

I want to estimate the classical simple linear regression parameters using a maximum likelihood estimation. However, I'm having trouble to program such a thing. Here's the data I generated:

 Clear[y, x, e]
SeedRandom["MV régression simple"]
e = RandomVariate[NormalDistribution[0, 4], 100];
x = RandomVariate[NormalDistribution[20, 3], 100];
y = 4 + x*2.1 + e;

And here's what I did to try and find the likelihood function:

Clear[L]
L[b0_, b1_, s_] = 
 LogLikelihood[NormalDistribution[b0 + b1*x, s], y]

I don't know how to take account of the fact that yi depends only on xi and not the whole x list of data.

Thanks for your future help

Answer 1

Taking your model (though with more random numbers to make confirmation of the result easier)

Clear[y, x, e]
SeedRandom["MV régression simple"]
e = RandomVariate[NormalDistribution[0, 4], 1000];
x = RandomVariate[NormalDistribution[20, 3], 1000];
y = 4 + x*2.1 + e;

Define a slightly modified likelihood function, using MapThread to account for the differences in x and y values for each sample and summing over all instances with Total

Clear[L]
L[b0_, b1_, s_] = 
  Total[MapThread[
    LogLikelihood[NormalDistribution[b0 + b1*#1, s], {#2}] &, {x, y}]];

A maximum likelihood estimate can be found, which is close to the specified parameters

Maximize[{L[b0, b1, s], s > 0}, {b0, b1, s}]
(* {-2777.3, {b0 -> 3.82871, b1 -> 2.10828, s -> 3.8898}} *)