Definition: In maximum likelihood estimation, the score vector is the gradient of the
likelihood function with respect to the parameters. So it has the same number
of elements as the parameter vector does (often denoted k). The score is a
random variable; it's a function of the data. It has expectation zero, and is
set to zero exactly for a given sample in the maximum likelihood estimation
process.
Denoting the score as S(q), and the likelihood function as L(q), where in both cases the data are also implied arguments:
S(q) = dL(q)/d(q)
Example: In OLS regression of Yt=Xtb+et, the score for each possible parameter value, b, is Xt'et(b).
The variance of the score is E[score2]-(E[score])2) which is E[score2] since E[score] is zero. E[score2] is also called the information matrix and is denoted I(q).
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