Moment Space

Considerable statistical analysis and probabilistic economic modelling deals just with the first and second moments of the random variables. The full probability distribution of the variables plays no role. Moment space is a Euclidean space of random variables that embodies the first and second moments. A vector x in the space represents a random variable. The inner product $$~\left \langle \boldsymbol {y},\boldsymbol {x}\right \rangle$$ of random variables y and x is the expected value of their product (the second noncentral cross-moment). The mean is a linear function of x. To work in moment space enables the application of convex optimization and duality theory. Moment space is convenient for modelling random variables in a problem depending only on the first and second moment. We apply moment space to econometrics and finance. Least-squares linear regression involves only the first and second moments of the random variables, so moment space is applicable. Maximum-likelihood estimation of a normal distribution involves only the first and second moments. In finance, using moment space to study mean/variance choice under uncertainty is effective. To embed the second moment in the inner product simplifies the derivation of relationships and the statement of results. The cross-moment transformation between subspaces is the unique linear transformation A:M→N such that the inner product of x∈M and y∈N is $$\left \langle \boldsymbol {y},{\textit {{\sf { {A}}}}}\boldsymbol {x}\right \rangle \!.$$ A second interpretation is valuable. Let M and N denote the idempotents corresponding to these subspaces. Alternatively see the cross-moment transformation as the square linear transformation NM, such that the domain and range are restricted to M and N.