BIOGEME Statistics
Biogeme reports statistics and "robust" statistics about the estimated parameters (standard errors, t-tests and p values). We describe here how there are computed.
Let L(β) be the likelihood function to be maximized, and H=∇2 L(β*) its second derivatives matrix calculated at the estimated parameters β*. The Rao-Cramer bound is defined as
RRao = - H-1. The standard errors reported in the Biogeme output files are the diagonal entries of this matrix RRao. The rest of the matrix is reported in the section Correlation of coefficients in the column Covariance. A normalized version is reported in the column Correlation, knowing that Corr(βi,βj) = Cov(βi,βj)/(Var(βi)Var(βj))1/2.
If β is an estimated parameter and σ its reported standard error, the reported t-test is calculated as t=β/σ. The reported p value is calculated as p = 2(1-Φ(t)), where Φ(t) is the Cumulative Distribution Function (CDF) of a standardized normal random variable, estimated at the value t.
For the Exogenous Maximum Likelihood Estimator (ESML), the log likelihood function to be maximized is defined as Σn log Pn(β). The score of an observation n is defined as sn = ∇ log Pn(β). The BHHH matrix is defined as B=Σnsns'n, where s'n is the transposed of sn, so that sns'n is a K by K matrix of rank 1. The robust variance-covariance matrix is calculated as Rrobust = RRao B RRao. The "robust" statistics reported in the Biogeme out put files are computed as described above, based on the matrix Rrobust is used instead of RRao.
For the Weighted Exogenous Maximum Likelihood Estimator (WESML), the log likelihood function to be maximized is defined as Σn wn log Pn(β). The score of an observation n is defined as above, and the BHHH matrix is defined as B=Σnwnsns'n.
Biogeme