Covariance Matrix of the Parameters

The elements of the covariance matrix of the parameter estimates are calculated when the minimization algorithm has converged with a zero value for the Marquardt s directional parameter. The covariance matrix of the parameters COV(k) is determined using Equation 11.1 where the degrees of freedom used 

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It was shown earlier that ifN experiments have been performed, the covariance matrix of the parameters is estimated by... 

The prior covariance matrix of the parameters (VB) is a diagonal matrix and since in the solution of the problem only the inverse of VB is used, it is preferable... 

Finally, we compute Glesjer s test statistics for the three models discussed in Section 14.3.5. We regress e2, e, and log e on 1, Xh and X2. We use the White estimator for the covariance matrix of the parameter estimates in these regressions as there is ample evidence now that the disturbances are heteroscedastic related to X2. To compute the Wald statistic, we need the two slope coefficients, which we denote q, and the 2x2 submatrix of the 3x3 covariance matrix of the coefficients, which we denote Vq. The statistic is W - q Vq 1q. For the three regressions, the values are 4.13, 6.51, and 6.60, respectively. The critical value from the chi-squared distribution with 2 degrees of freedom is 5.99, so the second and third are statistically significant while the first is not. 

Based on the (symmetric) variance-covariance matrix of the parameter estimates V(b), one can determine confidence limits of the parameter estimates. The diagonal elements v contain the parameter estimate variances, and the off-diagonal elements the covariances between the parameter estimates. The interval of parameter values that are statistically not significantly different from the estimated value b, at a selected probability level (1 - a) is defined by... 

In practical cases, it will probably be difficult to estimate the covariances within the 0Bw and even more difficult to estimate any correlations between the parameters 6 and Bw. If the latter are neglected, the covariance matrix of the parameters determined by the fit, which includes also the errors due to the fixed parameters, will then be the sum of Eqs. 21b and 30 ... 

In these last equations, P, a and r are now the expectation values (solutions) of the quantities defined above, and 0p is the covariance matrix of the parameters. is the number of observables and m the number of parameters. Because the functional dependence of the observables (rotational constants, principal moments of inertia or planar moments) on the structural parameters is strongly non-linear in most cases, an iterative process is essential. Typically, one begins with an assumed structure and expands the moment of inertia functions in terms of the parameters of this structure in a Taylor senes up to the linear term. 

The latter standard deviation is obtained as the corresponding diagonal element of the covariance matrix of the parameter estimates. In the case of a model, which is linear in the parameters (y = Xb), this matrix is calculated as... 

The covariance matrix of the parameters does not only provide the information on the individual significance of the parameters. A correlation coefficient between the parameter estimates can be calculated from the appropriate elements of this covariance matrix, that is, the covariance between the two parameters considered, element iij), and their corresponding variances, diagonal elements i andy ... 

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