Improved Regression Estimation of a Multivariate Relationship with Population Data on the Bivariate Relationship

Regression coefficients specify the partial effect of a regressor on the dependent variable. Sometimes the bivariate or limited multivariate relationship of that regressor variable with the dependent variable is known from population-level data. We show here that such population- level data can be used to reduce variance and bias about estimates of those regression coefficients from sample survey data. The method of constrained MLE is used to achieve these improvements. Its statistical properties are first described. The method constrains the weighted sum of all the covariate-specific associations (partial effects) of the regressors on the dependent variable to equal the overall association of one or more regressors, where the latter is known exactly from the population data. We refer to those regressors whose bivariate or limited multivariate relationships with the dependent variable are constrained by population data as being ‘‘directly constrained.’’ Our study investigates the improvements in the estimation of directly constrained variables as well as the improvements in the estimation of other regressor variables that may be correlated with the directly constrained variables, and thus ‘‘indirectly constrained’’ by the population data. The example application is to the marital fertility of black versus white women. The difference between white and black women’s rates of marital fertility, available from population-level data, gives the overall association of race with fertility. We show that the constrained MLE technique both provides a far more powerful statistical test of the partial effect of being black and purges the test of a bias that would otherwise distort the estimated magnitude of this effect. We find only trivial reductions, however, in the standard errors of the parameters for indirectly constrained regressors.

Tiered Pricing and Cost Functions: Are Equity, Cost Recovery and Economic Efficiency Compatible Goals?

Suppliers of water and energy are frequently natural monopolies, with their pricing regulated by governmental agencies. Pricing schemes are evaluated by the efficiency of the resource allocation they lead to, the capacity of the utilities to capture their costs and the distributional effects of the policies, in particular, impacts on the poor. One pricing approach has been average cost pricing, which guarantees cost recovery and allows utilities to provide their product at relatively low prices. However, average cost pricing leads to economically inefficient consumption levels, when sources of water and energy are limited and increasing the supply is costly. An alternative approach is increasing block rates (hereafter, IBR or tiered pricing), where individuals pay a low rate for an initial consumption block and a higher rate as they increase use beyond that block. An example of IBR is shown in Figure 1 (on next page), which shows a rate structure for residential water use. With the rates in Figure 1, a household would be charged $0.46 and $0.71 per hundred gallons for consumption below and above 21,000 gallons per month, respectively.