Unbiased predictor

We next describe the first steps in the derivation of the best linear unbiased predictor (BLUP) of Y x) at an untried input vector x (see, for example, Sacks et al 1989). Similar steps are used in Section 4 to estimate the effects of one, two, or more input variables. It is then apparent how to adapt results and computational methods for predicting Y(x) to the problem of estimating such effects. 

Following the random-function model (1), consider the prediction of T(x ) by f (.v) = a x)y, that is, a linear combination of the n values of the output variable observed in the experiment. The best linear unbiased predictor is obtained by minimizing the mean squared error of the linear predictor or approximator, Y(x). The mean squared error, MSIi K(x), is... 

This constraint is also sometimes motivated by unbiasedness, that is, from E[T(a )] = E[T(a )] for all /3. Thus, the best linear unbiased predictor, or optimal value of a(x), results from the following optimization problem,... 

Derivation of the Best Linear Unbiased Predictor of an Effect... 

The best linear unbiased predictor (BLUP) of Ye(xe) in (18) follows from the properties of Ze xe) in (17). Clearly, Ze(xe), like Z(x), has expectation zero. Its variance, however, differs from one effect to another ... 

Also under OLS assumptions, the regression parameter estimates have a number of optimal properties. First, 0 is an unbiased estimator for 0. Second, the standard error of the estimates are at a minimum, i.e., the standard error of the estimates will be larger than the OLS estimates given any other assumptions. Third, assuming the errors to be normally distributed, the OLS estimates are also the maximum likelihood (ML) estimates for 0 (see below). It is often stated that the OLS parameter estimates are BLUE (Best Linear Unbiased Predictors) in the sense that best means minimum variance. Fourth, OLS estimates are consistent, which in simple terms means that as the sample size increases the standard error of the estimate decreases and the bias of the parameter estimates themselves decreases. 

There are four distinct and incompatible versions of the expectations hypothesis. The unbiased version states that current forward rates are unbiased predictors of future spot rates. Let ft T,T +1) be the forward rate at time t for the period from Tto T+ 1 and tt be the one-period spot rate at time T. The unbiased expectations hypothesis states that /Rt,t + i) is the expected value of rj. This relationship is expressed in (3.36). 

Figure 8,2 Methodology of grapevine selection in Portugal. EBLUPs empirical best linear unbiased predictors.

Measurement error causes double-trouble attenuation of the slope and increased error about the regression line. However, when more complex error structures are assumed, such as when X is not an unbiased estimate of x or the variance of 8 depends on x, then it is possible for the opposite effect to occur, e.g., i is inflated (Car-roll, Ruppert, and Stefanski, 1995). Rarely are these alternative measurement error models examined, however. The bottom line is that measurement error in the predictors leads to biased estimates of the regression parameters, an effect that is dependent on the degree of measurement error relative to the distribution of the predictors. 

A covariate is a source of variation contributing to SSY that may not be of particular interest but whose effect must be removed (1) in order to get unbiased estimates of other predictors of interest and (2) in order to reduce the noise level of the system so that predictors of interest can be more clearly seen. It may be that a covariate is confounded with a predictor of interest. The use of the t ratio in evaluating the reality of that predictor s contribution will then quite properly be conservative— discounting the information held in common with the covariate. 

---