TAILIEUCHUNG - Class Notes in Statistics and Econometrics Part 14

CHAPTER 27 Best Linear Prediction. Best Linear Prediction is the second basic building block for the linear model, in addition to the OLS model. Instead of estimating a nonrandom parameter β about which no prior information is available, in the present situation one predicts a random variable z whose mean and covariance matrix are known. | CHAPTER 27 Best Linear Prediction Best Linear Prediction is the second basic building block for the linear model in addition to the OLS model. Instead of estimating a nonrandom parameter 3 about which no prior information is available in the present situation one predicts a random variable z whose mean and covariance matrix are known. Most models to be discussed below are somewhere between these two extremes. Christensen s Chr87 is one of the few textbooks which treat best linear prediction on the basis of known first and second moments in parallel with the regression model. The two models have indeed so much in common that they should be treated together. 703 704 27. BEST LINEAR PREDICTION . Minimum Mean Squared Error Unbiasedness Not Required Assume the expected values of the random vectors y and z are known and their joint covariance matrix is known up to an unknown scalar factor a2 0. We will write this as 2 yy yz zy zz a2 0. y z M v y is observed but z is not and the goal is to predict z on the basis of the observation of y . There is a unique predictor of the form z B y b . it is linear with a constant term the technical term for this is affine with the following two properties it is unbiased and the prediction error is uncorrelated with y . C z - z y O. The formulas for B and b are easily derived. Unbiasedness means v B y b the predictor has therefore the form z v B y p . Since z z B y p z v B I v V . MINIMUM MEAN SQUARED ERROR UNBIASEDNESS NOT REQUIRED 705 the zero correlation condition translates into B Qyy QZy which due to equation holds for B Q zyQyy. Therefore the predictor z V Qzyftyy y - m satisfies the two requirements. Unbiasedness and condition are sometimes interpreted to mean that z is an optimal predictor. Unbiasedness is often naively but erroneously considered to be a necessary condition for good estimators. And if the prediction error were correlated with the .

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