TAILIEUCHUNG - Class Notes in Statistics and Econometrics Part 21

CHAPTER 41 Interval Estimation. We will first show how the least squares principle can be used to construct confidence regions, and then we will derive the properties of these confidence regions. . A Basic Construction Principle for Confidence Regions The least squares objective function | CHAPTER 41 Interval Estimation We will first show how the least squares principle can be used to construct confidence regions and then we will derive the properties of these confidence regions. . A Basic Construction Principle for Confidence Regions The least squares ob jective function whose minimum argument gave us the BLUE naturally allows us to generate confidence intervals or higher-dimensional confidence regions. A confidence region for based on y X 3 e can be constructed as follows Draw the OLS estimate 3 into k-dimensional space it is the vector which minimizes SSE y X 3 T y X 3 . 921 922 41. INTERVAL ESTIMATION For every other vector 3 one can define the sum of squared errors associated with that vector as SSE y X 3 T y X 3 . Draw the level hypersurfaces if k 2 level lines of this function. These are ellipsoids centered on 3. Each of these ellipsoids is a confidence region for . Different confidence regions differ by their coverage probabilities. If one is only interested in certain coordinates of and not in the others or in some other linear transformation then the corresponding confidence regions are the corresponding transformations of this ellipse. Geometrically this can best be seen if this transformation is an orthogonal projection then the confidence ellipse of the transformed vector R is also a projection or shadow of the confidence region for the whole vector. Projections of the same confidence region have the same confidence level independent of the direction in which this pro jection goes. The confidence regions for with coverage probability n will be written here as Bp-n or if we want to make its dependence on the observation vector y explicit Bp n y . These confidence regions are level lines of the SSE and mathematically it is advantageous to define these level lines by their level relative to the minimum level . as as the set of all for which the quotient of the attained SSE . CONSTRUCTION OF CONFIDENCE REGIONS 923 y X T y X 3 .

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