Not clear about the concept of degrees of freedom? That's not embarassed at all, since many professors/book writers are not clearly defining them either. Read the following article by I.J. Good on American Statistician to clear the cloud.
The American Statistician, Vol. 27, No. 5 (Dec., 1973), pp. 227-228
Friday, February 12, 2010
Wednesday, February 10, 2010
Prior information
Making use of prior information in economic modelling serves as a major role in improving the predictability of economic models and its role in economic analysis. Nevertheless, prior information, which is usually not from the observed sample, is subjected to mispecification. Therefore, a valid test of this nonsample information using available sample deserves particular attention.
Hypothesis testing on prior in the form of equality constraints has a long history, dating back to 1920s by the work of Theil, Wald, Pearson, Newyman, etc. The test on inequality constraints starts in the statistics literature by Walat (1987 JASA, 1988 BKA, 1989 JoE), who transform the testing problem to be the one-sided hypothesis testing framework. Results of Kudo, Perlman, Nuesch, etc. were applied to derive the asymptotic/exact distributions of the LR, W, KT statistics. These statistics are shown to be (asymptotically) equivalent and have a distribution charactorized by a weighted sum of independent chi-squared distributions.
Testing inequality constraints has its own difficulty since the constrained estimator does not have an explicit expression as the original data matrix. The derivation of the test statistic needs finding a least favorable value of the parameter of interest. It turns out that the constraint imposed will be equality and those unbinding ones are not imposed.
Testing inequality constraints still has a long way to go, given its rather complexity. Yet, a lot of time economic priors come in the form of inequality. For example, the MPC is in between 0 and 1. The production function is increasing and concave, i.e. the first derivative is positive and the second derivative is negative. Equity premium should increase in some economic ratios, such as earning price ratiro, book to market ratio, etc. Given these examples, it is urgent to develop some convincing testing procedures that can accomodate all kinds of inequality prior information validation.
Hypothesis testing on prior in the form of equality constraints has a long history, dating back to 1920s by the work of Theil, Wald, Pearson, Newyman, etc. The test on inequality constraints starts in the statistics literature by Walat (1987 JASA, 1988 BKA, 1989 JoE), who transform the testing problem to be the one-sided hypothesis testing framework. Results of Kudo, Perlman, Nuesch, etc. were applied to derive the asymptotic/exact distributions of the LR, W, KT statistics. These statistics are shown to be (asymptotically) equivalent and have a distribution charactorized by a weighted sum of independent chi-squared distributions.
Testing inequality constraints has its own difficulty since the constrained estimator does not have an explicit expression as the original data matrix. The derivation of the test statistic needs finding a least favorable value of the parameter of interest. It turns out that the constraint imposed will be equality and those unbinding ones are not imposed.
Testing inequality constraints still has a long way to go, given its rather complexity. Yet, a lot of time economic priors come in the form of inequality. For example, the MPC is in between 0 and 1. The production function is increasing and concave, i.e. the first derivative is positive and the second derivative is negative. Equity premium should increase in some economic ratios, such as earning price ratiro, book to market ratio, etc. Given these examples, it is urgent to develop some convincing testing procedures that can accomodate all kinds of inequality prior information validation.
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