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Standard Data Structures

X 1 : 4.7 Unconditional Moments The previous sections derived the form of the conditional mean and variance of least-squares estimator, where we conditioned on the regressor matrix X. What about the unconditional mean and variance? Many authors and textbooks present unconditional results by either assuming or treating the regressor matrix X as ìÖxedî. Statistically, this is appropriate when the values of the regressors are determined by the experiment and the only randomness is through the realizations of y. Fixed regressors is not appropriate for observational data. Thus econometric results for Öxed regressors are better interpreted as cond CHAPTER 4. LEAST SQUARES REGRESSION 110 The core question is to state conditions under which the unconditional moments of the estimator are Önite. For example, if it determined that E b < 1, then applying the law of iterated expectations (Theorem 2.1), we Önd that the unconditional mean of b is also E  b  = E  E  b j X  = : A challenge is that b may not have Önite moments. Take the case of a single dummy variable regressor di with no intercept. Assume P (di = 1) = p < 1. Then b = Pn i=1 P diyi n i=1 di is well deÖned if Pn i=1 di > 0. However, P ( Pn i=1 di = 0) = (1 p) n > 0. This means that with positive (but small) probability, b does not exist. Consequently b has no Önite moments! We ignore this complication in practice but it does pose a conundrum for theory. This existence problem arises whenever there are discrete regressors. A solution can be obtained when the regressors have continuous distributions. A particularly clean statement was obtained by Kinal (1980) under the assumption of normal regressors and errors. While we introduce the normal regression model in Chapter 5 we present this result here for convenience. Theorem 4.3 (Kinal, 1980) In the linear regression model, if in addition (xi ,ei) have a joint normal distribution then for any r, E b r < 1 if and only if r < n k + 1. This shows that when the errors and regressors are normally distributed that the least-squares estimator possesses all moments up to n k, which includes all moments of practical interest. The normality assumption is not particularly critical for this result. What is key is the assumption that the regressors are continuously distribu