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**Additional info for A Guide to Modern Econometrics (2nd Edition)**

**Sample text**

Most estimators in econometrics can be shown to be asymptotically normally distributed (under weak regularity conditions). By√the asymptotic distribution of a consistent estimator βˆ we√mean the distribution of N (βˆ − β) as N goes to inﬁnity. The reason for the factor N is that asymptotically βˆ is equal to β with probability one for all consistent estimators. That is, βˆ − β has a degenerate distribution for N → ∞ with all probability mass at zero. 1; or Greene, 2003, Appendix D). 19 To be precise, E{εi |xi } = 0 implies that E{εi g(xi )} = 0 for any function g (see Appendix B).

XN yN . . xNK ... So, in the N × K matrix X the i-th row refers to observation i, and the k-th column refers to the k-th explanatory variable (regressor). 4), can be rewritten in matrix notation using that the inner product of a given vector a with itself (a a) is the sum of its squared elements (see Appendix A). 17) 3 The data for this example are available as WAGES1. ORDINARY LEAST SQUARES AS AN ALGEBRAIC TOOL 13 from which the least squares solution follows from differentiating4 with respect to β˜ and setting the result to zero: ˜ ∂S(β) ˜ = 0.

To do so, we need to make some assumptions about the error term and the explanatory variables xi . The ﬁrst set of assumptions we consider are the so-called Gauss–Markov assumptions. These assumptions are usually standard in the ﬁrst chapters of econometrics textbooks, although – as we shall see below – they are not all strictly needed to justify the use of the ordinary least squares estimator. They just constitute a simple case in which the small sample properties of b are easily derived. The standard set of Gauss–Markov assumptions is given by E{εi } = 0, i = 1, .