By Madhavan A.
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Extra resources for Market microstructure: A practitioner's guide
Example text
5) Also tr(Ik ) = k, where to highlight the dimensionality of an identity matrix, we write Ik instead of I. 5). 5). 3. 5)) t=1 = σ 2 tr (XT X)−1 (XT X) = σ 2 tr(Ip ) = σ 2 p. 30). In the case where xt omits input variables associated with nonzero βj ’s, T Eyt = E(β xt ) and therefore Eet = E(yt − βT xt ) = 0. 2 Principal component analysis (PCA) n n t=1 T E(yt − β xt ) Ee2t = E(RSS) = 2 41 + tr Cov(e) . 31). 1. Let V be a p × p matrix. A complex number λ is called an eigenvalue of V if there exists a p × 1 vector a = 0 such that Va = λa.
2. 30). 29) can be interpreted as a linear penalty for adding input variables to the regression. Next consider the case of omitting important input variables xj (with βj = 0) from the regression model. 2. Therefore RSSp s2K in this case tends to be substantially larger than in the regression model that includes all regressors corresponding to nonzero βj . Therefore overfitting or underfitting will tend to increase the value of Cp , leading to Mallows’ selection criterion. 3 Variable selection 15 Akaike’s information criterion Akaike (1973) introduced an information criterion (AIC) based on likelihood theory for parametric models.
30). In the case where xt omits input variables associated with nonzero βj ’s, T Eyt = E(β xt ) and therefore Eet = E(yt − βT xt ) = 0. 2 Principal component analysis (PCA) n n t=1 T E(yt − β xt ) Ee2t = E(RSS) = 2 41 + tr Cov(e) . 31). 1. Let V be a p × p matrix. A complex number λ is called an eigenvalue of V if there exists a p × 1 vector a = 0 such that Va = λa. Such a vector a is called an eigenvector of V corresponding to the eigenvalue λ. We can rewrite Va = λa as (V − λI)a = 0. Since a = 0, this implies that λ is a solution of the equation det(V − λI) = 0.