On Predictive Least Squares Principles

Testing for Cointegrating Rank Via Model Selection: Evidence From 165 Data Sets

“…where is the maximum likelihood estimator of Σ with given p and cointegrating rank r. Note that the penalty terms of AIC, HQ and SIC are a simple function of a parameter count (K), which increases with p and r. Wei (1992), Phillips and Ploberger (1996), and Chao and Phillips (1999) argue that the coefficients associated with regressors exhibiting a trend should be penalized more strongly than regressors without a trend. Phillips and Ploberger (1996) and Chao and Phillips (1999) propose the use of the posterior information criterion (PIC):…”

Asymptotic Log-Loss of Prequential Maximum Likelihood Codes

“…Examples are [22,10,13,25,17]. In all these papers it is shown that either the regret or the redundancy grows as k 2 ln n + o(ln n), either in expectation or almost surely.…”

“…To evaluate the term inside the expectation further we first Taylor approximate f (x n ) around µ n =μ(x n ), for given x n with (µ * −μ n ) 2 < a 2 n = 1/ √ n. We get f (x n ) = −(µ * −μ n ) d dµ ln Mμ n (x n ) + n 1 2 (µ * −μ n ) 2 I(µ n ), (25) where I is the Fisher information (as defined in Section 7) and µ n lies in between µ * andμ, and depends on the data x n . Since the first derivative of µ at the ML estimateμ is 0, the first-order term is 0.…”

Model selection and prediction: Normal regression

“…In the context of time series, Wax (1988) derived the weak consistency of an analogous estimator of the order of an autoregressive process without the Gaussian assumption, and Hemerly and Davis (1989) strengthened it to the a.s. consistency. Moreover, Wei (1992) obtained the a.s. consistency and asymptotic expansions of APE under stochastic regression models. Now we turn to selection rules based on the residual sum of squares, which is RSS~(k) = E~ rt,n(k) 2 where the ordinary residuals rt,~,(k) are defined above.…”

“…Moreover, the equivalence of BIC and APE has been shown by Hannah et el. (1989) for the finite-dimensional autoregressive models and by Wei (1992) for finite-dimensional stochastic regression models. (ii) If k* < K, and liminf(21oglogn)-lc~n > 2, we have, for some 7 > 2, pr(/~n > k*) _< O((logn) -'y) as n -~ cx~.…”