99 research outputs found
A consistent conditional moment test of functional form
Conditional moment (CM) tests of functional form exploit the property that for correctly specified models the conditional expectation of certain functions of the observations should be almost surely equal to zero. A chi-square misspecification test can then be based on weighted means of thes
Complex Unit Roots and Business Cycles: Are They Real?
In this paper the asymptotic properties of ARMA processes with complex- conjugate unit roots in the AR lag polynomial are studied. These processes behave quite differently from processes with a single root equal to 1. In particular, the asymptotic properties of a standardized version of the periodogram for such processes are analyzed, and a nonparametric test of the complex unit root hypothesis against the stationarity hypothesis is derived. This test is applied to the annual change of the monthly number of unemployed in the US, in order to see whether this time series has complex unit roots in the business cycle frequencies.
CONSISTENCY AND ASYMPTOTIC NORMALITY OF SIEVE ML ESTIMATORS UNDER LOW-LEVEL CONDITIONSâCORRIGENDUM TO SUPPLEMENTARY MATERIAL
The Wold Decomposition
In Chapter 7 in Bierens (2004) the Wold decomposition was motivated by claiming that every zero-mean covariance stationary process Xt can be written as Xt = Pâ j=1 ÎČjXtâj + Ut, where E[Ut.Xtâj] =0for all j â„ 1, and Pâ j=1 ÎČjXtâj is the projection of Xt on its past. However, in general this claim is incorrect. In this note I will give a more general (and hopefully correct) proof of the Wold decomposition. 1 Projections on spaces spanned by a sequence The fundamental projection theorem states that: Theorem 1. Given a sub-Hilbert space S of a Hilbert space H and an element y â H, there exists a unique element by â S such that ||y â by| | = infzâS ||y â z||. Moreover the residual u = y â by is orthogonal to any z â S: hu, zi =0. Proof: See for example Bierens (2004, Th. 7.A.3, p. 202). This result is the basis for the famous Wold (1938) decomposition for covariance stationary time series, which in its turn is the basis for time series analysis. Thanks to Peter Boswijk (University of Amsterdam) for pointing out an error in a previous version of this note. Moreover, the queries of the students in my graduate time series courses have led to substantial improvements of the proof of the Wold decomposition. 1 The proof of the Wold decomposition in Anderson (1994) is more transparent than the original proof by Wold (1938). However, rather than following Andersonâs proof, I will in this note derive first a general Wold decomposition for a regular sequence1 in a general Hilbert space, and then specialize this result to the Wold decomposition for covariance stationary time series. First, we need to define sub-Hilbert spaces spanned by a sequence in a Hilbert space, as follows. Let {xk} â k=1 be a sequence of elements of a Hilbert space H, and let Mm = span({xj} m j=1) be the space spanned by x1,..., xm, i.e., Mm consists of all linear combinations of x1,..., xm. Then Lemma 1. Mm is a Hilbert space. Proof: Without loss of generality we may assume that the m Ă m matrix ÎŁm with elements hxi,xji,i,j=1,..., m, is non-singular, as otherwise we can remove one or more xjâs from the list {xj} m j=1 and still span the same space. For example, suppose that rank(ÎŁm) =m â 1, and let c =(c1,..., cm) 0 be the eigenvector corresponding to the zero eigenvalue. Then ° Pm j=1 cjx
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