Notes on Economic Time Series Analysis: System Theoretic by Masanao Aoki

By Masanao Aoki

In seminars and graduate point classes i've got had numerous possibilities to debate modeling and research of time sequence with economists and fiscal graduate scholars prior to now a number of years. those reports made me conscious of a niche among what monetary graduate scholars are taught approximately vector-valued time sequence and what's on hand in fresh process literature. Wishing to fill or slender the distance that i believe is extra greatly unfold than my own reports point out, i've got written those notes to enhance and reor­ ganize fabrics i've got given in those classes and seminars. i've got endeavored to give, in as a lot a self-contained approach as workable, a physique of effects and strategies in process idea that I pass judgement on to be correct and worthy to economists drawn to utilizing time sequence of their examine. i've got primarily acted as an middleman and interpreter of process theoretic effects and views in time sequence via filtering out non-essential info, and providing coherent bills of what I deem to be very important yet no longer on hand, or available to economists. for that reason i've got excluded from the notes many effects on a number of estimation tools or their statistical homes simply because they're amply mentioned in lots of average texts on time sequence or on statistics.

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Equally allocated to W(z) and W' (z-l). Spectral Factorization theorem. A The zero on Izl =1 are This is the scalar version of the basic result for vector-valued process is: Theorem Let S(z) be a real rational full rank covariance generating function. Then it can be factored as S(z) stable, of minimum phase, and L' = W(Z)LW*(Z) =L where W(z) is real, ration,al, > O. Consequently, W-l(z) is analytic in Izl > 1. W-1 (z) = LOCkZ -k , co We can then write for Izl > 1, where the Taylor series exapnsion is valid in Iz I > 1.

The state vector dimension need not be an integer multiple of the dimension of y. Just because a component of Yt' say the third component, enters the state vector does not mean that the third component of Yt-l also enters into the state vector. , to span the predictor space. vector y~-N+l) I (y~, y~-l' ••• , and the future vector (Y~+l' ••• , y~+N)' we obtain an important example of the Hankel matrix Yt+l (6) E Yt+N [y~, y~-l·· ·y~_N+1J ... ~ ........ A2 ~+l Al A2 ~ ........ A2N - l This matrix is important for us because we construct a state-space model of the time series {Yt} by operating on this and related Hankel matrices.

E, S(z)* = s' (z- ). -1 of S(z), then so is zk ' if zk is real. So if zk is a zero If zk is complex, then zf is also a zero. h} be Let {Yh , Y a set of complex roots of mmS(z) = 0 with Iyhl > 1 and also containing half of those roots iYhl =l. Let Pj be a real root. I. I. I. -1 -1 ] 'IT (z - y,)]( (z - Y ) (z - Y* ) 'IT (z - P-1 j ) J k=l k j=l j=l k ~ m. Yk ) (z - Yf>'IT~=l (z - Pj ) . -1 -1 -Y = liz - Y= l-Yz/z =Y -z/z. or z[z - y] = -(z we see that S(z) has no zero in Izl < 1. , minimum phase filter.

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