Estimation of Transition Probabilities Using Median Absolute Deviations

Abstract

The probabtltty-constratned mtntmum abso­ lute demattons (MAD) esttmator appears to be supenor to the probab,ltty-constratned qundrattc programmtng esttmator tn esttmattng transttton probab,ltttes Wlth ltmtted aggregate t,me senes data Furthemwre, one can reduce the number of columns tn the probabtltty­ constratned MAD stmplex tableau by adopttng the medmn property KeylJJOrds_ Mtntmum absolute devtattons, transttwn probab,l,ttes, medmn absolute demattons, If!UUlrattc programmtng Markov processes are a speCIal class of mathematIcal models that,are often applIed to economIc declslOnmak­ Ing In stochastIC dynamIC programnung (5), structural changes of an Industry or changes m sIZe econonues (23), or internatIOnal trade (6) I To estImate a meamngful transItIon matnx, researchers need tIme-ordered data that reflect Intertemporal changes of mIcro umts over states (or classIficatIOns) However, tIme-ordered changes of mICroeconomlc umts are, generally not avail­ able for most econonuc vanables, therefore, researchers must often work WIth aggregate time senes data In an Ingemous artIcle, Lee, Judge, and Takayama (13) showed how one can estImate transItion probabIlitIes for a Markov process reflectIng the behaVIOr of nucro umts WIth only aggregate tIme senes data They con­ cluded from a IInuted tnal, based on the assumptIOn of normalIty of the error terms, that the probabllIty­ constraIned quadratIc programming (QP) estImator IS supenor to the probabIlIty-constrained mlmmum abso­ lute deVIatIOns (MAD) estImator In estImating transI­ tIOn probabIlItIes In a subsequent artICle, Lee, Judge, and Zellner concluded from theIr samplIng expenment that the probabIlIty-constrained MAD estImator IS mfenor to the probability-constramed QP estImator (14, p 135) We prove here that the probabIlIty-constrained MAD estImator IS supenor to the probabilIty-constrained QP KIm and Schruble are econonusts With the CommodIty EconOmICS DIVl sion and the Resources and Technology Dlvlslon, ER& Theyapprecl ate the helpful comments of V A SPOSito at Iowa State Uruverslty, T C Lee at the Uruverslty of Connecticut, and G G Judge at the Uruverslty of Cahforma Berkeley on an earher draft of thIS paper 1 ItahcIZed numbers m parentheses refer to Items In the References at the end of thIS artIcle estimator when estimating tranSItIOn probabllItles'Wlth lmuted aggregate time senes data Second, we present an alternatIve model, nummIZatlOn of medIan absolute deVIatIOns (MOMAD), based on the assumptIons that the error terms are nonnormally rustnbuted and that the researcher has a prwr! InformatIOn about the dynanuc nature of the Markov process ThIrd, we prove that the MOMAD estimator IS Identical WIth the probabllIty­ constramed MAD estunator, wluch-Bassett and Koenker (3) concluded IS a more efficIent estimator for any error rustnbutlOn for wluch the meruan IS supenor to the mean as an estimator of locatIOn Moreover, the constramt matnx assoc!ated WIth the MOMAD model mvolves fewer columns m the SImplex tableau