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Show Solutions of AR(1) Equations Ryan Johnston ILajos Horvath) Department of Mathematics Aclmeaerlet Ran ordered BEquenceofraluetoruarJablesaceqi^llyipacedtlrnelnceruah Time series models are most often used to ofc*aln an undemanding of the uncferlying fadora that produce observed data, then fit chew models to forecasted men rtcr. Time series analysis Is used i application* such as budgetary analysis, census analysis, economic Forecasi irg. I nventory studies, process a nc quality control stock market arrays, uulrty si udies, worWcad pro|ec1 ens. and yield projections. The most commonly used time series equation Is the a uioregresslve process. The autoregressive piocess IsadlF-rerence equation determined by random ua rabies. The cfctri burton of such random ua rabies Is the key comcc nenrt In modellrg time series. The time series considered In this pra)?a Is the First order autoregressme equation, written as Aft; 1J The Afll Inequation Is a siandard linear difference equal \on Xr = pXf. r + Et c =U, ± I. ±i ___ where the > i are called 1 he error terms cr I n novations and are what make the varlabll n y in the time series. For pradh^al reasensn rt is desirable to have a unique sdul en that is Independent ohime^iatloraryj and a rur^tlonoFthe past error terms. A solution that is independem c-f time albwsane to be able to avid an Inltla zmdikm, whkh may be dlffeult to find or a1 an inconvenient Location In a time series. A solution as a function of the pasi ernr tprm* h necessary in models used to forecast 1 i ii conditions are necessary In order to guaranieel he existence of a unique stationary sdul en orthepast error terms^ The I nerature lvplca lly assumes the error terms are an u nccM lelated sequence of ra nefcm ua rubles with a probability distnbuipn that has zero Tor the mea n ar>d a hnrte varbnee. Thl-s project alms toexpkre If weaker astumplons can be macteafcKut the error terms and still guarantee the existence ofasiationary solution Aside "rom the mathemaiK^lskpnlficanceorthis. theie are practical advamages Sometime series, sudi as stcck markeJ exhibit betauor thai dc^snot appear to E*a mocfel that assumes thecondnpns of lhe llteiature. It Is thai weaker condh c-ni abcut the error terms can be made, whkh can still guaraniee the existence of the deslred scJui&n These less restrldiuecondhcns albwmc<^ls1o havedfitrltxjtlonsthat area be11er£ffcf time series such as stock market prices. |