▁without ▁feedback ▁loops : ▁the ▁directed ▁graph ▁of ▁the ▁model ▁must ▁contain ▁no ▁cycles , ▁i . e . ▁it ▁is ▁a ▁directed ▁a cyc lic ▁graph , ▁which ▁has ▁been ▁extens ively ▁studied ▁in ▁the ▁caus al ▁analysis ▁framework ▁of ▁J ude a ▁Pear l . ▁ ▁Path ▁trac ing ▁in ▁un standard ized ▁models ▁If ▁the ▁mode led ▁variables ▁have ▁not ▁been ▁standard ized , ▁an ▁additional ▁rule ▁allows ▁the ▁expected ▁cov arian ces ▁to ▁be ▁calculated ▁as ▁long ▁as ▁no ▁paths ▁exist ▁connecting ▁dependent ▁variables ▁to ▁other ▁dependent ▁variables . ▁ ▁The ▁simplest ▁case ▁obtain s ▁where ▁all ▁resid ual ▁var ian ces ▁are ▁mode led ▁explicitly . ▁In ▁this ▁case , ▁in ▁addition ▁to ▁the ▁three ▁rules ▁above , ▁calculate ▁expected ▁cov arian ces ▁by : ▁ ▁Comput e ▁the ▁product ▁of ▁coefficients ▁in ▁each ▁route ▁between ▁the ▁variables ▁of ▁interest , ▁trac ing ▁backwards , ▁changing ▁direction ▁at ▁a ▁two - head ed ▁arrow , ▁then ▁trac ing ▁for wards . ▁ ▁Sum ▁over ▁all ▁distinct ▁routes , ▁where ▁path ways ▁are ▁considered ▁distinct ▁if ▁they ▁contain ▁different ▁coefficients , ▁or ▁encounter ▁those ▁coefficients ▁in ▁a ▁different ▁order . ▁ ▁Where ▁resid ual ▁var ian ces ▁are ▁not ▁explicitly ▁included , ▁or ▁as ▁a ▁more ▁general ▁solution , ▁at ▁any ▁change ▁of ▁direction ▁encountered ▁in ▁a ▁route ▁( except ▁for ▁at ▁two - way ▁ar rows ), ▁include ▁the ▁variance ▁of ▁the ▁variable ▁at ▁the ▁point ▁of ▁change . ▁That ▁is , ▁in ▁trac ing ▁a ▁path ▁from