In　many　applications,　a　single　Box-Cox　transformation　cannot　necessarily　produce　the　normality,　constancy　of　variance　and　linearity　of　systematic　effects.　In　this　paper,　by　establishing　a　heterogeneous　linear　regression　model　for　the　Box-Cox　transformed　response,　we　propose　a　hybrid　strategy,　in　which　variable　selection　is　employed　to　reduce　the　dimension　of　the　explanatory　variables　in　joint　mean　and　variance　models,　and　Box-Cox　transformation　is　made　to　remedy　the　response.　We　propose　a　unified　procedure　which　can　simultaneously　select　significant　variables　in　the　joint　mean　and　variance　models　of　Box-Cox　transformation　which　provide　a　useful　extension　of　the　ordinary　normal　linear　regression　models.　With　appropriate　choice　of　the　tuning　parameters,　we　establish　the　consistency　of　this　procedure　and　the　oracle　property　of　the　obtained　estimators.　Moreover,　we　also　consider　the　maximum　profile　likelihood　estimator　of　the　Box-Cox　transformation　parameter.　Simulation　studies　and　a　real　example　are　used　to　illustrate　the　application　of　the　proposed　methods.