[Verse 1] Stock prices whisper secrets from yesterday Each tick depends on what came before the break of day Like echoes bouncing through financial halls Today's return builds on history's calls AR one connects the dots in sequence Past performance leaves its blueprint [Chorus] Auto-regressive, looking backward Current equals past plus random factor Phi coefficient shows the power Yesterday's grip on this very hour Memory fading, coefficients tell the tale How much persistence will prevail [Verse 2] Take today's price minus yesterday's close Divide by yesterday, that's how the formula goes This return depends on prior returns we've seen Plus epsilon white noise, keeping data clean Phi between zero and one means stable ground Above one, explosive growth will be found [Chorus] Auto-regressive, looking backward Current equals past plus random factor Phi coefficient shows the power Yesterday's grip on this very hour Memory fading, coefficients tell the tale How much persistence will prevail [Bridge] Estimate phi through ordinary least squares Regress today on yesterday, see what it declares High phi means momentum carries strong Low phi means shocks don't linger long Unit root test checks if phi equals one Stationarity's where good models begun [Verse 3] AR two adds another lag in the mix Two periods back help current period fix Multiple coefficients sum must stay below One point zero or the model will explode and grow Volatility clustering shows in squares AR-GARCH captures what standard models miss upstairs [Final Chorus] Auto-regressive, looking backward Current equals past plus random factor Sum of phi's must stay inside the boundary Keep the time series foundationary Memory patterns tell tomorrow's story Built on yesterday's financial glory [Outro] When current values chase their history AR models solve the mystery
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