This　paper　investigates　a　partially　linear　spatial　autoregressive　panel　data　model　that　incorporates　fixed　effects,　constant　and　time-varying　regression　coefficients,　and　a　time-varying　spatial　lag　coefficient.　A　two-stage　least　squares　estimation　method　based　on　profile　local　linear　dummy　variables　(2SLS-PLLDV)　is　proposed　to　estimate　both　constant　and　time-varying　coefficients　without　the　need　for　first　differencing.　The　asymptotic　properties　of　the　estimator　are　derived　under　certain　conditions.　Furthermore,　a　residual-based　goodness-of-fit　test　is　constructed　for　the　model,　and　a　residual-based　bootstrap　method　is　used　to　obtain　p-values.　Simulation　studies　show　the　good　performance　of　the　proposed　method　in　various　scenarios.　For　illustration,　the　carbon　emission　data　from　Chinese　provinces　and　the　public　capital　productivity　data　from　the　United　States　are　analyzed.　©　2025