Change of Gate-Source Voltage of a NMOS Transistor due to Temperature
The gate-source voltage of a MOS transistor changes because of two things. An increase of the threshold voltage and a decrease of mobility with increasing temperature.

Which effect dominates depends on the bias point, so Vgs can either go up or down with temperature.

The mobility  \$\mu\$  is contained within 
$$
\beta = \mu C_{ox} \frac WL
$$
and it is a function of temperature given by
$$
  \mu \propto T^{BEX}
$$
where BEX depends on the technology (a typical value is -3/2).

For \$\beta\$ we find that
$$
\beta(T) = \beta(T_r) \left(\frac{T}{T_r}\right)^\text{BEX}
$$
wherer Tr is the  reference temperature where \$\beta(T_r)\$ was determined.

Including this dependence should give you the right results, don't forget to use the absolute temperature in your calculations (Kelvin).