The Mauna Loa monthly $C{O}_2$ data series is freely available from the NOAA. You may access this data from the hyperlink below.

Mauna Loa

Fit data to the model

$Y_t=+{}_{1}t+{}_2{t}^2+cos(\frac{2(t-)}{12})+lo{n}_t$

where $Y_t$ are the monthly average $C{O}_2$ observations. Assume $\{lo{n}_t\}$ are IID $N(0,{}^2)$ Let $t=1$ being January $1959,t=2$ February $1959,$dots, with ending date December $2020.$ Use the methods discussed in class to obtain estimates for the model parameters and hat$({)}^2.$ Plot $Y_t$ and hat$(Y{)}_t$ verses time on the same graph.

Solution: