The data set linked below contains a time series generated from the following univariate dynamic linear

model:

$y_t=F_t{}_t+v_t$

${}_t=G_t{}_{t-1}+w_t$

where

$F_t=1\mathrm{.}2$

$G_t=0\mathrm{.}8$

${}_{0}N(0,{}_{{}_0}^2=25)$

$v_{t}N(0,{}_v^2=9)$

$w_{t}N(0,{}_w^2=4)$

a$.$ Apply a Kalman filter to this data to make one$-$step ahead predictions of ${}_t$ given $y_{1\text{:}(t-1)}.$ Create a

times$-$series plot containing the observations $y_t$ and one$-$step ahead predictions of ${}_t.$ Include a $95\%$

confidence band around your ${}_t$ predictions. Report the numerical values found for $a_{40}$ and $R_{40}.$

Here is the data:

df$=$structure$($list$($yt $=$ c$(4\mathrm{.}775114157,6\mathrm{.}094965658,2\mathrm{.}054150731,5\mathrm{.}029304107,7\mathrm{.}208172081,-3\mathrm{.}908933262,5\mathrm{.}08879584,-1\mathrm{.}14161645,2\mathrm{.}872719715,7\mathrm{.}006694276,$ NA$,-4\mathrm{.}063644149,0\mathrm{.}690397699,-4\mathrm{.}057012627,-3\mathrm{.}158702097,-7\mathrm{.}666647236,-3\mathrm{.}629952752,0\mathrm{.}223249677,0\mathrm{.}871667252,-5\mathrm{.}297199802,0\mathrm{.}773616301,9\mathrm{.}30143461,$ NA$,$ NA$,$ NA$,$ NA$,$ NA$,-1\mathrm{.}392325651,-1\mathrm{.}139686324,-0\mathrm{.}614167816,-6\mathrm{.}790519245,-1\mathrm{.}484028604,-0\mathrm{.}782412553,4\mathrm{.}189780704,5\mathrm{.}671506803,5\mathrm{.}001537563,6\mathrm{.}090001064,5\mathrm{.}61670967,6\mathrm{.}988118491,-2\mathrm{.}816103618,-4\mathrm{.}733746488,-3\mathrm{.}562144952,1\mathrm{.}934662513,4\mathrm{.}050734641,2\mathrm{.}401936769,-4\mathrm{.}573637599,-6\mathrm{.}793321296,3\mathrm{.}009706686,-1\mathrm{.}357926176,1\mathrm{.}357093617,7\mathrm{.}897002401,0\mathrm{.}926244362,-6\mathrm{.}135790731,-8\mathrm{.}429459914,-9\mathrm{.}627914759,-8\mathrm{.}254841632,-8\mathrm{.}023314705,-8\mathrm{.}045145516,-8\mathrm{.}69992485,-3\mathrm{.}174832993,-2\mathrm{.}638945385,-4\mathrm{.}083487911,1\mathrm{.}189584726,$NA$,-0\mathrm{.}24675,-3\mathrm{.}66991685,0\mathrm{.}238810648,-0\mathrm{.}258878453,6\mathrm{.}013557818,0\mathrm{.}750807825,7\mathrm{.}440970001,2\mathrm{.}677808002,4\mathrm{.}680658862,3\mathrm{.}490279481,2\mathrm{.}118235201,-0\mathrm{.}116317841,9\mathrm{.}372043928,2\mathrm{.}638159801,1\mathrm{.}495362941,$NA$,-3\mathrm{.}203131784,1\mathrm{.}359388777,-5\mathrm{.}135485155,-2\mathrm{.}269723044,-4\mathrm{.}148135687,-7\mathrm{.}516736773,-2\mathrm{.}771367807,-2\mathrm{.}133860794,0\mathrm{.}119136303,-2\mathrm{.}457359522,-2\mathrm{.}834472268,0\mathrm{.}57665347,-0\mathrm{.}336411705,3\mathrm{.}845469155,-0\mathrm{.}329544257,-4\mathrm{.}58862382,0\mathrm{.}132723059,-1\mathrm{.}222440657,6\mathrm{.}279975203,3\mathrm{.}462652491),$ time $=$ c