$1.$ Vasicek Model of interest rates: In the Worksheet $$Prob $2$ you are provided the one$-$year deposit rate interest rates downloaded from Professor Robert Shiller$$s $($Nobel prize winner for Economics, Yale University$)$ website. The series starts from $1871$ and end in $2011.$ What are the parameters of the Vasicek process, that is$,$ a$)$ the long run mean b$)$ speed of mean reversion and c$)$ volatility of the process? $($Hint: Use $$ t $=1).$Year One year Interest rates r$($t$)$

$18716\mathrm{.}35$

$18727\mathrm{.}81$

$18738\mathrm{.}35$

$18746\mathrm{.}86$

$18754\mathrm{.}96$

$18765\mathrm{.}33$

$18775\mathrm{.}03$

$18784\mathrm{.}9$

$18794\mathrm{.}25$

$18805\mathrm{.}1$

$18814\mathrm{.}79$

$18825\mathrm{.}26$

$18835\mathrm{.}35$

$18845\mathrm{.}65$

$18854\mathrm{.}22$

$18864\mathrm{.}26$

$18876\mathrm{.}11$

$18885\mathrm{.}02$

$18894\mathrm{.}68$

$18905\mathrm{.}41$

$18915\mathrm{.}97$

$18923\mathrm{.}93$

$18938\mathrm{.}52$

$18943\mathrm{.}32$

$18953\mathrm{.}09$

$18965\mathrm{.}76$

$18973\mathrm{.}44$

$18983\mathrm{.}55$

$18993\mathrm{.}36$

$19004\mathrm{.}64$

$19014\mathrm{.}3$

$19024\mathrm{.}72$

$19035\mathrm{.}5$

$19044\mathrm{.}34$

$19054\mathrm{.}17$

$19065\mathrm{.}47$

$19076\mathrm{.}23$

$19085\mathrm{.}32$

$19093\mathrm{.}65$

$19105\mathrm{.}26$

$19114$

$19124\mathrm{.}35$

$19135\mathrm{.}65$

$19144\mathrm{.}64$

$19153\mathrm{.}65$

$19163\mathrm{.}64$

$19174\mathrm{.}25$

$19185\mathrm{.}98$

$19195\mathrm{.}56$

$19207\mathrm{.}3$

$19217\mathrm{.}44$

$19224\mathrm{.}58$

$19234\mathrm{.}96$

$19244\mathrm{.}34$

$19253\mathrm{.}87$

$19264\mathrm{.}28$

$19274\mathrm{.}26$

$19284\mathrm{.}64$

$19296\mathrm{.}01$

$19304\mathrm{.}15$

$19312\mathrm{.}43$

$19323\mathrm{.}36$

$19331\mathrm{.}46$

$19341\mathrm{.}01$

$19350\mathrm{.}75$

$19360\mathrm{.}75$

$19370\mathrm{.}88$

$19380\mathrm{.}88$

$19390\mathrm{.}56$

$19400\mathrm{.}56$

$19410\mathrm{.}53$

$19420\mathrm{.}63$

$19430\mathrm{.}69$

$19440\mathrm{.}72$

$19450\mathrm{.}75$

$19460\mathrm{.}76$

$19471\mathrm{.}01$

$19481\mathrm{.}35$

$19491\mathrm{.}58$

$19501\mathrm{.}32$

$19512\mathrm{.}12$

$19522\mathrm{.}39$

$19532\mathrm{.}58$

$19541\mathrm{.}8$

$19551\mathrm{.}81$

$19563\mathrm{.}21$

$19573\mathrm{.}86$

$19582\mathrm{.}54$

$19593\mathrm{.}74$

$19604\mathrm{.}28$

$19612\mathrm{.}91$

$19623\mathrm{.}39$

$19633\mathrm{.}5$

$19644\mathrm{.}09$

$19654\mathrm{.}46$

$19665\mathrm{.}44$

$19675\mathrm{.}55$

$19686\mathrm{.}17$

$19698\mathrm{.}05$

$19709\mathrm{.}11$

$19715\mathrm{.}66$

$19724\mathrm{.}62$

$19737\mathrm{.}93$

$197411\mathrm{.}03$

$19757\mathrm{.}24$

$19765\mathrm{.}7$

$19775\mathrm{.}28$

$19787\mathrm{.}78$

$197910\mathrm{.}88$

$198011\mathrm{.}37$

$198117\mathrm{.}63$

$198214\mathrm{.}6$

$19839\mathrm{.}37$

$198411\mathrm{.}11$

$19858\mathrm{.}35$

$19867\mathrm{.}31$

$19876\mathrm{.}25$

$19887\mathrm{.}63$

$19899\mathrm{.}29$

$19908\mathrm{.}43$

$19916\mathrm{.}92$

$19923\mathrm{.}91$

$19933\mathrm{.}44$

$19944\mathrm{.}35$

$19956\mathrm{.}45$

$19965\mathrm{.}68$

$19975\mathrm{.}78$

$19985\mathrm{.}68$

$19995\mathrm{.}31$

$20006\mathrm{.}61$

$20014\mathrm{.}63$

$20021\mathrm{.}85$

$20031\mathrm{.}18$

$20041\mathrm{.}49$

$20053\mathrm{.}41$

$20065\mathrm{.}32$

$20075\mathrm{.}34$

$20083\mathrm{.}42$

$20091\mathrm{.}02$

$20100\mathrm{.}46$

$20110\mathrm{.}37$