FIN $5309$ Homework $6$ Solution $-$ Fall $2021$

$(40$ points$)$ RStudio application

Consider the daily gold fixing price $10$:$30$ am $($London time$)$ in London Bullion Market in U$.$S$.$ dollars per Troy ounce from January $3,1995$ to Oct $31,2019.$ The data can be obtained from FRED using the quantmod package. Since there are some missing values, we need to remove them before analysis. Let $??$ri$??).$ See instructions below.

install.packages$("$quantmod$")$

require$($quantmod$)$

getSymbols$(\text{'}$GOLDAMGBD$228$NLBM$\text{'},$ src$=$'FRED'$)$

GOLD GOLDAMGBD$228$NLBM$[6982$:$13459]$

idx c$(1$:nrow$($GOLD$))[$

is$.$na$($GOLD$)]$

GOLD GOLD$[-$idx$]$

as$.$numeric$($GOLD$))$

$($a$)$ Obtain the time plots of $??$ and $r_{?}($in one page, using the command par $).$

$($b$)$ Compute the first $12$ lags of ACF of $??$ Based on the ACF, is there a unit root in $??$ Why?

$($c$)$ Let $r_{?}=100**(?{?}_{?}-?{?}_{?-1})$ be the return series of the gold prices, in percentages. Consider the $r_{?}$ series. Test $H_0$:${}_1=$dots$={}_{12}=0$ versus $H_a$:${}_{i}0$ for some $1i12.$ Draw your conclusion.

$($d$)$ Use the command ar$($rt$,$method$=$"mle",order.max$=20)$ to specify the order of an AR model for $r_{?}.$

$($e$)$ Build an AR model for $r_{?},$ including model checking. Refine the model by excluding all estimates with t$-$ratio less than $1\mathrm{.}645.$ Write down the fitted model.

$($f$)$ Use the fitted AR model to compute $1-$step to $4-$step ahead forecasts of $r_{?}$ at the forecast origin Oct $31,2019.$