STA$4807/014/0/2024$

$($b$)$ Comment on the adequacy of the Dummy$-$SARIMA coupled model with respect to the following:

Figure $3\mathrm{.}1$: Residual Analysis from fitting the Dummy $-$SARIMA $(3,0,0)(0,0,1{)}_{12}$ coupled model.

$($i$)$ Significance of parameters.

$($ii$)$ Standardized residuals.

$($iii$)$ ACF plot of residuals.

$(3)$

$($iv$)Q*Q$ plot of standardized residuals.

$($v$)$ The Ljung$-$Box statistics.

$($c$)$ What do you understand with this modelling approach?

$($d$)$ The auto.arima function was used to automatically fit a SARIMA model SARIMA $(p,d,q)(P,D,Q{)}_{12},$ i$.$e$.,$ using the commands:$>$ fit$\mathrm{.}3=$ auto.arima $($hotel$)$Series: hotel$)$ARIMA$(0,0,4)(0,1,1)[12]$ with drift

Coefficients:

sigma$^2=225\mathrm{.}2$: log likelihood $=-642\mathrm{.}56$

AIC$=1299\mathrm{.}11$ AICC$=1299\mathrm{.}87$ BIC$=1320\mathrm{.}46$

Training set error measures:

Training set $0\mathrm{.}04723565$

Further, the $t-$ratios were computed to test the significance of parameters using the commands:

$>$ fit. $3$ $coef

$9>$ Elt. $3\%$coefsmal $-6\mathrm{.}989027$e$-04-6\mathrm{.}311451$e$-056\mathrm{.}293860$e$-04$drift $2\mathrm{.}398724$e$-06-1\mathrm{.}086903$e$-05-2\mathrm{.}183012$e$-05-2\mathrm{.}245237$e$-05-3\mathrm{.}640869$e$-06$mal drift $\backslash$mal $2\mathrm{.}398724$e$-06$ma$3-2\mathrm{.}183012$e$-05$ma$4-2\mathrm{.}245237$e$-05$smal $-3\mathrm{.}640869$e$-06$drift $3\mathrm{.}754714$e$-04>$ std$.$errors$=($diag$($flt$\mathrm{.}3$&var.coef$))$mal$\backslash$ t$.$ratios $-$ fit$\mathrm{.}3\}$coef$/$std$.$errors$>$ t$.$ratiosmol mal$>|$

$($i$)$ Write down the SARIMA $(p,d,q)(P,D,Q{)}_{12}$ in the form