FORMULA SHEET

$c_i=\frac{1}{T}{\displaystyle \underset{t=1}{\overset{T}{}}}[(x_t-(\stackrel{}{x}))(x_{t+k}-(\stackrel{}{x}))]$;$k=1,2,$dots,$\stackrel{}{x}=\frac{1}{T}{\displaystyle \underset{t=1}{\overset{T}{}}}{x}_1,r_k=\frac{c_k}{c_0},$;$k=1,2,$dots

AIC$=loghat({)}_k^2+\frac{n+2k}{n},$BIC$=loghat({)}_k^2+\frac{klog(n)}{n}$

${}_{11}={}_1,{}_{22}=\frac{{}_2-{}_1^2}{1-{}_1^2},{}_{kk}=\frac{{}_k-{\displaystyle \underset{j=1}{\overset{k-1}{}}}{}_{k-1,j}{}_{k-j}}{1-{\displaystyle \underset{j=1}{\overset{k-1}{}}}{}_{k-1,j}{}_j},k=3,4,$dots

${}_k={}_{k-1,j}-{}_{uk-k-j}{}_{k-1,0},j=1,2,$dots,$k-1.$

$MA(q)$

${}_0={}^2{\displaystyle \underset{i=0}{\overset{q}{}}}{}_i^2,{}_k={}^2{\displaystyle \underset{j=0}{\overset{q}{}}}{}_j{}_{j+k}$;$kq$

$AR(p)$

${}_0=\frac{{}^2}{1-{}_1{}_1-{}_2{}_2-\text{d}\text{o}\text{t}\text{s}-{}_p{}_p},{}_k=\frac{O}{{?}_1}{}_{k-1}+\frac{O}{{?}_2}{}_{k-2}+$dots$+\frac{O}{{?}_p}{}_{k-p}$

ARMA$(1,1)$

${}_0=\frac{1-2-{}^2}{1-{}^2}{}^2,{}_1=\frac{(1-)(-)}{1-{}^2}{}^2,{}_k={}_{k-1}={}^{k-1}{}_1,k2.$

QUESTIONS

Q$1.$ Consider the monthly sales plot for a firm given below. Briefly discuss the plot.