Pembahasan Soal Ujian Profesi Aktuaris
Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
Mata Ujian | : | Metoda statistika |
Periode Ujian | : | Mei 2017 |
Nomor Soal | : | 24 |
SOAL
Sebuah regresi linear digunakan untuk mencocokkan suatu deret waktu dengan 30 pengamatan, diketahui:
\({\widehat \varepsilon _1} = – 7\)
\({\widehat \varepsilon _{30}} = 11\)
\(\sum\limits_{t = 1}^{t = 30} {\widehat \varepsilon _t^2} = 2422\)
\(\sum\limits_{t = 2}^{t = 30} {({{\widehat \varepsilon }_t}{\rm{ x }}{{\widehat \varepsilon }_{t – 1}})} = 801\)
Hitunglah statistik Durbin-Watson
- 1.31
- 1.27
- 1.23
- 1.19
- 1.15
Diketahui | \({\widehat \varepsilon _1} = – 7\)
\({\widehat \varepsilon _{30}} = 11\)
\(\sum\limits_{t = 1}^{t = 30} {\widehat \varepsilon _t^2} = 2422\)
\(\sum\limits_{t = 2}^{t = 30} {({{\widehat \varepsilon }_t}{\rm{ x }}{{\widehat \varepsilon }_{t – 1}})} = 801\) |
Rumus yang digunakan | \(d = \frac{{\sum\limits_{t = 2}^{t = 30} {{{({{\widehat \varepsilon }_t} – {{\widehat \varepsilon }_{t – 1}})}^2}} }}{{\sum\limits_{t = 1}^{t = 30} {\widehat \varepsilon _t^2} }}\) |
Proses pengerjaan | \(\sum\limits_{t = 2}^{t = 30} {{{({{\widehat \varepsilon }_t} – {{\widehat \varepsilon }_{t – 1}})}^2}} = \sum\limits_{t = 2}^{t = 30} {(\widehat \varepsilon _t^2 – 2{{\widehat \varepsilon }_t}{\rm{ x }}{{\widehat \varepsilon }_{t – 1}} + \widehat \varepsilon _{t – 1}^2)} \)
\(= \sum\limits_{t = 2}^{t = 30} {\widehat \varepsilon _t^2} – 2\sum\limits_{t = 2}^{t = 30} {{{\widehat \varepsilon }_t}{\rm{ x }}{{\widehat \varepsilon }_{t – 1}}} + \sum\limits_{t = 2}^{t = 30} {\widehat \varepsilon _{t – 1}^2} \)
\(= (2422 – (49)) – 2(801) + (2422 – (121))\)
\(= 3072\)
selanjutnya
\(d = \frac{{\sum\limits_{t = 2}^{t = 30} {{{({{\widehat \varepsilon }_t} – {{\widehat \varepsilon }_{t – 1}})}^2}} }}{{\sum\limits_{t = 1}^{t = 30} {\widehat \varepsilon _t^2} }} = \frac{{3072}}{{2422}} = 1.26837 = 1.27\) |
Jawaban | b. 1.27 |