Pembahasan Soal Ujian Profesi Aktuaris
Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
Mata Ujian |
: |
Metoda Statistika |
Periode Ujian |
: |
November 2017 |
Nomor Soal |
: |
27 |
SOAL
Sebuah regresi linier digunakan untuk mencocokkan suatu deret waktu dengan 30 pengamatan. Diketahui:
\({{\hat \varepsilon }_1} = – 8\)
\({{\hat \varepsilon }_{30}} = 10\)
\(\sum\limits_{t = 1}^{30} {\hat \varepsilon _t^2} = 3200\)
\(\sum\limits_{t = 2}^{30} {\left( {{{\hat \varepsilon }_t}{{\hat \varepsilon }_{t – 1}}} \right)} = 760\)
Hitunglah statistik Durbin-Watson (dibulatkan 2 desimal).
- 1,27
- 1,37
- 1,47
- 1,57
- 1,67
Diketahui |
regresi linier digunakan untuk mencocokkan suatu deret waktu dengan 30 pengamatan
\({{\hat \varepsilon }_1} = – 8\)
\({{\hat \varepsilon }_{30}} = 10\)
\(\sum\limits_{t = 1}^{30} {\hat \varepsilon _t^2} = 3200\)
\(\sum\limits_{t = 2}^{30} {\left( {{{\hat \varepsilon }_t}{{\hat \varepsilon }_{t – 1}}} \right)} = 760\) |
Rumus yang digunakan |
\(d = \frac{{\sum\limits_{t = 2}^n {{{\left( {{{\hat \varepsilon }_t} – {{\hat \varepsilon }_{t – 1}}} \right)}^2}} }}{{\sum\limits_{t = 1}^n {{{\hat \varepsilon }_t}^2} }}\)
\(= \frac{{\sum\limits_{t = 1}^n {{{\hat \varepsilon }_t}^2} + \sum\limits_{t = 2}^n {{{\hat \varepsilon }_{t – 1}}^2} – 2\sum\limits_{t = 2}^n {\left( {{{\hat \varepsilon }_t}{{\hat \varepsilon }_{t – 1}}} \right)} }}{{\sum\limits_{t = 1}^n {{{\hat \varepsilon }_t}^2} }}\) |
Proses pengerjaan |
\(d = \frac{{\sum\limits_{t = 1}^{30} {{{\hat \varepsilon }_t}^2} + \sum\limits_{t = 2}^{30} {{{\hat \varepsilon }_{t – 1}}^2} – 2\sum\limits_{t = 2}^{30} {\left( {{{\hat \varepsilon }_t}{{\hat \varepsilon }_{t – 1}}} \right)} }}{{\sum\limits_{t = 1}^{30} {{{\hat \varepsilon }_t}^2} }}\)
\(= \frac{{3200 + \left( {3200 – {{\left( { – 8} \right)}^2} – {{10}^2}} \right) – 2\left( {760} \right)}}{{3200}}\)
\(= 1,47375\) |
Jawaban |
c. 1,47 |