Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Metoda Statistika |
| Periode Ujian |
: |
April 2019 |
| Nomor Soal |
: |
8 |
SOAL
Dalam sebuah proses autoregresi yang terintegrasi (integrated autoregressive process), ARI (1,1,0) diberikan:
\({w_t} = 0,73{w_{t – 1}} + 1,4 + {\varepsilon _t}\)
Tentukan persamaan forecast 1 periode untuk \({y_t}\)
- \({\hat y_T}\left( 1 \right) = 1,73{y_T} – 0,73{y_{T – 1}} + 1,4\)
- \({\hat y_T}\left( 1 \right) = 1,73{y_T} + 0,73{y_{T – 1}} + 1,4\)
- \({\hat y_T}\left( 1 \right) = 1,73{y_T} + 0,73{y_{T – 1}} – 1,4\)
- \({\hat y_T}\left( 1 \right) = 0,73{y_T} – 1,73{y_{T – 1}} + 1,4\)
- \({\hat y_T}\left( 1 \right) = 0,73{y_T} + 0,73{y_{T – 1}} + 1,4\)
| Diketahui |
ARI (1,1,0) dengan \({w_t} = 0,73{w_{t – 1}} + 1,4 + {\varepsilon _t}\) |
| Rumus yang digunakan |
Bentuk umum ARI (1,1,0) adalah \({w_t} = {\phi _1}{w_{t – 1}} + \delta + {\varepsilon _t}\)
\({\hat y_T}\left( l \right) = {y_T} + {\hat w_T}\left( 1 \right) + \cdots + {\hat w_T}\left( l \right)\)
\({\hat w_T}\left( l \right) = \phi _1^l{y_T} – \phi _1^l{y_{T – 1}} + \left( {\phi _1^{l – 1} + \cdots + {\phi _1} + 1} \right)\delta \) |
| Proses pengerjaan |
\({{\hat y}_T}\left( 1 \right) = {y_T} + {{\hat w}_T}\left( 1 \right) = {y_T} + {\phi _1}{y_T} – {\phi _1}{y_{T – 1}} + \delta \)
\(= {y_T} + 0,73{y_T} – 0,73{y_{T – 1}} + 1,4\)
\(= 1,73{y_T} – 0,73{y_{T – 1}} + 1,4\) |
| Jawaban |
a. \({\hat y_T}\left( 1 \right) = 1,73{y_T} – 0,73{y_{T – 1}} + 1,4\) |
