Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Metoda Statistika |
| Periode Ujian |
: |
Mei 2018 |
| Nomor Soal |
: |
28 |
SOAL
Diberikan beberapa stokastik deret waktu di bawah:
- Proses Random Walk
- Proses AR(1) dengan \(0 < {\phi _1} < 1\)
- Proses AR(1) dengan \({\phi _1} = 1\)
Proses manakah di atas yang bukan proses Stationary?
- i dan ii
- i dan iii
- ii saja
- iii saja
- Semuanya proses Stationary
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| Diketahui |
Diberikan beberapa stokastik deret waktu di bawah:
- Proses Random Walk
- Proses AR(1) dengan \(0 < {\phi _1} < 1\)
- Proses AR(1) dengan \({\phi _1} = 1\)
|
| Rumus yang digunakan |
Stationary process is one whose joint distribution and conditional distribution both are invariant with respect to displacement time. In other words if \({y_T}\) is stationary
- \(p\left( {{y_t}} \right) = p\left( {{y_{t + m}}} \right)\)
- \(E\left( {{y_t}} \right) = E\left( {{y_{t + m}}} \right)\)
- \(E\left[ {{{\left( {{y_t} – {\mu _y}} \right)}^2}} \right] = E\left[ {{{\left( {{y_{t + m}} – {\mu _y}} \right)}^2}} \right]\)
- \(Cov\left( {{y_t},{y_k}} \right) = Cov\left( {{y_{t + m}},{y_{t + m + k}}} \right)\)
Sumber: Econometric Models and Economic Forecasts (Fourth Edition), 1998, by Pindyck, R.S. and Rubinfeld,D.L., Halaman 493-494
- Random Walk: \({y_t} = {y_{t – 1}} + {\varepsilon _t}\)
\({{\hat y}_{T + 1}} = E\left( {\left. {{y_{T + 1}}} \right|{y_T}, \ldots ,{y_1}} \right)\)
\({{\hat y}_{T + 1}} = {y_T} + E\left( {{\varepsilon _{T + 1}}} \right) = {y_T}\)
\({{\hat y}_{T + 2}} = E\left( {\left. {{y_{T + 2}}} \right|{y_T}, \ldots ,{y_1}} \right) = E\left( {{y_{T + 1}} + {\varepsilon _{T + 2}}} \right)\)
\({{\hat y}_{T + 2}} = E\left( {{y_T} + {\varepsilon _{T + 1}} + {\varepsilon _{T + 2}}} \right) = {y_T}\)
- Random Walk with Drift: \({y_t} = {y_{t – 1}} + \delta + {\varepsilon _t}\)
\({{\hat y}_{T + 1}} = E\left( {\left. {{y_{T + 1}}} \right|{y_T}, \ldots ,{y_1}} \right)\)
\({{\hat y}_{T + 1}} = {y_T} + d + E\left( {{\varepsilon _{T + 1}}} \right) = {y_T} + d\)
\({{\hat y}_{T + 2}} = E\left( {\left. {{y_{T + 2}}} \right|{y_T}, \ldots ,{y_1}} \right) = E\left( {{y_{T + 1}} + d + {\varepsilon _{T + 2}}} \right)\)
\({{\hat y}_{T + 2}} = E\left( {{y_T} + d + d + {\varepsilon _{T + 1}} + {\varepsilon _{T + 2}}} \right) = {y_T} + 2d\)
- Proses AR(1): \({y_t} = {\phi _1}{y_{t – 1}} + \delta + {\varepsilon _t}\)
\(\mu = \frac{\delta }{{1 – {\phi _1}}}\)
|
| Proses pengerjaan |
Proses Random Walk
Random walk merupakan proses stationary karena dengan model \({y_t} = {y_{t – 1}} + {\varepsilon _t}\) memiliki nilai rata-rata yang selalu sama setiap periodenya
Proses AR(1) dengan \(0 < {\phi _1} < 1\)
Merupakan proses stationary karena nilai rata-ratanya hanya terpenuhi untuk nilai \(0 < {\phi _1} < 1\)
Proses AR(1) dengan \({\phi _1} = 1\) dengan model \({y_t} = {y_{t – 1}} + \delta + {\varepsilon _t}\) merupakan proses random walk with drift dan bukan proses stationary karena memiliki nilai varians yang semakin besar setiap periodenya |
| Jawaban |
D. iii saja |
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