Pembahasan Ujian PAI: A50 - No. 27 - November 2017

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

Kunci Jawaban & Pembahasan

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

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