Pembahasan Ujian PAI: A50 – No. 28 – Mei 2018

1. Proses Random Walk
2. Proses AR(1) dengan \(0 < {\phi _1} < 1\)
3. Proses AR(1) dengan \({\phi _1} = 1\)

1. \(p\left( {{y_t}} \right) = p\left( {{y_{t + m}}} \right)\)
2. \(E\left( {{y_t}} \right) = E\left( {{y_{t + m}}} \right)\)
3. \(E\left[ {{{\left( {{y_t} – {\mu _y}} \right)}^2}} \right] = E\left[ {{{\left( {{y_{t + m}} – {\mu _y}} \right)}^2}} \right]\)
4. \(Cov\left( {{y_t},{y_k}} \right) = Cov\left( {{y_{t + m}},{y_{t + m + k}}} \right)\)

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 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