Math / Data & Statistics

QuestionQuestion 1. Assume that the following data set $\left\{\left(t, x_{t}\right)\right\}$ is from a stationary $\operatorname{AR}(1)$ time series with $\phi=0.78$ . \begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline $t$ & 1920 & 1925 & 1930 & 1935 & 1940 & 1945 & 1950 & 1955 \\ \hline $x_{t}$ & 0.112 & 0.88 & 0.68 & 0.53 & $?$ & 0.32 & $?$ & $?$ \\ \hline \end{tabular} a) Use the best linear predictor to estimate $x_{1940}$ using $x_{1935}$ . b) Use the best linear predictor to estimate $x_{1940}$ using $x_{1930}$ and $x_{1935}$ . c) Use the best linear predictor to estimate $x_{1940}$ using $x_{1935}$ and $x_{1945}$ . d) Use the best linear predictor to estimate $x_{1950}$ using $x_{1945}$ . e) Use the best linear predictor to estimate $x_{1955}$ .

Studdy Solution

STEP 1

1. The data set is from a stationary AR(1) time series with $\phi = 0.78$ .
2. The AR(1) model is given by $x_t = \phi x_{t-1} + \epsilon_t$ , where $\epsilon_t$ is a white noise error term.
3. The best linear predictor for an AR(1) process is based on the previous value(s) and the parameter $\phi$ .

STEP 2

1. Estimate $x_{1940}$ using $x_{1935}$ .
2. Estimate $x_{1940}$ using $x_{1930}$ and $x_{1935}$ .
3. Estimate $x_{1940}$ using $x_{1935}$ and $x_{1945}$ .
4. Estimate $x_{1950}$ using $x_{1945}$ .
5. Estimate $x_{1955}$ .

STEP 3

Use the AR(1) model to estimate $x_{1940}$ using $x_{1935}$ .
The formula is: $x_{1940} = \phi \cdot x_{1935}$
Given: $x_{1935} = 0.53$
Calculate: $x_{1940} = 0.78 \times 0.53 = 0.4134$

STEP 4

Use the AR(1) model to estimate $x_{1940}$ using $x_{1930}$ and $x_{1935}$ .
The formula is: $x_{1940} = \phi^2 \cdot x_{1930} + \phi \cdot x_{1935}$
Given: $x_{1930} = 0.68, \quad x_{1935} = 0.53$
Calculate: $x_{1940} = 0.78^2 \times 0.68 + 0.78 \times 0.53$ $x_{1940} = 0.6084 \times 0.68 + 0.4134$ $x_{1940} = 0.4137 + 0.4134 = 0.8271$

STEP 5

Use the AR(1) model to estimate $x_{1940}$ using $x_{1935}$ and $x_{1945}$ .
The formula is: $x_{1940} = \phi \cdot x_{1935} + \phi^2 \cdot x_{1945}$
Given: $x_{1935} = 0.53, \quad x_{1945} = 0.32$
Calculate: $x_{1940} = 0.78 \times 0.53 + 0.78^2 \times 0.32$ $x_{1940} = 0.4134 + 0.6084 \times 0.32$ $x_{1940} = 0.4134 + 0.1947 = 0.6081$

STEP 6

Use the AR(1) model to estimate $x_{1950}$ using $x_{1945}$ .
The formula is: $x_{1950} = \phi \cdot x_{1945}$
Given: $x_{1945} = 0.32$
Calculate: $x_{1950} = 0.78 \times 0.32 = 0.2496$

STEP 7

Use the AR(1) model to estimate $x_{1955}$ .
Since $x_{1955}$ is three steps ahead, we use: $x_{1955} = \phi^3 \cdot x_{1945}$
Given: $x_{1945} = 0.32$
Calculate: $x_{1955} = 0.78^3 \times 0.32$ $x_{1955} = 0.474552 \times 0.32 = 0.15105664$
The estimated values are: a) $x_{1940} = 0.4134$ b) $x_{1940} = 0.8271$ c) $x_{1940} = 0.6081$ d) $x_{1950} = 0.2496$ e) $x_{1955} = 0.1511$

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