Math Snap

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:
$$x_{1950} = 0.78 \times 0.32 = 0.2496$$

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) $x_{1955} = 0.1511$