Math  /  Data & Statistics

QuestionQuestion 1. Assume that the following data set {(t,xt)}\left\{\left(t, x_{t}\right)\right\} is from a stationary AR(1)\operatorname{AR}(1) time series with ϕ=0.78\phi=0.78. \begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hlinett & 1920 & 1925 & 1930 & 1935 & 1940 & 1945 & 1950 & 1955 \\ \hlinextx_{t} & 0.112 & 0.88 & 0.68 & 0.53 & ?? & 0.32 & ?? & ?? \\ \hline \end{tabular} a) Use the best linear predictor to estimate x1940x_{1940} using x1935x_{1935}. b) Use the best linear predictor to estimate x1940x_{1940} using x1930x_{1930} and x1935x_{1935}. c) Use the best linear predictor to estimate x1940x_{1940} using x1935x_{1935} and x1945x_{1945}. d) Use the best linear predictor to estimate x1950x_{1950} using x1945x_{1945}. e) Use the best linear predictor to estimate x1955x_{1955}.

Studdy Solution
Use the AR(1) model to estimate x1955x_{1955}.
Since x1955x_{1955} is three steps ahead, we use: x1955=ϕ3x1945 x_{1955} = \phi^3 \cdot x_{1945}
Given: x1945=0.32 x_{1945} = 0.32
Calculate: x1955=0.783×0.32 x_{1955} = 0.78^3 \times 0.32 x1955=0.474552×0.32=0.15105664 x_{1955} = 0.474552 \times 0.32 = 0.15105664
The estimated values are: a) x1940=0.4134x_{1940} = 0.4134 b) x1940=0.8271x_{1940} = 0.8271 c) x1940=0.6081x_{1940} = 0.6081 d) x1950=0.2496x_{1950} = 0.2496 e) x1955=0.1511x_{1955} = 0.1511

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