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midterm - PSTAT 174 274 Midterm Consider GDP from 1983 1Q...

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PSTAT 174 / 274 - Midterm Consider GDP from 1983 1Q to 2002 4Q. Figure 1 shows the linear trend over time and you plotted the deterended time series with differentiation. GDP Time Billion \$ 0 20 40 60 80 4000 6000 8000 10000 (a) Detrended Time Series with Differentiation Time Billion \$ 0 20 40 60 80 0 50 100 150 200 (b) Figure 1: GDP data from 1983 1Q to 2002 4Q 1. (30 points) You can consider the following model, x t = β 1 + β 2 t + w t , where β 1 and β 2 are known constants and w t is a white noise process with variance σ 2 . Prove that the first difference series x t = x t - x t - 1 is stationary by finding its mean and autocovariance function. 2. (30 points) Consider the following AR(1) time series model X t = 0 . 4 X t - 1 + w t , w t i.i.d N (0 , 1225) . (a) Find and sketch the ACF, ρ ( h ) . (b) Find and sketch the PACF, φ hh . 1

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3. (10 points) You used differentiation to remove the linear trend and plotted ACF/PACF. Identify the best linear time series model that you want to fit the detrended time series.
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midterm - PSTAT 174 274 Midterm Consider GDP from 1983 1Q...

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