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Case Study: Case-Shiller Home price index for Chicago from FRED (or via STATA ) https://fred.stlouisfed.org/series/CHXRSA and calculate the return associated with this series. (the original data source is here Note that CHXRNSA is the raw, unadjusted series and CHXRSA is the seasonally adjusted series. The seasonally adjusted series removes the cyclical variation that repeats itself year after year (i.e., summer tends to have higher prices than the winter). For this question, you want to use the returns of the seasonally adjusted series (i.e., CHXRSA).
Questions:
a. Examine the autocorrelation function (ACF) and partial autocorrelation function (PACF) of the adjusted returns. Do you see the same pattern?
b. Fit an AR(1) model to this data. Examine the residual autocorrelations.
c. Build the one-step ahead forecast for every observation in the sample using your fitted AR(1) model. Plot the forecast and the real data on the same plot.
d. What is one-step ahead out of sample forecast (i.e., the forecast of T+1) and find the 95% prediction interval.
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