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Midterm 1 for STAT 4363

1. (15 points) Generate 500 observations from each of these autoregressions:

where the error terms and are independent white noise series following N(0, 1).

a. Run a simple linear regression and comment on the summary of this regression fit.

b. Run regression diagnostics and comment on your observations.

2. (20 points) Consider the LakeHuron time series from the datasets package in R. This series has annual measurements of the depth of Lake Huron (in feet) from 1875-1972.

a. Fit a linear regression and obtain the fitted series  Plot the original series and the fitted series in the same graph.

b. Obtain the residuals series and plot its ACF and comment.

c. Plot the ACF of the differenced series and compare it to the ACF from part b.

d. Obtain the ACF of Lake Huron series for the years 1875-1923, and the ACF of the Lake Huron series for the years 1924-1972. Create a panel plot with these two ACF plots. By looking at this panel plot, comment whether the Lake Huron time series from 1875-1972 is stationary.

3. (20 points) Consider co2 time series from the datasets package in R. This series Xt has monthly atmospheric CO2 concentration, measured in ppm, from 1959 to 1997.

a. Apply a Loess smoother with span choices 0.01, 0.02, 0.03, 0.04. Create a panel plot containing 4 plots, with each plot containing the original and smoothed series at the corresponding span choice.

b. Apply the structural regression model to this time series using the decompose() function in R.

c. Plot the noise component from this regression and provide comments on the adequacy of the model fit.

4. (15 points) The figure below includes plots of three time series along with their ACF. For each series, identify a time series model that closely resembles that series.