Linear trends, periodicities, and extremes
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Time series regression models with periodically autocorrelated errors are studied for purposes of studying United States extreme temperatures trends. First we study properties of ordinary and generalized least squares estimates in a simple linear regression model with stationary, but autocorrelated, errors. Our primary focus is on deriving explicit expressions for the variances of the regres-sion parameter estimates for some common time series autocorrelation structures, including a first order autoregression and general moving-averages. We review large sample properties of the parameter estimates and suggest improvements to confi-dence intervals that are used in practice. An example where the variance of the trend slope estimate actually decreases with increasing autocorrelation in the errors is presented. We then move to the study of linear trends in monthly maximum and minimum temperatures observed during the past 150 years in the contiguous 48 United States. Regression and extreme value trend modeling methods are compared and contrasted. Issues of temporal autocorrelations, periodicities, extreme value modeling, station relocations (changepoints), and spatial smoothing algorithms arise. The seasonal aspect of the analysis allows for the issue of uniformity of temperature change over varying season to be investigated. Spatial contour maps of the estimated trends are drawn for each of the four seasons and the entire year. Finally, a conjecture on why the Southeastern United States is cooling is put forth.