You have decided to analyze the year-to-year variation in temperature data. Specifically you want to use this year's temperature to predict next year's temperature for certain cities. As a result, you collect the daily high temperature (Temp) for 100 randomly selected days in a given year for three United States cities: Boston, Chicago, and Los Angeles. You then repeat the exercise for the following year. The regression results are as follows (heteroskedasticity-robust standard errors in parentheses): Tempt BOS = 18.19 +0.75 < Temp 2-1 BOS R2 = 0.62, SER = 12.33 (6.46) (0.10) CHI Tempt I= 2.47 +0.95 x Temp -1, R2 = 0.93, SER = 5.85 (3.98) (0.05) LA Temp = 37.54 +0.44 x Temp 4; R2 = 0.18, SER = 7.17 (15.33) (0.22) (1) What is the prediction of the above regression for Los Angeles if the temperature in the previous year was 75 degrees? What would be the prediction for Boston and Chicago?
(2) Whether the intercept and the previous year's temperature are statistically significant at 95% level with two-tail hypothesis? Then, construct the corresponding confidence interval
(3) After reflecting on the results a bit, you consider the following explanation for the above results. Daily high temperatures on any given date are measured with error in the following sense: for any given day in any of the three cities, say January 28, there is a true underlying seasonal temperature (3), but each year there are different temporary weather patterns (V, w) which result in a temperature â different from X. For the two years in your data set, the situation can be described as follows: Xi1 = X + v; and X12 = X + wt Subtracting Xt1 from Xt2, you get X+2 = X+1 + wy – Vf. Hence the population parameter for the intercept and slope are zero and one, as expected. Show that the OLS estimator for the slope is inconsistent, where 8,91. 2 2 + d
(4) Use the formula above to explain the differences in the results for the three cities. Is your mathematical explanation intuitively plausible?