67. Consider a time series that is, a sequence of observations X1, X2, . . . on some response variable (e.g., concentration of a pollutant) over time with observed values x1, x2, . . . , xn over n time periods. Then the lag 1 autocorrelation coefficient is de ned as Autocorrelation coef cients r2, r3, . . . for lags 2, 3, . . . are de ned analogously.

a. Calculate the values of r1, r2, and r3 for the temperature data from Exercise 79 of Chapter 1.

b. Consider the n  1 pairs (x1, x2), (x2, x3), . . . ,

(xn1, xn). What is the difference between the formula for the sample correlation coef cient r applied to these pairs and the formula for r1? What if n, the length of the series, is large? What about r2 compared to r for the n  2 pairs (x1, x3),

(x2, x4), . . . , (xn2, xn)?

c. Analogous to the population correlation coef cient r, let ri (i  1, 2, 3, . . . ) denote the theoretical or long-run autocorrelation coef cients at the various lags. If all these r s are zero, there is no (linear) relationship between observations in the series at any lag. In this case, if n is large, each Ri has approximately a normal distribution with mean 0 and standard deviation and different Ri s are almost independent.
Thus H0: r  0 can be rejected at a signi cance level of approximately .05 if either or ri 2/1n. If n100 and r1.16, r2.09, ri  2/1n 1/1n r3.15, is there evidence of theoretical autocorrelation at any of the rst three lags?

d. If you are testing the null hypothesis in

(c) for more than one lag, why might you want to increase the cutoff constant 2 in the rejection region?
Hint: What about the probability of committing at least one type I error?

are displayed in the accompanying table. Construct a standardized residual plot and comment on its appearance.
x 1.50 1.50 2.00 2.50 2.50 e* .31 1.02 1.15 1.23 .23 x 3.00 3.50 3.50 4.00 e* .73 1.36 1.53 .07