By Falk M.

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**Example text**

3. The estimate βˆ2 defined above suffers from the drawback that all observations yt have to be strictly less than the estimate βˆ3 . 5). 4. Show that in a linear regression model yt = β1 xt + β2 , t = 1, . . , n, the squared multiple correlation coefficient R2 based on the least squares estimates βˆ1 , βˆ2 and yˆt := βˆ1 xt + βˆ2 is necessarily between zero and one with R2 = 1 if and only if yˆt = yt , t = 0, . . 12)). 5. (Population2 Data) The following table lists total population numbers of North RhineWestphalia between 1961 and 1979.

The ORDER option in the AXIS2 statement specifies the values to appear on the horizontal axis as well as their order, and the MINOR option determines the number of minor tick marks between two major ticks. VREF=0 generates a horizontal reference line through the value 0 on the vertical axis. The autocovariance function γ obviously satisfies γ(0) ≥ 0 and, by the CauchySchwarz inequality |γ(k)| = | E((Yt+k − E(Yt+k ))(Yt − E(Yt )))| ≤ E(|Yt+k − E(Yt+k )||Yt − E(Yt )|) ≤ Var(Yt+k )1/2 Var(Yt )1/2 = γ(0) for k ≥ 0.

Is called autocorrelation function. Let y1 , . . , yn be realizations of a time series Y1 , . . , Yn . The empirical counterpart of the autocovariance function is c(k) := 1 n n−k (yt+k − y¯)(yt − y¯) with bary = t=1 1 n n yt t=1 and the empirical autocorrelation is defined by r(k) := c(k) = c(0) n−k ¯)(yt t=1 (yt+k − y n (y − y¯)2 t=1 t − y¯) . See Exercise 8 (ii) in Chapter 2 for the particular role of the factor 1/n in place of 1/(n−k) in the definition of c(k). The graph of the function r(k), k = 0, 1, .