Investments, tenth edition

  scatter  diagram    in  Figure 8.3 , where the regression line is drawn through the scatter. The 

vertical distance of each point from the regression line is the value of HP’s residual,  e  

HP

 ( t ), 

corresponding to that particular month. The rates in  Figure  8.2  are not annualized, and 

the scatter diagram shows monthly swings of over  6  30% for HP, but returns in the range 

of  2 11% to 8.5% for the S&P 500. The 

regression analysis output obtained by 

using Excel is shown in  Table 8.1 .    

  The Explanatory Power 

of the SCL for HP 

 

Considering the top panel of  

Table 8.1 

 

first, we see that the correlation of HP 

with the S&P 500 is quite high (.7238), 

telling us that HP tracks changes in the 

returns of the S&P 500 fairly closely. The 

 R -square (.5239) tells us that variation in 

the S&P 500 excess returns explains about 

52% of the variation in the HP series. 

The adjusted  R -square (which is slightly 

smaller) corrects for an upward bias in 

 R -square that arises because we use the 

fitted values of two parameters,  

6

    the  slope 

6

 In general, the adjusted  R -square     (R

A

2

)  is derived from the unadjusted by    R

A

2

5 1 2 (1 2 R

2

) 

n

2 1

n

2 k 2 1

,  where  k

is the number of independent variables (here,  k  5 1). An additional degree of freedom is lost to the estimate of 

the intercept. 

 Figure 8.2 

Excess returns on HP and S&P 500  

−.4

−.3

−.2

−.1

.0

.1

60

10

0

20

30

40

50

.2

.3

.4

Observation Month

Excess Returns (%)

S&P 500

HP

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  Portfolio Theory and Practice

(beta) and  intercept (alpha), rather than 

their true, but  unobservable, values. With 

60 observations, this bias is small. The 

standard error of the regression is the 

standard deviation of the residual, which 

we discuss in more detail shortly. This is 

a measure of the slippage in the average 

relationship between the stock and the 

index due to the impact of firm-specific 

factors, and is based on  in-sample   data. 

A more severe test is to look at returns 

from periods after the one covered by the 

regression sample and test the power of 

the independent variable (the S&P 500) 

to predict the dependent variable (the 

return on HP). Correlation between 

regression forecasts and realizations 

of  out-of-sample  data is almost always 

considerably lower than in-sample 

correlation.