+ yMt + UU

R2t =

«2 + 021 R1t

+ Y2M, + U2t

Ru =

a3 + 031 R1t + 032 R2t

+ Y3Mt + U3t

R4t =

a4 + 041 Ru + 042 R2t + 043 R3t

+ Y4Mt + U4t

Rat =

aa + 051 R11 + 052 R2t + 053 R3t + 054 R4t

+ Y5Mt + U5t

Ret =

a6 + 061 R11 + 062 R2t + 063 R3t + 064 R4t + 065 R5t

+ Y6Mt + U6t

R?t =

a7 + 071 R11 + 072 R2t + 073 R3t + 074R4t + 075 R5t + 076 R6t + YjMt + Ujt

where R = rate of return on security 1 ( = Imperial Oil) R2 = rate of return on security 2 ( = Sun Oil)

R7 = rate of return on security 7 ( = Standard of Indiana) Mt = rate of return on the market index uit = disturbances (i = 1,2 7)

Before we present the results, the obvious question is: How do we choose which is security 1, which is security 2, and so on? Lee and Lloyd answer this question purely empirically. They regress the rate of return on security i on the rates of return of the remaining six securities and observe the resulting R2. Thus, there will be seven such regressions. Then they order the estimated R2 values, from the lowest to the highest. The security having the lowest R2 is designated as security 1 and the one having the highest R2 is designated as 7. The idea behind this is intuitively simple. If the R2 of the rate of return of, say, Imperial Oil, is lowest with respect to the other six securities, it would suggest that this security is affected least by the movements in the returns of the other securities. Therefore, the causal ordering, if any, runs from this security to the others and there is no feedback from the other securities.

Although one may object to such a purely empirical approach to causal ordering, let us present their empirical results nonetheless, which are given in Table 20.7.

In exercise 5.5 we introduced the characteristic line of modern investment theory, which is simply the regression of the rate of return on security i on the market rate of return. The slope coefficient, known as the beta coefficient, is a measure of the volatility of the security's return. What the Lee-Lloyd regression results suggest is that there are significant intra-industry relationships between security returns, apart from the common market influence represented by the market portfolio. Thus, Standard of Indiana's return depends not only on the market rate of return but also on the rates of return on Shell Oil, Phillips Petroleum, and

(Continued)

19"The Capital Asset Pricing Model Expressed as a Recursive System: An Empirical Investigation," Journal of Financial and Quantitative Analysis, June 1976, pp. 237-249.

782 PART FOUR: SIMULTANEOUS-EQUATION MODELS

EXAMPLE 20.3 (Continued)

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