## Regression Through The Origin

There are occasions when the two-variable PRF assumes the following form:

In this model the intercept term is absent or zero, hence the name regression through the origin.

CHAPTER SIX: EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODEL 165

As an illustration, consider the Capital Asset Pricing Model (CAPM) of modern portfolio theory, which, in its risk-premium form, may be expressed as1

where ERj = expected rate of return on security i

ERm = expected rate of return on the market portfolio as represented by, say, the S&P 500 composite stock index rf = risk-free rate of return, say, the return on 90-day Treasury bills fy = the Beta coefficient, a measure of systematic risk, i.e., risk that cannot be eliminated through diversification. Also, a measure of the extent to which the ith security's rate of return moves with the market. A fy > 1 implies a volatile or aggressive security, whereas a fy < 1 a defensive security. (Note: Do not confuse this fy with the slope coefficient of the two-variable regression, fy2.)

If capital markets work efficiently, then CAPM postulates that security is expected risk premium (= ERj- — rf) is equal to that security's fy coefficient times the expected market risk premium (= ERm — rf). If the CAPM holds, we have the situation depicted in Figure 6.1. The line shown in the figure is known as the security market line (SML).

For empirical purposes, (6.1.2) is often expressed as 1See Haim Levy and Marshall Sarnat, Portfolio and Investment Selection: Theory and Practice, Prentice-Hall International, Englewood Cliffs, N.J., 1984, Chap. 14.

166 PART ONE: SINGLE-EQUATION REGRESSION MODELS FIGURE 6.2 The Market Model of Portfolio Theory (assuming a, = 0).

The latter model is known as the Market Model.2 If CAPM holds, ai is expected to be zero. (See Figure 6.2.)

In passing, note that in (6.1.4) the dependent variable, Y, is (Ri — rf) and the explanatory variable, X, is hi, the volatility coefficient, and not (Rm — rf). Therefore, to run regression (6.1.4), one must first estimate hi, which is usually derived from the characteristic line, as described in exercise 5.5. (For further details, see exercise 8.28.)

As this example shows, sometimes the underlying theory dictates that the intercept term be absent from the model. Other instances where the zero-intercept model may be appropriate are Milton Friedman's permanent income hypothesis, which states that permanent consumption is proportional to permanent income; cost analysis theory, where it is postulated that the variable cost of production is proportional to output; and some versions of monetarist theory that state that the rate of change of prices (i.e., the rate of inflation) is proportional to the rate of change of the money supply.

How do we estimate models like (6.1.1), and what special problems do they pose? To answer these questions, let us first write the SRF of (6.1.1), namely,

Now applying the OLS method to (6.1.5), we obtain the following formulas for h2 and its variance (proofs are given in Appendix 6A, Section 6A.1):

2See, for instance, Diana R. Harrington, Modern Portfolio Theory and the Capital Asset Pricing Model: A User's Guide, Prentice Hall, Englewood Cliffs, N.J., 1983, p. 71.

CHAPTER SIX: EXTENSIONS OF THE TWO-VARIABLE LINEAR REGRESSION MODEL 167 