Page 1 : Regression through origin, There are occasions when the two-variable PRF assumes the following form:, Yi = β2 Xi + ui ..................(1), In this model the intercept term is absent or zero, hence the name regression, through the origin., A few 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., To estimate the intercept-less models like eq. (1), let us first write the SRF, of eq. (1), namely,, Yi = β̂2 Xi + ûi ...................(2), Now applying the OLS method to (2), we obtain the formulas for β̂2 and its, variance., We want to minimize, 2, X, X, û2i =, Yi − β̂2 Xi ................(3), with respect to β̂2 . Differentiating (3) with respect to β̂2 , we obtain, P, , X, d û2i, =2, Yi − β̂2 Xi (−Xi ) .................(4), dβ̂2, Setting (4) equal to zero and simplifying, we get, P, Xi Yi, β̂2 = P 2 .........................(5), Xi, Now substituting the PRF: Yi = β2 Xi + ui into this equation (eq. 5), we obtain, P, P, Xi (β2 Xi + ui ), Xi ui, P 2, β̂2 =, = β2 + P 2, Xi, Xi,  , i, [ Note: E β̂2 = β2 . Therefore,, , , E β̂2 − β2, , 2, , P, 2, Xi ui, =E P 2, ..........................(6), Xi, , Expanding the right-hand side of (6) and noting that the Xi are nonstochastic, and the ui are homoscedastic and uncorrelated, we obtain, P, Xi Yi, β̂2 = P 2, Xi, 1

Page 2 : and, ,  , , 2, σ2, var β̂2 = E β̂2 − β2 = P 2 .....................................(7), Xi, , where σ 2 is estimated by, σ̂ 2 =, , û2i, ....................................(8), n−1, P, , It is interesting to compare these formulas with those obtained when the intercept term is included in the model:, P, xi yi, β̂2 = P 2, xi,  , σ2, var β̂2 = P 2, xi, P 2, ûi, σ̂ 2 =, n−2, The differences between the two sets of formulas should be obvious:, In the model with the intercept term absent, we use raw sums of, squares and cross products but in the intercept-present model, we use adjusted (from mean) sums of squares and cross products., Second, the df for computing σ̂ 2 is (n − 1) in the first case and (n − 2) in, the second case., Although the intercept less or zero intercept model may be appropriate on, occasions, there are some features of this model that need to be noted., P, First,, ûi , which is always zero for the model with the intercept term (the, conventional model), need not be zero when that term is absent., In short,, , P, , ûi need not be zero for the regression through the origin., , Second, r2 , the coefficient of determination, which is always non-negative, for the conventional model, can on occasions turn out to be negative for the, intercept-less model. Therefore, the conventionally computed r2 may not be, appropriate for regression-through-the-origin models., , 2