"Figures in parentheses are standard errors. , advertised brake horsepower (hundreds); W, shipping weight (thousand of pounds); L, overall length in (tens of inches); V, 1 if the car has a V-8 engine, = 0 if it has a six-cylinder engine; T, 1 if the car is hard top, = 0 if not; A, 1 if automatic transmission is "standard" (i.e., included in price), = 0 if not; P, 1 if power steering is "standard," = 0 if not; B, 1 if power brakes are "standard," = 0 if not; C, 1 if car is designated as a "compact," = 0 if not.

ifications. The results are shown in Table 8.1. Since the dependent variable is the logarithm of price, the regression coefficients can be interpreted as the estimated percentage change in the price for a unit change in a particular quality, holding other qualities constant. For example, the coefficient of H indicates that an increase in 10 units of horsepower, ceteris paribus, results in a 1.2% increase in price. However, some of the coefficients have to be interpreted with caution. For example, the coefficient of P in the equation for 1960 says that the presence of power steering as "standard equipment" led to a 22.5% higher price in I960. In this case the variable P is obviously not measuring the effect

Table 8.1 Determinants of Prices of Automobiles"

8.3 Dummy Variables for Changes in Slope Coefficients

In Section 8.2 we considered dummy variables to allow for differences in the intercept term. These dummy variables assume the values zero or 1. Not all dummy variables are of this form. We can use dummy variables to allow for

R. C. Vogel and G. S. Maddala, "Cross-Section Estimates of Liquid Asset Demand by Manufacturing Corporations," The Journal of Finance, December 1967.

of power Steering alone but is measuring the effect of "luxuriousness" of the car. It is also picking up the effects of A and B. This explains why the coefficient of A is so low in 1960. In fact, A, P, and together can perhaps be replaced by a single dummy that measures "luxuriousness." These variables appear to be highly intercorrelated. Another coefficient, at first sight puzzling, is the coefficient of V, which, though not significant, is consistently negative. Though a V-8 costs more than a six-cylinder engine on a "comparable" car, what this coefficient says is that, holding horsepower and other variables constant, a V-8 is cheaper by about 4%. Since the V-8s have higher horsepower, what this coefficient is saying is that higher horsepower can be achieved more cheaply if one shifts to V-8 than by using the six-cyUnder engine. It measures the decline in price per horsepower as one shifts to V-8s even though the total expenditure on horsepower goes up. This example illustrates the use of dummy variables and the interpretation of seemingly wrong coefficients.

As another example consider the estimates of liquid-asset demand by manufacturing corporations. Vogel and Maddala computed regressions of the form log = a + p log S, where is the cash and S the sales, on the basis of data from the Internal Revenue Service, "Statistics of Income," for the year 1960-1961. The data consisted of 16 industry subgroups and 14 sizes classes, size being measured by total assets. When the regression equations were estimated separately for each industry, the estimates of p ranged from 0.929 to 1.077. The Rs were uniformly high, ranging from 0.985 to 0.998. Thus one might conclude that the sales elasticity of demand for cash is close to 1. Also, when the data were pooled and a single equation estimated for the entire set of 224 observations, the estimate of p was 0.992 and R = 0.987. When industry dummies were added, the estimate of P was 0.995 and R = 0.992. From the high Rs and relatively constant estimate of p one might be reassured that the sales elasticity is very close to 1. However, when asset-size dummies were introduced, the estimate of p fell to 0.334 with R~ of 0.996. Also, all asset-size dummies were highly significant. The situation is described in Figure 8.2. That the sales elasticity is significantly less than 1 is also confirmed by other evidence. This example illustrates how one can be very easily misled by high Rs and apparent constancy of the coefficients.

No asset dummies in regression

Asset dummies in the regression

Figure 8.2. Bias due to omission of dummy variables.

differences in slope coefficients as well. For example, if the regression equations are:

y, = a, + P,x, + M,

for the first group for the second group

2 = «2 + 2 2 + «2

we can write these equations together as

y, = a, + (a2 - a,) • 0 + p,x, + (p, - p,) 0 + , 2 = a, + (a2 - a,) • 1 + + (P2 ~ Pi) • 2 + "2

= a, + {a2 - a,)£), + p,x + (p2 - p,) +

where D, =

(8.4)

for all observations in the first group for all observations in the second group

for all observations in the first group

i.e., the respective value of jc for the second group

The coefficient of D, measures the difference in the intercept terms and the coefficient of A measures the difference in the slope. Estimation of equation (8.4) amounts to estimating the two equations separately if we assume that the errors have an identical distribution. If we delete A from equation (8.4), this amounts to allowing for different intercepts but not different slopes, and if we