delete D,, it amounts to allowing for different slopes but not different intercepts.

Suitable dummy variables can be defined when there are changes in slopes and intercepts at different times. Suppose that we have data for three periods and in the second period only the intercept changed (there was a parallel shift). In the third period the intercept and the slope have changed. Then we write

(8.5)

y, = a, p,x, + M, for period 1 2 = ttj + PiX2 + for period 2 = + + for period 3

Then we can combine these equations and write the model as

= a, + (02 - a,)£), -I- (a, - a,)A + P,x + (P2 - ,)£) + " (8.6)

where D, =

«-1

1 for observations in period 2 0 for other periods

for observations in period 3 for other periods

for observations in periods 1 and 2 or the respective value of x for i

all observations in period 3 Note that in all these examples we are assuming that the error terms in the different groups all have the same distribution. That is why we combine the data from the different groups and write an error term as in (8.4) or (8.6) and estimate the equation by least squares.

An alternative way of writing the equations (8.5), which is very general, is to stack the y-variables and the error terms in columns. Then write all the parameters a,, , , p,, P2 down with their multiplicative factors stacked in columns as follows:

= a,

+ a2

+ a.

+ P,

+ P2

/0 0

(8.7)

What this says is

y, = a,(l) + 02(0) + (0) + p,(x,) + p2(0) + = a,(0) + a2(l) + {0) + p,(x2) + P2(0) + "2 y, = a,(0) + 02(0) + (1) + p,(0) -I- 2( : ) + "

where ( ) is used for multiplication, e.g., 03(0) = a, x 0. Now we can write these equations as

= ,/) + aiDi + + p,D4 + P2D5 +

(8.8)

where the definitions of D,, D2, D3, D, are clear from (8.7). For instance,

for observations in the second group for all others

! 0 for all observations in group 3

Note that equation (8.8) has to be estimated without a constant term.

In this method we define as many dummy variables as there are parameters to estimate and we estimate the regression equation with no constant term. We will give an illustrative example in the next section, where this method is extended to take care of cross-equation constraints.

Note that equations (8.6) and (8.8) are equivalent. The method of writing the equation in terms of differences in the parameters is useful in tests for stability discussed in Section 8.5.

8.4 Dummy Variables for

Cross-Equation Constraints

The method described in Section 8.3 can be extended to the case where some parameters across equations are equal. As an illustration, consider the joint estimation of the demand for beef, pork, and chicken on the basis of data presented in Table 8.2." Waugh estimates a set of demand equations of the form

p, = a, + p„x, + . + , + +

2 = «2 + ,2, + 222 + + " + "2 (8.9)

= -I- , , + 23 2 + + + where , = retail price of beef P2 = retail price of pork = retail price of chicken JC = consumption of beef per capita jcj = consumption of pork per capita JC3 = consumption of chicken per capita = disposable income per capita

x,, X2, and Xj are given in Table 8.2. The prices in Table 8.2 are, however, retail prices divided by a consumer price index. Hence we multipUed them by the consumer price index p to get p,, p2, Pj. This index p and disposable income are as follows:

The data are from F. V. Waugh, Demand and Price Analysis: Some Examples from Agriculture, U.S.D.A. Techmcal Bulletin BI6, November 1964, Table 5-1, p. 39.

\x, corresponding values of jc for observations in groups 1 and 2

1 8.2 Per Capita Consumption and Deflated Prices of Selected Meats, 1948-1%3

"Carcass weight equivalent.

Divided by consumer price index (1957-1959 = 100). 1963 data are preliminary and were not used in the analysis.