fa, > 0,

if chicken and pork are competing products

< 0,

if chicken and pork are complementary products

= 0,

if chicken and pork are unrelated products

fa5 > 0,

if chicken and beef are competing products

< 0,

if they are complementary products

= 0,

if they are unrelated products

Suppose someone maintains that chicken and pork and beef are unrelated products in the sense that chicken consumption is not affected by the prices of pork and beef. In short,

Ho: fa = fa = 0 Therefore, the constrained regression becomes ln Yt = fa + fa ln X2, + fa ln X3, + ut

Equation (8.7.19) is of course the unconstrained regression. Using the data given in exercise 7.19, we obtain the following:

Unconstrained regression lnYt = 2.1898 + 0.3425 ln X2t — 0.5046 ln X3t + 0.1485 ln X4t + 0.0911 ln X5t (0.1557) (0.0833) (0.1109) (0.0997) (0.1007)

Constrained regression lnYt = 2.0328 + 0.4515ln X2t — 0.3772 ln X3t (0.1162) (0.0247) (0.0635)

(Continued)

CHAPTER EIGHT: MULTIPLE REGRESSION ANALYSIS: THE PROBLEM OF INFERENCE 273

EXAMPLE 8.4 (Continued)

where the figures in parentheses are the estimated standard errors. Note: The R2 values of (8.7.23) and (8.7.24) are comparable since the dependent variable in the two models is the same.

Now the F ratio to test the hypothesis (8.7.21) is

The value of m in the present case is 2, since there are two restrictions involved: p4 = 0 and p5 = 0. The denominator df, (n - k), is 18, since n = 23 and k = 5 (5 p coefficients). Therefore, the F ratio is

which has the F distribution with 2 and 18 df.

At 5 percent, clearly this F value is not statistically significant [F05(2,18) = 3.55]. The p value is 0.3472. Therefore, there is no reason to reject the null hypothesis—the demand for chicken does not depend on pork and beef prices. In short, we can accept the constrained regression (8.7.24) as representing the demand function for chicken.

Notice that the demand function satisfies a priori economic expectations in that the own-price elasticity is negative and that the income elasticity is positive. However, the estimated price elasticity, in absolute value, is statistically less than unity, implying that the demand for chicken is price inelastic. (Why?) Also, the income elasticity, although positive, is also statistically less than unity, suggesting that chicken is not a luxury item; by convention, an item is said to be a luxury item if its income elasticity is greater than one.

8.8 TESTING FOR STRUCTURAL OR PARAMETER STABILITY OF REGRESSION MODELS: THE CHOW TEST

When we use a regression model involving time series data, it may happen that there is a structural change in the relationship between the regressand Y and the regressors. By structural change, we mean that the values of the parameters of the model do not remain the same through the entire time period. Sometime the structural change may be due to external forces (e.g., the oil embargoes imposed by the OPEC oil cartel in 1973 and 1979 or the Gulf War of 1990-1991), or due to policy changes (such as the switch from a fixed exchange-rate system to a flexible exchange-rate system around 1973) or action taken by Congress (e.g., the tax changes initiated by President Reagan in his two terms in office or changes in the minimum wage rate) or to a variety of other causes.

How do we find out that a structural change has in fact occurred? To be specific, consider the data given in Table 8.9. This table gives data on disposable personal income and personal savings, in billions of dollars, for the United States for the period 1970-1995. Suppose we want to estimate a

274 PART ONE: SINGLE-EQUATION REGRESSION MODELS

Observation |
Savings |
Income |
Observation |
Savings |
Income |

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