Suppose you are given the following information: Model 1 lnYt = 4.99 +23.2D1 + 3

Question

Suppose you are given the following information:

Model 1 lnYt = 4.99 +23.2D1 + 36.5D2 + .732 lnX2 – 2.798 D1lnX2 + 4.251 D2ln X2 – .371ln X3 +.405 D1lnX3 – .236 D2lnX3 adjusted R2 = .921, RSS = .018645

Model 2 lnYt = 4.18 +.103D1 + .103D2 + .621 lnX2 – .201 lnX3 adjusted R2 = .852, RSS = .04195

Where n = 29 D1 = 1 for observations 12 to 20, (period 2) and 0 otherwise.

D2 = 1 for observations 21 to 29, (period 3) and 0 otherwise

(a) What are the elasticities of Y with respect to X2 and X3 in period 1, 2, and 3

(b) How can one test the Null Hypothesis that there has been no structural change in the elasticities of Y with respect to X2 and X3 over the three time periods, observations 1 – 11, 12 – 20, and 21 – 29?

Write out the null and alternative hypotheses in term of the B’s.

Calculate the test statistic, and its distribution.

Carry out the test and the 1% level. What is the result?

Explanation / Answer

a) In period1, D1=0 and D2=0

In period2, D1=1 and D2=0

In period3, D1=0 and D2=1

Consider model1, put the values of D1 and D2 ,

In period1, lnYt= 4.99 + .732lnX2 - 3.71lnX3

So, elasticity of Y w.r.t. X2 is .732 and X3 is -3.71 (since in log-log model, elasticity of dependent variable w.r.t. independent variable is the coeffient of independent variable itself)

In period2, lnYt= 4.99 + 23.2 + .732lnX2 - 2.798lnX2 - 3.71lnX3 + .405lnX3

So, elasticity of Y w.r.t. X2 is -2.066 and X3 is -3.305

In period3, lnYt= 4.99 + 36.5 + .732lnX2 + 4.251lnX2 - 3.71lnX3 - .236lnX3

So, elasticity of Y w.r.t. X2 is 4.983 and X3 is -3.946

Now, in model3, D1 and D2 does not affect the coefficients of X2 and X3, therefore, elasticity of Y w.r.t. X2 is .621 and X3 is -.201 in all the periods.

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