Dynamic Econometric Model Exercises

Exercise 10.2

a. Taking logarithms of the model we get
ln Yt =  +  ln Xt* + ut
Taking logarithms of the adaptive rule,
 ln Xt* -  ln Xt-1* =   ln Xt-1 -   ln Xt-1*
or
 ln Xt* =   ln Xt-1 - (1- ) ln Xt-1*
From the model, we have  ln Xt* = (ln Yt -  - ut) /  . Substituting this into the adaptive rule, we get
 (ln Yt -  - ut) /  =   ln Xt-1 - (1- ) (ln Yt-1 -  - ut-1) / 
or
ln Yt =  + (1- ) ln Yt-1 +   ln Xt-1 + ut - (1- ) ut-1
or
ln Yt = 0 + 1 ln Yt-1 + 2 ln Xt-1 + vt

b. Consistency requires that the disturbance of the estimating equation is uncorrelated with the explanatory variables. Therefore we require that E ( vt ) =  E ( vt  ln Yt-1 ) =  E ( vt  ln Xt-1 ) = 0.

c. Because  ln Yt-1 is a lagged dependent variable, OLS estimates are biased in small samples.

d. Under the specified properties,

E ( vt  ln Yt-1 ) =   E { [ ut - (1- ) ut-1]  ln Yt-1}
E ( vt  ln Yt-1 ) =  E [ ut  ln Yt-1 ] - (1- ) E [ ut-1 ln Yt-1]
E ( vt  ln Yt-1 ) =  E [ ut  ln Yt-1 ] - (1- ) E [ ut-1( +  ln Xt-1* + ut-1)]
Since ln Xt* is exogenous,
E ( vt  ln Yt-1 ) =  E [ ut  ln Yt-1 ] - (1- )  E(ut-1) - (1- )  ln Xt-1*E(ut-1) - (1- ) E(ut-12)
Since  ut ~ IND with mean zero and ln Yt-1 is predetermined, ut and  ln Yt-1 are uncorrelated so that E [ ut  ln Yt-1 ] = E ( ut) = 0. Therefore E ( vt  ln Yt-1 ) =  - (1- ) u2 0, and so the consistency conditions specified in (b) are not all satisfied, and OLS estimates are not consistent.
 

Exercise 10.3

a. We would expect inventories to be positive even if sales are zero, so  > 0. As sales increase, so will inventories, so  > 0. If desired inventories were to exceed actual inventories, we would expect inventories to rise, so  > 0. We also expect that  < 1 and  < 1, since otherwise the model would be explosive.

b.

It =  It-1*+ (1- ) It-1 + ut
It = ( +  St-1) + (1- ) It-1 + ut
It =  +  St-1 + (1- ) It-1 + ut
It = 0 + 1 St-1 + 2 It-1 + ut

c. Conisitency requires that the disturbance of the estimating equation is uncorrelated with the explanatory variables. Therefore we require that E ( ut ) =  E ( ut St-1 ) = E ( utIt-1 ) = 0. These conditions are not sufficient for BLUE because the model has a lagged dependent variable.

d. All parameters are estimable. First, regress It against a constant, St-1, and It-1 ,and get , , and . Then = 1 - , = / , and = /.
 

Exercise 10.13

a. False. While multicollinearity makes standard errors larger, OLS estimates are still unbiased, consistent, and efficient (BLUE). However, ignoring serial correlation, or adding an irrelevant variable, makes estimates inefficient.

b. True. Model B contains Yt-1 as a regressor, and  Yt-1 would normally be highly correlated with  Yt in time series data, resulting in a higher R2, even after adjusting for the increase in the number of regressors.

c. True. The presence of a lagged dependent variable generally results in a  DW statistic that is biased upward.when there is positive serial correlation in the disturbances. Therefore, If ven this upward-biased value of d is less than dL , serial correlation is indicated.

Exercise 10.17

a. Taking logarithms of Pt = KMt / Nt , we get ln Pt = ln K + ln Mt - ln Nt , or pt = k + mt - nt . This gives deltapt =  deltamt - deltant .
b.  pt = beta0 + beta1 pt-1 + beta2 mt + beta3 mt-1 + beta4 nt + beta5 nt-1 + ut . Substitute  beta1 = 1 - gamma, beta3 = gamma - beta2 , and beta5 = - gamma - beta4 . We have pt = beta0 +( 1 - gamma) pt-1 + beta2 mt + ( gamma - beta2 ) mt-1 + beta4 nt - ( gamma + beta4 ) nt-1 + ut , which gives deltapt = beta0 + beta2 deltamt + beta4 deltant - gamma (pt-1 - mt-1 +  nt-1 ) + ut . In the long run, the change in variables is zero. This reduces to p* = k0 + m* - n*, which is of the form in part (a).
c.  The model derived in part (b) is the error-correction model. If gamma neq 0 there is support for the error-correction mechanism.
d. The empirical results may be obtained using SHAZAM and the data dile DATA10-6 as follows:
|_READ (DATA10-6) YEAR N M P
 UNIT 88 IS NOW ASSIGNED TO: C:\ESL\USER\DATA10-6

 ...SAMPLE RANGE IS NOW SET TO:         1        36
 |_GENR DN=N-LAG(N)
 ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO
 |_GENR DM=M-LAG(M)
 ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO
 |_GENR DP=P-LAG(P)
 ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO
 |_GENR Z=LAG(P)-LAG(M)+LAG(N)
 ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO
 ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO
 ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO
 |_SAMPLE 2 36

 |_OLS DP DM DN Z

 REQUIRED MEMORY IS PAR=       5 CURRENT PAR=    2000
  OLS ESTIMATION
        35 OBSERVATIONS     DEPENDENT VARIABLE= DP
 ...NOTE..SAMPLE RANGE SET TO:      2,     36

  R-SQUARE =   0.4823     R-SQUARE ADJUSTED =   0.4322
 VARIANCE OF THE ESTIMATE-SIGMA**2 =   1.1915
 STANDARD ERROR OF THE ESTIMATE-SIGMA =   1.0916
 SUM OF SQUARED ERRORS-SSE=   36.937
 MEAN OF DEPENDENT VARIABLE =   2.3429
 LOG OF THE LIKELIHOOD FUNCTION = -50.6055


 VARIABLE   ESTIMATED  STANDARD   T-RATIO        PARTIAL STANDARDIZED ELASTICITY
   NAME    COEFFICIENT   ERROR      31 DF   P-VALUE CORR. COEFFICIENT  AT MEANS
 DM        0.71732E-02 0.2736E-02   2.622     0.013 0.426     0.4260     0.3503
 DN       -0.14795     0.7451     -0.1986     0.844-0.036    -0.0301    -0.1495
 Z        -0.42732E-03 0.1520E-03  -2.812     0.008-0.451    -0.4048     0.2848
 CONSTANT   1.2050      1.893      0.6366     0.529 0.114     0.0000     0.5143

We note that while the estimate of gamma is significantly different from zero, it is quite small numerically (0.0004). Thus while there is support for the error-correction mechanism, the estimate suggests that the error-correction process is a weak one. The change in the money supply term is statistically significant but is also small numerically. On the other hand, the change in population term is larger, but not statistically significant, so we cannot reject that it has no influence on changes in the price level.

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Updated February 20, 2006.
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