Suppose that the production technology in the agricultural sector of an economy can be represented by the following Cobb-Douglas function where Y t = real gross agricultural product in year t ($m); L...

Suppose that the production technology in the agricultural sector of an economy can be represented by the following Cobb-Douglas function


where

• 
  Y_t
  = real gross agricultural product in year t ($m);


• 
  L_t
  
  = labour days employed in the agricultural sector during year t (millions of days);


• 
  K_t
  = real capital input during year t ($m);


• 
  a,
  ß₂
  , and ß₃
  are parameters; and


• 
  e_t
  = random errors ~ N(0,s²
  ).

After taking the log of both the left hand side and the right hand side of the production above we obtain:


The data set is provided. The data is annual from 1990 to 2004.Y_t
= aL_t
^ß2K_t
^ß3e^et



QUESTION 1

Test against using a 5% significance level.

• Find the test statistic (3 decimal places)


• Find the critical value for the test (rounded to 3 decimal places)

QUESTION 2

• Find the point estimate for the sum of and (3 decimal places)

• From this point estimate would you conclude that this production function exhibits constant, decreasing or increasing returns to scale?

QUESTION 3

• What is the value of the test statistic to test whether the explanatory variables are jointly significant (3dp)? ?


• What is the critical value to test whether the explanatory variables are jointly significant at the 5% significance level (3dp)? ?


• Is the null hypothesis rejected in favour of the alternative? (Yes or no?)

QUESTION 4

• Use the regression output to construct an estimate of the elasticity of output with respect to labour.(3dp)


• Use the regression output to construct an estimate of the elasticity of output with respect to capital. (3dp)

QUESTION 5

• What is the value of the test statistic to test that the elasticity of output with respect to capital is equal to 1? (3dp) ?


• What is the critical value to test that the elasticity of output with respect to capital is equal to 1 against the alternative that it is less than one at the 5% significance level? (3dp) ?


• Is the null hypothesis rejected in favour of the alternative?

QUESTION 6

Construct a 90% confidence interval for the predicted real gross agricultural product in 2005 assuming that, in 2005, there are 300 millions labour days and 50000 millions in real capital input. You will need to obtain the 90% prediction interval for the estimated log of real gross agricultural product and then take the exponential of the upper and lower limits of this interval. Be careful when you add the observation for 2005. You add it for
L
and
K
and then have to take the log of these two variables to add the new observation to
l_L
and
l_K.

• What is the value of the
  t-critical value that should be used in calculating this prediction interval? (3 dp) ?


• What is the lower limit of this interval? (2 dp) ?


• What is the upper limit of this interval? (2 dp) ?