Assignment #2
Agricultural and Applied Economics 637
(Due: February 22, 2013)
1. (20 pts) Before you get started answering the following questions, I would like you to modify the basic Gauss-Newton estimation algorithm we estimated in class whereby you can choose between estimating a set of parameters via: (i) a fixed grid search, (ii) the Gauss-Newton method; and/or (iii) the Newton Raphson method.
You should develop this code so that it can be applied to any non-linear (in parameter) model. Once you have extended this code, I would like you to estimate the following Cobb-Douglas production function where you use the: (i) fixed grid search method so as to obtain reasonable starting values; (ii) Gauss-Newton algorithm for the intermediate iterations given the starting values obtained from (i); and (iii) Newton-Raphson for a final iteration from which the displayed results are based.
Quant t
 γCapital Labor + ε t
[1.1]
For this exercise I would like you to use Mizon (1977)’s data set. He was one of the first authors to provide an overview on statistical inference for nonlinear regression models. In the 1977 paper he estimated a variety of specifications for production functions including the Cobb-Douglas (C-D) and Constant Elasticity of Substitution
(CES) where they were allowed to have additive as well as multiplicative error terms.
He used U.K. data on capital, labor use and a common output measure for 24 industries encompassing the years 1954, 1957 and 1960 in his production function estimations. The following table provides a summary of the variables contained in the mizon_1977.xls
dataset.
Table 1: Variables Contained in the Mizon (1977) dataset
Variable Description Units
Quant Gross value-added at factor cost Mil. Lbs.
Capital Value of the stock of plant and machinery
LF Labor force available for work in the industry
Mil. Lbs.
1,000
Unemploy Number of workers unemployed in the industry 1,000
Hour Average hours per week worked by those employed Hours
Year Survey data year
Industry Industry ID Number
Year
#

(a) (15 pts) Assume that you would like to estimate the parameters of the above
Cobb-Douglas type production function where Labor is defined as the number of worker-hours of labor input (divided by 100 for scaling purposes). Present the typical regression output.
(b) (5 pts) At the optimal coefficient estimates check the convexity of the SSE function to determine whether you are at least at a local minimum. Again this should be written so that it can be applied to a model with any number of coefficients.
2. (25 pts) Consider the following consumption function where consumption, C t
, depends on income, Y t
, via the following model:
C t
 β
1
 β Y t
 e t e t
 θ e  θ e  ν t
[2.1]
[2.2] where υ t
is an iid random error with υ t
~ (0, σ
2
υ
).
(a) (5 pts) Using [2.1] and [2.2], reformulate the model such that:
C t
 
2

[2.3]
(b) (10 pts) Use the data, nl_cons_v2.xlsx
to obtain estimates of β
1
, β
2
, θ
1
, and θ
2
.
( Note : Given the number of parameters, use either the NR or GN. Do not do a grid search to obtain starting values.) Report the usual regression results. The consumption data is represented by the CONSUME variable and the income data is represented by the INC variable in the above dataset.
(c) (5 pts) Test whether the AR(1) model would have been adequate in describing the relationship among the regression errors versus the more extensive specification.
(d) (5 pts) Test the following null hypothesis: H
0
: θ
1
= θ
2
= 0. What is the implication of this hypothesis test?
3. (25 pts) An important ingredient underlying firm cost theory is the learning curve.
The idea behind the learning curve is that, as more items are produced, workers become more experienced and the average cost of production falls. Let’s represent the average cost (as represented by time) of producing X items by AC
X
. A typical learning curve model can be represented via the following: AC
X
=γX δ
where γ and δ
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are unknown parameters and X is the cumulative amount of output produced up to and including the current period (i.e., X t
=
 t s=1 x s
) where t is the current time period.
The total cost (time) of producing X t
items (TC
X
) is simply X t
*AC
Xt
=X t
γX t
δ
=
γX t
(δ+1)
. For estimation of the unknown parameters, cost is often represented by the amount of time taken to produce a given lot of items.
Suppose you have collected data pertaining to the time associated with the setup of spinning frames in a U.S. yarn producing firm. The data you collected and contained in the data set yarn-industry.xlsx
consists of:
TIME i
: Time (minutes) taken to set up the i th
lot of frames
FRAMES i
: Number of frames completed in the i th
lot [Note: This is not the cumulative number of frames ] and
LOT: Lot number.
Note, the total (cumulative) number of frames setup after i lots have been completed is: TOTFR i
  i j=1
FRAME j
. Therefore, the total time (TC) to setup TOTFR i frames can be represented by the following after incorporating the learning model:
TC
TOTFR i
TOTFR i
[3.1]
Given that you are interested in examining whether the learning curve model described above is relevant to the above firm you recognize that:
TOTFR i
- TC
TOTFR
[3.2]
(a) (10 pts) Using [3.2] as a base, formulate the statistical model to which you can apply nonlinear least squares to estimate learning curve parameters, γ and δ.
What is your underlying statistical model? Given this model, what is your null hypothesis as to the sign of δ if the above model of learning is appropriate to this firm? Using non-linear least squares, determine whether the data associated with this yarn-producing firm supports your null hypothesis?
(b) (5 pts) Estimate the average setup cost after 50, 400 and 800 cumulative frames have been produced. At each of these data points, test the null hypothesis that the associated average cost is different from 11.5 minutes.
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(c) (5 pts) Calculate the marginal cost, defined in terms of time as

TC

TOTFR
, of producing frames after 50, 400 and 800 cumulative frames have been produced.
At each of these data points, test the null hypothesis that the associated marginal effect is different from 11.5 minutes.
(d) (5 pts) At each of the above data points test the null hypothesis that MC-AC =
3.0 minutes .
4. (55 pts) Use the Mizon (1977) data accessed in Question 1 above and estimate the parameters of the following Constant Elasticity of Substitution (CES) production functions having both additive [4.1] and multiplicative [4.2] error terms.
1
CES: Multiplicative Disturbance
Quant t 

Capital t

  
)Labor t





 exp [4.1]
Additive Disturbance
Quant t 

Capital t

  
)Labor t






  t where: α ≡ efficiency parameter, α > 0
η ≡ the degree of homogeneity (scale parameter), η > 0
δ ≡ distribution parameter, 0 < δ < 1
ρ ≡ the substitution parameter, -1 < ρ < ∞, ρ ≠ 0
[4.2]
(a) (10 pts) Using the Mizon data set, estimate [4.1] and [4.2] via NLS. ( Note : Given the number of parameters, use either the NR or GN. Do not do a grid search to obtain starting values.) Present the typical regression statistics. Is the SSE function indeed at a (local) minimum at the parameter values you identified as minimizing the SSE function?
(b) (5 pts) For the CES model specifications what is the correlation between predicted and actual values of Quant (i.e., not ln[Quant])? Why would one be interested in determining such correlations? [ Note: When evaluating the relationship between predicted and actual values of Quant (versus ln(Quantity)), you should note that
E(ε)=0 but E[exp(ε t
)] need not equal 1.0 but in fact E[exp(ε t
)] = exp(σ 2 /2) ]. From
1 For more detail concerning the CES production function, refer to the following PDF: ces_2.pdf
.
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these results what specification would you say is preferred in terms of explaining the variance of Quant?
(d) (5 pts) Provide statistical evidence as to whether the production technology exhibits constant returns to scale using specification [4.1].
(e) (15 pts) Under the CES specification in [4.1] evaluate the marginal products of
Capital and Labor when these inputs are at their mean values. Are these marginal products positive from a statistical point of view? [ Note: Make sure you estimate these marginal products at a point on the predicted production functions and you include the variability of the predicted production function in your estimate of the variance of the MP’s.
]
(f) (5 pts) Using the functional form in [4.1] and at the mean values of Capital and
Labor, are the marginal products equal? [ Hint: There are at least two ways of doing this, one based on economic theory and one based on statistical relationships.]
(g) (5 pts) For the CES specification in [4.1] evaluate the elasticity of substitution. Is it different from 1.0?
(h) (10 pts) For the CES specification in [4.1] undertake a joint test that the intercept varies across survey year.
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