Monte Carlo Simulations

This section first proposes three models to be used to compare the performance of their estimators. The next subsection specifies an expenditure equation and addresses the data generation processes for all variables involved.

4.1 Design of Experiments

We plan to perform Monte Carlo simulations using three models and evaluate the properties of their estimates in terms of bias and mean square errors (MSE). Model A follows the five steps addressed by the previous section. Models B and C are adapted from Model A for the purpose of making comparisons among the three models. At the outset, all of the three models have to estimate the input share equations using the NISUR, i.e., carrying out the first step. The J 1 allocative parameter estimates are next exploited to estimate lnG and AE, denoted by G1 and AE1, respectively, _nt while the subsequent steps of the three models differ from one another. Note that the

1

J allocative parameters are treated as given thereafter. We now introduce them in details.

(i) Model A

This preferred model follows exactly the above five steps. Using the kernel estimates of ˆ (lnE E lnY_nt), ˆ (E X lnY_nt), and Eˆ (ln | lnG^ˆ Y_nt) from Step 2, we estimate equation (3-13) by the NISUR in Step 3 to obtain the estimates of . At the same time, nonlinear function lnG is assumed to be unknown, i.e., all of the _nt parameters shown in the parametric part of the cost function are jointly estimated, but exclude the parameters associated with the distribution of v and U. Estimates ˆ together with the J1 allocative parameter estimates are employed to calculate new estimates of lnG and AE, denoted by G2A and AE2A. The remaining parameters _nt embedded in the distributions of v and U are estimated in Steps 4 and 5.

(ii) Model B

This model is similar to Model A except that function lnG is treated in a _nt different way. Specifically, the estimated lnG , _nt lnG^ˆ_nt, derived from Step 1 is viewed as fixed so that it can be subtracted from the dependent variable. The new transformed equation becomes

lnE_nt lnG^ˆ_ntE(lnE_ntlnG^ˆ_nt lnY_nt)[X_ntE X( _nt lnY_nt)]  ˆ_nt (4-1) , where the notations are similarly defined to (3-13). After substituting the kernel estimates of Eˆ (lnElnGˆ lnY_nt) and ˆ (E X lnY_nt)for the corresponding conditional means in (4-1),  is estimated simply by ordinary least squares (OLS). This procedure is analogous to the one proposed by Kumbhakar and Lovell (2000, p.295-296) in spirit, while their underlying model is parametric. Estimates ˆ are next used to compute lnG and AE, denoted by G2B and AE2B. Finally, Steps 4 _nt and 5 are executed.

(iii) Model C

This model is further adapted from Model B and is similar to the one suggested by Kumbhakar and Lovell (2000, p.165) in essence. Again, their underlying model is parametric. Since the input share equations include vector  , their consistent estimate ˆ from Step 1 can be treated as fixed. In this manner, the new dependent

variable turns out to be lnE_nt lnG^ˆ_ntX_nt^ˆ with corresponding kernel estimate

ˆ ˆ

ˆ (ln ln ln _nt)

E E GX Y obtained by Step 2. Step 3 is no longer needed and Equation (3-14) of Step 4 is modified accordingly as:

ˆ ˆ ˆ ˆ ˆ

ˆ_nt lnE_nt lnG_nt X_nt E(lnE lnG X lnY_nt) _t

         (4-2)

After concentrating out ², we execute Step 5. This completes the entire procedure.

It is seen that the major differences among the three models stem from distinct treatments on lnG^ˆ_nt and ˆ. As a result, we can compare the performance of the resulting estimates among Models A to C, including the distribution parameters of v and U.

4.2 Model Specifications

This subsection specifies the expenditure equation and the data generation processes for all variables involved that will be used to carry out Monte Carlo simulations to investigate the finite-sample performance of the proposed estimators in the last subsection. Since we are also interested in the effects of the number of firms (N) and time periods (T) on the parameter estimates, we consider several (N, T) combinations. Specifically, we choose N = 50, 100, 200 with T = 6, 10, 20. Following Olson et al. (1980) and Fan et al. (1996), we consider three sets of variances and variance ratios, viz. (², ) = (1.88, 1.66), (1.35, 0.83), (1.63, 1.24). Finally,  = 0.025 and -0.025 are arbitrarily chosen.

The semiparametric cost frontier incorporating a single output and three inputs is formulated as: that a cost function is required to be linearly homogeneous in input prices and symmetrical by the microeconomic theory. We randomly pick W to normalize ₁

dependent variable E and the other two input prices to satisfy this requirement. The symmetry restriction is already imposed on (4-3). To understand whether the performance of the estimates is robust to changes in the functional form of M  , we ( ) specify an alternative form of M( ) 0.2y₁. We also extend (4-3) to a two-output and three-input case, assuming either M( ) 2 ln(1y₁) ln y₂ or

2 1 2

( ) 2 ln y + y y

M   ⁵. Note that in this extended case, the parametric part of (4-3) has to contain extra terms involving the cross products of ln y and (log) normalized ₂ input prices.

Input prices W , ₁ W , and ₂ W are randomly drawn from dissimilar uniform ₃ distributions (0,1)U , (0.5, 0.5)U , and U(0.5,1), respectively. The three-input and two -output quantities of x , ₁ x , ₂ x , ₃ y , and ₁ y are independently generated from ₂ normal distributions N(5, 0.5) , (3, 0.1)N , (5, 0.5)N , (31 , 10.1)N , and

(20 , 0.8),

N respectively. Two-sided error v is drawn from N(0,_v²) and

one-sided error u from a half-normal N^(0,_u²). The simulations are executed 1000 times for each model and the bias and the MSE are computed based on the 1000 replications. We set H₂/H = 0.8 and ₁ H₃/H =1.2. The true values of the ₁ coefficients are as follows: b = 0.3, ₂ b = 0.7, ₃ d = -0.05, ₂₂ d = -0.02, ₃₃ d = 0.5, ₂₃

e = 0.7, 12 e = 0.9, ₁₃ e = 0.3, and ₂₂ e = 0.5. ₂₃

The corresponding input share equations can be readily deduced by taking the first partial derivatives of lnE with respect to lnW , i = 1, 2, 3. Although the functional _i form of an expenditure equation is not unique, we recommend using those such as (4-3). A prominent feature of (4-3) consists in its smooth function being specified as a

5 Some of the nonparametric settings follow Fan et. al (1996) and Deng and Huang (2008).

function of (log) outputs only, i.e., the (shadow) input prices must be excluded.

Otherwise, one is confronted with a problem on how to properly disentangle the allocative parameters contained in M  . More importantly, the share equations are ( ) unable to be explicitly derived by taking partial derivatives due to the unknown smooth function dependent of shadow prices. This impedes a researcher from consequently identifying the allocative parameters.