## Nonlinear Least Squares with CES Production Function

This demo provides two data input options for variable estimation and reports regression statistics based on a Constant Elasticity of Substitution (CES) production function with *multiplicative error terms*. The reported statistics include estimators, standard errors, t-statistics and p-values (against non-significant coefficients assumption) at the estimated point. *For the best results, we recommend using Firefox for this interactive case study.*

Back to the Nonlinear Least Squares with Constant Elasticity of Substitution Production Function case study.

#### Option 1: Data in a text file

Users who have access to the data needed in the estimation should create a text file with the data, for example, the capital, labor, and production data collected in Mizon (1977). See mizon_ces_data.txt. User-provided data files must satisfy the following restrictions:

- The first column of the data file must be a column of output indexed by
*Q*, denoting the quantity of output. - The second and subsequent columns of the data file that contain input data may not contain any negative or zero input values. The last column of the data file must be a column of input indexed by
*v_end*and users are free to index any other input column as they wish.

The estimated variables in the CES model are indexed by *phi* for scale factor, *b_1* for value share of input 1, *b_2* for value share of input 2, *b_3* for output elasticity of input 3, ..., etc, and *rho* for substitution parameter. Note that the value share of the last input is calculated as a residual, $(1 - \sum^{m-1}_{n=1}b_n)$.

Users then can download a sample GAMS model file, ces_md_txt.gms (CES model with multiplicative disturbance, text input), and modify it to solve their own estimation problems. Users should specify their own set definitions (sets "t" and "m" in the sample), include their own table of data (as described above), and run the modified model to obtain the estimation results.

#### Option 2: Data in a GAMS data exchange (gdx) file

Users who have access to the data in a GAMS data exchange (gdx) file can use one of the following two methods.

**Method 1: Solve using the NEOS solver**Users can click on the "Solve with NEOS" button to find estimation results based on

*the default gdx file*, i.e., the file with the capital, labor, and production data collected in Mizon (1977). See mizon_ces.gdx. Alternatively, users can upload their own data by clicking on the button next to "Upload GDX File" and then "Solve with NEOS". User-provided gdx files must satisfy the same restrictions as listed above in Option 1.Clicking on the "Reset" button will clear the solution.

**Method 2: Calculate the regression statistics locally**Users who have access to GAMS can download the GAMS model file ces_md_gdx.gms and solve the model locally with the following command:

- "gams ces_md_gdx --in=mydata"

where mydata.gdx is a data file provided by the user. The gdx file must satisfy the same restrictions as in Option 1.