### Goals

This tutorial demonstrates the GMM estimation of a simple OLS model using the `gmmFit`

and `gmmFitIV`

procedures. After completing this tutorial you should be able to estimate an OLS model with exogenous regressors using:

## Introduction

In this example, we will estimate a simple OLS model using GMM. Because this model is a linear model, we can and will estimate the model using both `gmmFit`

and `gmmFitIV`

. The linear model we will estimate examines the relationship between gas mileage and vehicle weight and length:

$$mpg = \alpha + \beta_1*weight + \beta_2*length$$

The data for this model is stored in the dataset `auto2.dta`

, located in the GAUSS examples folder.

## Estimation with `gmmFitIV`

While the `gmmFit`

procedure minimizes the GMM objective function to estimate the model parameters, `gmmFitIV`

computes the analytic GMM estimates for instrumental variables. `gmmFitIV`

provides a compact method for estimating IV and OLS models. In fact, we can estimate the model using `gmmFitIV`

in one line:

```
//Create dataset file name with full path
dset_name = getGAUSShome() $+ "examples/auto2.dta";
//Perform estimation
call gmmFitIV(dset_name, "mpg ~ weight + length");
```

The output from our `gmmFitIV`

estimation reads:

Dependent Variable: mpg Number of Observations: 74 Number of Moments: 3 Number of Parameters: 3 Degrees of freedom: 71 Standard Prob Variable Estimate Error t-value >|t| ----------------------------------------------------------- CONSTANT 47.884873 7.506021 6.380 0.000 weight -0.003851 0.001947 -1.978 0.052 length -0.079593 0.067753 -1.175 0.244

The estimates from `gmmFitIV`

are the same as the estimates from `gmmFit`

, as you will see. However, note that the `gmmFitIV`

table includes variable names. This occurs because GAUSS is able to extract variable names from the formula string used to identify the model in `gmmFitIV`

.

## Estimation with `gmmFit`

### Load data

In order to estimate our model using `gmmFit`

we must first load our data into data matrices. For this example, we will use just three variables from the `auto2.dta`

dataset, `mpg`

, `weight`

, and `length`

.

```
//Create dataset file name with full path
dset_name = getGAUSShome() $+ "examples/auto2.dta";
//Load variables 'mpg', 'weight' and 'length'
//into matrix 'data'
data = loadd(dset_name , "mpg + weight + length");
```

**Note:**We are using the formula string method to load our data. Our formula string is constructed as a list of variables separated by a

`+`

, `variable_1 + variable_2 + ... + variable_k`

The columns in the matrix `data`

will be in the order the variables are specified in the formula string. We can use this information to create two separate data matrices, `y`

for our dependent variable and `X`

for or independent variables.

```
//Declare 'y' variable
y = data[., 1];
//'X' variables
X = data[., 2:3];
```

Finally, we want to include a constant in this model. This is not done automatically with the `gmmFit`

procedure and a column of ones must be concatenated to the beginning of the already defined data matrix `X`

:

```
//Concatenate a column of ones to the 'X' data
X = ones(rows(data), 1) ~ data[., 2:3];
```

### Write the moment equation

The next step for our `gmmFit`

estimation is to define our moment procedure. For this example, we will estimate a linear model with moments based on $E[x_tu_t(\theta_0)] = 0$ with $u_t(\theta_0) = y_t - \beta_t x_t$ :

```
proc meqn(b, yt, xt);
local ut,dt;
/** OLS resids **/
ut = yt - b[1] - b[2]*xt[., 2] - b[3]*xt[., 3];
/** Moment conditions **/
dt = ut.*xt;
retp(dt);
endp;
```

### Set Model Parameters

Model parameters are controlled using a `gmmControl`

structure. Therefore, prior to setting model parameters we must declare an instance of the `gmmControl`

structure and fill the instance with default values.

```
//Declare `gctl` to be a `gmmControl` struct
//and fill with default settings
struct gmmControl gctl;
gctl = gmmControlCreate();
```

The first thing we must set in the `gmmControl`

structure is the start values of the parameters, using `gctl.bStart`

.

```
//Set starting values
gctl.bStart = { 41, -0.005, -0.001 };
```

Finally, we will set up the initial weight matrix for the `gmmFit`

estimation so it will replicate the default model of the `gmmFitIV`

procedure. Because the variables `weight`

and `length`

are assumed to be exogenous in this model, the initial weight matrix used by `gmmFitIV`

will be equal to $\frac{1}{N}(X'X)^{-1}$. We can specify for `gmmFit`

to use the same matrix using the `gmmControl`

member `gctl.wInitMat`

:

```
//Set initial weight matrix
gctl.wInitMat = invpd((1/rows(X))*(X'X));
```

### Call `gmmFit`

We are finally ready to call `gmmFit`

. For this example, we will use the GAUSS keyword `call`

to run `gmmFit`

and print results directly to the input/output screen.

`call gmmFit(&meqn, y, x, gctl);`

The output from our `gmmFit`

estimation reads

Dependent Variable: Y Number of Observations: 74 Number of Moments: 3 Number of Parameters: 3 Degrees of freedom: 71 Standard Prob Variable Estimate Error t-value >|t| ----------------------------------------------------------- Beta1 47.884629 7.506023 6.379 0.000 Beta2 -0.003852 0.001947 -1.978 0.052 Beta3 -0.079591 0.067753 -1.175 0.244

which is the same, other than the variable names, as our results from `gmmFitIV`

earlier in this tutorial.

**Note:**We can add variable names to our

`gmmFit`

table using the `gmmControl`

structure member, `gctl.varNames`

. The `gctl.varNames`

structure member must be a string array that lists all independent variables first and includes the dependent variable as the last element. You do NOT have to name the constant, this is done automatically by GAUSS. ### Conclusion

Congratulations! You have:

- Estimated an OLS model using
`gmmFitIV`

. - Estimates an OLS model using
`gmmFit`

.

For convenience, the full program text is reproduced below.

Our next tutorial will demonstrate the estimation of an OLS model with endogenous variables.

```
//Create dataset file name with full path
dset_name = getGAUSShome() $+ "examples/auto2.dta";
//Perform estimation
call gmmFitIV(dset_name, "mpg ~ weight + length");
//Create dataset file name with full path
dset_name = getGAUSShome() $+ "examples/auto2.dta";
//Load variables 'mpg', 'weight' and 'length'
//into matrix 'data'
data = loadd(dset_name , "mpg + weight + length");
//Declare 'y' variable
y = data[.,1];
//'X' variables
X = data[.,2:3];
//Concatenate a column of ones to the 'X' data
X = ones(rows(data),1) ~ data[.,2:3];
//Declare `gctl` to be a `gmmControl` struct
//and fill with default settings
struct gmmControl gctl;
gctl = gmmControlCreate();
//Set starting values
gctl.bStart = { 41, -0.005, -0.001 };
//Set initial weight matrix
gctl.wInitMat = invpd((1/rows(X))*(X'X));
call gmmFit(&meqn, y, x, gctl);
proc meqn(b, yt, xt);
local ut,dt;
/** OLS resids **/
ut = yt - b[1] - b[2]*xt[.,2] - b[3]*xt[.,3];
/** Moment conditions **/
dt = ut.*xt;
retp(dt);
endp;
```