### Goals

This tutorial explores the estimation of a linear model. After this tutorial you should be able to

- Create linear data using the GAUSS random normal number generator and GAUSS matrix operations.
- Estimate the linear model using matrix operations.
- Estimate the linear model using the
`ols`

procedure.

## Introduction

The linear regression model is one of the fundamental workhorses of econometrics and is used to model a wide variety of economic relationships. The general model assumes a linear relationship between a dependent variable, `y`

, and one or more independent variables, `x`

.

$$ Y = \alpha + \beta X $$ $$ \alpha = \text{intercept} $$ $$\beta = \text{slope}$$

**Note:**The graph above was created using the

`plotScatter`

procedure. For more information on plotting data see our graph basics tutorial.

## Generate linear data

To create our linear data we use a simple univariate linear data generating process

$$ y_{i} = 1.3 + 5.7 x_{i} + \epsilon_{i} $$

where $ \epsilon_{i} $ is the random disturbance term. To generate our data we break the process into three simple steps:

### 1. Generate independent data

Our first step is to generate a vector with 100 random `x`

values drawn from $N(0,1)$ using the GAUSS command `rndn`

.

```
// Clear the workspace
new;
// Set seed to replicate results
rndseed 23423;
// Number of observations
num_obs = 100;
// Generate x ~ N(0,1), with
// 'num_obs' rows and 1 column
x = rndn(num_obs,1);
```

### 2. Generate the error term

We will use the same function, `rndn`

, to generate the random disturbances.

```
// Compute 100 observations of an error term ~ N(0,1)
error_term = rndn(num_obs,1);
```

### 3. Generate the dependent data

Finally, generate `y`

from `x`

and `error_term`

following the data generating process above.

```
// Simulate our dependent variable
y = 1.3 + 5.7*x + error_term;
```

**Note:**The multiplication operator in GAUSS is overloaded to compute matrix multiplication if both inputs are matrices or vectors, and to compute element-by-element (ExE) multiplication if one of the operands is a scalar. Use the dot multiplication operator .*, to compute ExE multiplication of matrices, vectors or multi-dimensional arrays.

## Estimate the model using matrix operations

Using `x`

and `y`

, we can estimate the model parameters and compare our estimates to the true parameters. In order to estimate the constant in our model, we must add a vector of ones to the `x`

matrix. This is easily done using the function `ones`

in GAUSS.

**Note:**The tilde operator, ~, horizontally concatenates two matrices or vectors into one larger matrix.

```
// Create a new (num_obs x 2) matrix, 'x_mat', where
// the first column is all ones
x_mat = ones(num_obs, 1) ~ x;
```

We can now estimate the two parameters of the model, the constant and slope coefficient, using our `x`

matrix and `y`

vector :
$$ \hat{\beta} = (X'X)^{-1}(X'y) $$

```
//Compute OLS estimates, using matrix operations
beta_hat = inv(x_mat'x_mat)*(x_mat'y);
print beta_hat;
```

The above `print`

statement should return the following output:

```
1.2795490
5.7218389
```

## Estimate the model using `ols`

function

Above we used matrix operations to calculate the parameters of the model. However, GAUSS includes a built-in function `ols`

which will perform the same estimation for us. The function will find the parameter models and provide several model diagnostics. `ols`

takes the following three inputs:

- dataset
- String, name dataset to use for regression. Use an empty string,
`""`

, if`x`

and`y`

are matrices - x
- Matrix of independent variable for regression, or string with variable names
- y
- Vector of dependent variables, or string with name of independent variable

In our case, we will use the `x`

and `y`

data which we created. As these are not related to any dataset, we can enter a blank string, `""`

for the dataset input. Furthermore, rather than sending the output to any variables, we will simply print the output to screen using the command `call`

.

`call ols("", y, x);`

Which should print a report similar to the following:

Valid cases: 100 Dependent variable: Y Missing cases: 0 Deletion method: None Total SS: 3481.056 Degrees of freedom: 98 R-squared: 0.969 Rbar-squared: 0.969 Residual SS: 107.205 Std error of est: 1.046 F(1,98): 3084.149 Probability of F: 0.000 Standard Prob Standardized Cor with Variable Estimate Error t-value >|t| Estimate Dep Var ----------------------------------------------------------------------------- CONSTANT 1.27955 0.105343 12.14651 0.000 --- --- X1 5.72184 0.103031 55.53512 0.000 0.984481 0.984481

### Conclusion

Congratulations! You have:

- Used matrix operations to generate linear data.
- Estimated an OLS model using matrix operations.
- Estimates an OLS model using
`ols`

.

For convenience, the full program text is reproduced below.

The next tutorial describes the using the estimated parameters to predict outcomes and compute residuals.

```
// Clear the workspace
new;
// Set seed to replicate results
rndseed 23423;
// Number of observations
num_obs = 100;
// Generate x ~ N(0,1), with
// 'num_obs' rows and 1 column
x = rndn(num_obs,1);
// Compute 100 observations of an error term ~ N(0,1)
error_term = rndn(num_obs,1);
// Simulate our dependent variable
y = 1.3 + 5.7*x + error_term;
// Create a new (num_obs x 2) matrix, 'x_mat', where
// the first column is all ones
x_mat = ones(num_obs, 1) ~ x;
// Compute OLS estimates, using matrix operations
beta_hat = inv(x_mat'x_mat)*(x_mat'y);
print beta_hat;
call ols("", y, x);
```