Chapter 5: Hypothesis Tests and Model Selection

Example 5.1 An Investment Equation

This example considers a linear regression model of painting auction prices such that:

\[\ln { \text{Price}} = \beta_1 + \beta_2 \ln {\text{Size}} + \beta_3 \text{AspectRatio} + \epsilon\]

In particular, it considers whether Size is a statistically significant determinant of Price. This is done by testing the hypothesis:

\[H_0: \beta_2 = 0\]
\[H_1: \beta_2 \neq 0\]

If Size is a statistically significant determinant of Price than the null hypothesis that \(\beta_2 = 0\) should be rejected.

Getting Started

To run this example on your own you will need to install the greeneLib package. This package houses all examples and associated data.

How to

Step One: Loading data

To start, load the relevant variables from Table 4.7 using loadd() and a formula string.

// Load data using loadd
fname = getGAUSShome() $+ "pkgs/GreeneLib/examples/TableF4-1.csv";
monet_data = loadd(fname, "HEIGHT+ ln(Price) + WIDTH");

The code above:

  1. Tranforms the raw data variable, Price into our dependent variable ln(Price).

  2. The raw data variables Height and Width are loaded so we can create our dependent variables, Aspect Ratio and the Size.

Step Two: Create dependent variables

Our dependent variables are created according to:

\[\text{Aspect Ratio} = \frac{Width}{Height}\]
\[\text{Size} = Width \times Height\]

First, we compute the new variables:

// Compute aspect ratio
aspect = monet_data[., "WIDTH"] ./ monet_data[., "HEIGHT"];

// Compute size
size = monet_data[., "WIDTH"] .* monet_data[., "HEIGHT"];

Next, we will use the function setColNames() to give our variables the correct names:

** Change assigned variable names, `width`,
** to match variables
aspect = setColNames(aspect, "Aspect Ratio");
size = setColNames(size, "Size");

Finally we will create a new dataframe containing our estimation data:

// Create regression data using the horizontal concatenation operator
reg_data = monet_data[., "ln_Price_"] ~ size ~ aspect;

Step Three: Estimate our linear model

Finally, we call olsmt() to run ordinary least squares estimation and store our results for later hypothesis testing.

Notice that we can transform our size variable to ln(Size) directly in the formula string.

** Calling olsmt
** Note that the print out includes
** coefficients along with the t-stats
** which test the hypothesis that
** the coefficients equal zero
struct olsmtOut o_out;
o_out = olsmt(reg_data, "ln_Price_ ~ ln(Size) + Aspect Ratio");

When we call olsmt() a complete set of results are printed to screen including:

  • Coefficient estimates.

  • The t-statistics testing the null hypothesis that the coefficient are equal to zero.

  • The p-values associated with the t-statistics.

                            Standard                 Prob   Standardized  Cor with
Variable         Estimate      Error      t-value     >|t|     Estimate    Dep Var
CONSTANT         -8.34236    0.678203    -12.3007     0.000       ---         ---
ln(Size)          1.31638   0.0920493     14.3009     0.000    0.573347    0.577572
Aspect Ratio   -0.0962332     0.15784   -0.609689     0.542  -0.0244435   -0.123553

These results confirm that:

  • The ln(Size) variable is statistically significant with a t-statistic equal to 14.3009.

  • The Aspect Ratio variable is not statistically significant with a t-statistic equal to -0.61.

Step Four: Additional testing

Following Greene, let’s also test

\[H_0: \beta_2 \leq 1\]
\[H_1: \beta_2 > 0\]
// Test hypothesis that beta_2 =< 1
t_stat_1 = (o_out.b[2] - 1) / o_out.stderr[2];

The t-statistic testing that \(\beta_2 \leq 1\) is 3.437.