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: .. math:: \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: .. math:: H_0: \beta_2 = 0 .. math:: H_1: \beta_2 \neq 0 If *Size* is a statistically significant determinant of *Price* than the null hypothesis that :math:\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 :func: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: .. math:: \text{Aspect Ratio} = \frac{Width}{Height} .. math:: \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 :func: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 :func: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 :func: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 .. math:: H_0: \beta_2 \leq 1 .. math:: 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 :math:\beta_2 \leq 1 is 3.437.