Consider two regression models estimated on a sample of 350 observations.
\[
y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \beta_4 x_4 + \beta_5 x_5 + u \quad \text{--------(1)}
\]
\[
y = \alpha_0 + \alpha_1 x_1 + \alpha_2 x_2 + v \quad \text{--------(2)}
\]
The \( R^2 \) in model (1) is \( R^2_1 = 0.3521 \) and in model (2) is \( R^2_2 = 0.2314 \). The value of the test statistic to test the \( H_0: \beta_3 = \beta_4 = \beta_5 = 0 \) is _________ (round off to three decimal places).
\[ y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \beta_4 x_4 + \beta_5 x_5 + u \quad \text{--------(1)} \] \[ y = \alpha_0 + \alpha_1 x_1 + \alpha_2 x_2 + v \quad \text{--------(2)} \] The \( R^2 \) in model (1) is \( R^2_1 = 0.3521 \) and in model (2) is \( R^2_2 = 0.2314 \). The value of the test statistic to test the \( H_0: \beta_3 = \beta_4 = \beta_5 = 0 \) is _________ (round off to three decimal places).
Show Hint
Correct Answer: 21.1
Solution and Explanation
- \( R^2_1 = 0.3521 \) is the \( R^2 \) value from model (1),
- \( R^2_2 = 0.2314 \) is the \( R^2 \) value from model (2), - \( p = 3 \) is the number of restrictions (since \( \beta_3, \beta_4, \beta_5 \) are being tested),
- \( n = 350 \) is the sample size,
- \( k_1 = 5 \) is the number of predictors in model (1).
Substituting these values into the formula: \[ F = \frac{(0.3521 - 0.2314)/3}{(1 - 0.3521)/(350 - 5)} = \frac{0.1207/3}{0.6479/345} \] \[ F = \frac{0.0402}{0.00188} \approx 21.34 \] Thus, the value of the test statistic is \( 21.340 \).
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