Question

To study the effect of $X_1$ and $X_2$ on Y, the following regression model is estimated using a large sample:
$ Y = a + \beta_1 X_1 + \beta_2X_2 + ∈ $
The OLS estimates and standard errors are presented below:

	$a$	$\beta_1$	$\beta_2$
Estimates	2.3	0.39	1.8
Standard errors	1.15	0.13	1

Given the above information, which of the following is CORRECT?

• A. $a$ is statistically significant at 1% level, $\beta_1$ is statistically significant at 5% level, and $\beta_2$ is statistically significant at 10% level
• B. $a$ is statistically significant at 1% level, $\beta_1$ is statistically significant at 10% level, and $\beta_2$ is statistically significant at 5% level
• C. $a$ is statistically significant at 5% level, $\beta _1$ is statistically significant at 1% level, and $\beta _2$ is statistically significant at 10% level
• D. $a$ is statistically significant at 5% level, $\beta_1$ is statistically significant at 10% level, and $\beta_2$ is statistically significant at 5% level

Answer 1

The Correct Option is C

Statistical Significance of Coefficients Using t-Values 

Statistical significance is determined by calculating the t-value for each coefficient, which is the ratio of the estimate to its standard error.

Step 1: Calculating t-Values

The t-values are computed as follows:

• t(α) = 2.30 / 1.15 ≈ 2.0
• t(β₁) = 0.39 / 0.13 ≈ 3.0
• t(β₂) = 1.80 / 1.00 = 1.8

Step 2: Comparing with Critical t-Values

The standard critical t-values for a two-tailed test are:

• 1.96 for the 5% significance level
• 2.58 for the 1% significance level

Step 3: Assessing Statistical Significance

• α (t ≈ 2.0): Significant at the 5% level but not at the 1% level.
• β₁ (t ≈ 3.0): Significant at the 1% level.
• β₂ (t = 1.8): Significant at the 10% level but not at the 5% level.

Conclusion:

Based on the t-values and significance levels:

• Coefficient α is statistically significant at 5% but not at 1%.
• Coefficient β₁ is statistically significant at 1%.
• Coefficient β₂ is statistically significant at 10% but not at 5%.