Question

With n=70 observations, we run the regression:

y=α+β₁x₁+β₂x₂+u,
and get SSR=46991.1951. We also know the sample variance of y is 866.0915. What is the Wald (F) statistic, and its degrees of freedom, for testing the null hypothesis that H₀: β₁=β₂=0?

Answer 1

To calculate the Wald (F) statistic for the given regression with 70 observations and testing the null hypothesis β₁ = β₂ = 0, use the sample variance, the given SSR, and the number of observations to perform the calculation. The degrees of freedom for the Wald (F) statistic are df_Numerator=2 and df_Denominator=n-k-1 where n is the number of observations and k is the number of predictors.

Step-by-step explanation:

The student asked how to find the Wald (F) statistic and its degrees of freedom for testing the null hypothesis that H0: β₁ = β₂ = 0 in a multiple regression scenario with n=70 observations and an SSR (sum of squared residuals) of 46991.1951.

To find the Wald (F) statistic, we use the following formula:

• F = (SSR for reduced model - SSR for full model) / (dfr - dff)
• Variance of y / n

In this case, the reduced model assumes both β₁ and β₂ are zero, which would imply the SSR for the reduced model is just the variance of y times n. We then subtract the SSR for the full model (given as 46991.1951) and divide by the difference in the degrees of freedom between the reduced (dfr=n-1) and full models (dff=n-k-1, where k is the number of predictors, here 2).

The F statistic's degrees of freedom are dfNumerator=k and dfDenominator=n-k-1, where k is the number of predictors being tested, so in this case, dfNumerator=2.