Consider the unrestricted model with k explanatory variables: Unrestricted Model
ID: 3216869 • Letter: C
Question
Consider the unrestricted model with k explanatory variables: Unrestricted Model: Y = beta_0 + beta_1 X_1 + ... + beta_kX_k + U. Using a random sample of n observations in which the assumption of no perfect holds, you are interested in testing hypothesis H_0 against H_1: H_0: beta_k - q + 1 = 0, beta_k - q + 2 = 0, ..., beta_k = 0, H_1: not(beta_k - q + 1 = 0, beta_k - q + 2 = 0, ..., beta_k = 0). Under H_0, the restricted model is given by: Restricted Model: Y = beta_0 + beta_1X_1 + ... + beta_k - q^X_k - q + U. Show that the F-statistic, which is given by: F = (SSR_r - SSR_ur)/(df_r - df_ur)/SSR_ur/(df_ur) = (SSR_r - SSR_ur)/q/SSR_ur/(n - k - 1), is also in this case equal to F = (R_ur^2 - R_r^2)/(df_r - df_ur)/(1 - R_ur^2)/(df_ur) = (R_ur^2 - R_r^2)/q/(1 - R_ur^2)/(n - k - 1), where SSR_ur, R_ur^2, SSR_r, and R_r^2 are the sum of squared residuals and R-squared from the unrestricted and restricted regressions respectively.Explanation / Answer
SSRr is the sum of squared residuals from the restricted model and SSRur is the sum of squared residuals from the unrestricted model.
Notice that, since SSRr can be no smaller than SSRur,
the F statistic is always nonnegative (and
almost always strictly positive). the order of
the SSRs in the numerator of F has usually been reversed. Also, the SSR in the
denominator of F is the SSR from the unrestricted model.
The difference in SSRs in the numerator of F is divided by q, which
is the number of restrictions imposed in moving from the unrestricted to the restricted model (q independent variables are dropped). Therefore, we can write
q = numerator degrees of freedom = dfr - dfur,
which also shows that q is the difference in degrees of freedom between the restricted and
unrestricted models. (Recall that df = number of observations - number of estimated
parameters.) Since the restricted model has fewer parameters—and each model is estimated using the same n observations—dfr is always greater than dfur.
The SSR in the denominator of F is divided by the degrees of freedom in the unrestricted model:
n - k - 1 = denominator degrees of freedom = dfur