F test and LM test

Problem

In a linear regression model \(y_i=β_1+β_2x_{i,2}+ \ldots +β_kx_{i,k}+ε_i\) we can test the null hypothesis \(H_0:β_2= \ldots =β_k=0\) using either an F-test or an LM test. The \(F\) -test and the LM test are “asymptotically equivalent”, which means that they will result in the same outcome as \(n→∞\) . Suppose that we have a hypothesis with \(r=5\) restrictions ( \(k=6\) ), n = 10000 and \(R^2=0.001\) .

  1. Calculate the value of the LM test
  2. Calculate the value of the F test
  3. Find \(p\) -value of the LM test
  4. Find \(p\) -value of the F test

Solution