# How to prove that the robust F statistic is asymptotically chi squared distributed?

The linear model is

$$y_{i}= x_{i}'\beta+u_{i}$$

When written in vector notation such that $$y_{i}$$ is a $$1$$ x $$1$$ matrix of outcomes, $$x_{i}'$$ is a $$1$$ x $$k$$ matrix of control variables, $$\beta$$ is a $$k$$ x $$1$$ matrix of population parameters for the coefficients of the control variables, and $$u_{i}$$ is a $$1$$ x $$1$$ matrix for the residual error term.

$$F_{R}$$ is the robust F statistic for the linear model.

$$F_{R}= (R\hat\beta_{ols}- c)'\Big[R[\sum{x_{i}x_{i}'}]^{-1}[\sum{\hat u^2_{i}x_{i}x_{i}'}][\sum{x_{i}x_{i}'}]^{-1}R'\Big]^{-1}(R\hat\beta_{ols}- c)/J$$

where $$R$$ is a $$j$$ x $$k$$ matrix of restrictions, $$c$$ is a $$j$$ x $$1$$ matrix of null hypothesis, $$\hat\beta_{ols}$$ is a $$k$$ x $$1$$ matrix of estimates of coefficients of $$x_{i}$$, $$\hat u_{i}$$ is a $$1$$ x $$1$$ matrix of the estimated residual error, $$J$$ is the degrees freedom, I think it is a scalar but I am not 100% certain.

How do you prove that $$F_{R}$$ is asymptotically approximately chi squared distributed with $$J$$ degrees of freedom.

$$F_{R} \sim \chi^{2}(J)$$

• Please tell us what asymptotics you have in mind and what procedure the F statistic is used for, for otherwise this result is not generally true.
– whuber
Commented Dec 28, 2018 at 15:13
• If the number of degrees of freedom in the denominator remains fixed while that in the numerator stays constant, then the distribution approaches a chi-square distribution. The question should get rephrased to make it clear that that is what is meant. Commented Dec 28, 2018 at 19:44
• Following the math writing, apart from $R$ and $c$, other symbols also need to be defined, otherwise, people need to guess what you say, which is not acceptable in scientific writing. Commented Dec 30, 2018 at 5:38

I won't give you the full proof, just some key hints:

• Recall that, if $$X \sim N(A,\Sigma_{l*l})$$, then $$(X-A)'\Sigma^{-1}_{l*l}(X-A) \sim \chi_l^2$$

• Under the appropiate assumptions we have asymptotic normality i.e. $$\sqrt n (b-\beta)\sim N(0,Var(b) )$$ where b is our OLS estimate

• $$H0:R\beta=c$$, then under $$H0$$ we have $$\sqrt n (Rb-R\beta) = R \sqrt n (b-\beta) \sim N(0,Rvar(b)R')$$

• Under the appropiate assumptions, in large samples the sample variance converges in probability to the true population variance so it's not random anymore