# Derivation of Wald Statistics with partitioned regression

For that regression:

$$y =X_1 \beta_1 +X_2 \beta_2 + u$$ with $$u \sim N(0,\sigma^2 I)$$

and I want to derive Wald for the hypothesis $$\beta_2 = 0$$

as you know, for the regression of $$y = y =X \beta + u$$ and hypothesis $$H_0:R\beta =r$$

Wald is equal to $$W$$ = $$\left( R\widehat{\beta }-r\right) ^{T} \left( R \cdot I^{-1} \left( \widehat{\beta }\right)\cdot R^{T}\right) \left( R\widehat{\beta }-r\right)$$

where "inverse of information matrix" $$I^{-1} ( \widehat{\beta })$$ $$= \sigma^2 (X^ T X) ^{-1}$$

so Wald becomes:

$$W$$ = $$\dfrac{1}{\widehat {\sigma} ^2} \cdot \left( R\widehat{\beta }-r\right) ^{T} \left( R\left( X^{T}X\right) ^{-1}R ^T\right) ^{-1} \left( R\widehat{\beta }-r\right)$$ has an asymptotically Chi-Square distribution $$\chi_q^2$$

but I want to derive this Wald in the form of $$n \cdot \dfrac{SSR_R - SSR_U}{SSR_U}$$ (with the help of LM test)

$$SSR_R$$ and $$SSR_U$$ stand for the restricted and the unrestricted.

because I want to build that relationship: $$W = \dfrac{n \cdot k_2}{n-k} \cdot F$$ where F is the F-test in the form of $$\dfrac{(SSR_R - SSR_U)/k_2}{SSR_U/(n-k)}$$

How can I do this from the perspective of Maximum Likelihood Estimation? I really couldn't understand how I should use the restriction.

• I think Chapter 5 of Johnston and DiNardo's Econometric Methods has this worked out. Commented Jan 2, 2023 at 20:23
• It covers the case $y = X \beta + u$ on the page of 148 as I see but but I couldn't understand how can I relate this to partitioned form Commented Jan 3, 2023 at 7:10
• I work out the result for the F-statistic here: stats.stackexchange.com/questions/258461/… Commented Jan 3, 2023 at 10:45
• Thanks I will check Commented Jan 3, 2023 at 12:54