# Likelihood ratio for partitioned regression

Normally we have $$y= X \beta + u$$ and we find $$\hat{\beta}_{MLE}$$

and $$\hat{\sigma^2}_{MLE}$$ from the function of:

$$l(\theta,y) = -\dfrac {n}{2}\cdot \ln (2 \pi) -\dfrac {n}{2} \cdot \ln (\sigma^2) -\dfrac {1}{2 \sigma^2} \cdot (y - X \beta)^T (y - X \beta)$$

and we take partial derivative w.r.t $$\beta$$ and $$\sigma^2$$ such that $$\dfrac {\partial l(\theta,y)}{\partial \beta}$$ and $$\dfrac {\partial l(\theta,y)}{\partial \sigma^2}$$

so we obtain $$2\times 1$$ matrix we take $$E(S(\theta,y))$$ of this matrix (as you know $$S(\theta,y) = \dfrac {\partial l(\theta,y)}{\partial \theta}$$ ) and we go on Variance too.

so for that regression $$y= X_1 \beta_1 + X_2 \beta_2 + u$$ should we write $$y -X_1 \beta_1 -X_2 \beta_2$$ instead of $$y - X \beta$$ and take partial derivative w.r.t $$\beta_1$$ and $$\beta_2$$ and get a $$3\times 1$$ dimension matrix ? or did I miss a point here? then in what form information matrix should be in this case?

Edit:

if I should process with this, then does it make sense to construct Information Matrix in this way:

$$-E\left[\dfrac{\partial^2 l\left( \theta | y\right) }{\partial \theta \partial \theta ^{T}}\right] = \begin{pmatrix} \dfrac{\partial^2 l\left( \theta | y\right) }{\partial \beta _{1}\partial \beta _{1}^{T}} & \dfrac{\partial^2 l\left( \theta | y\right) }{\partial \beta _{1}\partial \beta _{2}} & \dfrac{\partial^2 l\left( \theta | y\right) }{\partial \beta _{1}\partial{\sigma^2}} \\ \dfrac{\partial^2 l\left( \theta | y\right) }{\partial \beta _{2}\partial \beta _{1}} & \dfrac{\partial^2 l\left( \theta | y\right) }{\partial \beta _{2}\partial \beta _{2}^T} & \dfrac{\partial^2 l\left( \theta | y\right) }{\partial \beta _{2}\partial \sigma^2} \\ \dfrac{\partial^2 l\left( \theta | y\right) }{\partial \sigma^2\partial \beta_1} & \dfrac{\partial^2 l\left( \theta | y\right) }{\partial \sigma^2\partial \beta_2} & \dfrac{\partial^2 l\left( \theta | y\right) }{\partial \sigma^2\partial \sigma^2} \end{pmatrix}$$

if so, after taking partial derivatives, I will get $$3\times 3$$ matrix, so what is the variance of $$\beta_1, \beta_2$$?

• What is $In$? Is it supposed to be $\ln$? Commented Jan 6, 2023 at 6:58
• Oh yes, It is I thought It won't be a problem. I fixed. Commented Jan 6, 2023 at 7:01
• Your current derivative w.r.t. $\beta$ does the same, ie, collects the partial derivatives with respect to all elements of $\beta$, such as $\beta_1$ and $\beta_2$ Commented Jan 6, 2023 at 11:41
• I edited the question and tried an information matrix construction, is it correct? Commented Jan 6, 2023 at 12:57