Forming a consistent estimator for the area under the regression line

I am trying to solve the following problem:

Take the following simple linear regression model, where $$x_i \in \mathbb R$$:

$$y_i=\beta_0 + x_i \beta_1 + \epsilon_i$$

Given that:

• $$\mathbb E[\epsilon_i]=0$$
• $$\mathbb E[\epsilon_i|x_i]=0$$
• $$\beta_0 >0$$
• $$\beta_1 <0$$

Let $$\theta_0$$ represent the area under the regression line. Propose a consistent estimator of $$\theta_0$$.

I have began by finding the integral of $$y_i$$ with respect to $$x_i$$. That is,

$$\theta_0= \int \beta_0 + x_i \beta_1 + \epsilon_i$$ $$dx_i=\beta_0x_i + {x_i^2\over{2}}{\beta_1} + \epsilon_ix_i$$

I am considering proposing an MLE and proceeding to find the derivative of the log-likelihood of this expression. However, since this expression is rather complicated and I foresee the MLE computation turning incredibly thorny, I suspect I may be approaching this incorrectly. Any thoughts?

• The regression line is defined over $\mathbb{R}$; presumably you intend it to apply over some specific interval. It would be good to make it explicit what interval is intended (do you mean just the part where the regression function is positive in the first quadrant? That is over $0<x,<-\beta_0/\beta_1$?) Nov 19 '21 at 1:17
• Yes, it would be over the first quadrant I believe Nov 19 '21 at 2:27

The indefinite integral you have provided, $$\theta_0$$, involves two parameters $$\beta_0$$ and $$\beta_1$$ and the random term $$\epsilon_i$$. Hence $$\theta_0$$ is random. Then you say you want to find the MLE or a consistent estimator of $$\theta_0$$. You cannot find the MLE of an random quantity, you can only find the MLE of parameters (unknown constants) in a model. So you would have to add $$\epsilon_i$$ as an unknown parameter in the model, and you would find that its MLE is zero. It makes much more sense to find a consistent estimator for the mean of the area under the regression curve. In fact, the two solutions are the same since the only reasonable estimate of $$\epsilon_i$$ is 0 and then $$\theta_0$$ equals the mean area under the regression curve.
Finally, you can estimate the MLEs of $$\beta_0$$ and $$\beta_1$$ using simple linear regression (least squares) and replace them with their population values to obtain the MLE of the mean area under the regression curve.