Forming a consistent estimator for the area under the regression line

Question

I am trying to solve the following problem:

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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 $\theta_0$.

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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?

Answer 1

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.