Modeling linear regression with covariate dependent error

Question

I have a model of the form $y_i = a*x_i + b + e_i$. The error terms $e_i$ are independently drawn from a distribution that depends on $x$ as well as on global parameters; however, the noise has conditional mean $0$ given $x$. The goal is to formulate models and derive good estimators for the parameters $a$ and $b$ based on sample $(x; y)$.

Can anyone suggest 1/2 models for this problem?

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Edit: Since $E[e|x]=0$ implies $E[e]=0$. So the only two models I can think of are:

1. $e \sim N(0, x^2 \sigma^2)$, and
2. $e \sim t(0,x^2,v)$, where in the 2nd model, I am considering non-standardized $t$ distribution with scale parameter $x^2$ and $df=v$.

My question is: Are the 2 models I suggested correct and can there be any more models?

Answer 1

You could use iterative feasible generalized least squares.

Start by setting weights for each datapoint to 1, i.e. no weighting.

1. Fit a weighted regression model for each dataset using weights.
2. Create a single dataset combining residuals/errors and their respective x values.
3. Fit $e_i^2 = a\cdot x_i + b$. If the noise is zero mean, $e_i^2$ is equal to the variance of the error at $x_i$.
4. Update your weights with the squared errors prediction model
5. Go back to 1 until convergence.

Answer 2

If the distribution of your error term can be described by two independent parameters, for example a Gaussian distribution with mean independent of the variance then your problem is familiar. Since $E(error|x)=0$ we can only have the variance/scale of your error depending on $x$. In that case your regression model has heteroskedascity. GLS (Generalized Least Squares) are suitable for this problem as opposed to LS (Least Squares) who assume homoskedascity.