Consider a simple exponential employment duration model where $t_j$ is length of the employment and $x_j$ is a suitable economic explanatory variable. The distribution of $t_j$ is assumed to be independently exponential with parameter $\lambda = e^{x_j\beta}$.
$\text{pdf}(t_j) = \lambda_j e^{−\lambda_j t_j }$ where $\lambda_j = e^{x_jβ}$
I'm confused as to how to write a likelihood function and objective function for this model. It seemed like it would be simple with only 1 explanatory variable but I don't even know how to start.
Lets say you collect a simple random sample of $N$ employment durations, $t_i$ and the associated economic variables $x_i$.
Then, for a single observation, say $(x_1,t_1)$, the likelihood will be a function of $\beta$:
$$L_1(\beta;x_1,t_1)=\beta x_1 \exp(-x_1 \beta t_i)$$
Now, since you have $N$ independent samples, you simply multiply the likelihood functions to get the likelihood function for the entire sample ($L_S$):
$$L_S(\beta)=\prod_{i=1}^N L_i(\beta;x_i,t_i)=\beta^Ne^{-\beta\sum t_ix_i} \prod_{i=1}^N x_i$$
Can you take it the rest of the way to get the MLE for $\beta$?