Cross$-$validation is a technique used to assess how well a predictive model will perform on unseen data. Here's a breakdown of the options:

a$.**$With cross$-$validation, we carefully estimate a variety of models and choose the one that yields the lowest BIC.$**$

$-$ This is not the primary purpose of cross$-$validation. While cross$-$validation can be used to compare different models, the goal is typically to assess the model's performance on unseen data rather than selecting based on BIC.

b$.**$With cross$-$validation, models are selected on the basis of their in$-$sample fit.$**$

$-$ This is not correct. Cross$-$validation is used to evaluate the model's performance on out$-$of$-$sample data, not for selecting models based on their in$-$sample fit.

c$.**$With cross$-$validation, models are evaluated based on their out$-$of$-$sample forecasting accuracy.$**$

$-$ This is correct. Cross$-$validation involves splitting the data into training and testing sets multiple times. The model is trained on the training set and then tested on the testing set to evaluate its forecasting accuracy on unseen data.

d$.**$With cross$-$validation, we select models that yield both low BIC values and no evidence of residual correlation.$**$

$-$ This is not the primary purpose of cross$-$validation. While cross$-$validation can help identify models with good fit, its main goal is to assess how well the model generalizes to unseen data.