If you're only concerned about linear regression, the nested F test is a great too. This is the process:
- Come up with a linear model and estimate it. This is the "restricted" model."
- Decide which other predictors you'd like to add and estimate the model with them added. This is the "full" model.
- Compute the RSS for each model. (RSS = sum of squares of residuals)
- Find RSS(restricted) - RSS(full) / # of predictors you added in step 2.
- Find RSS(full) / (N - total # of predictors + 1)
- Your F statistic is step 5 divided by step 6. Your null hypothesis is that the full model contributes nothing to prediction, against an alternative of contributing something.
- If F > F(# of added predictors, N - total # predictors + 1), reject your null. Thereby you are justified in adding those parameters.
The issue with this test is that you need it only operates pairwise. That is, it won't tell you which parameters to include, which parameters to include separately or as groups, or generally which comparisons to make. So you have to have a somewhat reasonable set of additional parameters in mind before you jump into nesting models. There's also the Ramsey test for misspecification, which can tell you whether your linear model is appropriate.