8: Fixing the Violations
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In Chapter 6: Dood! Check the Requirements, we examined the assumptions of ordinary least squares and how to check that they are not violated by your model. The requirements (assumptions) have different importance to our estimation method. The most important requirement is that the model uniformly fits the data (constant expected value of the residuals). In this chapter, we see some ways to fix those violations.
Much of this chapter will deal with transforming the dependent variable, because misidentified models is the greatest problem in modeling. Frequently, fixing this problem also fixes other problems with assumption violations.
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In the previous chapters, we introduced the ordinary least squares (OLS) estimation method for the classical linear model (CLM) — and its assumptions (requirements). In the previous chapter, we looked at how to test that the requirements are sufficiently met in our data and model. We also looked at the importance of the assumptions. In this chapter, we determine some methods for dealing with some violations of those requirements. Hopefully, this extends the usefulness of this simple and straightforward estimation method.
Recall that the ordinary least squares estimation method (OLS) requires that the error terms have a constant expected value, have a constant variance, and are generated from a Normal (Gaussian) process. But, what happens when these requirements are not met?
There are essentially three ways of handling violations depending on the type and the severity: First, you can ignore it. Ignoring the violations is usually not too bad when you are dealing with predicting within the domain of the observed data, as the increase in bias and the loss of efficiency are usually minor. However, if it is important to estimate parameters, you definitely should not ignore this violation. Furthermore, if the assumption of a constant expected value is practically violated, you need to fix it.
Second, we can use other methods (and modeling paradigms) to perform regression. Two popular alternatives to the Classical Linear Model paradigm are the Generalized Linear Model (GLM) and the Generalized Additive Model (GAM). The former paradigm will be covered in Chapter 14: Generalized Linear Models and beyond. The latter is well examined in Wood (2006). The strength of these models (and estimation methods) is that they extend the CLM to include (for instance) discrete dependent variables and non-linear relationships (Nelder and Wedderburn 1972; Wood 2006). These unified paradigms allow the computer to estimate the effect coefficients using a very powerful method (called Maximum Likelihood Estimation). The drawback is that not all problems lend themselves to fitting using Maximum Likelihood Estimation. Luckily, most do. Even more luckily, new estimation methods are developed frequently.
However, if we desire to stay within the realm of the classical linear model, estimating the parameters using ordinary least squares, we can adjust for many violations simply by transforming the dependent variable — especially if the violations are minor.
These transformations are very flexible. Once you get used to working in two different systems of units, you can easily use transformation methods to "Normalize" many restricted dependent variables. Unfortunately, one cannot transform an arbitrary dependent variable; there are types that cannot be fit using this technique, such as categorical. To handle these types of dependent variables, we will need to introduce a new modeling paradigm.
Finally, you can make adjustments to the estimates and their standard errors to "fix" or "adjust for" the violation. This is a common practice in the presence of heteroskedasticity and of multicollinearity.
Unfortunately, these do not work for violations of model fit (non-constant expected residuals)... which is the most important of the requirements.