10: Other Least Squares
- Page ID
- 57749
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In Chapter 6: Dood! Check the Requirements, we examined the three assumptions and how to check that they are not violated by your model. In the previous chapter, we saw how to fix our model to handle some of those violations. We were left, however, with some violations we could not fix. In this chapter, we jettison ordinary least squares regression and examine other types of regressions that rely on minimizing the sum of squared residuals. Each of these techniques allows you to specify a different covariance matrix. The requirement is that you actually know its structure without having to estimate it.
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In the past several chapters, we have examined the classical linear model (CLM) and how to estimate the parameters using ordinary least squares (OLS). That introduction came in these chapters:
- Chapter 3: Intro to Linear Regression
- Chapter 4: Matrices and Linear Regression
- Chapter 5: Improved! Now with Probabilities
In Chapter 6: Dood! Check the Requirements, we discovered how to check the requirements (assumptions) of the ordinary least squares method. Chapter 8: Fixing the Violations gave us some options for dealing with violations of the requirements.
However, it may be that those fixes do not fully succeed — or cannot fully succeed. This chapter provides two estimation methods that offer advantages over ordinary least squares, as long as you have sufficient knowledge (science) of the structure of the problem — also known as the data-generation process.
This chapter reintroduces ordinary least squares. It then focuses on the covariance matrix of the residuals. As we reduce requirements on that matrix, we move from ordinary least squares to weighted least squares to generalized least squares.