Assumptions of Ordinary Least Squares

II.I.3 Assumptions of Ordinary Least Squares (OLS)

The assumptions of Ordinary Least Squares (OLS) can be divided into two different groups

	the weak set of assumptions
	the strong set of assumptions.

Both sets of assumptions can be written in function of Y_t and e_t.

The weak set of assumptions

(II.I.3-1)

Here it is assumed that the hypothesized model represents the true (linear) relationship such that after extraction of the RHS influences from the endogenous variable, the residual component has an expectation of zero.

(II.I.3-2)

This assumption implies that there is no "tracking" in the endogenous variable, nor in the error component. The concept of tracking can, intuitively, be defined as the fact that variable values at any specified index t can be predicted by previous variable values t-k.

Example of uncorrelated residuals

(II.I.3-3)

It is assumed that the endogenous and residual variable are homoskedastic (c.q. constant variance over the complete range).

Example of homoskedastic residuals

Example of heteroskedastic residuals

The strong set of assumptions

The strong set of assumptions include assumptions (II.I.3-1), (II.I.3-2), and (II.I.3-3). Above this, it is assumed that

(II.I.3-4)

Only if the weak assumptions, which the researcher is always advised to investigate after a linear regression model has been fitted, are satisfied, the use of the OLS method is justified.

If, above this, the normality assumption is valid as well, confidence intervals and tests for the estimated value of a and ß are easily computed.

An exception to this last rule holds only true for quite large samples where the generalized central limit theorem is valid. In this case the confidence intervals of the parameters can also be computed appropriately.