2.1 The model and its assumptions

Definition 2.1.1. The two-variable linear regression model is \[ Y_i = \alpha + \beta X_i + u_i , \qquad i = 1, \dots , n , \] with the following assumptions.

(A1): Exogeneity: \(E(u_i \mid X_1, \dots , X_n) = 0\) for every \(i\).
(A2): Homoscedasticity: \(\Var (u_i \mid X_1, \dots , X_n) = \sigma ^2\), the same for every \(i\).
(A3): No autocorrelation: \(\Cov (u_i, u_j \mid X_1, \dots , X_n) = 0\) for \(i \neq j\).
(A4): Variation in the regressor: the \(X_i\) are not all equal, so \(\sum x_i^2 > 0\).
(A5): Normality (needed only for exact small-sample inference): conditionally on the \(X\)’s, the \(u_i\) are normally distributed.

Johnston, like many texts, states the assumptions for fixed (non-stochastic) \(X\): the regressor values are treated as chosen in advance, as in an experiment, and only \(u\) is random. Conditioning on the \(X\)’s, as above, gives the same results for random regressors: every expectation below is conditional on \(X_1, \dots , X_n\), and the \(X\)’s are treated as constants within it. The conditional formulation is the honest one for economic data, where the regressors are as random as the dependent variable.

Each assumption has an economic meaning, and each is the subject of a later section that asks what happens when it fails.

(A1) says that the disturbance contains nothing systematically related to \(X\). It fails when a relevant variable correlated with \(X\) is omitted, when \(X\) is measured with error, or when \(X\) and \(Y\) are determined jointly, as price and quantity are in a market. Its failure biases least squares, and that is the most serious problem in applied econometrics.
(A2) fails when the spread of \(Y\) around the line changes with \(X\): the food spending of rich households varies more than that of poor ones. Least squares stays unbiased, but its usual standard errors are wrong.
(A3) fails in time series, where a shock to this quarter’s consumption persists into the next. Again least squares stays unbiased but its standard errors are wrong.
(A4) is needed for the estimator to exist at all: a line cannot be fitted through points that all share the same \(X\).

[next] [front] [up]

Questions on this section

Stuck on something here? Ask below and it stays attached to this topic.