Assumptions of the CLRM

A1

Linearity: Y is a linear function of the X's and u (linear in the parameters)

The true relationship can be written as a sum of β·X terms. Note that X² or ln X may appear, since the model is still linear in the β's.

Misspecified functional form. The line is the wrong shape, so estimates are biased and predictions are systematically off (for example, fitting a line to a curve).

A2

Non-stochastic regressors: X's are fixed in repeated samples

Imagine re-running the study: the X values stay put and only u changes. This is a simplifying assumption.

Usually relaxed to "X is independent of u" (A3). If X is random and related to u, you get bias.

A3a

Zero conditional mean: E[u | X] = 0

Given any X, the errors average out to zero. Nothing systematically pushes Y up or down at particular X values. This also means E[Y|X] = β₁ + β₂X₂ + ..., i.e., the line is the average of Y at each X.

Bias and inconsistency: b no longer centers on β. This is the most important assumption. Omitted variables, measurement error in X and simultaneity all break it.

A3b

u and X uncorrelated: cov(u, X) = 0

Nothing hiding in the error is related to the regressors.

Same as A3a. This is the form you use for omitted variable bias.

A4

Homoskedasticity: var(uᵢ) = σ² for all i

The scatter around the line is equally wide everywhere. Heteroskedasticity means the spread depends on X (for example, high-income households vary more in spending).

b is still unbiased, but no longer efficient (not minimum variance). Worse, the usual standard errors are wrong, so t-tests and confidence intervals are unreliable. Fix: robust standard errors or GLS.

A5

No autocorrelation: cov(uᵢ, uⱼ) = 0 for i ≠ j

Knowing one observation's error tells you nothing about another's. Violated in time series (this quarter's shock persists) and in spatial data (neighboring states share shocks).

b is still unbiased, but not efficient, and the standard errors are wrong (typically too small, so you find false significance). Fix: robust or clustered SEs, model the dynamics.

A6

No perfect collinearity: no X is an exact linear function of other X's

Each X must bring some new information. If X₃ = 2X₂, you cannot separate their effects.

With perfect collinearity, OLS cannot be computed (dividing by 0, since 1 − r₂₃² = 0 in the formula in Part E). With high but imperfect collinearity, OLS works but standard errors become huge and estimates unstable.

A7

No specification bias: the model is correctly specified. No relevant variable omitted, no irrelevant one included, right functional form

The model you estimate is the true model. Key form: u isn't correlated with Y or the X's because nothing important is left in u.

Omitted variable bias (the theme of Problems 7 to 9): the included variable picks up the effect of the omitted one. Including irrelevant variables does not bias but inflates the variance.

A8

Normal errors: u ~ N(0, σ²)

Errors are bell-shaped around zero.

Not needed for unbiasedness or Gauss-Markov. It is what makes t and F tests exact in small samples. In large samples, the central limit theorem rescues you.