Model Misspecification

A regression that violates its assumptions can produce confident-looking but wrong conclusions. This module covers the main specification errors and the three classic violations, conditional heteroskedasticity, serial correlation and multicollinearity: what each does to the estimates, how to test for it with the Breusch-Pagan, Durbin-Watson, Breusch-Godfrey and VIF diagnostics, and how to correct it. Level II vignettes typically present diagnostic output and ask candidates to draw the right inference.

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What you will learn

1. Identify common specification errors, such as omitted variables, wrong functional form and inappropriate pooling, and their consequences.

A good specification rests on economic reasoning, parsimony, out-of-sample performance, suitable functional form and the regression assumptions.
Omitting a relevant variable correlated with included ones biases their coefficients; the bias does not shrink with more data.
Omitted-variable slope on X1 is roughly b1 + b2 x d, where d is the slope of X2 on X1.
Nonlinear relationships need transformations such as logs; raw amounts may need scaling by size.
Pooling data from different regimes imposes one set of coefficients on populations that behave differently.

2. Explain conditional heteroskedasticity, how to detect it with the Breusch-Pagan test, and how robust standard errors correct inference.

Conditional heteroskedasticity, error variance linked to the X variables, is the harmful type.
Coefficients stay unbiased and consistent; standard errors are usually understated, so t-statistics are inflated and Type I errors rise.
Breusch-Pagan: regress squared residuals on the X variables; BP = n x R-squared, chi-square with k df, one-tailed.
Correct with robust (White-corrected) standard errors and recompute the t-statistics; GLS is an alternative.
The F-test of overall significance is also unreliable under conditional heteroskedasticity.

3. Explain serial correlation, how the Durbin-Watson and Breusch-Godfrey tests detect it, and how to correct for it.

Positive serial correlation is typical; it understates standard errors and inflates t-statistics (Type I errors).
Coefficients stay consistent unless a lagged dependent variable is an independent variable, in which case they become inconsistent.
DW is about 2(1 - r): near 2 no first-order correlation; below d_l reject no positive correlation; between d_l and d_u inconclusive.
Breusch-Godfrey tests up to p lags jointly and stays valid with lagged dependent variables.
Correct with Newey-West (serial-correlation consistent) standard errors or by respecifying the model.

4. Diagnose multicollinearity with variance inflation factors and describe practical remedies.

Multicollinearity leaves coefficients unbiased but inflates their standard errors, causing Type II errors.
Symptom: high R-squared and significant F with few significant t-statistics.
VIF_j = 1/(1 - R_j squared); above 5 investigate, above 10 serious.
Pairwise correlations can miss multicollinearity among three or more variables.
Remedies: drop or combine variables, use a better proxy, add data, or tolerate it for pure forecasting.

Module at a glance

Omitting a relevant correlated variable biases the included coefficients (slope about b1 + b2 x d); more data do not fix it.
Wrong functional form, poor scaling and pooling across regimes are other common specification errors.
Conditional heteroskedasticity leaves coefficients unbiased but understates standard errors; detect with BP = n x R-squared (chi-square, k df); fix with robust standard errors.
Positive serial correlation understates standard errors; DW is about 2(1 - r); Breusch-Godfrey tests several lags; fix with Newey-West standard errors.
With a lagged dependent variable, serial correlation makes coefficients inconsistent and DW is invalid.
Multicollinearity inflates standard errors (Type II errors); VIF = 1/(1 - R_j squared), above 5 investigate, above 10 serious.
High R-squared and significant F with insignificant t-statistics is the signature of multicollinearity.

This free module has 4 lessons, 15 practice questions with full explanations.

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