Model Misspecification: formula sheet

Key formulas

Features of a correctly specified model

Economic reasoning + parsimony + good out-of-sample fit + appropriate functional form + no assumption violations

Use this as a checklist when a vignette asks whether a model is well specified.

Log transformation for a non-linear relationship

ln(Y) = b0 + b1X + ε or Y = b0 + b1 ln(X) + ε

Use when the relationship is curved or when the variable grows proportionally. Check the residuals for a pattern first.

Consequence of regressors correlated with the error

Cov(Xj, ε) ≠ 0 ⇒ coefficient estimates are biased and inconsistent

Hypothesis tests and confidence intervals are then unreliable. This is the most serious type of misspecification.

Adjusted R-squared

Adjusted R² = 1 − [(n − 1) ÷ (n − k − 1)] × (1 − R²)

Penalises extra variables, so it supports parsimony. R² never falls when you add a variable.

Omitted variable bias condition

Bias in b₁ ≠ 0 when the omitted variable affects Y and is correlated with X₁

Both conditions are needed. If the omitted variable is uncorrelated with the included regressors, no bias in their slopes arises from it.

Direction of bias (general econometric rule of thumb)

Sign of bias = sign(effect of omitted variable on Y) × sign(correlation of omitted variable with X₁)

This is a general econometric rule of thumb for one omitted variable, not a formula from the CFA curriculum. Same signs overstate the included coefficient; opposite signs understate it. It does not carry over cleanly when several variables are omitted.

Four types of functional form error

Omitted variable | Wrong transformation | Inappropriate pooling | Inappropriate scaling

Learn these as a checklist. Most questions ask you to name one.

Typical consequence

Misspecification → biased, inconsistent coefficients; unreliable standard errors and tests

Adding more observations does not cure it.

Lagged dependent variable model

Y(t) = b0 + b1·X(t) + b2·Y(t-1) + ε(t)

Problem arises when ε(t) is serially correlated: Y(t-1) is then correlated with ε(t), so estimates are inconsistent.

Measurement error in one regressor (simple regression)

Estimated slope is biased toward zero (attenuation)

Holds for one regressor measured with classical noise. With multiple regressors the direction of bias is not predictable.

Random walk (unit root)

x(t) = x(t-1) + ε(t), so b1 = 1 in x(t) = b0 + b1·x(t-1) + ε(t)

Nonstationary: mean-reversion level undefined, variance grows over time. First difference is stationary.

First difference

Δx(t) = x(t) - x(t-1)

Common fix for a unit root before regressing.

Breusch-Pagan test statistic

BP = n × R² (from regressing squared residuals on the independent variables); chi-square with k degrees of freedom

One-tailed, right-tail test. H0: no conditional heteroskedasticity. Reject if BP exceeds the critical value. R² is from the auxiliary regression, not the original model.

Durbin-Watson statistic

DW ≈ 2(1 − r), where r is the first-order correlation of residuals

DW ≈ 2 means no serial correlation. DW below 2 suggests positive correlation; above 2 suggests negative. Compare with lower and upper critical values dl and du: below dl reject H0; between dl and du inconclusive; above du fail to reject (for positive correlation).

Breusch-Godfrey test

BG = (n − p) × R² from regressing residuals on the independent variables and p lagged residuals; chi-square with p degrees of freedom

Tests serial correlation at higher orders. Using it, you can test more than one lag. Check the exact statistic form given in the question.

Variance inflation factor

VIF_j = 1 ÷ (1 − R_j²)

R_j² is from regressing variable j on the other independent variables. A VIF above 5 warrants investigation and above 10 is a serious concern. These are rules of thumb, not strict laws.

Corrections

Heteroskedasticity: robust (White) standard errors. Serial correlation: Newey-West (HAC) standard errors. Multicollinearity: remove or combine variables, or use more data.

Newey-West also corrects for heteroskedasticity. Robust errors change standard errors, not coefficients.

Quick revision

• A well-specified model is based on economic reasoning, has a parsimonious set of variables, and performs well out of sample.
• Omitted variables can bias coefficients and make estimates inconsistent when the omitted variable is correlated with included ones.
• Wrong functional form, such as not transforming a variable when the relationship is nonlinear, shows up as patterns in the residuals.
• Heteroskedasticity means error variance is not constant; coefficient estimates stay consistent, but standard errors are unreliable.
• Conditional heteroskedasticity is the form that matters most, because it is related to the independent variables; the Breusch-Pagan test checks for it.
• Fix heteroskedasticity with robust (White-corrected) standard errors, or with generalised least squares.
• Serial correlation means errors are correlated across observations; the Durbin-Watson test and the Breusch-Godfrey test detect it.
• Positive serial correlation makes standard errors too small, so t-statistics are too large and Type I errors are more likely. This holds when no regressor is a lagged dependent variable. If a lagged dependent variable is a regressor, serial correlation makes the coefficient estimates inconsistent, not just the standard errors.
• Fix serial correlation with Newey-West (HAC) standard errors or by modifying the model.
• Multicollinearity means independent variables are highly correlated: a high R² with insignificant individual t-statistics is the classic sign.
• A VIF above 5 warrants investigation and above 10 is a serious concern; remedies include dropping or combining variables.
• Nonstationary time series with a unit root can produce spurious regressions; test with a Dickey-Fuller test and consider differencing.

Common mistakes

• Choosing the model with the highest R-squared as the best specified. Fix: Remember that a good model also needs economic logic, parsimony and out-of-sample performance. Use adjusted R-squared and out-of-sample results to judge.
• Treating all misspecification as having the same consequence. Fix: Link each error to its effect. Any misspecification that makes a regressor correlated with the error, including omitted variables, wrong form or scale, and pooling, gives biased and inconsistent estimates, so you cannot trust tests.
• Saying any omitted variable causes bias. Fix: Check that the omitted variable affects Y and is correlated with an included regressor. If not, slope bias from it is not expected.
• Confusing pooling with omission. Fix: Pooling is about combining different regimes or groups in one equation. Omission is about a missing explanatory variable.
• Saying a lagged dependent variable always causes misspecification. Fix: Remember the trouble comes from the lag combined with serially correlated errors. If the errors are not serially correlated, including a lag does not create this problem.
• Saying measurement error in a regressor only inflates standard errors. Fix: Measurement error makes the regressor correlated with the error, so the coefficient itself is biased and inconsistent.
• Saying multicollinearity biases the coefficients. Fix: Coefficients remain unbiased and consistent. Only standard errors inflate, so significance is understated.
• Using the R² of the original regression in the Breusch-Pagan statistic. Fix: Use the R² from the regression of squared residuals on the independent variables.

Exam tips

• Read each analyst statement in the vignette as a possible specification claim and test it against the checklist.
• Do not choose an option just because it cites a higher R-squared; look for out-of-sample or economic support.
• Know which failures make estimates biased and inconsistent, since options often hinge on that wording.
• Keep specification errors and error-assumption violations distinct, but remember they can interact; options often blur them deliberately.
• Remember that perfect multicollinearity violates the assumption and prevents estimation. High multicollinearity leaves estimates unbiased but inflates standard errors, so t-statistics can be insignificant despite a high R-squared.
• If a question asks for a fix, match it: transform for curvature, drop or justify weak variables for parsimony, and remove the correlation between regressor and error.
• Most questions ask you to name the error from a short description, so memorize the four types and their clues.
• For omitted variable questions, always test the correlation condition before claiming bias.