Questions: Model Specification Testing and Diagnostics

Question 1

A researcher runs a Ramsey RESET test on their OLS regression and rejects the null hypothesis at the 5% level. What can they conclude?

• A. At least one of the included regressors is statistically insignificant and should be dropped
• B. The model has a specification problem — possibly a missing nonlinear term, interaction, or omitted variable — but the test doesn't identify what is wrong
• C. The error term is heteroskedastic and robust standard errors should be used
• D. The model is correctly specified but the sample size is too small to estimate precisely

Answer: (B)

RESET is a diagnostic, not a prescription. A rejection means the fitted values' powers (Ŷ², Ŷ³) have additional explanatory power beyond the original regressors, which signals that the linear model is missing something systematic. But the test cannot identify whether the problem is a nonlinear functional form, a missing interaction term, or an omitted variable that enters nonlinearly — the researcher must investigate separately. RESET does not test for heteroskedasticity (that's Breusch-Pagan), individual regressor significance (that's t-tests), or sample size adequacy.

Question 2

A researcher finds that their regression residuals show a clear fan shape — small residuals at low fitted values, large residuals at high fitted values. What is the primary concern?

• A. The residuals are autocorrelated, indicating a missing lag variable
• B. The model has omitted variable bias, inflating coefficient estimates
• C. Heteroskedasticity is present, meaning standard errors are wrong even though coefficient estimates may be unbiased
• D. The sample has outliers that should be removed before re-estimating

Answer: (C)

A fan shape in residuals vs. fitted values is the classic signature of heteroskedasticity — the error variance grows with fitted values. The coefficients themselves may still be unbiased (OLS is unbiased under heteroskedasticity), but the standard errors are incorrect, making all t-statistics, p-values, and confidence intervals invalid. This is why residual diagnostics are essential before trusting any inferential output. The fix is typically robust (heteroskedasticity-consistent) standard errors or weighted least squares.

Question 3

If you run a Ramsey RESET test and fail to reject the null hypothesis, you have confirmed that your model is correctly specified.

• T. True
• F. False

Answer: False

False. Failing to reject RESET means the test did not detect evidence of functional form misspecification using powers of fitted values as auxiliary regressors — it is not a confirmation of correct specification. RESET has limited power against certain alternatives (e.g., particular omitted variables), and other specification problems (heteroskedasticity, autocorrelation, wrong regressors) are not tested by RESET at all. Specification testing is a collection of diagnostics; passing one test does not certify the model.

Question 4

If a regression model is misspecified — for example, if it omits a variable that belongs in the equation — then the t-statistics on included coefficients can be misleading even if they appear highly significant.

• T. True
• F. False

Answer: True

True. Omitting a relevant variable typically causes omitted variable bias: the included coefficients absorb the effect of the missing variable, and their estimates are biased. More subtly, the standard errors and t-statistics are computed assuming the model is correctly specified. If the true error includes the omitted variable's variation, the error structure assumed by OLS is wrong. A t-statistic may appear large not because a true effect exists but because a correlated omitted variable is driving both. This is why specification testing must precede inferential interpretation.

Question 5

Why must specification testing logically precede hypothesis testing in a regression analysis, rather than being performed afterward as a check?

Model answer: Hypothesis testing (t-tests, F-tests, confidence intervals) assumes the model is correctly specified — that the functional form is appropriate, the right variables are included, and the error term satisfies OLS assumptions. If any of these assumptions fail, the distribution of test statistics is no longer what the formulas assume, and p-values and confidence intervals are invalid. Specification testing examines whether these prerequisites hold. Running hypothesis tests first and then checking specification afterward treats the conclusions as valid before establishing that the framework producing them is sound. The logical order is: check model validity first, then trust what the model tells you.

The practical consequence is that researchers who skip specification diagnostics may confidently report 'significant' results that are artifacts of misspecification — nonlinear relationships modeled as linear, autocorrelated errors treated as white noise, or heteroskedastic variance producing inflated or deflated standard errors. Each of these failures makes the reported statistics unreliable as evidence. Specification testing is the discipline of checking your instrument before trusting its readings.