Unsure How to Interpret Your Regression Results?

Autocorrelation occurs when the error terms are correlated with their own past values, violating the assumption of independent observations. Heteroscedasticity means the variance of the error term is not constant across all levels of the independent variables. Both conditions violate the classical Gauss-Markov assumptions and render standard OLS standard errors invalid. Identifying which issue affects your dataset dictates whether you must apply Newey-West standard errors or a White correction to restore valid statistical inference for your hypothesis testing.

The interpretation depends entirely on whether the dependent variable is also logged. In a log-log model, the coefficient represents an elasticity, meaning a one percent increase in the independent variable causes a percentage change in the dependent variable equal to the coefficient. In a level-log model, you must divide the coefficient by one hundred to find the absolute unit change in the dependent variable resulting from a one percent increase in the regressor. Failing to apply this mathematical conversion produces an impossible economic interpretation.

The standard approach involves estimating the initial OLS regression, extracting the residuals, and squaring them. You then regress these squared residuals against the original independent variables to perform a Breusch-Pagan test. If the resulting p-value is less than your chosen significance level, you reject the null hypothesis of homoscedasticity. You must then re-estimate the original model using White standard errors. This mathematical adjustment widens the confidence intervals to account for the non-constant variance, ensuring your final t-statistics accurately reflect the true precision of the estimates.

The Hausman test evaluates whether the unique unobserved errors correlate with the regressors in your panel dataset. You first estimate both the fixed effects and random effects models and save their respective covariance matrices. The test computes a chi-square statistic based on the difference between the two sets of coefficients. A significant p-value indicates the random effects assumption fails, mandating the use of the fixed effects estimator to prevent omitted variable bias. Ignoring this diagnostic step invalidates the theoretical justification for your chosen specification.

Testing for non-stationarity requires running an Augmented Dickey-Fuller procedure on the raw data. This test regresses the first difference of the variable on its lagged level and lagged differences. If the t-statistic on the lagged level is less negative than the critical value, the series contains a unit root. You must apply first differencing to transform the data into a stationary process before estimating any OLS coefficients. Regressing variables that contain unit roots generates spurious outputs with artificially inflated R-squared values and meaningless significance tests.

A mathematically sound submission follows a strict chronological sequence of statistical operations. Begin by clearly stating the theoretical model and the expected signs of the coefficients. Next, present the baseline estimation alongside a complete table of diagnostic tests evaluating the Gauss-Markov assumptions. Document every mathematical correction applied, such as instrumental variables or heteroscedasticity-consistent standard errors. Conclude by interpreting the final, valid coefficients in the context of the original economic problem. This linear structure proves you understand the relationship between diagnostic testing and valid inference.

Assessment typically separates the computational execution from the economic meaning. Producing the correct coefficient matrix and test statistics accounts for part of the credit; the rest depends on translating those outputs into precise economic terms. Reporting a statistically significant coefficient without explaining its marginal effect dimensionally leaves the interpretation incomplete. A correct estimation still needs a written section that explains what the variance metrics imply about model reliability.