Arbitrary Constants in Log-Transformed Regressions Distort Elasticities Across Scale Changes, Requiring Estimation in Levels via Poisson Pseudo-Maximum Likelihood

When a data analyst adds a single unit to a dataset to avoid taking the logarithm of zero, they can inadvertently alter their final elasticity estimates by more than 50 percent simply by changing the currency denomination. The basis of this distortion is the unit of measurement itself.

A constant of one means a single dollar in one dataset, but a million dollars if the column header is changed to millions. This mathematical vulnerability silently corrupts regression models across economics and public health. Researchers have historically ignored this scale distortion to force their data into linear models.

The logarithmic transformation is the workhorse of applied statistics because it allows analysts to interpret coefficients as elasticities, or percentage changes. However, the logarithm of zero is mathematically undefined. This creates a structural problem for datasets where zero is a valid and common outcome.

In international trade, for example, bilateral trade flows are frequently zero because many country pairs simply do not exchange goods. Dropping these zero observations entirely discards critical information and introduces selection bias. To retain them, researchers traditionally added a small arbitrary constant to all observations before applying the logarithm.

This log(y+1) workaround appears harmless on the surface, but it fundamentally breaks the scale invariance required for elasticity estimation. If a researcher measures trade in single dollars, adding one alters a zero value to a one, which is a negligible shift for large trade volumes.

The mechanics of scale distortion

The distortion becomes catastrophic when the exact same data is measured in different units. If the researcher measures trade in millions of dollars, adding one to a zero observation is equivalent to adding one million dollars to the baseline. The arbitrary constant's weight shifts entirely depending on the chosen scale.

Because the constant does not scale with the data, the resulting elasticity estimates become a function of the units rather than the underlying economic reality. Identical underlying data will yield statistically distinct coefficients purely based on whether the analyst recorded the dependent variable in single units or thousands.

This flaw was definitively exposed in 2006 when econometricians J.M.C. Santos Silva and Silvana Tenreyro published The Log of Gravity in The Review of Economics and Statistics. They demonstrated that under heteroskedasticity, the parameters of log-linearized models estimated by ordinary least squares lead to biased estimates of true elasticities.[1][3]

"Although economists have long been aware of Jensen's inequality, many econometric applications have neglected an important implication of it," Santos Silva and Tenreyro wrote. They proved that the expected value of a logarithm is not equal to the logarithm of the expected value, a divergence that corrupts log-linear models.[1]

Jensen's inequality states that the function of an average is not equal to the average of a function for non-linear transformations. Because the logarithm is a strictly concave function, the expected value of the log-transformed error term depends heavily on its variance.[1]

The heteroskedasticity bias

If the variance of the error term changes across observations, a condition known as heteroskedasticity, the expected value of the error term also changes. This violates the core assumption of ordinary least squares regression, causing the estimator to attribute the shifting variance to the independent variables, thereby biasing the elasticities.[1]

To solve both the zero-value problem and the heteroskedasticity bias, the authors proposed abandoning the logarithmic transformation entirely. Instead, they advocated for the Poisson Pseudo-Maximum Likelihood estimator. PPML estimates the multiplicative model directly in its native levels.[1]

Rather than transforming the data to fit a linear model, PPML fits an exponential conditional mean to the raw data. The model takes the form E[y|x] = exp(xβ), which naturally accommodates zero outcomes without requiring any arbitrary constants or data manipulation.[2]

Because PPML does not log-transform the dependent variable, a zero remains a zero. The estimator solves the Poisson score equations for the conditional mean, retaining the full sample intact. Crucially, the resulting coefficients can still be interpreted exactly as they would be in a log-linear model.[2]

The estimator does not require the data to follow a Poisson distribution, nor does it require the dependent variable to be an integer. It is a pseudo-maximum likelihood estimator, meaning it remains consistent as long as the conditional mean is correctly specified, regardless of the underlying data distribution.[2][6]

Estimating directly in levels

The implementation of PPML has been democratized by modern statistical software. The estimator is natively supported in standard econometrics packages, allowing researchers to fit models with millions of observations and multiple high-dimensional fixed effects in seconds.[2]

These software packages utilize an iterated reweighted least squares algorithm to solve the Poisson likelihood function. This computational efficiency has removed the final barrier to adoption, as estimating non-linear models was historically too computationally intensive for large panel datasets.[2]

The impact of adopting PPML is highly quantifiable. In a standard gravity model dataset with 200 observations where 46.5 percent of the outcomes are zero, dropping the zeros for a standard log-OLS regression shrinks the group slope estimate from 0.62 down to 0.37.[2]

That 40 percent reduction in the estimated effect size is purely an artifact of the chosen statistical method. By retaining the 93 zero observations and correcting for heteroskedasticity, PPML recovers the true magnitude of the relationship that the log-linear model obscures.[2]

Beyond handling zeros and heteroskedasticity, PPML possesses a unique mathematical property that makes it indispensable for structural modeling. It is the only quasi-maximum likelihood estimator that preserves total flows between the actual and estimated matrices.[4]

Solving the adding-up problem

This property, known as the adding up problem, plagues other estimators. When using ordinary least squares, the sum of the predicted trade flows across all partners does not equal the sum of the actual trade flows. PPML guarantees that these totals remain identical.[4]

The World Bank has highlighted this feature, noting that PPML produces estimates in which actual and estimated total trade flows sum perfectly across all destination markets. This structural consistency is vital for researchers simulating the global impacts of tariffs or free trade agreements.[4]

The adoption of PPML has fundamentally reshaped empirical trade literature. The Log of Gravity has amassed over 2,500 citations in the EconPapers database, making it one of the most influential methodological contributions to applied economics in the twenty-first century.[3]

Its use is now expanding far beyond international trade. Researchers in health economics and business administration are increasingly deploying PPML to handle positively skewed dependent variables. Any field that models non-negative outcomes with a high density of zeros stands to benefit from the estimator.[6]

In health economics, for instance, medical expenditure data is heavily right-skewed and contains many zeros for patients who did not seek care. A standard log-linear model either drops healthy patients or distorts their data with an arbitrary constant, whereas PPML models the true distribution of healthcare costs accurately.[6]

Expanding beyond the baseline

Recent econometric research has continued to refine the PPML approach. In 2022, researchers at the CESifo network proposed a Generalized Poisson-Pseudo Maximum Likelihood estimator. This advancement relaxes the baseline assumption that the conditional variance must be strictly proportional to the conditional mean.[5]

By employing an iterated Generalized Method of Moments, the generalized estimator calculates the conditional variance directly from the data. This makes the coefficient estimates even more efficient and robust to different underlying data generating processes, while retaining all the zero-handling benefits of standard PPML.[5]

Despite these advancements, the core insight from 2006 remains the foundation of modern multiplicative modeling. Transforming data to fit a model introduces structural biases that are often invisible to the analyst. The model must instead be adapted to fit the true shape of the data.[7]

Replicating older studies using PPML frequently overturns established consensus, suggesting a vast archive of literature may require re-estimation. It remains unclear how many historical empirical findings in economics and public health were driven entirely by the arbitrary choice of units in a log(y+1) transformation.[7]

For modern data analysts, the directive is clear. When the outcome is non-negative, the relationship is multiplicative, and the data contains zeros, the logarithmic transformation is a liability. Estimating directly in levels via PPML is the only mathematically consistent path forward.[7]