Why Functional Form Is a Substantive Choice
A structural model translates assumptions about behavior and unobserved heterogeneity into estimates of parameters. Those assumptions include more than which variables enter an equation. They also include the functional form: how variables combine, how a response changes over its range, and what distribution is assigned to unobserved shocks. Two specifications can describe the same observed data while implying different parameter values or predictions.
Sensitivity analysis makes this dependence visible. The goal is not to find the specification that produces the preferred answer, nor to declare that any difference proves one model wrong. It is to compare a defensible set of alternatives while holding the data, estimation target, and other maintained assumptions as constant as possible. Then we can say which conclusions persist and which are conditional on a particular functional form.
A comparison is interpretable only when its alternatives are clearly described. For example, replacing a normal error distribution with a logistic one is a distributional comparison. Replacing a linear response equation with a quadratic one is a behavioral comparison. Changing several features at once may be useful as a broader robustness exercise, but it does not reveal which change accounts for a difference.
Distributional Form in a Threshold Model
Consider a binary response generated by a latent index:
Here \(Y_i\) equals one when the latent index is positive and zero otherwise. Suppose the conditional distribution of \(u_i\) given \(x_i\) has cumulative distribution function \(F\). The model implies a choice probability directly from the threshold event. No symmetry assumption is needed for this first expression.
If \(F\) is symmetric about zero, then \(1-F(-t)=F(t-)\), where \(F(t-)=P(u_i<t\mid x_i)\) for \(t=\alpha+\beta x_i\); this equals \(F(t)\) when there is no atom at \(t\), and the probability then simplifies to \(F(\alpha+\beta x_i)\). Without that symmetry, the simplification does not follow. In particular, setting the intercept \(\alpha\) to zero does not itself make an asymmetric error distribution symmetric.
Worked Example: Normal Versus Logistic Errors
Suppose a hypothetical threshold model has \(\alpha=0\), and a constructed population target is \(P(Y_i=1\mid x_i=1)=0.70\). Compare the implied slope under two symmetric, fixed-scale error distributions: a standard normal error and a standard logistic error. This calculation isolates the distributional choice; it is not a claim about a real study.
Standard normal specification. With \(u_i\sim N(0,1)\), the standard normal CDF is \(\Phi\), so symmetry gives \(P(Y_i=1\mid x_i)=\Phi(\beta x_i)\). At \(x_i=1\), solve \(0.70=\Phi(\beta)\):
At \(x_i=2\), this specification predicts \(\Phi(2(0.5244))=\Phi(1.0488)\approx 0.8529\).
Standard logistic specification. The standard logistic CDF is \(\Lambda(t)=1/(1+\exp(-t))\), which is also symmetric about zero. Solving \(0.70=\Lambda(\beta)\) gives:
At \(x_i=2\), the prediction is \(\Lambda(1.6946)\approx 0.8448\). Both specifications reproduce the target probability at \(x_i=1\), but their predicted probabilities at \(x_i=2\) differ: about 0.8529 versus 0.8448. Their coefficient magnitudes are not directly comparable as if they shared the same latent-error scale; the distributions impose different scales. The fitted probabilities and their differences at specified values of \(x_i\) are often more immediately interpretable.
The example illustrates a general point: matching a particular data feature does not ensure that two models agree elsewhere. The chosen distribution determines how the probability changes as the index moves away from the point used to match the target. A useful sensitivity report therefore states both the distributional alternatives and the values of the observed variables at which resulting probabilities or other implications are compared.
Behavioral Form and Response Curvature
Behavioral functional forms can differ even when they use the same variables. A linear equation imposes a constant marginal response to \(x_i\). A quadratic equation allows that response to change with \(x_i\). To interpret either equation as a conditional-mean model, suppose the structural disturbance has conditional mean zero, \(E(\varepsilon_i\mid x_i)=0\). For estimation by ordinary least squares, the design must also have sufficient variation to identify the coefficients in the chosen equation.
Worked Example: Linear Versus Quadratic Response
Consider hypothetical observations of an outcome \(q_i\), measured in units, at three values of an input \(x_i\). Assume, for this illustration, that \(E(\varepsilon_i\mid x_i)=0\). Compare a linear conditional mean with a quadratic one using the same observations.
| \(x_i\) | \(q_i\) |
|---|---|
| 0 | 10 |
| 1 | 7 |
| 2 | 6 |
Linear form. Let \(q_i=a+bx_i+\varepsilon_i\). The OLS slope is the sum of centered cross-products divided by the sum of squared centered \(x_i\) values. Here \(\bar{x}=1\) and \(\bar{q}=23/3\):
The intercept is \(\hat a=\bar q-\hat b\bar x=23/3-(-2)(1)=29/3\approx9.667\). The estimated linear response at \(x_i=3\) is \(29/3-2(3)=11/3\approx3.667\) outcome units. Its marginal response is \(-2\) outcome units per input unit at every \(x_i\).
Quadratic form. Now let \(q_i=a+bx_i+cx_i^2+\varepsilon_i\). The three observations imply \(a=10\), \(b+c=-3\), and \(2b+4c=-4\). Subtracting twice the second equation from the third gives \(2c=2\), so \(\hat c=1\), \(\hat b=-4\), and \(\hat a=10\). The estimated response at \(x_i=3\) is \(10-4(3)+1(3^2)=7\) outcome units. The marginal response is \(b+2cx_i\), which equals \(-4+2(1)(2)=0\) outcome units per input unit at \(x_i=2\).
Thus, the linear form estimates a constant negative marginal response, while the quadratic form allows the response to flatten and predicts a different outcome at \(x_i=3\). The quadratic equation uses three parameters for three observations and fits them exactly; that fact does not establish that the quadratic form is correct. With no residual degrees of freedom here, the example demonstrates dependence on specification, not evidence for a preferred model. Nor does the assumed conditional-mean equation alone establish a causal effect.
A Short Derivation: Why Error Distributions Can Change a Location Estimate
The effect of a distributional choice can be seen in a simple structural location model \(Y_i=\theta+u_i\), with independent observations from a common specified error distribution. If the errors are normal with mean zero and variance \(\sigma^2\), the log likelihood as a function of \(\theta\), omitting terms that do not depend on \(\theta\), is:
Differentiating and setting the score equal to zero yields \(\sum_i(Y_i-\theta)/\sigma^2=0\), hence the normal maximum-likelihood estimate is \(\hat\theta=\bar Y\). If instead the errors have a centered Laplace distribution with scale \(b>0\), the log likelihood, again omitting constants, is \(-\sum_i|Y_i-\theta|/b\). Maximizing it is equivalent to minimizing the sum of absolute deviations. A minimizer is a sample median: moving \(\theta\) from below the middle observation toward it reduces the total absolute distance, while moving past it increases that distance. Thus the distributional assumption changes which feature of the observations determines the estimate.
Worked Example: Normal and Laplace Location Estimates
For a hypothetical sample \(1,2,2,3,12\), estimate the common location \(\theta\) under the two independent, identically distributed error models just described. Treat each distribution's shape as specified; the calculation compares location estimates and does not assert which distribution generated the data.
Under the normal model, the maximum-likelihood estimate is the sample mean:
Under the Laplace model, the estimate is a sample median. The ordered observations are \(1,2,2,3,12\), so \(\hat\theta_{\mathrm{Laplace}}=2\). The total absolute deviation at 2 is \(1+0+0+1+10=12\); the median minimizes this quantity. The large observation 12 pulls the normal location estimate farther than the Laplace estimate. Here the choice of error distribution changes the location estimate from 4 to 2 outcome units.
How to Conduct and Report a Sensitivity Comparison
A disciplined comparison begins by identifying the conclusion of interest: a structural coefficient, a marginal response at a stated value, a probability, or a prediction. Then specify a small set of substantively plausible functional forms and estimate each on the same data using a method appropriate to that specification. Record the maintained assumptions alongside the results. If a comparison changes the error distribution, for example, state its location and scale convention; if it changes behavioral curvature, state the range of \(x_i\) over which the response is interpreted.
Report more than whether coefficient signs agree. Coefficients may have different scales or refer to different features of a response. Compare quantities with a common interpretation and units, such as predicted choice probabilities at specified covariate values or marginal responses at specified input levels. State whether the substantive conclusion is stable across the alternatives, and identify exactly what changes when it is not. A comparison is informative even when the alternatives produce similar estimates: it shows that the reported conclusion is not especially dependent on the particular forms examined.
Functional-form sensitivity is distinct from sampling uncertainty. A set of alternative point estimates does not by itself provide a confidence interval, a probability that a model is correct, or a formal test of one specification against another. Likewise, stable estimates across a few alternatives do not prove that all relevant assumptions are valid. The result is conditional on the alternatives actually examined.
Common Pitfall
- Wrong reasoning: “The estimated coefficient is larger under the logistic model, so the behavioral effect is stronger.” This conclusion treats coefficients from different latent-error scales as directly comparable.
- Correction: State the scale conventions and compare interpretable quantities, such as predicted probabilities at the same covariate values. More generally, isolate which functional-form change was made and avoid treating a specification-dependent estimate as assumption-free.
Check Your Understanding
Answer each question using the assumptions and distinctions developed above.
- For \(Y_i=1\{\alpha+\beta x_i+u_i>0\}\), write \(P(Y_i=1\mid x_i)\) using the conditional CDF \(F\). What additional condition permits simplifying it to \(F(\alpha+\beta x_i)\)?
- Why are the normal and logistic slope estimates in the threshold example not automatically comparable as measures of effect magnitude?
- In the quadratic response model, what is the marginal response at a general value \(x_i\), and why does it differ conceptually from the linear model's slope?
- What feature of the sample determines the normal location estimate, and what feature determines the Laplace location estimate?
- Why does agreement across several tested functional forms not prove that a structural conclusion is assumption-free?