Neglected Heterogeneity in Tobit Model

In a Tobit model y=y^*1 [ y^* > 0] where y^* = xβ + u and u| x ~ N (0, σ²), if a heterogeneity component v is neglected, i.e. the true model should be y^* = xβ + γv + u , the neglected heterogeneity issue will arise. For instance, if y=y^* is the profit of a firm, y is the observed profit of the firm, and x only includes the variables about demand side, then the model neglects the variables about the supply side v , which should be included in the true model. In this example, the neglected heterogeneity issue arises.

If the neglected heterogeneity satisfies the independent condition v| x ~ N (0, s²) and v is independent with u , for instance v is the price of the raw materials which are unrelated with the demand side features, the true model can be rewritten as:

y=y^*1 [ y^* > 0] , where y ∗ = x β + u ~ , u ~ | x ~ ♦ (0, γ² s² + σ² )

Then, if run the Tobit model of y on x , the estimate for β will still be consistent but the error variance estimate will be for γ² s² + σ² rather than σ² . In this simple case, the estimation Average Partial Effect (APE) ∂E [ y | x ] / ∂x_i can be computed based on those estimates.

If v is correlated with x, for instance, v denotes the advertisement cost which has strong interactive relationship with the demand side, then estimation through the Tobit model without considering the neglected heterogeneity will cause an endogeneity issue implicitly. Now, rewrite the model as:

y^* = x₁β₁ + x₂β₂ + γv + u ; x₂ = x₁δ₁ + z δ₂ + η .

where η is a normal error and only correlated only with v . Then v can be represented as v = θη + ε where ε is independent with u . Then, the model can be rewritten as:

y^* = x₁β₁ + x₂β₂ + γθη + γε + u  ;

x₂ = x₁δ₁ + z δ₂ + η .

or more succinctly

y^* = x₁β₁ + x₂β₂ + γθη + ũ ;

x₂ = x₁δ₁ + z δ₂ + η .

where u ~ ∣ x , z ∼ ( 0 , σ ~ 2 ) . Then the coefficients β, δ , γθ and σ ~ 2 can be consistently estimated by a 2-stage procedure:

(1) Regress x₂ on x₁ and z , obtain the estimate for the coefficient δ ^ and the residual η ^  ;

(2) Run the Tobit model of y on x and η ^ , obtain the estimate for the coefficients β , γθ as well as the σ ~ 2 .

The APE can be computed correspondingly. The key idea here is to use the first-stage regression to purify the error term and then estimate the model without the endogeneity issue based on the assumption about the correlation structure. Another important assumption here is the normality distributional assumption, which cannot be relaxed under the Tobit model framework.