Table 1 Time Parameter, Correlation Coefficient, and Shift in WTP Parameter Estimates for Bivariate Probit Models
| Random Time Parametera (RTPw/o) | Random Time Parameter with Observed Characteristics for Time Parameterb (RTPw) | Fixed Time Parameter with Observed Characteristics for Time Parameter (FTPw) | |
|---|---|---|---|
| Mean Time Parameter Estimate | |||
Exponential  | 0.068 (0.18) | 0.128 (0.03)** | 0.195 (0.00)* |
Hyperbolic  | 0.171 (0.79) | 0.269 (0.25) | 0.455 (0.06)*** |
| Standard Deviation of Time Parameter Estimate from Unobserved Heterogeneity | |||
| Exponential (σρ) | 0.378 (0.00)* | 0.131 (0.00)* | — |
| Hyperbolic (σμ) | 1.128 (0.07)** | 0.119 (0.00)* | — |
| Correlation Coefficient between the Initial and Follow-Up Questions (τ) | |||
| Exponential | −0.263 (0.10)*** | 0.126 (0.50) | 0.319 (0.05)** |
| Hyperbolic | −0.238 (0.20) | 0.078 (0.69) | 0.261 (0.09)*** |
| Structural Shift in Utility with the Follow-Up Question (δ) | |||
| Exponential | −0.315 (0.05)** | −0.337 (0.02)** | −0.281 (0.03)** |
| Hyperbolic | −0.338 (0.05)** | −0.381 (0.02)** | −0.272 (0.04)** |
| Number of observed characteristics for the time parameter | 0 | 8 | 17 |
| LRI | |||
| Exponential | 0.173 | 0.214 | 0.287 |
| Hyperbolic | 0.173 | 0.224 | 0.287 |
| AIC | |||
| Exponential | 558.68 | 552.96 | 529.26 |
| Hyperbolic | 558.78 | 547.26 | 529.49 |
Note: p-values for the estimates shown in parentheses next to the estimates. We round p-values < 0.001–0.00.
↵a The distribution of the random time parameter is normal. The FTPw/o mean time parameter estimate is zero, and the p-value is one in the exponential and hyperbolic models. The estimate of the time parameter in the FTPw/o model is only noise around zero without observed heterogeneity.
↵b A RTPw model with more than eight observed characteristics for the time parameter cannot solve with a negative semi-definite Hessian to recover the standard errors of the coefficients.
↵* p < 0.1;
↵** p < 0.05;
↵*** p < 0.01.