Criteria for Selecting the Optimal Number of Classes by Discounting Model
| Discounting Model | No. of Classes | Log-Likelihood | AIC | BIC | Parameter |
|---|---|---|---|---|---|
| Exponential | 1 | –4,185.30 | 8,388.61 | 8,445.00 | 9 |
| 2 | –3,571.86 | 7,195.73 | 7,358.61 | 18 | |
| 3 | –3,557.42 | 7,200.84 | 7,470.23 | 27 | |
| 4 | –3,482.19 | 7,084.39 | 7,460.29 | 36 | |
| 5 | –3,654.94 | 7,463.88 | 7,946.28 | 45 | |
| Hyperbolic (HM) | 1 | –4,199.30 | 8,416.61 | 8,472.99 | 9 |
| 2 | –3,579.80 | 7,211.60 | 7,374.48 | 18 | |
| 3 | –3,939.42 | 7,964.85 | 8,234.24 | 27 | |
| 4 | –4,011.48 | 8,142.96 | 8,518.86 | 36 | |
| 5 | –4,022.11 | 8,198.23 | 8,680.63 | 45 | |
| Hyperbolic (Harvey) | 1 | –4,197.39 | 8,412.78 | 8,469.17 | 9 |
| 2 | –3,623.60 | 7,299.21 | 7,462.09 | 18 | |
| 3 | –4,115.93 | 8,317.87 | 8,587.26 | 27 | |
| 4 | –4,038.28 | 8,196.57 | 8,572.46 | 36 | |
| 5 | –3,639.52 | 7,433.04 | 7,915.44 | 45 | |
| Quasi-hyperbolic | 1 | –4,185.31 | 8,390.62 | 8,453.27 | 10 |
| 2 | –3,571.87 | 7,199.73 | 7,375.15 | 20 | |
| 3 | –3,489.97 | 7,071.94 | 7,360.12 | 30 | |
| 4 | –3,479.66 | 7,087.33 | 7,488.28 | 40 | |
| 5 | –3,762.20 | 7,688.41 | 8,202.13 | 50 |
Note: N = 777 in total (3 alternatives per question) × (5 questions per person) = 11,655 observations. Bootstrap standard errors using 250 bootstrap samples for the exponential form: l (log-likelihoods are 10.31 for one class and 9.22 for two classes); Akaike information criteria (AIC) are 21.35 for one class and 19.45 for two classes; Bayesian information criteria (BIC) are 20.87 for one class and 19.70 for two classes.