On the Appropriate Measure to Estimate Hyperbolic Discounting Rate (K) using the Method of Least Squares

The estimation of the rate at which value declines with delay requires identifying the correct discounting model, applying the appropriate parameter estimation method, and choosing the dependent measure from which parameters are estimated. The simplest adequate discounting model is the hyperbolic model; the simplest method to estimate its sole free parameter, K, is the method of least squares. Estimates of K based on relative subjective values (RSV), although typical, are not necessarily more reliable than those obtained through other measures. We consider an alternative measure termed immediacy premium (IP): the ratio of value lost over value preserved due to outcome delay. According to hyperbolic discounting, IP is a linear function of delay. As a result, estimates of K obtained from IP circumvent the divergence between individual and aggregate estimates obtained with the RSV method. Moreover, published data suggests that estimates of K based on RSV and IP differ systematically in humans and in pigeons. Regardless of the dependent measure adopted, estimates of K obtained from nonhuman animals, but not from humans, yield residuals that conform with error-distribution assumptions of the method of least squares. Although residuals obtained using the IP method in human data diverged more from normality than those obtained using the RSV method, the sequential dependence over delays of the former was weaker than that of the latter. We therefore recommend adopting both RSV and IP when estimating hyperbolic K using the method of least squares, resorting to more elaborate estimation methods when inferences drawn from these estimates are inconsistent.