# INTERTEMPORAL CHOICE AND THE CROSS-SECTIONAL VARIANCE OF MARGINAL UTILITY: Regression results

We regress the variance of log consumption on its own lag (see equation 9) for the three countries for which we have data. Table 2 reports the point estimates, the standard errors and the p-value of a test that the coefficient of the lagged variance equals one. The basic specification for the variance of marginal utility also included a full set of cohort dummies. However, an F-test for the significance of the cohort dummies was never statistically different from zero, regardless of the measure of marginal utility. The cohort dummies are therefore dropped in the final specifications.

As is shown in Section 2, OLS estimates of equation (9) are potentially subject to measurement error and endogeneity bias. Thus, for each specification we report three regressions, each using a different set of instruments. The first set includes only the second lag of the cross-sectional variance. The second replaces the lagged variance with the (median) age and age square of the cohort. The final set includes both age, age square and the second lag of the variance.

Table 2

Dependent variable: variance of log(consumption)

Country | Coefficient | Standard error | P-value | Instrument set | Number of observations |

United Kingdom | 0.945 | 0.037 | 0.14 | lag2 variance | 766 |

0.996 | 0.039 | 0.91 | age, age” | 776 | |

0.969 | 0.035 | 0.38 | age, age^{2}, lag2 variance |
766 | |

United States | 1.029 | 0.173 | 0.87 | lag2 variance | 500 |

0.978 | 0.099 | 0.82 | age, age^{2} |
510 | |

0.977 | 0.101 | 0.82 | age, age^{2}, lag2 variance |
500 | |

Italy | 0.798 | 0.123 | 0.11 | lag2 variance | 28 |

0.830 | 0.106 | 0.12 | age, age^{2} |
38 | |

0.773 | 0.108 | 0.05 | age, age^{2}, lag2 variance |
28 |

None of the regressions reported in Table 2 rejects the null hypothesis that the coefficient of the lagged variance equals one, with the exception of the last regression for Italy, where the test rejects the null hypothesis at the 5 percent level. The coefficient of the lagged variance are generally estimated with small standard errors, particularly in the case of the UK. Furthermore, the point estimates for the US and the UK are remarkably close to one and similar across different sets of instruments. The results are not affected by excluding the youngest and oldest cohorts or cohorts whose cells are not very large. For instance, excluding retired households (older than 57 years) does not affect the estimate of /Г, regardless of the instruments used in the estimation.

In Table 3 the dependent variable is the cross-sectional variance of log consumption per adult equivalent, as in equation (10). The estimates are not as precise as in Table 2, particularly for the US, but the general pattern is confirmed. The magnitude of the estimated coefficients for the UK and the US is hardly affected, while in Italy the point estimates are much closer to one. In no case can the hypothesis that /р= 1 be rejected at conventional significance levels. The results are again robust to the exclusion from the sample of the oldest cohorts.

Table 3

Dependent variable: variance of log(consumption per adult equivalent)

Country | Coefficient | Standard error | P-value | Instrument set | Number of observations |

United Kingdom | 0.912 | 0.075 | 0.24 | lag2 variance | 766 |

1.029 | 0.112 | 0.80 | age, age” | 776 | |

0.942 | 0.070 | 0.41 | age, age~, lag2 variance | 766 | |

United States | 0.941 | 0.480 | 0.90 | lag2 variance | 500 |

0.975 | 0.229 | 0.91 | age, age^{2} |
510 | |

1.014 | 0.232 | 0.95 | age, age^{2}, lag2 variance |
500 | |

Italy | 1.121 | 0.362 | 0.74 | lag2 variance | 28 |

0.930 | 0.190 | 0.71 | age, age^{2} |
38 | |

0.987 | 0.185 | 0.94 | age, age^{2}, lag2 variance |
28 |