Estimating the causal effect of quitting smoking on time to death or first myocardial infarction requires that one control for the differences in risk factors between individuals who elect to quite at each time t versus those who elect to continue smoking at time t. In this paper we examine the limitations of standard time varying Cox proportional hazards models to yield tests and estimates of this effect. Implementing the method of G-estimation proposed by Robins, we perform an observational analysis of data from the Multiple Risk Factor Intervention Trial (MRFIT) and estimate the causal effect of cigarette cessation while controlling for such time varying confounders as angina. We reject the null hypothesis of no effect of quitting on time to failure, and estimate that by quitting smoking, an individual increases by 50 per cent his time to death or first myocardial infarction (MI).