  This paper intends to meet recent claims for the attainment of more rigorous
statistical methodology within the econophysics literature. To this end, we
consider an econometric approach to investigate the outcomes of the
log-periodic model of price movements, which has been largely used to forecast
financial crashes. In order to accomplish reliable statistical inference for
unknown parameters, we incorporate an autoregressive dynamic and a conditional
heteroskedasticity structure in the error term of the original model, yielding
the log-periodic-AR(1)-GARCH(1,1) model. Both the original and the extended
models are fitted to financial indices of U. S. market, namely S&P500 and
NASDAQ. Our analysis reveal two main points: (i) the
log-periodic-AR(1)-GARCH(1,1) model has residuals with better statistical
properties and (ii) the estimation of the parameter concerning the time of the
financial crash has been improved.
