Time series studies estimating multiple-period changes can use overlapping data in orderto achieve greater efficiency (Gilbert). A common example is using annual returns whenmonthly data are available. A one-year change could be calculated from January to December,another from February to January, and so on. In this example the January to December andFebruary to January changes would overlap for eleven months.

The overlapping of observationscreates a moving average (MA) error term and thus ordinary least squares (OLS) parameterestimates would be inefficient and hypothesis tests biased (Hansen and Hodrick). Past literaturehas recognized the presence of the moving average error term.

In general, the bootstrap is used in statistics as a resampling method to approximate standard errors, confidence intervals, and p-values for test statistics, based on the sample data. This method is significantly helpful when the theoretical distribution of the test statistic is unknown. In Stata, you can use the bootstrap command or the vce(bootstrap) option (available for many estimation commands) to bootstrap the standard errors of the parameter estimates. We recommend using the vce() option whenever possible because it already accounts for the specific characteristics of the data. This adjustment is particularly relevant for panel data where the randomly selected observations for the bootstrap cannot be chosen by individual record but must be chosen by panel.