The autocorrelation in population density may result from the effect of solar activity or from other factors, e.g., interaction with predators. In our null hypothesis we consider that solar activity has no effect on the population density, but we allow autocorrelations in the population density. Thus, to test for significance of the correlation between processes X and Y we need to estimate the standard error using the equation above. However, both variables are autocorrelated, and thus, we will use the correlogram product method. Correlation between the area defoliated by the gypsy moth and solar activity may be coincidental.
More data is needed to check the relationship.
Another possible way of the analysis is autoregression combined with external factors. However, coefficients b i do not necessary have biological meaning. For example, the equation above assums that current defoliation depends on defoliation in previous years, but there may be no functional relationship between areas defoliated in different years. Thus, it is necessary to use caution in the interpretation of autoregressive models. People often select arbitrarily some combination of factors that will maximize statistical significance of the factor tested.