[39] The general formulation of the SIC to select between models *j* = 1,2,*…*, *m* is given by:

where *SIC*_{j} is the Schwarz Information Criterion for model *j*, *L*( ) is the maximum likelihood function for the estimated model *j*, *k*_{j} is the number of parameters to be estimated for model *j* and *n* is the number of observations.

[40] In the regression model with no change point (model iii) and assuming the residuals are independent and normally distributed, the SIC can be expressed as:

with , and being the maximum likelihood estimates of the intercept *λ*, the linear trend *β* and the error variance *σ*^{2} in the regression model. The number 3 represents the number of parameters to estimate (intercept, trend and variance). By plugging equation (A4) and equation (A3) into equation (A2) and making some simplifications, we obtain:

where is the residual sum of squares of the model. Under the hypothesis that there is a shift in the intercept only (model iv), there is one additional parameter to estimate and the SIC becomes:

where and are the maximum likelihood estimates of the intercept before (*λ*_{1}) and after (*λ*_{2}) the shift at time *k* in the regression model. Again, by plugging equation (A8) and equation (A7) into equation (A6) and doing simplifications, we get:

where . The SIC is evaluated for each possible time for a change point, from time 3 to *n* − 2, to ensure that there are at least as many observations as parameters to estimate on each side of the shift. Finally, under the hypothesis that there is a shift in both the intercept and trend (model v), the SIC is:

where and are the maximum likelihood estimates of the trend before (*β*_{1}) and after (*β*_{2}) the shift at time *k* in the regression model. By plugging equation (A12) and equation (A11) into equation (A10) and simplifying, we get this expression:

where *RSS*_{3} is the residual sum of squares of the model with a shift both in the intercept and trend and is given by . The time of the most likely shift (*p*) is determined by *SIC*_{v}(*p*) = min{*SIC*_{v}(*k*), *k* = 2,…,*n* − 2}. Then, a model with a change after time *p* is selected if *SIC*_{v}(*p*) < *SIC*_{iii} or if *SIC*_{iv}(*p*) < *SIC*_{iii}. Otherwise, it seems more likely that there is no shift in the model. Similarly, the SIC can be computed for the different models with or without a change point that are presented in Table 1. The SIC formulation has to be modified according to the number of parameters to estimate and to the residual sum of squares of each respective model. Furthermore, the time for which the SIC is computed has to be modified to make sure there are at least as many observations as parameters to estimate on each side of the shift. The SIC equations for all the models used in this study are presented in Table A1.

[41] There is no significance level involved with the decision rule presented above. To assess significance, a critical value can be included in the decision rule. For example, when comparing model i and ii, model ii will be selected if *SIC*_{ii}(*p*) + *c*_{α} < *SIC*_{i}, with *c*_{α} being the critical value at the *α* significance level. In this paper, we obtain the critical values and associated significance levels by Monte Carlo simulations. More specifically, we use the set of synthetic series mimicking a constant mean net land uptake presented in section 3.3.2. For each synthetic series, we fit models i and ii, compute the associate SIC values (*SIC*_{i} and *SIC*_{ii}(*p*)) and their difference. These differences give an estimate of the distribution of *c*_{α}under a model with no-shift in the mean net land uptake. The value that is larger than 95% of the other values is the critical value associated to within a 5% significance level. The significance for a shift in the net land uptake observations can be obtained by comparing the difference between the SIC of a model with a shift and a model with no-shift to the critical value at a desired significance level (e.g., the common 5% level). If this difference is larger than the critical value for the 5% significance level approximated by Monte Carlo, the associated p-value is smaller than 0.05 and the hypothesis that there is no shift in the mean net land uptake can be rejected. In this context, the p-value represents the probability of observing a difference between the models with a shift and models with no-shift at least as extreme as the observed difference.