Supplemental Material
Modeling climate effects on hip fracture rate by multivariate GARCH model in Montreal region, Canada
Reza Modarresa, Taha B. M. J. Ouardaa,b, Alain Vanassec, Maria Gabriela Orzancoc, Pierre Gosselind
aCanada Research Chair on the Estimation of Hydrometeorological Variables, INRS-ETE, 490 de la Couronne, Quebec, Qc, Canada, G1K 9A9
b Masdar Institute of Science and Technology, P.O. Box 54224, Abu Dhabi, UAE
cGroupe de recherche PRIMUS, Université de Sherbrooke 3001, 12e avenue Nord, Sherbrooke, QC, Canada, J1H 5N4
dChangements climatiques, Unité Santé et environnement, Institut National de santé Publique du Québec (INSPQ), 945 rue Wolf, Québec, Qc, Canada, G1V 5B3
Appendix A. Statistical methods
The univariate linear ARMAX time series model has the following general form (1)
where Yt is the observed (hip fracture rate) time series, is a polynomial of order p, is a polynomial of order q, B is the backward operator, and is an independent identically distributed (i.i.d) normal error with a zero mean and standard deviation . The is the parameter of the model and X is the independent, input, regressor or eXogeneous (climate) variable. The order of the ARMA model is identified by the Autocorrelation function (ACF) of the sample.
The nonlinear GARCH model is then used to model the conditional variance remaining in the residuals () of the ARMAX model. The conditional variance () is estimated by the following equation (Hamilton, 1994):
(2)
Where and are the parameters of the model and V and M are the order of the GARCH(V,M) model.
The extension from univariate GARCH model to an n-variate model requires allowing the conditional variance-covariance matrix of n-dimensional zero mean random variables, , to depend on the elements of the sigma field. Letting be measurable with respect to the sigma field, the multivariate GARCH is where N indicates a normal distribution.
The conditional variance in the CCC model is defined as the following for two time series, i and j,
(3)
Where
(4)
and can be defined as any univariate GARCH model and
(5)
is the constant conditional correlation.
Appendix B. Additional Analyses
Appendix Table 1. Correlation coefficients between daily HFr and climate variables
Climate variables / F1r / F2r / M1r / M2rMaximum temperature / -0.027 / -0.064*** / -0.028 / -0.029
Minimum temperature / -0.038 / -0.070*** / -0.034 / -0.032
Mean Temperature / -0.032 / -0.067*** / -0.031 / -0.031
Rainfall depth (mm) / -0.016 / -0.028 / -0.010 / -0.013
Snow depth (mm) / 0.031 / 0.054*** / -0.004 / 0.008
Precipitation depth (mm) / -0.008 / -0.004 / -0.012 / -0.012
Maximum snow depth (mm) / 0.029 / 0.048*** / 0.015 / 0.000
Mean Wind speed (km/s) / 0.003 / 0.018 / 0.021 / -0.012
Day Length (hr) / -0.023 / -0.038 / -0.31 / -0.031
Maximum pressure (hp) / 0.020 / 0.018 / -0.012 / -0.05
Minimum pressure (hp) / 0.019 / -0.002 / -0.012 / 0.004
Mean pressure (hp) / 0.022 / 0.014 / -0.013 / 0.000
*** p<0.01,** p<0.05,* p<0.10
Appendix Figure 1. Autocorrelation functions of daily HFr time series of different age and gender groups
Appendix Figure 2. Autocorrelation functions of 3-day HFr time series of different age and gender groups
Appendix Figure 3. Autocorrelation functions of 5-day HFr time series of different age and gender groups
Appendix Figure 4. Conditional variance of 3-day HFr time series
Appendix Figure 5. Conditional variance of 5-day HFr time series
Appendix Figure 6. Conditional Covariance against 3-day hip fracture rate. Example for F1 and F2 groups
Appendix Figure 7. Conditional Covariance against 3-day hip fracture rate. Example for M1 and M2 groups
Appendix Figure 8. Conditional Covariance between 5-day F1 time series and climate variables
Appendix Figure 9. Conditional Covariance between 5-day F2 time series and climate variables
Appendix Figure 10. Conditional Covariance between 5-day M1 time series and climate variables
Appendix Figure 11. Conditional Covariance between 5-day M2 time series and climate variables
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