Fig. 108. Subsurface temperature anomaly for Atlantic Canada. Thick line represents averaged temperature-depth profile for the region. Inset shows the 1000 years' long simulation of SAT temperature for three sets of forcings used in the ECHO-g model (see text). (Redrawn from the data by Beltrami et al. (2006).)

the work by Jones et al. (2003), combination of different kinds of climatic information, e.g., the SAT and the free atmosphere temperature records, has permitted the detection of weak signals that otherwise would not be found using some of these datasets alone. It is clear that as much as possible paleoclimatic reconstructions should be compiled for reliable detection/attribution research. The comparison of the measured versus simulated borehole temperature-depth profiles could represent a valuable independent technique for such studies. The above-described investigations by Beltrami et al. (2006) may be completed/accomplished by the quantitative analysis using traditional statistical techniques.

3.4.5 Granger causality to investigate the human influence on climate

Until quite recently the investigations of the occurrence of the global warming due to greenhouse effect were restricted to the estimates obtained using Global Circulation Models and to statistical tests of the measured data. The former method implies well-defined theoretical

Fig. 109. Comparison of measured (TP) and calculated temperature-depth profles (CTRL, FORI, FOR2; for details see text) for four regions in Canada: (A) British Columbia/Yukon, (B) Manitoba/Saskatchewan, (C) Quebec/Ontario, and (D) Atlantic Canada. (Redrawn from the data by Beltrami et al. (2006).)

constraints on the observed data, while the latter one represents a useful tool to detect changes in global temperatures and to ascertain the variables responsible for these changes. Most of the univariate/multivariate statistical tests used for the latter studies represent regressions that reflect simple correlations. Granger (1969,1980) (the year 2003 Nobel Prize laureate in Economic Sciences) has argued that exclusively an interpretation of a set of tests is able to reveal reliable information about causality. In the beginning the Granger causality test was applied only in economics. He has analyzed economic time series with common trends (co-integration). For example, it was proved that a significant increase in the petroleum price has preceded almost all of the post-war economic regressions. Recently the Granger causality analysis has acquired great popularity among the researchers from other scientific branches, especially in climatology. A usual question that arises in time series analysis is whether or not one variable can forecast another. One of the ways to address this question is the Granger Causality Test (GCT). The description of this method and computer codes are presented on numerous web sites (e.g., Statistics & Operation Research (SAS), or in the lmtest library of the R package, Thus, below we present only a principal description of the method.

According to Granger (1969), causality can be defined as follows. A variable Y is causal for another variable X if the knowledge of the past variations of Y can assist in prediction of the future state of X over and above the knowledge of the past variations of

X alone. In other words, if inclusion Y as predictor improves the prediction of X, then Y is said to be Granger causal for X.

Granger causality testing applies only to statistically stationary time series. There are many ways to apply the test of the Granger causality. Below we describe the simple approach that uses the autoregressive specification of the bivariate vector autoregression. The procedure looks like as follows.

(1) We select an autoregressive lag p to the time series X (x1, x2, ..., xT) and Y (yp y2, ..., yT), and estimate by the ordinary least squares technique following restricted p x = ct + X x'-i + e' (48)

i=i and unrestricted equations p

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