Cross-Correlations between Energy and Emissions Markets: New Evidence from Fractal and Multifractal Analysis

<table>Log-log plots of the mean return interval <svg height="16.5625" id="M203" style="vertical-align:-3.50804pt" version="1.1" viewbox="0 0 44.549999 16.5625" width="44.549999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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z" id="x1D70F"></path></g>
<g transform="matrix(.012,-0,0,-.012,8.275,16.212)"><path d="M433 39l-85 -51q-27 0 -47 20q-16 16 -24 44q-5 -3 -24 -16.5t-27 -18.5t-23 -13.5t-25 -12t-19 -3.5q-52 0 -86 36.5t-34 85.5q0 71 81 99q128 43 155 65v17q0 54 -23 83.5t-62 29.5q-28 0 -45 -19t-29 -66q-7 -23 -29 -23q-15 0 -29 13t-14 30t31 40q25 19 68.5 40.5
t81.5 29.5q49 0 82 -27q45 -39 45 -123v-185q0 -60 35 -60q18 0 36 11zM275 84v156q-12 -6 -49 -23t-41 -19q-31 -14 -46 -31t-15 -43q0 -35 22.5 -55.5t48.5 -20.5q19 0 42 10.5t38 25.5z" id="x61"></path></g><g transform="matrix(.012,-0,0,-.012,13.284,16.212)"><path d="M478 437v-26q-39 -5 -53 -17.5t-35 -57.5q-68 -149 -142 -347h-28q-47 128 -138 350q-17 43 -30.5 54.5t-53.5 17.5v26h205v-26q-40 -4 -46.5 -14.5t6.5 -42.5q41 -115 95 -247q43 101 89 238q12 36 3.5 48.5t-50.5 17.5v26h178z" id="x76"></path></g><g transform="matrix(.012,-0,0,-.012,18.698,16.212)"><path d="M380 106l15 -22q-42 -52 -87 -74t-76 -22q-89 0 -142 62t-53 151q0 103 66 175q67 73 146 73h1q61 0 105.5 -46.5t44.5 -105.5q0 -16 -4.5 -22.5t-18.5 -9.5q-54 -10 -262 -17q1 -95 46 -143.5t104 -48.5q61 0 115 50zM225 412q-35 0 -64.5 -32.5t-40.5 -90.5q83 0 170 3
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<g transform="matrix(.017,-0,0,-.017,24.375,12.138)"><path d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" id="x28"></path></g><g transform="matrix(.017,-0,0,-.017,30.257,12.138)"><path d="M474 429q-19 -69 -47 -222l-65 -347q-8 -49 0 -60t49 -16l26 -3l-4 -26l-246 -12l4 25l17 3q37 6 50 20.5t23 60.5l67 321h-2q-65 -79 -150 -138q-69 -47 -104 -47q-28 0 -48.5 32t-20.5 81q0 78 35 148t90 117q67 57 161 77q23 5 58 5q43 0 90 -15zM387 387
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</svg> versus threshold <svg height="12.1125" id="M204" style="vertical-align:-3.50804pt" version="1.1" viewbox="0 0 8.4750004 12.1125" width="8.4750004" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path d="M474 429q-19 -69 -47 -222l-65 -347q-8 -49 0 -60t49 -16l26 -3l-4 -26l-246 -12l4 25l17 3q37 6 50 20.5t23 60.5l67 321h-2q-65 -79 -150 -138q-69 -47 -104 -47q-28 0 -48.5 32t-20.5 81q0 78 35 148t90 117q67 57 161 77q23 5 58 5q43 0 90 -15zM387 387
q-30 16 -75 16q-58 0 -92 -27q-49 -39 -78.5 -112t-29.5 -136q0 -34 8.5 -52.5t21.5 -18.5q40 0 109.5 63.5t103.5 115.5q16 53 32 151z" id="x1D45E"></path></g>
</svg> (in units of <svg height="8.3000002" id="M205" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 9.7250004 8.3000002" width="9.7250004" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,8.038)"><path d="M548 455q-32 -77 -74 -77q-37 0 -103 10l-3 -2q36 -30 48.5 -62t12.5 -80q0 -103 -75 -179.5t-175 -76.5q-70 0 -113 45t-43 128q0 115 83.5 200.5t209.5 85.5q31 0 77.5 -4t67.5 -4q36 0 62 30zM350 274q0 54 -17 86q-19 35 -57 35q-50 0 -90 -37.5t-59 -91.5t-19 -109
q0 -61 24.5 -96.5t65.5 -35.5q51 0 87.5 46.5t50.5 100.5t14 102z" id="x1D70E"></path></g>
</svg>) for oil (a), gas (b), and <svg height="15.5125" id="M206" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 30.575001 15.5125" width="30.575001" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.363)"><path d="M614 175l29 -10q-33 -109 -57 -154q-121 -26 -184 -26q-90 0 -160.5 29t-112.5 77t-63.5 105.5t-21.5 119.5q0 157 108 253t277 96q36 0 71.5 -5t69 -13.5t36.5 -8.5q15 -102 20 -150l-29 -8q-20 79 -66.5 114t-128.5 35q-119 0 -187.5 -86t-68.5 -207
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<g transform="matrix(.012,-0,0,-.012,24.188,15.45)"><path d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" id="x32"></path></g>
</svg> (c).</table>

Mathematical Problems in Engineering

fig3

Figure 3

Figure 3: Cross-Correlations between Energy and Emissions Markets: New Evidence from Fractal and Multifractal Analysis 