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edit approved Nov 28, 2019 at 14:03
T Goose
edit approved Nov 28, 2019 at 14:03
T Goose
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$$(1-\phi_1B-\phi_2B^2)(1-B)(1-B^{12})^2Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$$$(1-\phi_1B-\phi_2B^2)(1-B)(1-B^{12})^2Y_t=(1+\Theta_1B^{12}+\Theta_2B^{24})e_t$$ $$(1-\phi_1B-\phi_2B^2)(1-B)(1-2B^{12}+B^{24})Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$$$(1-\phi_1B-\phi_2B^2)(1-B)(1-2B^{12}+B^{24})Y_t=(1+\Theta_1B^{12}+\Theta_2B^{24})e_t$$ $$(1-\phi_1B-\phi_2B^2-B+\phi_1B^2+\phi_2B^3)(1-2B^{12}+B^{24})Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$$$(1-\phi_1B-\phi_2B^2-B+\phi_1B^2+\phi_2B^3)(1-2B^{12}+B^{24})Y_t=(1+\Theta_1B^{12}+\Theta_2B^{24})e_t$$

$$(1-\phi_1B-\phi_2B^2-B+\phi_1B^2+\phi_2B^3-2B^{12}+2\phi_1B^{13}+2\phi_2 B^{14}+2B^{13}-2\phi_1B^{14}-2\phi_2B^{15}+B^{24}-\phi_1B^{25}-\phi_2B^{26}-B^{25}+\phi_1B^{26}+\phi_2B^{27})Y_t=(1-\Theta_1 B^{12}-\Theta_2B^{24})e_t$$$$(1-\phi_1B-\phi_2B^2-B+\phi_1B^2+\phi_2B^3-2B^{12}+2\phi_1B^{13}+2\phi_2 B^{14}+2B^{13}-2\phi_1B^{14}-2\phi_2B^{15}+B^{24}-\phi_1B^{25}-\phi_2B^{26}-B^{25}+\phi_1B^{26}+\phi_2B^{27})Y_t=(1+\Theta_1 B^{12}+\Theta_2B^{24})e_t$$

$$Y_t-\phi_1 Y_{t-1} -\phi_2 Y_{t-2}-Y_{t-1}+\phi_1Y_{t-2}+\phi_2Y_{1-3}-2Y_{t-12}+2\phi_1Y_{t-13}+2\phi_2Y_{t-14}+2Y_{t-13}-2\phi_1Y_{t-14}-2\phi_2Y_{t-15}+Y_{t-24}-\phi_1Y_{t-25}-\phi_2Y_{t-26}-Y_{t-25}+\phi_1Y_{t-26}+\phi_2Y_{t-27}=e_t-\Theta_1 e_{t-12}-\Theta_2 e_{t-24}$$$$Y_t-\phi_1 Y_{t-1} -\phi_2 Y_{t-2}-Y_{t-1}+\phi_1Y_{t-2}+\phi_2Y_{1-3}-2Y_{t-12}+2\phi_1Y_{t-13}+2\phi_2Y_{t-14}+2Y_{t-13}-2\phi_1Y_{t-14}-2\phi_2Y_{t-15}+Y_{t-24}-\phi_1Y_{t-25}-\phi_2Y_{t-26}-Y_{t-25}+\phi_1Y_{t-26}+\phi_2Y_{t-27}=e_t+\Theta_1 e_{t-12}+\Theta_2 e_{t-24}$$

$$(1-\phi_1B-\phi_2B^2)(1-B)(1-B^{12})^2Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$ $$(1-\phi_1B-\phi_2B^2)(1-B)(1-2B^{12}+B^{24})Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$ $$(1-\phi_1B-\phi_2B^2-B+\phi_1B^2+\phi_2B^3)(1-2B^{12}+B^{24})Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$

$$(1-\phi_1B-\phi_2B^2-B+\phi_1B^2+\phi_2B^3-2B^{12}+2\phi_1B^{13}+2\phi_2 B^{14}+2B^{13}-2\phi_1B^{14}-2\phi_2B^{15}+B^{24}-\phi_1B^{25}-\phi_2B^{26}-B^{25}+\phi_1B^{26}+\phi_2B^{27})Y_t=(1-\Theta_1 B^{12}-\Theta_2B^{24})e_t$$

$$Y_t-\phi_1 Y_{t-1} -\phi_2 Y_{t-2}-Y_{t-1}+\phi_1Y_{t-2}+\phi_2Y_{1-3}-2Y_{t-12}+2\phi_1Y_{t-13}+2\phi_2Y_{t-14}+2Y_{t-13}-2\phi_1Y_{t-14}-2\phi_2Y_{t-15}+Y_{t-24}-\phi_1Y_{t-25}-\phi_2Y_{t-26}-Y_{t-25}+\phi_1Y_{t-26}+\phi_2Y_{t-27}=e_t-\Theta_1 e_{t-12}-\Theta_2 e_{t-24}$$

$$(1-\phi_1B-\phi_2B^2)(1-B)(1-B^{12})^2Y_t=(1+\Theta_1B^{12}+\Theta_2B^{24})e_t$$ $$(1-\phi_1B-\phi_2B^2)(1-B)(1-2B^{12}+B^{24})Y_t=(1+\Theta_1B^{12}+\Theta_2B^{24})e_t$$ $$(1-\phi_1B-\phi_2B^2-B+\phi_1B^2+\phi_2B^3)(1-2B^{12}+B^{24})Y_t=(1+\Theta_1B^{12}+\Theta_2B^{24})e_t$$

$$(1-\phi_1B-\phi_2B^2-B+\phi_1B^2+\phi_2B^3-2B^{12}+2\phi_1B^{13}+2\phi_2 B^{14}+2B^{13}-2\phi_1B^{14}-2\phi_2B^{15}+B^{24}-\phi_1B^{25}-\phi_2B^{26}-B^{25}+\phi_1B^{26}+\phi_2B^{27})Y_t=(1+\Theta_1 B^{12}+\Theta_2B^{24})e_t$$

$$Y_t-\phi_1 Y_{t-1} -\phi_2 Y_{t-2}-Y_{t-1}+\phi_1Y_{t-2}+\phi_2Y_{1-3}-2Y_{t-12}+2\phi_1Y_{t-13}+2\phi_2Y_{t-14}+2Y_{t-13}-2\phi_1Y_{t-14}-2\phi_2Y_{t-15}+Y_{t-24}-\phi_1Y_{t-25}-\phi_2Y_{t-26}-Y_{t-25}+\phi_1Y_{t-26}+\phi_2Y_{t-27}=e_t+\Theta_1 e_{t-12}+\Theta_2 e_{t-24}$$

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mpiktas
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$$(1-\phi_1B-\phi_2B^2)(1-B)(1-B^{12})^2Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$ $$<=> (1-\phi_1B-\phi_2B^2)(1-B)(1-2B^{12}+B^{24})Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$$$(1-\phi_1B-\phi_2B^2)(1-B)(1-2B^{12}+B^{24})Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$ $$<=>(1-\phi_1B-\phi_2B^2-B+\phi_1B^2+\phi_2B^3)(1-2B^{12}+B^{24})Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$$$(1-\phi_1B-\phi_2B^2-B+\phi_1B^2+\phi_2B^3)(1-2B^{12}+B^{24})Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$

$$<=>(1-\phi_1B-\phi_2B^2-B+\phi_1B^2+\phi_2B^3-2B^{12}+2\phi_1B^{13}+2\phi_2 B^{14}+2B^{13}-2\phi_1B^{14}-2\phi_2B^{15}+B^{24}-\phi_1B^{25}-\phi_2B^{26}-B^{25}+\phi_1B^{26}+\phi_2B^{27})Y_t=(1-\Theta_1 B^{12}-\Theta_2B^{24})e_t$$$$(1-\phi_1B-\phi_2B^2-B+\phi_1B^2+\phi_2B^3-2B^{12}+2\phi_1B^{13}+2\phi_2 B^{14}+2B^{13}-2\phi_1B^{14}-2\phi_2B^{15}+B^{24}-\phi_1B^{25}-\phi_2B^{26}-B^{25}+\phi_1B^{26}+\phi_2B^{27})Y_t=(1-\Theta_1 B^{12}-\Theta_2B^{24})e_t$$

$$<=>Y_t-\phi_1 Y_{t-1} -\phi_2 Y_{t-2}-Y_{t-1}+\phi_1Y_{t-2}+\phi_2Y_{1-3}-2Y_{t-12}+2\phi_1Y_{t-13}+2\phi_2Y_{t-14}+2Y_{t-13}-2\phi_1Y_{t-14}-2\phi_2Y_{t-15}+Y_{t-24}-\phi_1Y_{t-25}-\phi_2Y_{t-26}-Y_{t-25}+\phi_1Y_{t-26}+\phi_2Y_{t-27}=e_t-\Theta_1 e_{t-12}-\Theta_2 e_{t-24}$$$$Y_t-\phi_1 Y_{t-1} -\phi_2 Y_{t-2}-Y_{t-1}+\phi_1Y_{t-2}+\phi_2Y_{1-3}-2Y_{t-12}+2\phi_1Y_{t-13}+2\phi_2Y_{t-14}+2Y_{t-13}-2\phi_1Y_{t-14}-2\phi_2Y_{t-15}+Y_{t-24}-\phi_1Y_{t-25}-\phi_2Y_{t-26}-Y_{t-25}+\phi_1Y_{t-26}+\phi_2Y_{t-27}=e_t-\Theta_1 e_{t-12}-\Theta_2 e_{t-24}$$

$$(1-\phi_1B-\phi_2B^2)(1-B)(1-B^{12})^2Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$ $$<=> (1-\phi_1B-\phi_2B^2)(1-B)(1-2B^{12}+B^{24})Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$ $$<=>(1-\phi_1B-\phi_2B^2-B+\phi_1B^2+\phi_2B^3)(1-2B^{12}+B^{24})Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$

$$<=>(1-\phi_1B-\phi_2B^2-B+\phi_1B^2+\phi_2B^3-2B^{12}+2\phi_1B^{13}+2\phi_2 B^{14}+2B^{13}-2\phi_1B^{14}-2\phi_2B^{15}+B^{24}-\phi_1B^{25}-\phi_2B^{26}-B^{25}+\phi_1B^{26}+\phi_2B^{27})Y_t=(1-\Theta_1 B^{12}-\Theta_2B^{24})e_t$$

$$<=>Y_t-\phi_1 Y_{t-1} -\phi_2 Y_{t-2}-Y_{t-1}+\phi_1Y_{t-2}+\phi_2Y_{1-3}-2Y_{t-12}+2\phi_1Y_{t-13}+2\phi_2Y_{t-14}+2Y_{t-13}-2\phi_1Y_{t-14}-2\phi_2Y_{t-15}+Y_{t-24}-\phi_1Y_{t-25}-\phi_2Y_{t-26}-Y_{t-25}+\phi_1Y_{t-26}+\phi_2Y_{t-27}=e_t-\Theta_1 e_{t-12}-\Theta_2 e_{t-24}$$

$$(1-\phi_1B-\phi_2B^2)(1-B)(1-B^{12})^2Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$ $$(1-\phi_1B-\phi_2B^2)(1-B)(1-2B^{12}+B^{24})Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$ $$(1-\phi_1B-\phi_2B^2-B+\phi_1B^2+\phi_2B^3)(1-2B^{12}+B^{24})Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$

$$(1-\phi_1B-\phi_2B^2-B+\phi_1B^2+\phi_2B^3-2B^{12}+2\phi_1B^{13}+2\phi_2 B^{14}+2B^{13}-2\phi_1B^{14}-2\phi_2B^{15}+B^{24}-\phi_1B^{25}-\phi_2B^{26}-B^{25}+\phi_1B^{26}+\phi_2B^{27})Y_t=(1-\Theta_1 B^{12}-\Theta_2B^{24})e_t$$

$$Y_t-\phi_1 Y_{t-1} -\phi_2 Y_{t-2}-Y_{t-1}+\phi_1Y_{t-2}+\phi_2Y_{1-3}-2Y_{t-12}+2\phi_1Y_{t-13}+2\phi_2Y_{t-14}+2Y_{t-13}-2\phi_1Y_{t-14}-2\phi_2Y_{t-15}+Y_{t-24}-\phi_1Y_{t-25}-\phi_2Y_{t-26}-Y_{t-25}+\phi_1Y_{t-26}+\phi_2Y_{t-27}=e_t-\Theta_1 e_{t-12}-\Theta_2 e_{t-24}$$

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$$(1-\phi_1B-\phi_2B^2)(1-B)(1-B^{12})^2Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$ $$<=> (1-\phi_1B-\phi_2B^2)(1-B)(1-2B^{12}+B^{24})Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$ $$<=>(1-\phi_1B-\phi_2B^2-B+\phi_1B^2+\phi_2B^3)(1-2B^{12}+B^{24})Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$

$$<=>(1-\phi_1B-\phi_2B^2-B+\phi_1B^2+\phi_2B^3-2B^{12}+2\phi_1B^{13}+2\phi_2 B^{14}+2B^{13}-2\phi_1B^{14}-2\phi_2B^{15}+B^{24}-\phi_1B^{25}-\phi_2B^{26}-B^{25}+\phi_1B^{26}+\phi_2B^{27})Y_t=e_t-\Theta_1 e_{t-12}-\Theta_2 e_{t-24}$$$$<=>(1-\phi_1B-\phi_2B^2-B+\phi_1B^2+\phi_2B^3-2B^{12}+2\phi_1B^{13}+2\phi_2 B^{14}+2B^{13}-2\phi_1B^{14}-2\phi_2B^{15}+B^{24}-\phi_1B^{25}-\phi_2B^{26}-B^{25}+\phi_1B^{26}+\phi_2B^{27})Y_t=(1-\Theta_1 B^{12}-\Theta_2B^{24})e_t$$

$$<=>Y_t-\phi_1 Y_{t-1} -\phi_2 Y_{t-2}-Y_{t-1}+\phi_1Y_{t-2}+\phi_2Y_{1-3}-2Y_{t-12}+2\phi_1Y_{t-13}+2\phi_2Y_{t-14}+2Y_{t-13}-2\phi_1Y_{t-14}-2\phi_2Y_{t-15}+Y_{t-24}-\phi_1Y_{t-25}-\phi_2Y_{t-26}-Y_{t-25}+\phi_1Y_{t-26}+\phi_2Y_{t-27}=e_t-\Theta_1 e_{t-12}-\Theta_2 e_{t-24}$$

$$(1-\phi_1B-\phi_2B^2)(1-B)(1-B^{12})^2Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$ $$<=> (1-\phi_1B-\phi_2B^2)(1-B)(1-2B^{12}+B^{24})Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$ $$<=>(1-\phi_1B-\phi_2B^2-B+\phi_1B^2+\phi_2B^3)(1-2B^{12}+B^{24})Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$

$$<=>(1-\phi_1B-\phi_2B^2-B+\phi_1B^2+\phi_2B^3-2B^{12}+2\phi_1B^{13}+2\phi_2 B^{14}+2B^{13}-2\phi_1B^{14}-2\phi_2B^{15}+B^{24}-\phi_1B^{25}-\phi_2B^{26}-B^{25}+\phi_1B^{26}+\phi_2B^{27})Y_t=e_t-\Theta_1 e_{t-12}-\Theta_2 e_{t-24}$$

$$(1-\phi_1B-\phi_2B^2)(1-B)(1-B^{12})^2Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$ $$<=> (1-\phi_1B-\phi_2B^2)(1-B)(1-2B^{12}+B^{24})Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$ $$<=>(1-\phi_1B-\phi_2B^2-B+\phi_1B^2+\phi_2B^3)(1-2B^{12}+B^{24})Y_t=(1-\Theta_1B^{12}-\Theta_2B^{24})e_t$$

$$<=>(1-\phi_1B-\phi_2B^2-B+\phi_1B^2+\phi_2B^3-2B^{12}+2\phi_1B^{13}+2\phi_2 B^{14}+2B^{13}-2\phi_1B^{14}-2\phi_2B^{15}+B^{24}-\phi_1B^{25}-\phi_2B^{26}-B^{25}+\phi_1B^{26}+\phi_2B^{27})Y_t=(1-\Theta_1 B^{12}-\Theta_2B^{24})e_t$$

$$<=>Y_t-\phi_1 Y_{t-1} -\phi_2 Y_{t-2}-Y_{t-1}+\phi_1Y_{t-2}+\phi_2Y_{1-3}-2Y_{t-12}+2\phi_1Y_{t-13}+2\phi_2Y_{t-14}+2Y_{t-13}-2\phi_1Y_{t-14}-2\phi_2Y_{t-15}+Y_{t-24}-\phi_1Y_{t-25}-\phi_2Y_{t-26}-Y_{t-25}+\phi_1Y_{t-26}+\phi_2Y_{t-27}=e_t-\Theta_1 e_{t-12}-\Theta_2 e_{t-24}$$

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