However they can be approximated using the “zero” function from the “calc” menu. The other four roots are more difficult to find. Zooming into the \(x\)-axis, and checking the table shows that the only obvious root is \(x=3\). At this point, we can only approximate the root with the “zero” function from the “calc” menu: Finding the exact value of this second root can be quite difficult, and we will say more about this in section 2 below. The roots can be seen by zooming into the graph.įrom the table and the graph we see that there is a root at \(x=-2\) and another root at between \(-3\) and \(-2\). Since this is a polynomial of degree \(4\), all of the essential features are already displayed in the above graph. The graph of \(f(x)=x^4+3x^3-x+6\) in the standard window is displayed as follows.We say that \(x=3\) is a root of multiplicity \(2\). Indeed, since \(3\) is a root, we can divide \(f(x)\) by \(x-3\) without remainder and factor the resulting quotient to see that that This is due to the fact that \(x=3\) appears as a multiple root. Note, that the root \(x=3\) only “touches” the \(x\)-axis. Zooming into the graph reveals that there are in fact two roots, \(x=2\) and \(x=3\), which can be confirmed from the table. Graphing \(f(x)=-x^3+8x^2-21x+18\) with the calculator shows the following display.Since the polynomial is of degree \(3\), there cannot be any other roots. This may easily be checked by looking at the function table. The graph suggests that the roots are at \(x=1\), \(x=2\), and \(x=4\). # analysis using AR(1) model for means and ARCH(1) model for variancesĬan <- th + th * (y - th)Ĭondit.var <- th * ( 1 + th * (y - th) ^ 2) UBS <- ts(UBSCreditSuisse $UBS_LAST, start = c( 2000, 1), frequency = 365.25) Perform a likelihood ratio test to test whether the AR(2) coefficient is significative.
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