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It does so by finding the \ values of \[Tau] that minimise the objective function:\n\n\t\[ScriptCapitalL] \ = ", Cell[BoxData[ $TraditionalForm \`\[Sum]\+\(t = 1$\%T$(y\_t - \[Tau]\_t)$\^2 + \ \[Lambda] $\[Sum]\+\(t = 2$\%$T - 1$$( \((\[Tau]\_\(t + 1$ - \[Tau]\_t)\) - $(\[Tau]\_t - \[Tau]\_\(t - 1$)\))\)\^2\)\)]], "\n\nwhere the weight on smoothness (\[Lambda]) is typically defined to be \ 1600 for quarterly data. It turns out that the first-order conditions of \ this minimisation problem are linear equations that can be re-used for any \ series. They are also quite simple: other than the first two and last two \ conditions, they have the same form. It is straightforward to write a \ function that finds HP-filtered trend of a series in ", StyleBox["Mathematica ", FontSlant->"Italic"], "by using the general forms of these first-order conditions \[Dash] the \ loss function does not have to be evaluated or differentiated each time, and \ the same function can be used for any value of \[Lambda]. " }], "Text", FontSize->13], Cell[BoxData[ $HPFilter[data_?VectorQ, lambda_Integer] := \n\t Module[{\[Tau], focs, taulist, answer}, \n\t\t With[{\[DoubleStruckCapitalT] = Length[data], \[Lambda] = N[lambda]}, \n\t\t\ttaulist\ = \ Table[\[Tau][t], {t, 1, \[DoubleStruckCapitalT]}]; \n focs = \tThread[ Flatten[Join[{ \(-2. $\ data\[LeftDoubleBracket]1\[RightDoubleBracket]\ + 2. \ \[Tau][1] + 2. \ \[Lambda]\ \[Tau][1] - 4. \ \[Lambda]\ \[Tau][2] + 2. \ \[Lambda]\ \[Tau][3], $-2. $\ data\[LeftDoubleBracket]2\[RightDoubleBracket] - 4. \ \[Lambda]\ \[Tau][1] + 2. \ \[Tau][2] + 10. \ \[Lambda]\ \[Tau][2] - 8. \ \[Lambda]\ \[Tau][3] + 2. \ \[Lambda]\ \[Tau][4]}, \n Table[\t$-2. $\ \ data\[LeftDoubleBracket]t\[RightDoubleBracket] + 2. \ \[Lambda]\ \[Tau][t - 2] - 8. \ \[Lambda]\ \[Tau][t - 1] + 2. \ \[Tau][t] + 12. \ \[Lambda]\ \[Tau][t] - 8. \ \[Lambda]\ \[Tau][t + 1] + 2. \ \[Lambda]\ \[Tau][t + 2], {t, 3, \[DoubleStruckCapitalT] - 2}], \n \t\t{$-2$\ data\[LeftDoubleBracket]\[DoubleStruckCapitalT] - 1 \[RightDoubleBracket] + 2. \ \[Lambda]\ \[Tau][\[DoubleStruckCapitalT] - 3] - 8. \ \[Lambda]\ \[Tau][\[DoubleStruckCapitalT] - 2] + 2. \ \[Tau][\[DoubleStruckCapitalT] - 1] + 10. \ \[Lambda]\ \[Tau][\[DoubleStruckCapitalT] - 1] - 4. \ \[Lambda]\ \[Tau][\[DoubleStruckCapitalT]], 2. \ \[Lambda]\ \[Tau][\[DoubleStruckCapitalT] - 2] - 4. \ \[Lambda]\ \[Tau][\[DoubleStruckCapitalT] - 1] + 2. \ \[Lambda]\ \[Tau][\[DoubleStruckCapitalT]]}]] == 0]; \n\t\tanswer = Solve[focs, taulist]; \n\t\t Flatten[Expand[N[taulist /. answer]]]\n\t\t\t\t\t]]\)], "Input", FontSize->12], Cell[TextData[{ "Sometimes, however, a univariate approach isn\[CloseCurlyQuote]t enough: \ we may also want to use conditioning information from the series\ \[CloseCurlyQuote] relationships with other variables to determine the true \ trend series. In that case, we need a ", StyleBox["multivariate Hodrick-Prescott filter", FontWeight->"Bold"], ". These are also quite simple to calculate using ", StyleBox["Mathematica", FontSlant->"Italic"], ": however, the structure of the first-order conditions will depend on the \ structure and number of the conditioning equations, so there is no simple \ way to automate their determination. Fortunately, the structure of the \ first-order conditions does ", StyleBox["not", FontSlant->"Italic"], " depend on the numerical parameter values in the conditioning equations, \ so you don\[CloseCurlyQuote]t have to re-write your HP filter functions each \ time you re-estimate the underlying equations and change the parameter \ values." }], "Text", FontSize->13] }, Open ]], Cell[CellGroupData[{ Cell["Calculating a Multivariate Hodrick-Prescott Filter", "Subtitle"], Cell[TextData[{ "The loss function underlying a multivariate HP filter is a simple \ extension of the univariate case:\n\n\t\[ScriptCapitalL] = ", Cell[BoxData[ $TraditionalForm \`\[Sum]\+\(i = 1$\%n$ \[Beta]\_i$ $\[Sum]\+\(t = 1$\%T \[Epsilon]\_$i, t$\^2\) + $\[Beta]\_trend$ $\[Sum]\+\(t = 1$\%T$(y\_t - \[Tau]\_t)$\^2\) + \ \[Lambda] $\[Sum]\+\(t = 2$\%$T - 1$$( \((\[Tau]\_\(t + 1$ - \[Tau]\_t)\) - $(\[Tau]\_t - \[Tau]\_\(t - 1$)\))\)\^2\)\)]], "\n\nWhere ", StyleBox["n", FontSlant->"Italic"], " is the number of conditioning equations, the \[Beta]\[CloseCurlyQuote]s \ are the weights on the deviations from trend and the conditioning equations, \ and the ", Cell[BoxData[ $TraditionalForm\`\[Epsilon]\_\(i, t$\)]], " \[CloseCurlyQuote]s are the residuals from the ", Cell[BoxData[ $TraditionalForm\`i\^th$]], "conditioning equation at time ", StyleBox["t", FontSlant->"Italic"], ".\n\nFinding the first-order conditions is as simple as defining a loss \ function like this, with algebraic symbols for the weights and the parameters \ in the conditioning equations, and differentiating it with respect to each of \ the ", Cell[BoxData[ $TraditionalForm\`\[Tau]\_t$]], ". Because the first-order conditions of a Hodrick-Prescott filter \[Dash] \ other than the first two and last two \[Dash] have a common structure, we can \ use a small horizon (T) in the loss function to find the correct structure \ for data series of any length. The following section goes through this \ process step-by-step for a multivariate HP filter for output with three \ conditioning equations (prices, unemployment and capacity utilisation)." }], "Text", FontSize->13], Cell[CellGroupData[{ Cell["1. The Residuals", "Subsubtitle"], Cell[TextData[{ "Firstly, we have to define the residuals, obtained by subtracting the body \ of the relavent behavioural equation from its dependent variable. This can be \ done with a standard ", StyleBox["Mathematica", FontSlant->"Italic"], " function definition. Note the [t_] construct instead of a subscript. The \ \[OpenCurlyDoubleQuote]t_\[CloseCurlyDoubleQuote] translates as \ \[OpenCurlyDoubleQuote]anything, but for the purposes of the definition we\ \[CloseCurlyQuote]ll call it t\[CloseCurlyDoubleQuote]. Also, ", StyleBox["this is all done in algebra", FontWeight->"Bold"], ": at this stage you don\[CloseCurlyQuote]t need data or numerical \ coefficients. We also define the residual to be zero for values of less \ than one: we assume that t=1 is the start of any series." }], "Text", FontSize->13], Cell[CellGroupData[{ Cell[TextData[{ "Definition for ", StyleBox["\[Epsilon]", FontFamily->"Symbol"], StyleBox["\[Pi]", FontFamily->"Symbol", FontVariations->{"CompatibilityType"->"Subscript"}] }], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ $\[Epsilon]\_p[t_] := 0 /; t < 1$, $\[Epsilon]\_p[t_] := p[t] - \[Delta]\_1\ \((y[t - 1] - \[Tau][t - 1])$ - \[Delta]\_2\ $(y[t - 2] - \[Tau][t - 2])$\ \)}], "Input", PageWidth->PaperWidth, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Definition for ", StyleBox["\[Epsilon]", FontFamily->"Symbol"], StyleBox["\[Upsilon]", FontFamily->"Symbol", FontVariations->{"CompatibilityType"->"Subscript"}] }], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ $\[Epsilon]\_\[Upsilon][t_] := 0 /; t < 1$, $\[Epsilon]\_\[Upsilon][t_] := \ u[t] - \[Theta]\ \((y[t - 1] - \[Tau][t - 1])$ - \[Xi]\)}], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Definition for ", Cell[BoxData[ $TraditionalForm\`\[Epsilon]\_c$]] }], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ $\[Epsilon]\_c[t_] := 0 /; t < 1$, $\[Epsilon]\_c[t_] := c[t]\ - \ \[Alpha]\_1\ \((y[t - 1] - \[Tau][t - 1])$ - \[Alpha]\_2\ $(y[t - 2] - \[Tau][t - 2])$\)}], "Input", AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["Test these definitions", "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "As these cells show, function definitions such as the ones above are \ \[OpenCurlyDoubleQuote]rewrite rules\[CloseCurlyDoubleQuote]: they tell ", StyleBox["Mathematica", FontSlant->"Italic"], " to substitute the right-hand side of the function definition whereever it \ encounters something matching the left-hand side of the function definition." }], "Text", FontSize->13], Cell[CellGroupData[{ Cell[BoxData[ $\[Epsilon]\_p[\(-3$]\)], "Input", AspectRatioFixed->True], Cell[BoxData[ $0$], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ $\[Epsilon]\_c[3]$], "Input", AspectRatioFixed->True], Cell[BoxData[ $c[3] - \[Alpha]\_2\ \((y[1] - \[Tau][1])$ - \[Alpha]\_1\ $(y[2] - \[Tau][2])$\)], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["2. The Loss Function ", "Subsubtitle", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ The horizon over which to calculate the first-order conditions: 10 \ is usually enough, depending on the lag structure\ \>", "Subsubsection"], Cell[BoxData[ $\(\[DoubleStruckCapitalT] = 10; $\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Define the list of taus into a little vector", "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ $taulist = Table[\[Tau][t], {t, 1, \[DoubleStruckCapitalT]}]$], "Input",\ AspectRatioFixed->True], Cell[BoxData[ ${\[Tau][1], \[Tau][2], \[Tau][3], \[Tau][4], \[Tau][5], \[Tau][6], \[Tau][7], \[Tau][8], \[Tau][9], \[Tau][10]}$], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["This is the definition of the loss function", "Subsubsection"], Cell[TextData[{ "You can see how ", StyleBox["Mathematica", FontSlant->"Italic"], " substitutes the definitions for the \[Epsilon]\[CloseCurlyQuote]s \ wherever they occur. Note that the terms in brackets are not expanded out \ automatically: you need to wrap an expression in the function ", StyleBox["ExpandAll[ ]", FontFamily->"Courier"], " for that to happen." }], "Text", FontSize->13], Cell[CellGroupData[{ Cell[BoxData[ $\[ScriptCapitalL] = \[Mu]\ \(\[Sum]\+\(t = 1$\%\[DoubleStruckCapitalT] \[Epsilon]\_p[t]\^2 \) + \[Beta]\ $\[Sum]\+\(t = 1$\%\[DoubleStruckCapitalT] \[Epsilon]\_\[Upsilon][ t]\^2\) + \[Psi]\ $\[Sum]\+\(t = 1$\%\[DoubleStruckCapitalT] \[Epsilon]\_c[t] \^2\) + \[Gamma]\ $\[Sum]\+\(t = 1$\%\[DoubleStruckCapitalT]$( y[t - 1] - \[Tau][t - 1])$\^2\) + \[Lambda]\ $\[Sum]\+\(t = 2$\%$\[DoubleStruckCapitalT] - 1$$( \[Tau][t + 1]\^2 + 4\ \[Tau][t]\^2 + \[Tau][t - 1]\^2 - 4\ \[Tau][t + 1]\ \[Tau][t] + 2\ \[Tau][t + 1]\ \[Tau][t - 1] - 4\ \[Tau][t]\ \[Tau][t - 1]) $\)\)], "Input", PageWidth->PaperWidth, AspectRatioFixed->True], Cell[BoxData[ $\[Beta]\ \((\((\(-\[Xi]$ + u[1] - \[Theta]\ $(y[0] - \[Tau][0])$)\)\^2 + $(\(-\[Xi]$ + u[2] - \[Theta]\ $(y[1] - \[Tau][1])$)\)\^2 + $(\(-\[Xi]$ + u[3] - \[Theta]\ $(y[2] - \[Tau][2])$)\)\^2 + $(\(-\[Xi]$ + u[4] - \[Theta]\ $(y[3] - \[Tau][3])$)\)\^2 + $(\(-\[Xi]$ + u[5] - \[Theta]\ $(y[4] - \[Tau][4])$)\)\^2 + $(\(-\[Xi]$ + u[6] - \[Theta]\ $(y[5] - \[Tau][5])$)\)\^2 + $(\(-\[Xi]$ + u[7] - \[Theta]\ $(y[6] - \[Tau][6])$)\)\^2 + $(\(-\[Xi]$ + u[8] - \[Theta]\ $(y[7] - \[Tau][7])$)\)\^2 + $(\(-\[Xi]$ + u[9] - \[Theta]\ $(y[8] - \[Tau][8])$)\)\^2 + $(\(-\[Xi]$ + u[10] - \[Theta]\ $(y[9] - \[Tau][9])$)\)\^2) \) + \[Psi]\ $(\((c[1] - \[Alpha]\_2\ \((y[\(-1$] - \[Tau][$-1$])\) - \[Alpha]\_1\ $(y[0] - \[Tau][0])$)\)\^2 + $(c[2] - \[Alpha]\_2\ \((y[0] - \[Tau][0])$ - \[Alpha]\_1\ $(y[1] - \[Tau][1])$)\)\^2 + $(c[3] - \[Alpha]\_2\ \((y[1] - \[Tau][1])$ - \[Alpha]\_1\ $(y[2] - \[Tau][2])$)\)\^2 + $(c[4] - \[Alpha]\_2\ \((y[2] - \[Tau][2])$ - \[Alpha]\_1\ $(y[3] - \[Tau][3])$)\)\^2 + $(c[5] - \[Alpha]\_2\ \((y[3] - \[Tau][3])$ - \[Alpha]\_1\ $(y[4] - \[Tau][4])$)\)\^2 + $(c[6] - \[Alpha]\_2\ \((y[4] - \[Tau][4])$ - \[Alpha]\_1\ $(y[5] - \[Tau][5])$)\)\^2 + $(c[7] - \[Alpha]\_2\ \((y[5] - \[Tau][5])$ - \[Alpha]\_1\ $(y[6] - \[Tau][6])$)\)\^2 + $(c[8] - \[Alpha]\_2\ \((y[6] - \[Tau][6])$ - \[Alpha]\_1\ $(y[7] - \[Tau][7])$)\)\^2 + $(c[9] - \[Alpha]\_2\ \((y[7] - \[Tau][7])$ - \[Alpha]\_1\ $(y[8] - \[Tau][8])$)\)\^2 + $(c[10] - \[Alpha]\_2\ \((y[8] - \[Tau][8])$ - \[Alpha]\_1\ $(y[9] - \[Tau][9])$)\)\^2)\) + \[Mu]\ $( \((p[1] - \[Delta]\_2\ \((y[\(-1$] - \[Tau][$-1$])\) - \[Delta]\_1\ $(y[0] - \[Tau][0])$)\)\^2 + $(p[2] - \[Delta]\_2\ \((y[0] - \[Tau][0])$ - \[Delta]\_1\ $(y[1] - \[Tau][1])$)\)\^2 + $(p[3] - \[Delta]\_2\ \((y[1] - \[Tau][1])$ - \[Delta]\_1\ $(y[2] - \[Tau][2])$)\)\^2 + $(p[4] - \[Delta]\_2\ \((y[2] - \[Tau][2])$ - \[Delta]\_1\ $(y[3] - \[Tau][3])$)\)\^2 + $(p[5] - \[Delta]\_2\ \((y[3] - \[Tau][3])$ - \[Delta]\_1\ $(y[4] - \[Tau][4])$)\)\^2 + $(p[6] - \[Delta]\_2\ \((y[4] - \[Tau][4])$ - \[Delta]\_1\ $(y[5] - \[Tau][5])$)\)\^2 + $(p[7] - \[Delta]\_2\ \((y[5] - \[Tau][5])$ - \[Delta]\_1\ $(y[6] - \[Tau][6])$)\)\^2 + $(p[8] - \[Delta]\_2\ \((y[6] - \[Tau][6])$ - \[Delta]\_1\ $(y[7] - \[Tau][7])$)\)\^2 + $(p[9] - \[Delta]\_2\ \((y[7] - \[Tau][7])$ - \[Delta]\_1\ $(y[8] - \[Tau][8])$)\)\^2 + $(p[10] - \[Delta]\_2\ \((y[8] - \[Tau][8])$ - \[Delta]\_1\ $(y[9] - \[Tau][9])$)\)\^2)\) + \[Gamma]\ $( \((y[0] - \[Tau][0])$\^2 + $(y[1] - \[Tau][1])$\^2 + $(y[2] - \[Tau][2])$\^2 + $(y[3] - \[Tau][3])$\^2 + $(y[4] - \[Tau][4])$\^2 + $(y[5] - \[Tau][5])$\^2 + $(y[6] - \[Tau][6])$\^2 + $(y[7] - \[Tau][7])$\^2 + $(y[8] - \[Tau][8])$\^2 + $(y[9] - \[Tau][9])$\^2)\) + \[Lambda]\ $(\[Tau][1]\^2 - 4\ \[Tau][1]\ \[Tau][2] + 5\ \[Tau][2]\^2 + 2\ \[Tau][1]\ \[Tau][3] - 8\ \[Tau][2]\ \[Tau][3] + 6\ \[Tau][3]\^2 + 2\ \[Tau][2]\ \[Tau][4] - 8\ \[Tau][3]\ \[Tau][4] + 6\ \[Tau][4]\^2 + 2\ \[Tau][3]\ \[Tau][5] - 8\ \[Tau][4]\ \[Tau][5] + 6\ \[Tau][5]\^2 + 2\ \[Tau][4]\ \[Tau][6] - 8\ \[Tau][5]\ \[Tau][6] + 6\ \[Tau][6]\^2 + 2\ \[Tau][5]\ \[Tau][7] - 8\ \[Tau][6]\ \[Tau][7] + 6\ \[Tau][7]\^2 + 2\ \[Tau][6]\ \[Tau][8] - 8\ \[Tau][7]\ \[Tau][8] + 6\ \[Tau][8]\^2 + 2\ \[Tau][7]\ \[Tau][9] - 8\ \[Tau][8]\ \[Tau][9] + 5\ \[Tau][9]\^2 + 2\ \[Tau][8]\ \[Tau][10] - 4\ \[Tau][9]\ \[Tau][10] + \[Tau][10]\^2)$\)], "Output", FontSize->9] }, Open ]], Cell[TextData[{ "Finding the first-order conditions is as simple as telling ", StyleBox["Mathematica", FontSlant->"Italic"], " to take the partial derivative of the loss function above (\ \[ScriptCapitalL]) with respect to each of the \[Tau][t]\[CloseCurlyQuote]s. \ Note the use of the Expand[ ] function to expand out all the brackets in \ these derivatives. The first-order conditions are then just each of these \ derivatives equated to zero. " }], "Text", FontSize->13], Cell[BoxData[ $\(focs = Table[{t, Expand[\[PartialD]\_\(\[Tau][t]$\[ScriptCapitalL]]}, {t, 1, \[DoubleStruckCapitalT]}]; \)\)], "Input", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ $TableForm[focs]$], "Input", AspectRatioFixed->True], Cell[BoxData[ TagBox[GridBox[{ {"1", $\(-2$\ \[Beta]\ \[Theta]\ \[Xi] + 2\ \[Psi]\ c[2]\ \[Alpha]\_1 + 2\ \[Psi]\ c[3]\ \[Alpha]\_2 + 2\ \[Mu]\ p[2]\ \[Delta]\_1 + 2\ \[Mu]\ p[3]\ \[Delta]\_2 + 2\ \[Beta]\ \[Theta]\ u[2] - 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ y[0] - 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ y[0] - 2\ \[Gamma]\ y[1] - 2\ \[Beta]\ \[Theta]\^2\ y[1] - 2\ \[Psi]\ \[Alpha]\_1\%2\ y[1] - 2\ \[Psi]\ \[Alpha]\_2\%2\ y[1] - 2\ \[Mu]\ \[Delta]\_1\%2\ y[1] - 2\ \[Mu]\ \[Delta]\_2\%2\ y[1] - 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ y[2] - 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ y[2] + 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ \[Tau][0] + 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ \[Tau][0] + 2\ \[Gamma]\ \[Tau][1] + 2\ \[Beta]\ \[Theta]\^2\ \[Tau][1] + 2\ \[Lambda]\ \[Tau][1] + 2\ \[Psi]\ \[Alpha]\_1\%2\ \[Tau][1] + 2\ \[Psi]\ \[Alpha]\_2\%2\ \[Tau][1] + 2\ \[Mu]\ \[Delta]\_1\%2\ \[Tau][1] + 2\ \[Mu]\ \[Delta]\_2\%2\ \[Tau][1] - 4\ \[Lambda]\ \[Tau][2] + 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ \[Tau][2] + 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ \[Tau][2] + 2\ \[Lambda]\ \[Tau][3]\)}, {"2", $\(-2$\ \[Beta]\ \[Theta]\ \[Xi] + 2\ \[Psi]\ c[3]\ \[Alpha]\_1 + 2\ \[Psi]\ c[4]\ \[Alpha]\_2 + 2\ \[Mu]\ p[3]\ \[Delta]\_1 + 2\ \[Mu]\ p[4]\ \[Delta]\_2 + 2\ \[Beta]\ \[Theta]\ u[3] - 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ y[1] - 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ y[1] - 2\ \[Gamma]\ y[2] - 2\ \[Beta]\ \[Theta]\^2\ y[2] - 2\ \[Psi]\ \[Alpha]\_1\%2\ y[2] - 2\ \[Psi]\ \[Alpha]\_2\%2\ y[2] - 2\ \[Mu]\ \[Delta]\_1\%2\ y[2] - 2\ \[Mu]\ \[Delta]\_2\%2\ y[2] - 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ y[3] - 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ y[3] - 4\ \[Lambda]\ \[Tau][1] + 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ \[Tau][1] + 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ \[Tau][1] + 2\ \[Gamma]\ \[Tau][2] + 2\ \[Beta]\ \[Theta]\^2\ \[Tau][2] + 10\ \[Lambda]\ \[Tau][2] + 2\ \[Psi]\ \[Alpha]\_1\%2\ \[Tau][2] + 2\ \[Psi]\ \[Alpha]\_2\%2\ \[Tau][2] + 2\ \[Mu]\ \[Delta]\_1\%2\ \[Tau][2] + 2\ \[Mu]\ \[Delta]\_2\%2\ \[Tau][2] - 8\ \[Lambda]\ \[Tau][3] + 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ \[Tau][3] + 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ \[Tau][3] + 2\ \[Lambda]\ \[Tau][4]\)}, {"3", $\(-2$\ \[Beta]\ \[Theta]\ \[Xi] + 2\ \[Psi]\ c[4]\ \[Alpha]\_1 + 2\ \[Psi]\ c[5]\ \[Alpha]\_2 + 2\ \[Mu]\ p[4]\ \[Delta]\_1 + 2\ \[Mu]\ p[5]\ \[Delta]\_2 + 2\ \[Beta]\ \[Theta]\ u[4] - 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ y[2] - 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ y[2] - 2\ \[Gamma]\ y[3] - 2\ \[Beta]\ \[Theta]\^2\ y[3] - 2\ \[Psi]\ \[Alpha]\_1\%2\ y[3] - 2\ \[Psi]\ \[Alpha]\_2\%2\ y[3] - 2\ \[Mu]\ \[Delta]\_1\%2\ y[3] - 2\ \[Mu]\ \[Delta]\_2\%2\ y[3] - 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ y[4] - 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ y[4] + 2\ \[Lambda]\ \[Tau][1] - 8\ \[Lambda]\ \[Tau][2] + 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ \[Tau][2] + 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ \[Tau][2] + 2\ \[Gamma]\ \[Tau][3] + 2\ \[Beta]\ \[Theta]\^2\ \[Tau][3] + 12\ \[Lambda]\ \[Tau][3] + 2\ \[Psi]\ \[Alpha]\_1\%2\ \[Tau][3] + 2\ \[Psi]\ \[Alpha]\_2\%2\ \[Tau][3] + 2\ \[Mu]\ \[Delta]\_1\%2\ \[Tau][3] + 2\ \[Mu]\ \[Delta]\_2\%2\ \[Tau][3] - 8\ \[Lambda]\ \[Tau][4] + 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ \[Tau][4] + 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ \[Tau][4] + 2\ \[Lambda]\ \[Tau][5]\)}, {"4", $\(-2$\ \[Beta]\ \[Theta]\ \[Xi] + 2\ \[Psi]\ c[5]\ \[Alpha]\_1 + 2\ \[Psi]\ c[6]\ \[Alpha]\_2 + 2\ \[Mu]\ p[5]\ \[Delta]\_1 + 2\ \[Mu]\ p[6]\ \[Delta]\_2 + 2\ \[Beta]\ \[Theta]\ u[5] - 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ y[3] - 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ y[3] - 2\ \[Gamma]\ y[4] - 2\ \[Beta]\ \[Theta]\^2\ y[4] - 2\ \[Psi]\ \[Alpha]\_1\%2\ y[4] - 2\ \[Psi]\ \[Alpha]\_2\%2\ y[4] - 2\ \[Mu]\ \[Delta]\_1\%2\ y[4] - 2\ \[Mu]\ \[Delta]\_2\%2\ y[4] - 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ y[5] - 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ y[5] + 2\ \[Lambda]\ \[Tau][2] - 8\ \[Lambda]\ \[Tau][3] + 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ \[Tau][3] + 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ \[Tau][3] + 2\ \[Gamma]\ \[Tau][4] + 2\ \[Beta]\ \[Theta]\^2\ \[Tau][4] + 12\ \[Lambda]\ \[Tau][4] + 2\ \[Psi]\ \[Alpha]\_1\%2\ \[Tau][4] + 2\ \[Psi]\ \[Alpha]\_2\%2\ \[Tau][4] + 2\ \[Mu]\ \[Delta]\_1\%2\ \[Tau][4] + 2\ \[Mu]\ \[Delta]\_2\%2\ \[Tau][4] - 8\ \[Lambda]\ \[Tau][5] + 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ \[Tau][5] + 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ \[Tau][5] + 2\ \[Lambda]\ \[Tau][6]\)}, {"5", $\(-2$\ \[Beta]\ \[Theta]\ \[Xi] + 2\ \[Psi]\ c[6]\ \[Alpha]\_1 + 2\ \[Psi]\ c[7]\ \[Alpha]\_2 + 2\ \[Mu]\ p[6]\ \[Delta]\_1 + 2\ \[Mu]\ p[7]\ \[Delta]\_2 + 2\ \[Beta]\ \[Theta]\ u[6] - 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ y[4] - 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ y[4] - 2\ \[Gamma]\ y[5] - 2\ \[Beta]\ \[Theta]\^2\ y[5] - 2\ \[Psi]\ \[Alpha]\_1\%2\ y[5] - 2\ \[Psi]\ \[Alpha]\_2\%2\ y[5] - 2\ \[Mu]\ \[Delta]\_1\%2\ y[5] - 2\ \[Mu]\ \[Delta]\_2\%2\ y[5] - 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ y[6] - 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ y[6] + 2\ \[Lambda]\ \[Tau][3] - 8\ \[Lambda]\ \[Tau][4] + 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ \[Tau][4] + 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ \[Tau][4] + 2\ \[Gamma]\ \[Tau][5] + 2\ \[Beta]\ \[Theta]\^2\ \[Tau][5] + 12\ \[Lambda]\ \[Tau][5] + 2\ \[Psi]\ \[Alpha]\_1\%2\ \[Tau][5] + 2\ \[Psi]\ \[Alpha]\_2\%2\ \[Tau][5] + 2\ \[Mu]\ \[Delta]\_1\%2\ \[Tau][5] + 2\ \[Mu]\ \[Delta]\_2\%2\ \[Tau][5] - 8\ \[Lambda]\ \[Tau][6] + 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ \[Tau][6] + 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ \[Tau][6] + 2\ \[Lambda]\ \[Tau][7]\)}, {"6", $\(-2$\ \[Beta]\ \[Theta]\ \[Xi] + 2\ \[Psi]\ c[7]\ \[Alpha]\_1 + 2\ \[Psi]\ c[8]\ \[Alpha]\_2 + 2\ \[Mu]\ p[7]\ \[Delta]\_1 + 2\ \[Mu]\ p[8]\ \[Delta]\_2 + 2\ \[Beta]\ \[Theta]\ u[7] - 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ y[5] - 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ y[5] - 2\ \[Gamma]\ y[6] - 2\ \[Beta]\ \[Theta]\^2\ y[6] - 2\ \[Psi]\ \[Alpha]\_1\%2\ y[6] - 2\ \[Psi]\ \[Alpha]\_2\%2\ y[6] - 2\ \[Mu]\ \[Delta]\_1\%2\ y[6] - 2\ \[Mu]\ \[Delta]\_2\%2\ y[6] - 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ y[7] - 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ y[7] + 2\ \[Lambda]\ \[Tau][4] - 8\ \[Lambda]\ \[Tau][5] + 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ \[Tau][5] + 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ \[Tau][5] + 2\ \[Gamma]\ \[Tau][6] + 2\ \[Beta]\ \[Theta]\^2\ \[Tau][6] + 12\ \[Lambda]\ \[Tau][6] + 2\ \[Psi]\ \[Alpha]\_1\%2\ \[Tau][6] + 2\ \[Psi]\ \[Alpha]\_2\%2\ \[Tau][6] + 2\ \[Mu]\ \[Delta]\_1\%2\ \[Tau][6] + 2\ \[Mu]\ \[Delta]\_2\%2\ \[Tau][6] - 8\ \[Lambda]\ \[Tau][7] + 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ \[Tau][7] + 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ \[Tau][7] + 2\ \[Lambda]\ \[Tau][8]\)}, {"7", $\(-2$\ \[Beta]\ \[Theta]\ \[Xi] + 2\ \[Psi]\ c[8]\ \[Alpha]\_1 + 2\ \[Psi]\ c[9]\ \[Alpha]\_2 + 2\ \[Mu]\ p[8]\ \[Delta]\_1 + 2\ \[Mu]\ p[9]\ \[Delta]\_2 + 2\ \[Beta]\ \[Theta]\ u[8] - 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ y[6] - 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ y[6] - 2\ \[Gamma]\ y[7] - 2\ \[Beta]\ \[Theta]\^2\ y[7] - 2\ \[Psi]\ \[Alpha]\_1\%2\ y[7] - 2\ \[Psi]\ \[Alpha]\_2\%2\ y[7] - 2\ \[Mu]\ \[Delta]\_1\%2\ y[7] - 2\ \[Mu]\ \[Delta]\_2\%2\ y[7] - 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ y[8] - 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ y[8] + 2\ \[Lambda]\ \[Tau][5] - 8\ \[Lambda]\ \[Tau][6] + 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ \[Tau][6] + 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ \[Tau][6] + 2\ \[Gamma]\ \[Tau][7] + 2\ \[Beta]\ \[Theta]\^2\ \[Tau][7] + 12\ \[Lambda]\ \[Tau][7] + 2\ \[Psi]\ \[Alpha]\_1\%2\ \[Tau][7] + 2\ \[Psi]\ \[Alpha]\_2\%2\ \[Tau][7] + 2\ \[Mu]\ \[Delta]\_1\%2\ \[Tau][7] + 2\ \[Mu]\ \[Delta]\_2\%2\ \[Tau][7] - 8\ \[Lambda]\ \[Tau][8] + 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ \[Tau][8] + 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ \[Tau][8] + 2\ \[Lambda]\ \[Tau][9]\)}, {"8", $\(-2$\ \[Beta]\ \[Theta]\ \[Xi] + 2\ \[Psi]\ c[9]\ \[Alpha]\_1 + 2\ \[Psi]\ c[10]\ \[Alpha]\_2 + 2\ \[Mu]\ p[9]\ \[Delta]\_1 + 2\ \[Mu]\ p[10]\ \[Delta]\_2 + 2\ \[Beta]\ \[Theta]\ u[9] - 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ y[7] - 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ y[7] - 2\ \[Gamma]\ y[8] - 2\ \[Beta]\ \[Theta]\^2\ y[8] - 2\ \[Psi]\ \[Alpha]\_1\%2\ y[8] - 2\ \[Psi]\ \[Alpha]\_2\%2\ y[8] - 2\ \[Mu]\ \[Delta]\_1\%2\ y[8] - 2\ \[Mu]\ \[Delta]\_2\%2\ y[8] - 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ y[9] - 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ y[9] + 2\ \[Lambda]\ \[Tau][6] - 8\ \[Lambda]\ \[Tau][7] + 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ \[Tau][7] + 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ \[Tau][7] + 2\ \[Gamma]\ \[Tau][8] + 2\ \[Beta]\ \[Theta]\^2\ \[Tau][8] + 12\ \[Lambda]\ \[Tau][8] + 2\ \[Psi]\ \[Alpha]\_1\%2\ \[Tau][8] + 2\ \[Psi]\ \[Alpha]\_2\%2\ \[Tau][8] + 2\ \[Mu]\ \[Delta]\_1\%2\ \[Tau][8] + 2\ \[Mu]\ \[Delta]\_2\%2\ \[Tau][8] - 8\ \[Lambda]\ \[Tau][9] + 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ \[Tau][9] + 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ \[Tau][9] + 2\ \[Lambda]\ \[Tau][10]\)}, {"9", $\(-2$\ \[Beta]\ \[Theta]\ \[Xi] + 2\ \[Psi]\ c[10]\ \[Alpha]\_1 + 2\ \[Mu]\ p[10]\ \[Delta]\_1 + 2\ \[Beta]\ \[Theta]\ u[10] - 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ y[8] - 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ y[8] - 2\ \[Gamma]\ y[9] - 2\ \[Beta]\ \[Theta]\^2\ y[9] - 2\ \[Psi]\ \[Alpha]\_1\%2\ y[9] - 2\ \[Mu]\ \[Delta]\_1\%2\ y[9] + 2\ \[Lambda]\ \[Tau][7] - 8\ \[Lambda]\ \[Tau][8] + 2\ \[Psi]\ \[Alpha]\_1\ \[Alpha]\_2\ \[Tau][8] + 2\ \[Mu]\ \[Delta]\_1\ \[Delta]\_2\ \[Tau][8] + 2\ \[Gamma]\ \[Tau][9] + 2\ \[Beta]\ \[Theta]\^2\ \[Tau][9] + 10\ \[Lambda]\ \[Tau][9] + 2\ \[Psi]\ \[Alpha]\_1\%2\ \[Tau][9] + 2\ \[Mu]\ \[Delta]\_1\%2\ \[Tau][9] - 4\ \[Lambda]\ \[Tau][10]\)}, {"10", $2\ \[Lambda]\ \[Tau][8] - 4\ \[Lambda]\ \[Tau][9] + 2\ \[Lambda]\ \[Tau][10]$} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], (TableForm[ #]&)]], "Output", FontSize->9] }, Open ]], Cell[TextData[{ "You can see that these first-order conditions are still algebraic \ expressions. ", StyleBox["That\[CloseCurlyQuote]s okay", FontWeight->"Bold"], ": because they are in algebraic form, you can use them as general \ definitions for any dataset. With a little editing, they form the basis of \ the function we will construct to perform the filtering." }], "Text", FontSize->13] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["3. Constructing the function", "Subsubtitle"], Cell[TextData[ "This is the final function. The structure is almost identical to the \ univariate HP filter function \[Dash] not surprising since they are doing the \ same thing essentially, solving some linear equations given some data and \ parameter values. The only differences are the structure of the first-order \ condition equations and the number of parameters that the user must supply \ (shown in the square brackets next to the function name at the beginning of \ the definition)."], "Text", FontSize->13], Cell[TextData[{ "There are, however, a couple of catches to avoid when copying over the \ definitions for the first-order conditions into the ", StyleBox["Join", FontFamily->"Courier"], " statement in the function." }], "Text", FontSize->13], Cell[TextData[{ "Firstly, function definitions don\[CloseCurlyQuote]t work if you have a \ variable with a subscript representing one of the parameters provided by the \ user: therefore, wherever ", Cell[BoxData[ $TraditionalForm\`\(\[Alpha]\_1, \ \[Alpha]\_2\ \ \ $\)]], "and the like appear in the derivatives above, something simpler (eg \ \[Alpha]1, \[Alpha]2) must be substituted. Alternatively, one could use \ \[Alpha]1, \[Alpha]2 etc as the labels for those parameters in the original \ definitions of the residuals. " }], "Text", FontSize->13], Cell[TextData[{ "Secondly, the derivatives obtained above use single square brackets to \ denote time subscripts. However, if you are dealing with vectors of numbers, \ you need to refer to elements of that vector (or ", StyleBox["Parts", FontFamily->"Courier"], " in ", StyleBox["Mathematica", FontSlant->"Italic"], " terminology), by using the part specification double-brackets instead of \ single brackets. For example Part", StyleBox[" i", FontSlant->"Italic"], " of vector X is referred to like this:\n\tX[[i]], or if you want it to \ look pretty (and be easier to read) you can use the special double-brackets \ provided in ", StyleBox["Mathematica\[CloseCurlyQuote]s ", FontSlant->"Italic"], "special mathematical fonts: X\[LeftDoubleBracket]i\[RightDoubleBracket]. \ Therefore, some editing of brackets to double-brackets will be required. " }], "Text", FontSize->13], Cell[TextData[{ "Finally, don\[CloseCurlyQuote]t forget that the ", StyleBox["Table[] ", FontFamily->"Courier"], "statement covers the middle group of first-order conditions, so you need \ to re-edit the relevant first-order condition to be in a general form with \ the part specification in terms of the Table\[CloseCurlyQuote]s iterator \ variable (in this case, t)." }], "Text", FontSize->13], Cell[BoxData[ $multiHPFilter[y_?VectorQ, p_?VectorQ, u_?VectorQ, c_?VectorQ, {\[Lambda]_, \[Mu]_, \[Psi]_, \[Beta]_, \[Gamma]_, \[Delta]1_, \[Delta]2_, \[Alpha]1_, \[Alpha]2_, \[Theta]_, \[Xi]_}] := \n\tModule[{\[Tau], focs, taulist, answer}, \n\t\t With[{\[DoubleStruckCapitalT] = Length[y]}, \n\t\t\t taulist\ = \ Table[\[Tau][t], {t, 1, \[DoubleStruckCapitalT]}]; \n focs = \tThread[ Flatten[Join[{ \(-2.$\ \[Beta]\ \[Theta]\ \[Xi] + 2.\ \[Psi]\ c\[LeftDoubleBracket]2\[RightDoubleBracket]\ \[Alpha]1 + 2.\ \[Psi]\ c\[LeftDoubleBracket]3\[RightDoubleBracket]\ \[Alpha]2 + 2.\ \[Mu]\ p\[LeftDoubleBracket]2\[RightDoubleBracket]\ \[Delta]1 + 2.\ \[Mu]\ p\[LeftDoubleBracket]3\[RightDoubleBracket]\ \[Delta]2 + 2.\ \[Beta]\ \[Theta]\ u\[LeftDoubleBracket]2\[RightDoubleBracket] - 2.\ \[Gamma]\ y\[LeftDoubleBracket]1\[RightDoubleBracket] - 2.\ \[Beta]\ \[Theta]\^2\ y\[LeftDoubleBracket]1\[RightDoubleBracket] - 2.\ \[Psi]\ \[Alpha]1\^2\ y\[LeftDoubleBracket]1\[RightDoubleBracket] - 2.\ \[Psi]\ \[Alpha]2\^2\ y\[LeftDoubleBracket]1\[RightDoubleBracket] - 2.\ \[Mu]\ \[Delta]1\^2\ y\[LeftDoubleBracket]1\[RightDoubleBracket] - 2.\ \[Mu]\ \ \[Delta]2\^2\ y\[LeftDoubleBracket]1\[RightDoubleBracket] - 2.\ \[Psi]\ \[Alpha]1\ \[Alpha]2\ y\[LeftDoubleBracket]2\[RightDoubleBracket] - 2.\ \[Mu]\ \[Delta]1\ \[Delta]2\ y\[LeftDoubleBracket]2\[RightDoubleBracket] + 2.\ \[Gamma]\ \[Tau][1] + 2.\ \[Beta]\ \[Theta]\^2\ \[Tau][1] + 2.\ \[Lambda]\ \[Tau][1] + 2.\ \[Psi]\ \[Alpha]1\^2\ \[Tau][1] + 2.\ \[Psi]\ \[Alpha]2\^2\ \[Tau][1] + 2.\ \[Mu]\ \[Delta]1\^2\ \[Tau][1] + 2.\ \[Mu]\ \[Delta]2\^2\ \[Tau][1] - 4.\ \[Lambda]\ \[Tau][2] + 2.\ \[Psi]\ \[Alpha]1\ \[Alpha]2\ \[Tau][2] + 2.\ \[Mu]\ \[Delta]1\ \[Delta]2\ \[Tau][2] + 2.\ \[Lambda]\ \[Tau][3], \n\t\t\t\t\t\t $-2$\ \[Beta]\ \[Theta]\ \[Xi] + \t 2.\ \[Psi]\ c\[LeftDoubleBracket]3\[RightDoubleBracket]\ \[Alpha]1 + 2.\ \[Psi]\ c\[LeftDoubleBracket]4\[RightDoubleBracket]\ \[Alpha]2 + 2.\ \[Mu]\ p\[LeftDoubleBracket]3\[RightDoubleBracket]\ \[Delta]1 + 2.\ \[Mu]\ p\[LeftDoubleBracket]4\[RightDoubleBracket]\ \[Delta]2 + 2.\ \[Beta]\ \[Theta]\ u\[LeftDoubleBracket]3\[RightDoubleBracket] - 2.\ \[Psi]\ \[Alpha]1\ \[Alpha]2\ y\[LeftDoubleBracket]1\[RightDoubleBracket] - 2.\ \[Mu]\ \[Delta]1\ \[Delta]2\ y\[LeftDoubleBracket]1\[RightDoubleBracket] - 2.\ \[Gamma]\ y\[LeftDoubleBracket]2\[RightDoubleBracket] - 2.\ \[Beta]\ \[Theta]\^2\ y\[LeftDoubleBracket]2\[RightDoubleBracket] - 2\ \[Psi]\ \[Alpha]1\^2\ y\[LeftDoubleBracket]2\[RightDoubleBracket] - 2.\ \[Psi]\ \[Alpha]2\^2\ y\[LeftDoubleBracket]2\[RightDoubleBracket] - 2.\ \[Mu]\ \[Delta]1\^2\ y\[LeftDoubleBracket]2\[RightDoubleBracket] - 2.\ \[Mu]\ \[Delta]2\^2\ y\[LeftDoubleBracket]2\[RightDoubleBracket] - 2.\ \[Psi]\ \[Alpha]1\ \[Alpha]2\ y\[LeftDoubleBracket]3\[RightDoubleBracket] - 2.\ \[Mu]\ \[Delta]1\ \[Delta]2\ y\[LeftDoubleBracket]3\[RightDoubleBracket] - 4.\ \[Lambda]\ \[Tau][1] + 2.\ \[Psi]\ \[Alpha]1\ \[Alpha]2\ \[Tau][1] + 2.\ \[Mu]\ \[Delta]1\ \[Delta]2\ \[Tau][1] + 2.\ \[Gamma]\ \[Tau][2] + 2.\ \[Beta]\ \[Theta]\^2\ \[Tau][2] + 10.\ \[Lambda]\ \[Tau][2] + 2.\ \[Psi]\ \[Alpha]1\^2\ \[Tau][2] + 2.\ \[Psi]\ \[Alpha]2\^2\ \[Tau][2] + 2.\ \[Mu]\ \[Delta]1\^2\ \[Tau][2] + 2.\ \[Mu]\ \[Delta]2\^2\ \[Tau][2] - 8.\ \[Lambda]\ \[Tau][3] + 2.\ \[Psi]\ \[Alpha]1\ \[Alpha]2\ \[Tau][3] + 2.\ \[Mu]\ \[Delta]1\ \[Delta]2\ \[Tau][3] + 2.\ \[Lambda]\ \[Tau][4]}, \n Table[$-2$\ \[Beta]\ \[Theta]\ \[Xi] + 2.\ \[Psi]\ c\[LeftDoubleBracket]t + 1\[RightDoubleBracket]\ \[Alpha]1 + 2.\ \[Psi]\ c\[LeftDoubleBracket]t + 2\[RightDoubleBracket]\ \[Alpha]2 + 2.\ \[Mu]\ p\[LeftDoubleBracket]t + 1\[RightDoubleBracket]\ \[Delta]1 + 2.\ \[Mu]\ p\[LeftDoubleBracket]t + 2\[RightDoubleBracket]\ \[Delta]2 + 2.\ \[Beta]\ \[Theta]\ u\[LeftDoubleBracket]t + 1\[RightDoubleBracket] - 2.\ \[Psi]\ \[Alpha]1\ \[Alpha]2\ y\[LeftDoubleBracket]t - 1\[RightDoubleBracket] - 2.\ \[Mu]\ \[Delta]1\ \[Delta]2\ y\[LeftDoubleBracket]t - 1\[RightDoubleBracket] - 2.\ \[Gamma]\ y\[LeftDoubleBracket]t\[RightDoubleBracket] - 2.\ \[Beta]\ \[Theta]\^2\ y\[LeftDoubleBracket]t\[RightDoubleBracket] - 2.\ \[Psi]\ \[Alpha]1\^2\ y\[LeftDoubleBracket]t\[RightDoubleBracket] - 2.\ \[Psi]\ \[Alpha]2\^2\ y\[LeftDoubleBracket]t\[RightDoubleBracket] - 2.\ \[Mu]\ \[Delta]1\^2\ y\[LeftDoubleBracket]t\[RightDoubleBracket] - 2.\ \[Mu]\ \[Delta]2\^2\ y\[LeftDoubleBracket]t\[RightDoubleBracket] - 2.\ \[Psi]\ \[Alpha]1\ \[Alpha]2\^2\ y\[LeftDoubleBracket]t + 1\[RightDoubleBracket] - 2.\ \[Mu]\ \[Delta]1\ \[Delta]2\ y\[LeftDoubleBracket]t + 1\[RightDoubleBracket] + 2.\ \[Lambda]\ \[Tau][t - 2] - 8.\ \[Lambda]\ \[Tau][t - 1] + 2.\ \[Psi]\ \[Alpha]1\ \[Alpha]2\ \[Tau][t - 1] + 2.\ \[Mu]\ \[Delta]1\ \[Delta]2\ \[Tau][t - 1] + 2.\ \[Gamma]\ \[Tau][t] + 2.\ \[Beta]\ \[Theta]\^2\ \[Tau][t] + 12.\ \[Lambda]\ \[Tau][t] + 2.\ \[Psi]\ \[Alpha]1\^2\ \[Tau][t] + 2.\ \[Psi]\ \[Alpha]2\^2\ \[Tau][t] + 2.\ \[Mu]\ \[Delta]1\^2\ \[Tau][t] + 2.\ \[Mu]\ \[Delta]2\^2\ \[Tau][t] - 8.\ \[Lambda]\ \[Tau][t + 1] + 2.\ \[Psi]\ \[Alpha]1\ \[Alpha]2\ \[Tau][t + 1] + 2.\ \[Mu]\ \[Delta]1\ \[Delta]2\ \[Tau][t + 1] + 2.\ \[Lambda]\ \[Tau][t + 2], {t, 3, \[DoubleStruckCapitalT] - 2}], \n \t\t{$-2$\ \[Beta]\ \[Theta]\ \[Xi] + 2.\ \[Psi]\ c\[LeftDoubleBracket]\[DoubleStruckCapitalT]\ \[RightDoubleBracket]\ \[Alpha]1 + 2.\ \[Mu]\ p\[LeftDoubleBracket]\[DoubleStruckCapitalT]\ \[RightDoubleBracket] \[Delta]1 + 2.\ \[Beta]\ \[Theta]\ u\[LeftDoubleBracket]\[DoubleStruckCapitalT]\ \[RightDoubleBracket] - 2.\ \[Psi]\ \[Alpha]1\ \[Alpha]2\ y\[LeftDoubleBracket]\[DoubleStruckCapitalT] - 2 \[RightDoubleBracket] - 2.\ \[Mu]\ \[Delta]1\ \[Delta]2\ y\[LeftDoubleBracket]\[DoubleStruckCapitalT] - 2 \[RightDoubleBracket] - 2.\ \[Gamma]\ y\[LeftDoubleBracket]\[DoubleStruckCapitalT] - 1 \[RightDoubleBracket] - 2.\ \[Beta]\ \[Theta]\^2\ y\[LeftDoubleBracket]\[DoubleStruckCapitalT] - 1 \[RightDoubleBracket] - 2.\ \[Psi]\ \[Alpha]1\^2\ y\[LeftDoubleBracket]\[DoubleStruckCapitalT] - 1 \[RightDoubleBracket] - 2.\ \[Mu]\ \[Delta]1\^2\ y\[LeftDoubleBracket]\[DoubleStruckCapitalT] - 1 \[RightDoubleBracket] + 2.\ \[Lambda]\ \[Tau][\[DoubleStruckCapitalT] - 3] - 8.\ \[Lambda]\ \[Tau][\[DoubleStruckCapitalT] - 2] + 2.\ \[Psi]\ \[Alpha]1\ \[Alpha]2\ \[Tau][\[DoubleStruckCapitalT] - 2] + 2.\ \[Mu]\ \[Delta]1\ \[Delta]2\ \[Tau][\[DoubleStruckCapitalT] - 2] + 2.\ \[Gamma]\ \[Tau][\[DoubleStruckCapitalT] - 1] + 2.\ \[Beta]\ \[Theta]\^2\ \[Tau][\[DoubleStruckCapitalT] - 1] + 10.\ \[Lambda]\ \[Tau][\[DoubleStruckCapitalT] - 1] + 2.\ \[Psi]\ \[Alpha]1\^2\ \[Tau][\[DoubleStruckCapitalT] - 1] + 2.\ \[Mu]\ \[Delta]1\^2\ \[Tau][\[DoubleStruckCapitalT] - 1] - 4.\ \[Lambda]\ \[Tau][\[DoubleStruckCapitalT]], 2.\ \[Lambda]\ \[Tau][\[DoubleStruckCapitalT] - 2] - 4.\ \[Lambda]\ \[Tau][\[DoubleStruckCapitalT] - 1] + 2.\ \[Lambda]\ \[Tau][\[DoubleStruckCapitalT]]}]] == 0]; \n\ \ \ \ \ \ \ \ answer = Solve[focs, taulist]; \n\t\t Flatten[Expand[N[taulist /. answer]]]\n\t\t\t\t\t]]\)], "Input", PageWidth->PaperWidth, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["4. A numerical example", "Subsubtitle"], Cell[CellGroupData[{ Cell["A very important caveat!", "Section"], Cell[TextData[{ "A multivariate HP filter requires the user to choose the loss-function \ weights on the squared residuals from the conditioning equations (ie ", Cell[BoxData[ $TraditionalForm\`\[Epsilon]\_p, \ \[Epsilon]\_u, \ and\ so\ on$]], "). 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Cell[StyleData["Subsubtitle", "Printout"], CellMargins->{{2, 10}, {8, 10}}, FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Section"], CellDingbat->"\[FilledSquare]", CellMargins->{{25, Inherited}, {8, 24}}, CellGroupingRules->{"SectionGrouping", 30}, PageBreakBelow->False, CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, FontFamily->"Helvetica", FontSize->16, FontWeight->"Bold"], Cell[StyleData["Section", "Presentation"], CellMargins->{{40, 10}, {11, 32}}, LineSpacing->{1, 0}, FontSize->24], Cell[StyleData["Section", "Condensed"], CellMargins->{{18, Inherited}, {6, 12}}, FontSize->12], Cell[StyleData["Section", "Printout"], CellMargins->{{0, 0}, {7, 22}}, FontSize->14] }, Open ]], Cell[CellGroupData[{ Cell[StyleData["Subsection"], CellDingbat->"\[FilledSmallSquare]", CellMargins->{{22, Inherited}, {8, 20}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, CounterIncrements->"Subsection", 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Cell[CellGroupData[{ Cell["Styles for Input/Output", "Section"], Cell["\<\ The cells in this section define styles used for input and output \ to the kernel. 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PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, CellLabelMargins->{{11, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Output"], Cell[StyleData["Output", "Presentation"], CellMargins->{{72, Inherited}, {10, 8}}, LineSpacing->{1, 0}], Cell[StyleData["Output", "Condensed"], CellMargins->{{41, Inherited}, {3, 2}}], Cell[StyleData["Output", "Printout"], CellMargins->{{12, 0}, {6, 4}}, FontSize->9] }, Open ]], Cell[CellGroupData[{ Cell[StyleData["Message"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{11, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Message", StyleMenuListing->None, FontColor->RGBColor[0, 0, 1]], Cell[StyleData["Message", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, Inherited}}, LineSpacing->{1, 0}], Cell[StyleData["Message", "Condensed"], CellMargins->{{41, Inherited}, {Inherited, Inherited}}], Cell[StyleData["Message", "Printout"], CellMargins->{{39, Inherited}, {Inherited, Inherited}}, FontSize->8, FontColor->GrayLevel[0]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{11, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Print", StyleMenuListing->None], Cell[StyleData["Print", "Presentation"], CellMargins->{{72, Inherited}, {Inherited, 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Cell[StyleData["DisplayFormula"], CellMargins->{{42, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling->True, ScriptLevel->0, SingleLetterItalics->True, StyleMenuListing->None, UnderoverscriptBoxOptions->{LimitsPositioning->True}], Cell[StyleData["DisplayFormula", "Presentation"], LineSpacing->{1, 5}], Cell[StyleData["DisplayFormula", "Condensed"], LineSpacing->{1, 1}], Cell[StyleData["DisplayFormula", "Printout"]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headers and Footers", "Section"], Cell[StyleData["Header"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontSize->10, FontSlant->"Italic"], Cell[StyleData["Footer"], CellMargins->{{0, 0}, {0, 4}}, StyleMenuListing->None, FontSize->9, FontSlant->"Italic"], Cell[StyleData["PageNumber"], CellMargins->{{0, 0}, {4, 1}}, StyleMenuListing->None, FontFamily->"Times", FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell["Palette Styles", "Section"], Cell["\<\ The cells below define styles that define 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