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Notebook[{

Cell[CellGroupData[{
Cell["Econometric Functions", "Title"],

Cell[TextData[{
  "A Warning Note!!!:  ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "'s built-in ",
  StyleBox["Regress[]",
    FontWeight->"Bold"],
  " function assumes data in the form ",
  Cell[BoxData[
      \(TraditionalForm
      \`{{\(x\_1\)\_1, \(x\_2\)\_1, ..., \(x\_k\)\_1, y\_1}, {\(x\_1\)\_2, 
          \(x\_2\)\_2, ..., \(x\_k\)\_2, y\_2}, ..., {\(x\_1\)\_T, 
          \(x\_2\)\_T, ..., \(x\_k\)\_T, y\_T}}\)]],
  " but the ",
  StyleBox["simpleLinRegress[ ]",
    FontWeight->"Bold"],
  " here and used in some of the other functions assumes data of the form: ",
  Cell[BoxData[
      \(TraditionalForm
      \`{{y\_1, \(x\_1\)\_1, \(x\_2\)\_1, ..., \(x\_k\)\_1}, {y\_2, 
          \(x\_1\)\_2, \(x\_2\)\_2, ..., \(x\_k\)\_2}, ..., {y\_T, 
          \(x\_1\)\_T, \(x\_2\)\_T, ..., \(x\_k\)\_T}}\)]],
  ".  This seemed to me to make more sense than the other presentation \
because we normally think of equations as reading from left to right with the \
regressand on the left-hand side and the regressors on the right.  It is also \
in line with the data ordering used in packages like Shazam."
}], "Text"],

Cell[CellGroupData[{

Cell["\<\
simpleLinRegress[ ]: A Simple Linear Regression function built for \
speed not features\
\>", "Section"],

Cell[CellGroupData[{

Cell[BoxData[
    \(simpleLinRegress::usage = "\<simpleLinRegress[data_MatrixQ] takes a \
rectangular matrix of real numbers and performs a linear regression or \
weighted linear regression of the first column on the other columns. The \
output is a non-rectangular list of three lists -- the estimated \
coefficients, the fitted values and the residuals.\>"\)], "Input"],

Cell[BoxData[
    \("simpleLinRegress[data_MatrixQ] takes a rectangular matrix of real \
numbers and performs a linear regression or weighted linear regression of the \
first column on the other columns. The output is a non-rectangular list of \
three lists -- the estimated coefficients, the fitted values and the \
residuals."\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(GLSWeights::usage\  = \ "\<GLSWeights is an option to simpleLinRegress \
that takes either the value Automatic, corresponding to OLS, or a list of \
real numbers corresponding to the weights to be used in a WLS \
procedure.\>"\)], "Input"],

Cell[BoxData[
    \("GLSWeights is an option to simpleLinRegress that takes either the \
value Automatic, corresponding to OLS, or a list of real numbers \
corresponding to the weights to be used in a WLS procedure."\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(IncludeConstant::usage\  = \ "\<IncludeConstant is an option to \
simpleLinRegress that indicates whether or not a constant should be added to \
the data matrix.\>"\)], "Input"],

Cell[BoxData[
    \("IncludeConstant is an option to simpleLinRegress that indicates \
whether or not a constant should be added to the data matrix."\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(simpleLinRegress::wtnum = "\<Either the vector of weights is not a \
vector of real numbers with the same number of element as the data sample \
size, or you have tried to apply WLS to a model with a single explanator and \
no constant.\>"\)], "Input"],

Cell[BoxData[
    \("Either the vector of weights is not a vector of real numbers with the \
same number of element as the data sample size, or you have tried to apply \
WLS to a model with a single explanator and no constant."\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(simpleLinRegress::notnum\  = \ "\<Some of the data supplied is \
non-numeric.\>"\)], "Input"],

Cell[BoxData[
    \("Some of the data supplied is non-numeric."\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Options[simpleLinRegress] = {GLSWeights -> Automatic, 
        IncludeConstant \[Rule] True}\)], "Input"],

Cell[BoxData[
    \({GLSWeights \[Rule] Automatic, 
      IncludeConstant \[Rule] True}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(SetOptions[simpleLinRegress, GLSWeights -> Automatic, 
      IncludeConstant \[Rule] True]\)], "Input"],

Cell[BoxData[
    \({GLSWeights \[Rule] Automatic, 
      IncludeConstant \[Rule] True}\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(simpleLinRegress[data_?MatrixQ, \ opts___?OptionQ] := \ 
      If[And\  @@ \ \((NumericQ\  /@ \ 
              Flatten[data])\), \n\t\tModule[{\[ScriptW], \[ScriptC]}, \
\[ScriptW]\  = \ \(GLSWeights\  /. {opts}\)\  /. 
              Options[simpleLinRegress]; \n\t\t\[ScriptC]\  = \ \
\(IncludeConstant\  /. {opts}\)\  /. 
              Options[simpleLinRegress]; \n\t\tWith[{\[ScriptY] = 
                If[Not[\[ScriptC]] && \ \(Dimensions[
                          data]\)\[LeftDoubleBracket]2\[RightDoubleBracket] == 
                      2, Flatten[First /@ data], 
                  First\  /@ \ data], \n\t\t\t\t\t\t\t\ \ \[ScriptX]\  = \ 
                If[\[ScriptC], 
                  Transpose[
                    Join[{Table[1, {Length[data]}]}, 
                      Transpose[Rest\  /@ \ data]]], 
                  If[\(Dimensions[
                          Rest\  /@ 
                            data]\)\[LeftDoubleBracket]2\[RightDoubleBracket] \
== 1, Flatten[Rest\  /@ \ data], Transpose[Rest\  /@ \ data]]], \[ScriptN] = 
                Length[data]}, \n\t\t\tModule[{betas, yhat, \ 
                resids}, \n\t\t\t\tWhich[\[ScriptW] === 
                  Automatic, \(betas\  = \ 
                    If[Not[\[ScriptC]] && \(Dimensions[
                              Rest\  /@ 
                                data]\)\[LeftDoubleBracket]2\
\[RightDoubleBracket] == 
                          1, {\((1/\((\[ScriptX] . \[ScriptX])\))\)*\((\
\[ScriptX] . \[ScriptY])\)}, 
                      Flatten[
                        Inverse[\((Transpose[\ \[ScriptX]] . \ \[ScriptX])\)] \
. \((Transpose[\ \[ScriptX]] . \[ScriptY])\)]];\), \n\t\t\t\t\t\((Length[\
\[ScriptW]] == \[ScriptN]\ )\) && \((Length[\[ScriptW]] > 
                      1\ )\) && \ \((And\  @@ \ \((NumericQ\  /@ \ \
\[ScriptW])\))\), \ \(betas\  = 
                    Inverse[
                        Transpose[\((\[ScriptW]*\[ScriptX])\)] . \
\((\[ScriptX])\)] . \((\((Transpose[\[ScriptW]*\[ScriptX]])\) . \
\((\[ScriptY])\))\);\), True, \ 
                Message[
                  simpleLinRegress::wtnum]; \ $Failed]; \n\t\t\t\tyhat\  = 
                If[Not[\[ScriptC]] && \ \(Dimensions[
                          data]\)\[LeftDoubleBracket]2\[RightDoubleBracket] == 
                      2, \[ScriptX]*
                    betas\[LeftDoubleBracket]1\[RightDoubleBracket], \ \
\[ScriptX] . betas]; \n\t\t\t\tresids\  = \ \[ScriptY] - 
                  yhat; \n\t\t\t\t{betas, yhat, resids}]]], 
        Message[simpleLinRegress::notnum]]\)], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["ARLagOrderSelectionTable[ ]", "Section"],

Cell["\<\
This function has been updated thanks to helpful comments by Virgil \
Stokes.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(ARLagOrderSelectionTable::usage = \
"\<ARLagOrderSelectionTable[data,maxp] takes a time series (vector of real \
numbers) and generates a table of various Information Criteria used for \
selecting the optimal lag order in an AR model describing the time series. It \
also presents the estimated coefficients of each implied AR model (with the \
constant first).  This function depends on the simpleLinRegress[] \
function.\>"\)], "Input"],

Cell[BoxData[
    \("ARLagOrderSelectionTable[data,maxp] takes a time series (vector of \
real numbers) and generates a table of various Information Criteria used for \
selecting the optimal lag order in an AR model describing the time \
series."\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(ARLagOrderSelectionTable[data : {__Real}, \ 
          maxp_Integer]\  /; \ \((maxp\  < \ Length[data])\)\  := \n\t
      Module[{\[Sigma]2, \[Beta]s\ , \ T = Length[data]}, \n\t\tTable[
          With[{result = 
                simpleLinRegress[
                  Transpose[
                    Table[Drop[RotateRight[data, k], j], \ {k, 0, j}]], 
                  IncludeConstant \[Rule] True]}, \[Sigma]2[j] = 
              Plus\  @@ \
\((\((result\[LeftDoubleBracket]3\[RightDoubleBracket])\)^2)\); \[Beta]s[j] = 
              First[result];], {j, maxp}]; \ \n\t
        Join[{{"\<p\>", \*"\"\<\!\(\[Sigma]\^2\)\>\"", "\<AIC(p)\>", \
"\<AICC(p)\>", "\<SC(p)\>", \ "\<Hannan-Quinn(p)\>", "\<coefs\>"}}, 
          Table[{i, \[Sigma]2[i], Log[\((\[Sigma]2[i])\)^2] + \ 2\ i\ /T, \ 
              T*\ Log[\((\[Sigma]2[i])\)^2] + 
                T\ \((1\  + \ i/T)\)/\((1 - \((i + 2)\)\ T)\), 
              Log[\[Sigma]2[i]] + i\ Log[T]/T, \ 
              Log[\[Sigma]2[i]]\  + \ 
                2\ i\ *Log[Log[T]]/T, {\[Beta]s[i]}}, {i, 1, 
              maxp}]\ ]\ ]\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["LjungBoxStatistic[ ]", "Section"],

Cell[BoxData[
    \(LjungBoxStatistic::usage = 
      "\<LjungBoxStatistic[data,k] takes a vector of real numbers and \
generates the Ljung-Box(k) test statistic and critical value. The Ljung-Box \
test statistic is used for establishing serial correlation and is distributed \
as Chi-square(k).\>"\)], "Input"],

Cell[BoxData[
    \(Needs["\<Statistics`NormalDistribution`\>"]\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(ljungboxcompiledteststat = 
      Compile[{{x, _Real, 1}, {k, _Integer}}, \ 
        \[Sum]\+\(i = 1\)\%k\((
            \((Plus@@\((Drop[x, \(-i\)]\[Times]Drop[x, i])\)/
                    \((\(\@Plus@@\((Drop[x, \(-i\)]\^2)\)\) 
                        \@Plus@@\((Drop[x, i]\^2)\))\))\)^2\ 
              \(Length[x] \((Length[x] - 2)\)\)\/\(Length[x] - i\))\)]\)], 
  "Input"],

Cell[BoxData[
    TagBox[
      \(CompiledFunction[{x, k}, 
        \[Sum]\+\(i = 1\)\%k
            \(\((Plus@@
                      \((Drop[x, \(-i\)]\ Drop[x, i])
                        \)\/\(\@Plus@@\(Drop[x, \(-i\)]\^2\)\ 
                      \@Plus@@\(Drop[x, i]\^2\)\))\)\^2\ 
              \((Length[x]\ \((Length[x] - 2)\))\)\)\/\(Length[x] - i\), 
        "-CompiledCode-"]\),
      False,
      Editable->False]], "Output"]
}, Open  ]],

Cell[BoxData[
    \(LjungBoxStatistic[data : {__Real}, k_Integer] := \n
      TableForm[
        \t{{"\<test stat\>", "\<5% critical value\>"}, {
            ljungboxcompiledteststat[data, k], 
            Quantile[ChiSquareDistribution[k], 0.95]}}]\)], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["CochraneOrcutt[ ]", "Section"],

Cell[CellGroupData[{

Cell[BoxData[
    \(CochraneOrcutt::usage = 
      "\<CochraneOrcutt[data] takes a matrix of real numbers and performs the \
Cochrane-Orcutt iterative procedure for estimating linear models in the \
presence of AR(1) serial correlation in the errors.\>"\)], "Input"],

Cell[BoxData[
    \("CochraneOrcutt[data] takes a matrix of real numbers and performs the \
Cochrane-Orcutt iterative procedure for estimating linear models in the \
presence of AR(1) serial correlation in the errors."\)], "Output"]
}, Open  ]],

Cell["\<\
Needs[\"Statistics`NormalDistribution`\"];
Needs[\"Statistics`ContinuousDistributions`\"];
Needs[\"Statistics`DataManipulation`\"];
Needs[\"Statistics`DescriptiveStatistics`\"];
Needs[\"Statistics`MultiDescriptiveStatistics`\"];
Needs[\"Statistics`LinearRegression`\"];
Needs[\"Statistics`DataSmoothing`\"];\
\>", "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(CochraneOrcutt[data_?MatrixQ] := \ 
      FixedPoint[
        With[{e = 
                \(simpleLinRegress[#, IncludeConstant \[Rule] True]
                    \)\[LeftDoubleBracket]3\[RightDoubleBracket]}, 
            With[{rhohat = 
                  Rest[e].Drop[e, \(-1\)]/Plus@@\((Drop[e, \(-1\)]^2)\)}, 
              Join[{\@\(1 - rhohat\^2\)\[Times]First[#]}, 
                Rest[#] - rhohat\ Drop[#, \(-1\)]]]]&, data]\)], "Input"]
}, Closed]]
}, Open  ]]
},
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]


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***********************************************************************)