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Notebook[{
Cell[BoxData[
    \(<< Statistics`ContinuousDistributions`\)], "Input"],

Cell[BoxData[
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Cell[TextData[{
  "The function 'p' generates the next price.  The function path generates a \
price path with N observations, starting from an initial price of p",
  StyleBox["0 = 60.  The function 'cEsttStat' finds estimates of 'c' and the \
t-statistic for a linear regression model and for a Dickey-Fuller test using \
a price path generated by the function 'path'.  This function returns an \
estimate of 'c', its standard error, and its t-statistic for the OLS model \
(as its first three output values), and the same three statisitics from the \n\
Dickey-Fuller model (as its fourth through sixth output values).", 
    "InlineFormula"]
}], "Subsection"],

Cell[BoxData[
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          lhs, \ rhs, \ d2}, \[IndentingNewLine]pr\  = \ 
          path[N]; \[IndentingNewLine] (*\ 
          OLS\ estimates\ *) \ \[IndentingNewLine]pr1\  = \ 
          Drop[pr, \ \(-1\)]; \[IndentingNewLine]pr2\  = \ 
          Drop[pr, \ 1]; \[IndentingNewLine]d1\  = \ 
          Transpose[{pr1, \ pr2}]; \[IndentingNewLine]r1\  = \ 
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            x]; \[IndentingNewLine]c1\  = \ \(\(\(\(r1[\([1]\)]\)[\([2]\)]\)[\
\([1]\)]\)[\([2]\)]\)[\([1]\)]; \[IndentingNewLine]se1\  = \ \(\(\(\(r1[\([1]\
\)]\)[\([2]\)]\)[\([1]\)]\)[\([2]\)]\)[\([2]\)]; \[IndentingNewLine]t1\  = \ \
\((1\  - \ c1)\)/se1; \[IndentingNewLine] (*\ 
          Dickey - Fuller\ estimates\ *) \ \[IndentingNewLine]diff\  = \ 
          pr\  - \ 91; \[IndentingNewLine]lhs\  = \ 
          Drop[diff, \ 1]\  - \ 
            Drop[diff, \ \(-1\)]; \[IndentingNewLine]rhs\  = \ \(-\ 
            Drop[diff, \ \(-1\)]\); \[IndentingNewLine]d2\  = \ 
          Transpose[{rhs, \ lhs}]; \[IndentingNewLine]r2\  = \ 
          Regress[d2, \ {x}, \ x, \ 
            IncludeConstant\  \[Rule] \ 
              False]; \[IndentingNewLine]c2\  = \ \(\(\(\(r2[\([1]\)]\)[\([2]\
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\(\(\(\(r2[\([1]\)]\)[\([2]\)]\)[\([1]\)]\)[\([1]\)]\)[\([2]\)]; \
\[IndentingNewLine]t2\  = \
\(\(\(\(r2[\([1]\)]\)[\([2]\)]\)[\([1]\)]\)[\([1]\)]\)[\([3]\)]; \
\[IndentingNewLine]{c1, \ se1, \ t1, \ c2, \ se2, \ t2}]\)], "Input"]
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Cell["Simulations with 46 price observations ", "Subsection"],

Cell[BoxData[
    \(\(data\  = \ 
        Transpose[Table[cEsttStat[45], \ {i, \ 1, \ 10000}]];\)\)], "Input"],

Cell[CellGroupData[{

Cell["\<\
The first vector in 'data' is  a list of values of the estimates '1 - a' for \
OLS.  The fourth vector in 'data' is a list of estiamtes of 'a' from the \
Dickey-Fuller test.  The mean of the estimate from OLS is almost twice the \
actual value (a = 0.02), whereas the mean of the estimates from the \
Dicley-Fuller test is close to the actual parameter value.\
\>", "Subsubsection"],

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Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
The 95th percentile of the t distribution with 44 degrees of freedom is \
1.68023.\
\>", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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