Injecting cross-sectional and temporal correlations into a random matrix











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Let $X$ be a random matrix, constituted of 4 vectors of time series $X_i,t$, $i=1,ldots,4$, $t=1,ldots,3$.



$$X=left(
begin{array}{cccc}
x_{1,1} & x_{2,1} & x_{3,1} & x_{4,1} \
x_{1,2} & x_{2,2} & x_{3,2} & x_{4,2} \
x_{1,3} & x_{2,3} & x_{3,3} & x_{4,3} \
end{array}
right).$$
I am trying to inject both temporal correlation between $X_{i,t}$ and $X_{i,t+1}$ and cross-sectional between $X_{i,t}$ and $X_{i+1,t}$.



We start with temporal correlation, with $epsilon$ as random noise, $mathbb{E}(epsilon_{i,j})=0$.
Rewriting
$$X=left(
begin{array}{cccc}
x_{1,1} & x_{2,1} & x_{3,1} & x_{4,1} \
rho _1 x_{1,1}+epsilon _{1,1} & rho _2 x_{2,1}+epsilon _{2,1} & rho _3 x_{3,1}+epsilon _{3,1} & rho _4 x_{4,1}+epsilon _{4,1} \
rho _1 left(rho _1 x_{1,1}+epsilon _{1,1}right)+epsilon _{1,2} & rho _2 left(rho _2 x_{2,1}+epsilon _{2,1}right)+epsilon _{2,2} & rho _3 left(rho _3 x_{3,1}+epsilon _{3,1}right)+epsilon _{3,2} & rho _4 left(rho _4 x_{4,1}+epsilon _{4,1}right)+epsilon _{4,2} \
end{array}
right)$$
and add cross-sectional correlation, rewriting
$X=left(
begin{array}{cccc}
x_{1,1} & frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,1} & frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,1} & frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,1} \
rho _1 x_{1,1}+epsilon _{1,1} & rho _2 left(frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,2}right)+epsilon _{2,1} & rho _3 left(frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,2}right)+epsilon _{3,1} & rho _4 left(frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,2}right)+epsilon _{4,1} \
rho _1 left(rho _1 x_{1,1}+epsilon _{1,1}right)+epsilon _{1,2} & rho _2 left(rho _2 left(frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,1}right)+epsilon _{2,1}right)+epsilon _{2,2} & rho _3 left(rho _3 left(frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,1}right)+epsilon _{3,1}right)+epsilon _{3,2} & rho _4 left(rho _4 left(frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,1}right)+epsilon _{4,1}right)+epsilon _{4,2} \
end{array}
right)$
Am I doing it right?






probability random-matrices







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    up vote
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    down vote

    favorite












    Let $X$ be a random matrix, constituted of 4 vectors of time series $X_i,t$, $i=1,ldots,4$, $t=1,ldots,3$.



    $$X=left(
    begin{array}{cccc}
    x_{1,1} & x_{2,1} & x_{3,1} & x_{4,1} \
    x_{1,2} & x_{2,2} & x_{3,2} & x_{4,2} \
    x_{1,3} & x_{2,3} & x_{3,3} & x_{4,3} \
    end{array}
    right).$$
    I am trying to inject both temporal correlation between $X_{i,t}$ and $X_{i,t+1}$ and cross-sectional between $X_{i,t}$ and $X_{i+1,t}$.



    We start with temporal correlation, with $epsilon$ as random noise, $mathbb{E}(epsilon_{i,j})=0$.
    Rewriting
    $$X=left(
    begin{array}{cccc}
    x_{1,1} & x_{2,1} & x_{3,1} & x_{4,1} \
    rho _1 x_{1,1}+epsilon _{1,1} & rho _2 x_{2,1}+epsilon _{2,1} & rho _3 x_{3,1}+epsilon _{3,1} & rho _4 x_{4,1}+epsilon _{4,1} \
    rho _1 left(rho _1 x_{1,1}+epsilon _{1,1}right)+epsilon _{1,2} & rho _2 left(rho _2 x_{2,1}+epsilon _{2,1}right)+epsilon _{2,2} & rho _3 left(rho _3 x_{3,1}+epsilon _{3,1}right)+epsilon _{3,2} & rho _4 left(rho _4 x_{4,1}+epsilon _{4,1}right)+epsilon _{4,2} \
    end{array}
    right)$$
    and add cross-sectional correlation, rewriting
    $X=left(
    begin{array}{cccc}
    x_{1,1} & frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,1} & frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,1} & frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,1} \
    rho _1 x_{1,1}+epsilon _{1,1} & rho _2 left(frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,2}right)+epsilon _{2,1} & rho _3 left(frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,2}right)+epsilon _{3,1} & rho _4 left(frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,2}right)+epsilon _{4,1} \
    rho _1 left(rho _1 x_{1,1}+epsilon _{1,1}right)+epsilon _{1,2} & rho _2 left(rho _2 left(frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,1}right)+epsilon _{2,1}right)+epsilon _{2,2} & rho _3 left(rho _3 left(frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,1}right)+epsilon _{3,1}right)+epsilon _{3,2} & rho _4 left(rho _4 left(frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,1}right)+epsilon _{4,1}right)+epsilon _{4,2} \
    end{array}
    right)$
    Am I doing it right?






    probability random-matrices







    share|cite|improve this question
























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Let $X$ be a random matrix, constituted of 4 vectors of time series $X_i,t$, $i=1,ldots,4$, $t=1,ldots,3$.



      $$X=left(
      begin{array}{cccc}
      x_{1,1} & x_{2,1} & x_{3,1} & x_{4,1} \
      x_{1,2} & x_{2,2} & x_{3,2} & x_{4,2} \
      x_{1,3} & x_{2,3} & x_{3,3} & x_{4,3} \
      end{array}
      right).$$
      I am trying to inject both temporal correlation between $X_{i,t}$ and $X_{i,t+1}$ and cross-sectional between $X_{i,t}$ and $X_{i+1,t}$.



      We start with temporal correlation, with $epsilon$ as random noise, $mathbb{E}(epsilon_{i,j})=0$.
      Rewriting
      $$X=left(
      begin{array}{cccc}
      x_{1,1} & x_{2,1} & x_{3,1} & x_{4,1} \
      rho _1 x_{1,1}+epsilon _{1,1} & rho _2 x_{2,1}+epsilon _{2,1} & rho _3 x_{3,1}+epsilon _{3,1} & rho _4 x_{4,1}+epsilon _{4,1} \
      rho _1 left(rho _1 x_{1,1}+epsilon _{1,1}right)+epsilon _{1,2} & rho _2 left(rho _2 x_{2,1}+epsilon _{2,1}right)+epsilon _{2,2} & rho _3 left(rho _3 x_{3,1}+epsilon _{3,1}right)+epsilon _{3,2} & rho _4 left(rho _4 x_{4,1}+epsilon _{4,1}right)+epsilon _{4,2} \
      end{array}
      right)$$
      and add cross-sectional correlation, rewriting
      $X=left(
      begin{array}{cccc}
      x_{1,1} & frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,1} & frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,1} & frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,1} \
      rho _1 x_{1,1}+epsilon _{1,1} & rho _2 left(frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,2}right)+epsilon _{2,1} & rho _3 left(frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,2}right)+epsilon _{3,1} & rho _4 left(frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,2}right)+epsilon _{4,1} \
      rho _1 left(rho _1 x_{1,1}+epsilon _{1,1}right)+epsilon _{1,2} & rho _2 left(rho _2 left(frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,1}right)+epsilon _{2,1}right)+epsilon _{2,2} & rho _3 left(rho _3 left(frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,1}right)+epsilon _{3,1}right)+epsilon _{3,2} & rho _4 left(rho _4 left(frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,1}right)+epsilon _{4,1}right)+epsilon _{4,2} \
      end{array}
      right)$
      Am I doing it right?






      probability random-matrices







      share|cite|improve this question













      Let $X$ be a random matrix, constituted of 4 vectors of time series $X_i,t$, $i=1,ldots,4$, $t=1,ldots,3$.



      $$X=left(
      begin{array}{cccc}
      x_{1,1} & x_{2,1} & x_{3,1} & x_{4,1} \
      x_{1,2} & x_{2,2} & x_{3,2} & x_{4,2} \
      x_{1,3} & x_{2,3} & x_{3,3} & x_{4,3} \
      end{array}
      right).$$
      I am trying to inject both temporal correlation between $X_{i,t}$ and $X_{i,t+1}$ and cross-sectional between $X_{i,t}$ and $X_{i+1,t}$.



      We start with temporal correlation, with $epsilon$ as random noise, $mathbb{E}(epsilon_{i,j})=0$.
      Rewriting
      $$X=left(
      begin{array}{cccc}
      x_{1,1} & x_{2,1} & x_{3,1} & x_{4,1} \
      rho _1 x_{1,1}+epsilon _{1,1} & rho _2 x_{2,1}+epsilon _{2,1} & rho _3 x_{3,1}+epsilon _{3,1} & rho _4 x_{4,1}+epsilon _{4,1} \
      rho _1 left(rho _1 x_{1,1}+epsilon _{1,1}right)+epsilon _{1,2} & rho _2 left(rho _2 x_{2,1}+epsilon _{2,1}right)+epsilon _{2,2} & rho _3 left(rho _3 x_{3,1}+epsilon _{3,1}right)+epsilon _{3,2} & rho _4 left(rho _4 x_{4,1}+epsilon _{4,1}right)+epsilon _{4,2} \
      end{array}
      right)$$
      and add cross-sectional correlation, rewriting
      $X=left(
      begin{array}{cccc}
      x_{1,1} & frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,1} & frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,1} & frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,1} \
      rho _1 x_{1,1}+epsilon _{1,1} & rho _2 left(frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,2}right)+epsilon _{2,1} & rho _3 left(frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,2}right)+epsilon _{3,1} & rho _4 left(frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,2}right)+epsilon _{4,1} \
      rho _1 left(rho _1 x_{1,1}+epsilon _{1,1}right)+epsilon _{1,2} & rho _2 left(rho _2 left(frac{sigma _2 rho _{1,2} x_{1,1}}{sigma _1}+epsilon _{2,1}right)+epsilon _{2,1}right)+epsilon _{2,2} & rho _3 left(rho _3 left(frac{sigma _3 rho _{1,3} x_{1,1}}{sigma _1}+epsilon _{3,1}right)+epsilon _{3,1}right)+epsilon _{3,2} & rho _4 left(rho _4 left(frac{sigma _3 rho _{1,4} x_{1,1}}{sigma _1}+epsilon _{4,1}right)+epsilon _{4,1}right)+epsilon _{4,2} \
      end{array}
      right)$
      Am I doing it right?






      probability random-matrices




      probability random-matrices






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      share|cite|improve this question











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      asked Nov 15 at 18:51









      Nero

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