User:Olyagnilova

Random variables

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  The projection of a random variable $Y$ onto the line spanned by a random variable $X$.
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  Geometric representation of random variables.
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  The law of iterated expectations. Equivalence of the two-step projecttion and direct projection of $Y$ onto $\mathbf {1}$.
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  Decomposition of mean squred error into the variance and the bias squared $\left(a={\sqrt {\lVert {\hat {\theta }}-E({\hat {\theta }})\rVert ^{2}}}\right.$, $b={\sqrt {\lVert \theta -E({\hat {\theta }})\rVert ^{2}}}$, $\left.c={\sqrt {\lVert {\hat {\theta }}-\theta \rVert ^{2}}}\right)$.
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  The Pythagorean theorem for random variables $X$ and $Y$.

Regression

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  Regression of $y$ and ${\hat {y}}$ onto $\mathbf {1}$.
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  Regression of $y$ on $Lin(\mathbf {1} ,x)$ and decomposition of ${\hat {y}}$ into a sum of ${\hat {\beta }}_{1}\mathbf {1}$ and ${\hat {\beta }}_{2}x$.
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  Vector $y$ projected onto vector $x$.
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  Centred vectors $x^{c}$ and $y^{c}$.
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  Proof of $sCorr(x+\alpha \mathbf {1} ,y)=sCorr(x,y)$.
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  OLS formula illustrated in $\mathbb {R} ^{k}$.
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  Determination coefficient as squared $\cos \varphi$ where $a={\sqrt {RSS}}$,
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  New residuals translated to the origin of the unit circle.
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  Residuals translated to the orgin of the unit circle.
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  Residuals ${\hat {u}}_{1}$ and ${\hat {u}}_{2}$.
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  Example of inversion for vector $a$.
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  Regressors $v_{1}$, $v_{2}$ obtained from inversion of the residuals ${\hat {u}}_{1}$, ${\hat {u}}_{2}$.
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  Regressions of $v_{1}$ onto $v_{2}$ and of $v_{2}$ onto $v_{1}$.
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  Original regressors in unit circle.
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  FWL: regression of $y$ on $z$ and of $x$ on $z$.
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  FWL: $Lin(x,z)$.
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  FWL: point $A$ stands for the origin, $B$ — ${\hat {\gamma }}z$, $C$ — $x$ , $D$ — ${\hat {\alpha }}z$, $E$ — intersection of vector $x$ and line parallel to ${\tilde {x}}$ , $F$ — ${\hat {\beta }}_{1}^{A}x$ , $G$ — ${\hat {\beta }}_{1}^{B}{\tilde {x}}$.
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  FWL: translation of ${\tilde {x}}$.
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  FWL: $Lin(x,z)$.
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  FWL: regression of y on $Lin(x,z)$.
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  FWL: “cleansed” ${\tilde {y}}$ regressed on “cleansed” ${\tilde {x}}$.
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  FWL: “cleansed” variables ${\tilde {x}}$ and ${\tilde {y}}$.
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  Alternative proof for the Frisch-Waugh-Lovell theorem.
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  FWL: similar triangles △ABC∼△EDC.
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  Gauss-Markov theorem for the case of three regressors.
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  Geometry of instrumental variables. $A$ stands for ${\hat {\beta }}_{IV}{\hat {x}}$, $B$ — ${\hat {x}}$, $C$ — $x$, $D$ — ${\hat {\beta }}_{IV}x$.
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  Geometry of proxy variables.
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  Illustration of the equality TSS = RSS + ESS where $a$ stands for ${\sqrt {RSS}}$, $b$ — ${\sqrt {TSS}}$, $c$ — ${\sqrt {ESS}}$.
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  Illustartion of ESS.
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  Illustration of RSS.
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  Total sum of squares and residual sum of squares.
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  Residual sum of squares.
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  Starting pictutre for regression illustrations.
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  “Centred” $x$ and $y$, i.e., projected onto $Lin(\mathbf {1} )$.
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  Proof of $Corr(y,{\hat {y}})=sign({\hat {\beta }}_{2})Corr(y,x)$ for the case of ${\hat {\beta }}_{2}<0$.
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  “Centred” ${\hat {y}}$, i.e., projected onto $Lin(\mathbf {1} )$.

Partial correlation

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  Partial correlation between $X$ and $Y$ while $Z$ is fixed.
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  Partial correlation definition through geometric mean / regression residuals, $Lin\perp (Z)$.
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  Alternative definition of the partial correlation through regressions on two variables.
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  Residuals ${\hat {u}}$ form regression of $X$ onto $Y$ and $Z$, ${\hat {u}}$ projected.
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  The residuals of both regressions.
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  Partial correlation definition through residuals, $Lin\perp (Z)$.
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  Residuals ${\hat {v}}$ from regression of $Y$ onto $X$ and $Z$, ${\hat {v}}$ projected.

Probability distributions

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  A 3-dimensional example for the chi-squared distribution.
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  F-distribution as the ratio of the projection lengths squared adjusted to the dimensions of the subspaces.
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  F-statistic as the cotangent squared of $\varphi$ where $a$ stands for ${\sqrt {RSS_{UR}}}$, $b$ — ${\sqrt {RSS_{R}-RSS_{UR}}}$, $c$ — ${\sqrt {RSS_{R}}}$.
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  Black dots represent the gas molecules. The red dot stands for the one we catch. Its speed along the horizontal axis is $v_{1}$, i.e., the first component of the velocity vector, and its speed along the vertical axis is $v_{2}$.
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  T-statistic as cotangent of $\varphi$, $Lin(x,\mathbf {1} )$.
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  T-statistic as cotangent of $\varphi$. Regression of $y$ onto $Lin(x,\mathbf {1} )$ and appropriate projections.