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Questions of R Square and Corr Coef

$y\sim(1,\boldsymbol{x})$, regress $y$ on $x$ with intercept. $y\sim(\boldsymbol{x})$, regress $y$ on $x$ without intercept. In the context of Statistics, SSE (Sum of Squares due to Error) and SSR (Sum of Squares due to Regression) are used more frequently. But in Economitrics, ESS (Explained Sum of Squares) and RSS (Residual Sum of Squares) are prefered. 0.1 Bivariate Regression Denote the $R^2$ of $y\sim(1,x_1)$ as $R_1^2$, $y\sim(1,x_2)$ as $R_2^2$, $y\sim(1,x_1,x_2)$ as $R_3^2$. And we have $corr(x_1,x_2)=\rho$.

Tuesday, November 26, 2024 | 3 minutes Read
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Questions of R Square

$y\sim(1,\boldsymbol{x})$, regress $y$ on $x$ with intercept. $y\sim(\boldsymbol{x})$, regress $y$ on $x$ without intercept. In the context of Statistics, SSE (Sum of Squares due to Error) and SSR (Sum of Squares due to Regression) are used more frequently. But in Economitrics, ESS (Explained Sum of Squares) and RSS (Residual Sum of Squares) are prefered. 0.1 Definition $$ R^2 = \frac{SSR}{SST} = \frac{||\hat{Y} - \overline{Y}||^2}{||Y - \overline{Y}||^2} = 1 - \frac{SSE}{SST} = 1 - \frac{||\hat{\epsilon}||^2}{||Y - \overline{Y}||^2} $$

Tuesday, November 26, 2024 | 3 minutes Read
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Questions of Assumptions

$y\sim(1,\boldsymbol{x})$, regress $y$ on $x$ with intercept. $y\sim(\boldsymbol{x})$, regress $y$ on $x$ without intercept. In the context of Statistics, SSE (Sum of Squares due to Error) and SSR (Sum of Squares due to Regression) are used more frequently. But in Economitrics, ESS (Explained Sum of Squares) and RSS (Residual Sum of Squares) are prefered. 0.1 Heteroskedasticity and Autocorrelation If the residuals ($\epsilon$) in a linear regression model exhibit heteroskedasticity (non-constant variance) or autocorrelation (correlation between residuals across observations), how will it impact the estimation and inference of $\beta$? How to test and solve these problems?

Tuesday, November 26, 2024 | 7 minutes Read
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Questions of Coefficients

$y\sim(1,\boldsymbol{x})$, regress $y$ on $x$ with intercept. $y\sim(\boldsymbol{x})$, regress $y$ on $x$ without intercept. In the context of Statistics, SSE (Sum of Squares due to Error) and SSR (Sum of Squares due to Regression) are used more frequently. But in Economitrics, ESS (Explained Sum of Squares) and RSS (Residual Sum of Squares) are prefered. 0.1 Product of $\beta$ Denote $\beta_1$ as the least squares optimal solution of $y=\beta x+\epsilon$, $\beta_2$ as the least squares optimal solution of $x=\beta y+\epsilon$. Find the min and max values of $\beta_1\beta_2$. $$ \beta_1 = \frac{Cov(X,Y)}{Var(X)},\quad \beta_2 = \frac{Cov(X,Y)}{Var(Y)}\Rightarrow \beta_1\beta_2 = \rho_{XY}^2 \in [-1,1] $$

Saturday, November 16, 2024 | 4 minutes Read
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Questions of Conceptions

With reference to donggua. I complete the answers of these questions. 0.1 Notations $y\sim(1,\boldsymbol{x})$, regress $y$ on $x$ with intercept. $y\sim(\boldsymbol{x})$, regress $y$ on $x$ without intercept. In the context of Statistics, SSE (Sum of Squares due to Error) and SSR (Sum of Squares due to Regression) are used more frequently. But in Economitrics, ESS (Explained Sum of Squares) and RSS (Residual Sum of Squares) are prefered. 0.2 Conceptions and Basic Definitions 0.2.1. The assumptions of LR Gauss-Markov Theory: Under the assumptions of classical linear regression, the ordinary least squares (OLS) estimator is the linear unviased estimator with the minimum variance. (BLUE)

Saturday, November 16, 2024 | 5 minutes Read