Correlation and Regression

Summaries

Correlation:

1)      remember to divide by (n-1) when accessing sample not population

2)      Outliners can result in significant correlation when there is actually none

3)      Spurious correlation: mathematiccally correlated but no casual relationship!

4)      Correlation is only for linear relationship

Significant test for correlation: r*sqrt(n-2)/sqrt(1-r^2) (DOF=n-2)

Autocorrelation => error is dependent on other observations

Heteroskedastic: non constant of error variance

Parameter instability: regression relationship changes over time

Ordinary Least Square (OLS) regression: Sum of squared error (SSE) Sum(Yi – Yi^)^2 is minimized

Y^ = a^ + b^ * X^

b^ = cov(XY)/var(X)

a^ = mean(Y) – mean(X) b^ (meaning the line pass thorough the means of X, Y

Intercept a^ is just the ex-post alpha (after the fact, vs. ex ante – before the fact)

This regression line is the security characteristic line (SCL)

Market model uses market index and is non-equilibrium

CAPM is equilibrium and uses market portfolio

So, betas are the same in the 2 models. But alphas are not.

Use student-t with DOF=n-2 to determine if b is significantly different from proposed value b0: (b-b0)/sigma_b

Confidence interval of predicted value t*sigma, where

Sigma² = SEE² (1+ 1/n + (X-X_mean)²/(n-1)S_x²)

Therefore, as n increases, predicted value variance equals to SEE² (variance of residuals)