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Multiple regression - Coggle Diagram
Multiple regression
Summary
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Multiple regression allows you to estimate how Y is affected by one of the X's holding all other X's constant (careful not to confuse causality)
Don't use too many variables, can fall into overfitting --> leads to very bad performance of model
Exercise, judgement to determine which variables belong ion a regression
OLS estimator
- minimizes the average squared difference between actual values of Y and the predicted ones based on the estimation line
- line through scatter plot is the OLS estimate
SE and CI
- Compare SE to coefficient, if it's comparable then the means that it is not a precise measurement
- CI, check width of CI to analyse whether it is a precise measurement
Magnitude of coefficient
just because co-efficient is small, doesn't necessarily mean that the effect will be small, depends on the values - make sure to analyse this
Measures of fit
Actual = predicted + residual
RMSE = standard deviation of predicted error term
R2 = fraction of variance of Y explained by variance of X
Adjusted R2 = this is the R2 with DOF adjustment for estimation uncertainty
Multicollinearity
imperfect multicollinearity
- when 2+ variables are very highly correlated
- states that one or more of the regression co-efficients will be imprecisely estimated
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Examining multiple regression results graphically
When there are more regressors it becomes impossible to to use a graph to analyse so must change our approach
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Assumptions
- all OOS observations are drawn from the same data distribution as estimated sample
- all samples are independantly and identically distributed
- large outliers are rare
- no perfect multicollinearity
Under these assumptions:
- beat hat has same mean as beta
- variance of beta hat is inversely proportional to n
- exact distribution of beta hat very complicated but law of large numbers holds
Interpretation of coefficient
- Taking partial derivatives will allow you to asses and interpret the data
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