Assumptions (formula)
Assumptions (in words)
Simple Regress Formulas
Multiple Regress Formulas
Miscellaneous
100
Assumption 1
Yi= Bo + B1Xi +ei
100
Assumption 1 (words)
The relationship between variables is linear
100
β̂1
∑(xi-xbar)(yi-ybar)
__________________
∑ xi-xbar) ^2
100
β̂1=
∑ (r1i-tilda * Yi)
_______________
∑(r1i-tilda)
100
Assumptions needed for OLS
1, 2, 3
200
Assumption 2
EV(ei l Xi) = 0
Cov(ei l Xi) = 0
200
Assumption 2 (words)
No remaining systematic unobserved factors influencing Yi
200
β̂ 0
β̂ = Ybar- β̂1xbar
200
Assumption 6 (words and formula)
No exact linear relationships among regressors
X2= a + bX1
200
Assumptions needed for efficient estimator
4 and 5
300
Assumption 3
∑ (xi -xbar) ^2 > 0
300
Assumption 3 (words)
Variation in X must be present
300
Alternative formula for β̂1
cov(x,y)
______
var(x)
300
Difference between RSS in multiple regression vs simple
Multiple regression's residual is ∑ (e-hat i) ^2, but in simple regression RSS = ∑(yi-yhat) ^2
300
Bias formula
EV(β̂1) - β1
400
Assumption 4 (formula)
Var(ei l X) = σ^2
400
Assumption 4 (words)
Variance of the error term given X = σ^2
400
σ^2
var(ei l Xi)
400
Multiple Regression Formula
EV(Yi l X1i, X2i) = B0+B1X1i +B2X2i
400
What changes the variance?
1. fewer observations (+)
2. More covariates (+)
3. larger error term (+)
4. R^2k grows (-)
500
Assumption 5 (formula)
Cov(ei, ej l X) = 0
500
Assumption 5
Errors for observation i says nothing about the errors observation j
500
TSS
RSS
ESS
TSS=∑(yi-ybar)^2
RSS=∑(yi-yhat) ^2
ESS = ∑(yhat-ybar) ^2
500
Alternative Beta 1 Formula
B1-bar= B1-hat + B2-tilda *α1-tilda
500
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