Formulas/definitons Flashcards
Rank
How many columns/ rows are linearly independent
B
(X’X)^-1(X’Y)
Y^
XB^
E^
Y-y^
Error var
E^’ E^ /(n-k)
Var B
Error var (X’X)^-1
E(B^-B)(B^-B)’)
fitted y y^
Py
P= X(X’X)-1X’
e residual with m
M=I-P
e^=My
RSS
e^’ e
s^2
RSS/n-k
R^2
1-RSS/TSS = 1- e^’e^/sum(y-y_)^2
=TSS-RSS/RSS
adjusted R^2
1-RSS/(n-k) / TSS/(n-1)
assumptions for multiple linear model
Linear model
errors have 0 mean E(e)=0
Homoskedasticity & no autocorrelation Var(e)= E(e’e)= var x I
no collinearity in x guarantees X is non singular
Normal errors
type 1 error
rejecting H0 when its true
type 2 error
accepting H0 when its false
t or z test
if population error is known do z test - B^-b/s x root var
t test- B^-b/s (sample ) root var
calculating confidence intervals
coefficienct of variable +- ttable value x se
hypothesis test
make hypothesis
calculate t stat coefficient/se
compare to table value
reject h0 if calculated is bigger than table
p value
compare to area (find on table using t value and df) , if value is smaller than eg 5% level, reject h0
calculating f stat multiple regression
w= r’B-c/ s x root r’(X’X)-1r
where H0=r’B=c etc
H0: B1+B2=1 what’s the t stat
B1+B2-1/se(B1+B2)
se(B1+B2)= root varB1+varB2+2covB1,B2
B2=0
(0,1,0,0)B=0
B2+B3=0
(0,1,1,0)B=0
B2=B3=0
(0,1,0,0)B=(0)
(0,0,1,0) (0)
wald test
(RB^-C)’(R(X’X)-1R’)-1(RB^-C)/S^2 (if population mean is known)
(RB^-C)’(R(X’X)-1R’)-1(RB^-C)/J/S^2
URSS
E^’E=(Y-XB^)(Y-XB^)
RRSS
E‘E=Y-XB(Y-XB)
TSS=sum(y-y_)^2
F STAT FOR OMITTED COEFFICIENT
(RRSS-URSS)/J/URSS/(N-K)
testing joint significance f stat
TSS-RSS/(k-1) / RSS/(n-k)
R^2/(k-1) / (1-R^2)/(n-k)
