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The Rules of Summation
å
n
i¼ 1
x (^) i ¼ x 1 þ x 2 þ þ x (^) n
å
n
i¼ 1
a ¼ na
å
n
i¼ 1
axi ¼ a å
n i¼ 1
x (^) i
å
n
i¼ 1
ðx (^) i þ yiÞ ¼ å
n i¼ 1
x (^) i þ å
n i¼ 1
y (^) i
å
n
i¼ 1
ðaxi þ byiÞ ¼ a å
n i¼ 1
x (^) i þ b å
n i¼ 1
y (^) i
å
n
i¼ 1
ða þ bxiÞ ¼ na þ b å
n i¼ 1
x (^) i
x ¼
å n i¼ 1 x (^) i n ¼^
x 1 þ x 2 þ þ x (^) n n å
n
i¼ 1
ðx (^) i xÞ ¼ 0
å
2 i¼ 1
å
3 j¼ 1
f ðx (^) i; y (^) jÞ ¼ å
2 i¼ 1
½ f ðx (^) i; y 1 Þ þ f ðxi; y 2 Þ þ f ðx (^) i; y 3 Þ ¼ f ðx 1 ; y 1 Þ þ f ðx 1 ; y 2 Þ þ f ðx 1 ; y 3 Þ þ f ðx 2 ; y 1 Þ þ f ðx 2 ; y 2 Þ þ f ðx 2 ; y 3 Þ
Expected Values & Variances
EðXÞ ¼ x 1 f ðx 1 Þ þ x 2 f ðx 2 Þ þ þ x (^) n f ðx (^) nÞ
¼ å
n i¼ 1
x (^) i f ðxiÞ ¼ å x
x f ðxÞ
E g½ ðXÞ ¼ å x
gðxÞ f ðxÞ
E g½ 1 ðXÞ þ g 2 ðXÞ ¼ å x
½g 1 ðxÞ þ g 2 ðxÞ f ðxÞ
¼ å x g 1 ðxÞ f ðxÞ þ å x g 2 ðxÞ f ðxÞ
¼ E g½ 1 ðXÞ þ E g½ 2 ðXÞ E(c) ¼ c E(cX ) ¼ cE(X ) E(a þ cX ) ¼ a þ cE(X ) var(X ) ¼ s^2 ¼ E[X E(X )] 2 ¼ E(X^2 ) [E(X )]^2 var(a þ cX ) ¼ E[(a þ cX) E(a þ cX)]^2 ¼ c^2 var(X )
Marginal and Conditional Distributions
f ðxÞ ¼ å y
f ðx; yÞ for each value X can take
f ðyÞ ¼ å x f ðx; yÞ for each value Y can take
f ðxjyÞ ¼ P X½ ¼ xjY ¼ y ¼
f ðx; yÞ f ðyÞ
If X and Y are independent random variables, then f (x,y) ¼ f (x)f ( y) for each and every pair of values x and y. The converse is also true.
If X and Y are independent random variables, then the conditional probability density function of X given that
Y ¼ y is f ðxjyÞ ¼ f ðx; yÞ f ðyÞ
f ðxÞ f ðyÞ f ðyÞ
¼ f ðxÞ
for each and every pair of values x and y. The converse is also true.
Expectations, Variances & Covariances
covðX; YÞ ¼ E½ðXE½XÞðYE½YÞ ¼ å x
å y
½x EðXÞ ½ y EðYÞ f ðx; yÞ
r ¼ covðX;YÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi varðXÞvarðYÞ
p
E(c 1 X þ c 2 Y ) ¼ c 1 E(X ) þ c 2 E(Y ) E(X þ Y ) ¼ E(X ) þ E(Y ) var(aX þ bY þ cZ ) ¼ a^2 var(X) þ b^2 var(Y ) þ c^2 var(Z ) þ 2 abcov(X,Y ) þ 2 accov(X,Z ) þ 2 bccov(Y,Z ) If X, Y, and Z are independent, or uncorrelated, random variables, then the covariance terms are zero and: varðaX þ bY þ cZÞ ¼ a^2 varðXÞ þ b^2 varðYÞ þ c^2 varðZÞ
Normal Probabilities
If X N(m, s^2 ), then Z ¼ X m s
Nð 0 ; 1 Þ If X N(m, s^2 ) and a is a constant, then
PðX aÞ ¼ P Z a m s
If X Nðm; s^2 Þ and a and b are constants; then
Pða X bÞ ¼ P am s
Z
b m s
Assumptions of the Simple Linear Regression Model SR1 The value of y, for each value of x, is y ¼ b 1 þ b 2 x þ e SR2 The average value of the random error e is E(e) ¼ 0 since we assume that E(y) ¼ b 1 þ b 2 x SR3 The variance of the random error e is var(e) ¼ s^2 ¼ var(y) SR4 The covariance between any pair of random errors, ei and e (^) j is cov(ei, e (^) j) ¼ cov(y (^) i, yj) ¼ 0 SR5 The variable x is not random and must take at least two different values. SR6 (optional) The values of e are normally dis- tributed about their mean e N(0, s^2 )
Least Squares Estimation If b 1 and b 2 are the least squares estimates, then ^y (^) i ¼ b 1 þ b 2 x (^) i ^e (^) i ¼ yi ^y (^) i ¼ yi b 1 b 2 xi
The Normal Equations Nb 1 þ Sx (^) i b 2 ¼ Sy (^) i Sx (^) i b 1 þ Sx^2 i b 2 ¼ Sx (^) i y (^) i
Least Squares Estimators
b 2 ¼
Sðx (^) i xÞðy (^) i yÞ S ðx (^) i xÞ^2 b 1 ¼ y b 2 x
Elasticity
h ¼ percentage change in y percentage change in x
Dy=y Dx=x
Dy Dx
x y
h ¼ DEðyÞ=EðyÞ Dx=x
DEðyÞ Dx
x EðyÞ
¼ b 2 x EðyÞ
Least Squares Expressions Useful for Theory
b 2 ¼ b 2 þ Swi e (^) i
wi ¼
x (^) i x Sðx (^) i xÞ^2
Swi ¼ 0 ; Swi x (^) i ¼ 1 ; Sw^2 i ¼ 1 =Sðx (^) i xÞ^2
Properties of the Least Squares Estimators
varðb 1 Þ ¼ s^2 Sx^2 i NSðx (^) i xÞ^2
varðb 2 Þ ¼ s^2 Sðx (^) i xÞ^2
covðb 1 ; b 2 Þ ¼ s^2 x Sðx (^) i xÞ^2
Gauss-Markov Theorem: Under the assumptions SR1–SR5 of the linear regression model the estimators b 1 and b 2 have the smallest variance of all linear and unbiased estimators of b 1 and b 2. They are the Best Linear Unbiased Estimators (BLUE) of b 1 and b 2.
If we make the normality assumption, assumption SR6, about the error term, then the least squares esti- mators are normally distributed.
b 1 N b 1 ;
s^2 å x^2 i NSðx (^) i xÞ^2
; b 2 N b 2 ;
s^2 Sðxi xÞ^2
Estimated Error Variance
^s^2 ¼ S^e^2 i N 2
Estimator Standard Errors
seðb 1 Þ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bvarðb 1 Þ
q ; seðb 2 Þ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bvarðb 2 Þ
q
t-distribution
If assumptions SR1–SR6 of the simple linear regression model hold, then
t ¼ bk bk seðbk Þ
tðN 2 Þ; k ¼ 1 ; 2
Interval Estimates
P[b 2 tcse(b 2 ) b 2 b 2 þ t (^) cse(b 2 )] ¼ 1 a
Hypothesis Testing
Components of Hypothesis Tests
- A null hypothesis, H 0
- An alternative hypothesis, H 1
- A test statistic
- A rejection region
- A conclusion If the null hypothesis H 0 : b 2 ¼ c is true, then
t ¼ b 2 c seðb 2 Þ
tðN 2 Þ
Rejection rule for a two-tail test: If the value of the test statistic falls in the rejection region, either tail of the t-distribution, then we reject the null hypothesis and accept the alternative. Type I error: The null hypothesis is true and we decide to reject it. Type II error: The null hypothesis is false and we decide not to reject it. p-value rejection rule: When the p-value of a hypoth- esis test is smaller than the chosen value of a, then the test procedure leads to rejection of the null hypothesis. Prediction y 0 ¼ b 1 þ b 2 x 0 þ e 0 ; ^y 0 ¼ b 1 þ b 2 x 0 ; f ¼ ^y 0 y 0
var^ bð f Þ ¼ ^s^2 1 þ 1 N þ ðx 0 xÞ^2 Sðx (^) i xÞ^2
; seð f Þ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi varbð f Þ
q
A (1 a) 100% confidence interval, or prediction interval, for y 0 ^y 0 tcseð f Þ Goodness of Fit Sðy (^) i yÞ^2 ¼ Sð^y (^) i yÞ^2 þ S^e^2 i SST ¼ SSR þ SSE
R^2 ¼
SSR
SST
SSE
SST
¼ ðcorrðy; ^yÞÞ^2
Log-Linear Model lnðyÞ ¼ b 1 þ b 2 x þ e; blnð yÞ ¼ b 1 þ b 2 x 100 b 2 % change in y given a one-unit change in x: ^y (^) n ¼ expðb 1 þ b 2 xÞ ^y (^) c ¼ expðb 1 þ b 2 xÞexpðs^^2 = 2 Þ Prediction interval:
exp blnðyÞ tcseð f Þ
h i ; exp blnð yÞ þ tcseð f Þ
h i
Generalized goodness-of-fit measure R^2 g ¼ ðcorrðy;^y (^) nÞÞ^2 Assumptions of the Multiple Regression Model MR1 y (^) i ¼ b 1 þ b 2 x (^) i 2 þ þ bK xiK þ e (^) i MR2 E(y (^) i) ¼ b 1 þ b 2 xi 2 þ þ bK xiK , E(e (^) i) ¼ 0. MR3 var(y (^) i) ¼ var(ei) ¼ s^2 MR4 cov(y (^) i, y (^) j) ¼ cov(e (^) i, ej) ¼ 0 MR5 The values of x (^) ik are not random and are not exact linear functions of the other explanatory variables. MR6 y (^) i N½ðb 1 þ b 2 x (^) i 2 þ þ bK x (^) iK Þ; s^2 , ei Nð 0 ; s^2 Þ
Least Squares Estimates in MR Model Least squares estimates b 1 , b 2 ,... , bK minimize Sðb 1 , b 2 ,... , bKÞ ¼ åðyi b 1 b 2 x (^) i 2 bKx (^) iKÞ^2
Estimated Error Variance and Estimator Standard Errors
^s^2 ¼
å ^e^2 i N K seðbkÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi varbðbk Þ
q
