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Professor Rojas. Formula Sheet Econometrics...

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The Rules of Summation

Expectations, Variances & Covariances

n

! xi ¼ x1 þ x2 þ # # # þ xn

covðX; YÞ ¼ E ½ðX &E ½X (ÞðY &E ½Y (Þ(

i¼1 n

¼ ! ! ½x & EðXÞ(½ y & EðYÞ( f ðx; yÞ

! a ¼ na

x y

i¼1 n

covðX;Y Þ r ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi varðX ÞvarðY Þ

n

! axi ¼ a ! xi

i¼1 n

i¼1

n

n

E(c1 X þ c2 Y ) ¼ c1 E(X ) þ c2 E(Y ) E(X þ Y ) ¼ E(X ) þ E(Y )

! ðxi þ yi Þ ¼ ! xi þ ! yi

i¼1 n

i¼1

i¼1

n

n

i¼1

i¼1

2

! ðaxi þ byi Þ ¼ a ! xi þ b ! yi

i¼1 n

n

! ða þ bxi Þ ¼ na þ b ! xi

i¼1

x¼

! xi

x1 þ x2 þ # # # þ xn ¼ n

n

n

2

If X, Y, and Z are independent, or uncorrelated, random variables, then the covariance terms are zero and:

i¼1

n

i¼1

2

var(aX þ bY þ cZ ) ¼ a var(X ) þ b var(Y ) þ c var(Z ) þ 2abcov(X,Y ) þ 2accov(X,Z ) þ 2bccov(Y,Z )

varðaX þ bY þ cZÞ ¼ a 2 varðX Þ þ b2 varðYÞ þ c2 varðZÞ

! ðxi & xÞ ¼ 0

i¼1 2

2

3

! ! f ðxi ; yj Þ ¼ ! ½ f ðxi ; y1 Þ þ f ðxi ; y2 Þ þ f ðxi ; y3 Þ( i¼1

i¼1 j¼1

¼ f ðx1 ; y1 Þ þ f ðx1 ; y2 Þ þ f ðx1 ; y3 Þ þ f ðx2 ; y1 Þ þ f ðx2 ; y2 Þ þ f ðx2 ; y3 Þ Expected Values & Variances EðXÞ ¼ x1 f ðx1 Þ þ x2 f ðx2 Þ þ # # # þ xn f ðxn Þ n

¼ ! xi f ðxi Þ ¼ ! x f ðxÞ x

i¼1

E ½gðX Þ( ¼ ! gðxÞ f ðxÞ x

E ½g 1 ðXÞ þ g 2 ðX Þ( ¼ !½ g 1ðxÞ þ g 2 ðxÞ( f ðxÞ x

¼ ! g 1ðxÞ f ðxÞ þ ! g 2 ðxÞ f ðxÞ

Normal Probabilities X&m ) Nð0; 1Þ s If X ) N(m, s ) and a is a constant, then " a & m# PðX * a Þ ¼ P Z * s If X ) Nðm; s2 Þ and a and b are constants; then $ % b&m a&m +Z+ Pða + X + bÞ ¼ P s s If X ) N(m, s2), then Z ¼ 2

Assumptions of the Simple Linear Regression Model SR1

x

x

¼ E½ g1 ðX Þ( þ E ½g2 ðX Þ( 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(X2) & [E(X )]2 var(a þ cX ) ¼ E [(a þ cX ) & E(a þ cX )]2 ¼ c 2var(X )

SR2 SR3 SR4 SR5

Marginal and Conditional Distributions f ðxÞ ¼ ! f ðx; yÞ for each value X can take

SR6

The value of y, for each value of x, is y ¼ b1 þ b2 x þ e The average value of the random error e is E(e) ¼ 0 since we assume that E(y) ¼ b1 þ b 2x The variance of the random error e is var(e) ¼ s2 ¼ var(y) The covariance between any pair of random errors, ei and ej is cov(e i, e j) ¼ cov(y i, yj ) ¼ 0 The variable x is not random and must take at least two different values. (optional) The values of e are normally distributed about their mean e ) N(0, s2)

y

f ðyÞ ¼ ! f ðx; yÞ for each value Y can take x

f ðx; yÞ f ðxjyÞ ¼ P½ X ¼ xjY ¼ y( ¼ 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 ðxÞ f ðyÞ ¼ f ðxÞ ¼ f ðyÞ f ðyÞ

for each and every pair of values x and y. The converse is also true.

Least Squares Estimation If b1 and b2 are the least squares estimates, then yi ¼ b1 þ b2 xi ^ ^ei ¼ yi & ^yi ¼ yi & b1 & b2 xi The Normal Equations Nb1 þ Sxi b2 ¼ Syi Sxi b1 þ Sxi2b2 ¼ Sxi yi Least Squares Estimators b2 ¼

Sðxi & xÞðyi & yÞ S ðxi & xÞ2

b1 ¼ y & b2 x

Elasticity percentage change in y Dy=y Dy x ¼ ¼ h¼ # percentage change in x Dx=x Dx y h¼

x DEðyÞ=EðyÞ DEðyÞ x ¼ # ¼ b2 # EðyÞ Dx=x Dx EðyÞ

Type I error: The null hypothesis is true and we decide to reject it.

Least Squares Expressions Useful for Theory b2 ¼ b2 þ Swi ei wi ¼

Swi xi ¼ 1;

2

Swi2 ¼ 1=Sðxi & xÞ

Properties of the Least Squares Estimators " # s2 Sx2i varðb2 Þ ¼ varðb1 Þ ¼ s2 2 NSðxi & xÞ Sðxi & xÞ2 " # &x covðb1 ; b2 Þ ¼ s2 2 Sðxi & xÞ Gauss-Markov Theorem: Under the assumptions SR1–SR5 of the linear regression model the estimators b1 and b2 have the smallest variance of all linear and unbiased estimators of b1 and b2 . They are the Best Linear Unbiased Estimators (BLUE) of b1 and b2 . If we make the normality assumption, assumption SR6, about the error term, then the least squares estimators are normally distributed. ! ! s2 s2 ! xi2 ; b2 ) N b 2 ; b1 ) N b 1 ; 2 2 NSðxi & xÞ Sðxi & xÞ Estimated Error Variance s ^2 ¼

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 hypothesis test is smaller than the chosen value of a, then the test procedure leads to rejection of the null hypothesis.

xi & x Sðxi & xÞ2

Swi ¼ 0;

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.

S^ei2 N&2

Estimator Standard Errors q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi seðb1 Þ ¼ b varðb1 Þ ; seðb2 Þ ¼ b varðb2 Þ

Prediction y0 ¼ b1 þ b2 x0 þ e0 ; y^0 ¼ b1 þ b2 x0 ; f ¼ y^0 & y0 " # q ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðx & xÞ2 b b ; seð f Þ ¼ varð v arð f Þ ¼ s ^2 1 þ þ 0 fÞ N Sðxi & xÞ2

A (1 & a) , 100% confidence interval, or prediction interval, for y0 y0 - tc seð f Þ ^ Goodness of Fit ei2 Sðyi & yÞ2 ¼ Sð^yi & yÞ2 þ S^ SST ¼ SSR þ SSE SSE SSR ¼ 1& ¼ ðcorrðy; ^yÞÞ2 R2 ¼ SST SST Log-Linear Model lnðyÞ ¼ b1 þ b2 x þ e; b lnð yÞ ¼ b1 þ b2 x

100 , b2 . % change in y given a one-unit change in x: yn ¼ expðb1 þ b2 xÞ ^ yc ¼ expðb1 þ b2 xÞexpð^ s2 =2Þ ^

Prediction interval: h i h i lnðyÞ & tc seð f Þ ; exp b lnð yÞ þ tc seð f Þ exp b

Generalized goodness-of-fit measure Rg2 ¼ ðcorrðy;^ yn ÞÞ2 Assumptions of the Multiple Regression Model

t-distribution

MR1 y i ¼ b 1 þ b2xi2 þ # # # þ bK xiK þ ei

If assumptions SR1–SR6 of the simple linear regression model hold, then

MR2 E(yi) ¼ b 1 þ b2 xi2 þ # # # þ b K xiK , E(e )i ¼ 0. 2 MR3 var(y i) ¼ var(ei) ¼ s

t¼

bk & b k ) tðN&2Þ ; k ¼ 1; 2 seðbk Þ

Interval Estimates P[b2 & tc se(b2 ) + b2 + b 2 þ tc se(b 2 )] ¼ 1 & a Hypothesis Testing Components of Hypothesis Tests 1. A null hypothesis, H 0 2. An alternative hypothesis, H1 3. A test statistic 4. A rejection region 5. A conclusion If the null hypothesis H0 : b 2 ¼ c is true, then t¼

b2 & c ) tðN&2Þ seðb2 Þ

MR4 cov(yi , yj ) ¼ cov(ei , ej) ¼ 0 MR5 The values of xik are not random and are not exact linear functions of the other explanatory variables. MR6 yi ) N½ðb1 þ b2 xi2 þ # # # þ bK xiK Þ; s2 ( , ei ) Nð0; s2 Þ Least Squares Estimates in MR Model Least squares estimates b1 , b2 , . . . , bK minimize Sðb1 , b 2, . . . , bKÞ ¼ !ðy i & b 1 & b2 xi2 & # # # & bK xiK Þ2 Estimated Error Variance and Estimator Standard Errors s ^2 ¼

! e^2i N&K

seðbk Þ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b varðbk Þ

Hypothesis Tests and Interval Estimates for Single Parameters bk & b k Use t-distribution t ¼ ) tðN&KÞ seðbk Þ t-test for More than One Parameter H0 : b2 þ cb3 ¼ a b2 þ cb3 & a ) tðN&KÞ seðb2 þ cb3 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2b b b seðb2 þ cb3 Þ ¼ varðb 2 Þ þ c varðb3 Þ þ 2c , covðb2 ; b3 Þ t¼

When H0 is true

Joint F-tests

Regression with Stationary Time Series Variables Finite distributed lag model yt ¼ a þ b0 xt þ b1 xt&1 þ b2 xt&2 þ # # # þ bq xt&q þ vt Correlogram rk ¼ ! ðyt & yÞðyt&k & yÞ= ! ðyt & yÞ2 p ffiffiffiffi For H0 : rk ¼ 0; z ¼ Trk ) Nð0; 1Þ LM test yt ¼ b1 þ b2 xt þ r^et&1 þ ^vt

Test H 0 : r ¼ 0 with t-test

et ¼ g1 þ g2 xt þ r^ ^ et&1 þ v^t Test using LM ¼ T , R2 AR(1) error yt ¼ b1 þ b2 xt þ et et ¼ ret&1 þ vt

To test J joint hypotheses,

Nonlinear least squares estimation

ðSSER & SSEU Þ=J SSEU =ðN & K Þ To test the overall significance of the model the null and alternative hypotheses and F statistic are

ARDL( p, q) model yt ¼ d þ d0 xt þ dl xt&1 þ # # # þ dq xt&q þ ul yt&1

F¼

þ # # # þ up yt&p þ vt AR( p) forecasting model yt ¼ d þ ul yt&1 þ u2 yt&2 þ # # # þ up yt&p þ vt

H0 : b2 ¼ 0; b3 ¼ 0; : : : ; bK ¼ 0 H 1 : at least one of the bk is nonzero F¼

ðSST & SSEÞ=ðK & 1Þ SSE=ðN & KÞ

RESET: A Specification Test ^yi ¼ b1 þ b2 xi2 þ b3 xi3

yi ¼ b1 þ b2 xi2 þ b3 xi3 þ ei yi ¼ b1 þ b2 xi2 þ b3 xi3 þ g1^yi2 þ ei ;

H0 : g 1 ¼ 0

yi ¼ b1 þ b2 xi2 þ b3 xi3 þ g1^yi 2þ g2^y3i þ ei ;

H0 : g 1 ¼ g 2 ¼ 0

AIC ¼ ln(SSE=N) þ 2K=N SC ¼ ln(SSE=N) þ K ln(N)=N

When x3 is omitted;

¼

Heteroskedasticity var(yi ) ¼ var(e i) ¼ s2i General variance function

Augmented Dickey-Fuller Tests:

b covðx2 ; x3 Þ & b2 ¼ b3 b varðx 2Þ

s2i ¼ expða1 þ a2 zi2 þ # # # þ aS ziS Þ x2 ¼ N , R2 ) xð2S&1Þ

2 6¼ Goldfeld-Quandt test for H0 : sM2 ¼ s2R versus H1 : sM sR2 ) FðNM &KM ;NR &KR Þ When H0 is true F ¼ s ^ 2M =^ Transformed model for varðei Þ ¼ si2 ¼ s2 xi

s2R

pffiffiffiffi pffiffiffiffi pffiffiffiffi pffiffiffiffi yi = xi ¼ b1 ð1= xi Þ þ b2ðxi = xi Þ þ ei = xi

Estimating the variance function

lnð^ e2i Þ ¼ lnðsi2Þ þ vi ¼ a1 þ a2 zi2 þ # # # þ aS ziS þ vi Grouped data varðei Þ ¼ s2i ¼

(

2 sM s2R

Unit Root Test for Stationarity: Null hypothesis: H0 : g ¼ 0 Dickey-Fuller Test 1 (no constant and no trend): Dyt ¼ gyt&1 þ vt

Dickey-Fuller Test 3 (with constant and with trend): Dyt ¼ a þ gyt&1 þ lt þ vt

Breusch-Pagan and White Tests for H0: a 2 ¼ a3 ¼ # # # ¼ a S ¼ 0 When H0 is true

, ðb0 þ b1 L þ b2 L2 þ # # #Þ

Dickey-Fuller Test 2 (with constant but no trend): Dyt ¼ a þ gyt&1 þ vt

Collinearity and Omitted Variables yi ¼ b1 þ b2 xi2 þ b3 xi3 þ ei s2 varðb2 Þ ¼ ð1 & r232 Þ ! ðxi2 & x2 Þ2 Eðb2/ Þ

Exponential smoothing ^yt ¼ ayt&1 þ ð1 & aÞ^ yt&1 Multiplier analysis d0 þ d1 L þ d2 L2 þ # # # þ dq Lq ¼ ð1 & u1 L & u2 L2 & # # # & up Lp Þ Unit Roots and Cointegration

Model Selection

biasðb/2 Þ

yt ¼ b1 ð1 & rÞ þ b2 xt þ ryt&1 & b2 rxt&1 þ vt

i ¼ 1; 2; . . . ; NM i ¼ 1; 2; . . . ; NR

Transformed model for feasible generalized least squares " .p ffiffiffiffi# " .pffiffiffiffiffi # .pffiffiffiffiffi . pffiffiffiffiffi ffi yi s ^ i þ b2 xi s ^ i þ ei s ^i s ^ i ¼ b1 1

m

Dyt ¼ a þ gyt&1 þ ! a s Dyt&s þ vt s¼1

Test for cointegration D^ et ¼ g^et&1 þ vt Random walk: yt ¼ yt&1 þ vt Random walk with drift: yt ¼ a þ yt&1 þ vt Random walk model with drift and time trend: yt ¼ a þ dt þ yt&1 þ vt Panel Data Pooled least squares regression yit ¼ b1 þ b2 x2it þ b3 x3it þ eit Cluster robust standard errors cov(eit, e is) ¼ cts Fixed effects model yit ¼ b1i þ b2 x2it þ b3 x3it þ eit b1i not random yit & yi ¼ b2 ðx2it & x2i Þ þ b3 ðx3it & x3i Þ þ ðeit & ei Þ Random effects model yit ¼ b1i þ b2 x2it þ b3 x3it þ eit

bit ¼ b1 þ ui random

yit & ayi ¼ b1 ð1 & aÞ þ b2 ðx2it & ax2i Þ þ b3 ðx3it & ax3i Þ þ vit/ &qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ¼ 1 & se Ts2u þ se2

Hausman test

&h i 1=2 b b t ¼ ðbFE;k & bRE;k Þ varðb FE;k Þ & varðbRE;k Þ

Additional Equations

JB = ˆ = var(λ)

◆ ✓ (K − 3)2 Jarque-Bera Test S2 + 4 ◆✓ ◆ ✓ ✓ ◆ ◆2 ∂λ ∂λ ∂λ 2 ∂λ cov(b1 , b 2 ) var(b2 ) + 2 var(b1 ) + ∂β1 ∂β2 ∂β2 ∂β1

N 6 ✓

ˆ δDD = (¯ yT reatment, After − y¯Control, After ) − (¯ yT reatment, Bef ore − y¯Control, Bef ore ) SST

= (N − 1)sy2

(1) (2) (3) (4)

Appendix

D

Standard Normal Distribution

Example: P(Z ≤ 1.73) ! Φ (1.73) ! 0.9582

"4 "3 "2 "1

0 z

1

2

3

4

T a b l e 1 Cumulative Probabilities for the Standard Normal Distribution F(z) ¼ P(Z " z) z 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

0.00

0.01

0.02

0.03

0.5000 0.5398 0.5793 0.6179 0.6554 0.6915 0.7257 0.7580 0.7881 0.8159 0.8413 0.8643 0.8849 0.9032 0.9192 0.9332 0.9452 0.9554 0.9641 0.9713 0.9772 0.9821 0.9861 0.9893 0.9918 0.9938 0.9953 0.9965 0.9974 0.9981 0.9987

0.5040 0.5438 0.5832 0.6217 0.6591 0.6950 0.7291 0.7611 0.7910 0.8186 0.8438 0.8665 0.8869 0.9049 0.9207 0.9345 0.9463 0.9564 0.9649 0.9719 0.9778 0.9826 0.9864 0.9896 0.9920 0.9940 0.9955 0.9966 0.9975 0.9982 0.9987

0.5080 0.5478 0.5871 0.6255 0.6628 0.6985 0.7324 0.7642 0.7939 0.8212 0.8461 0.8686 0.8888 0.9066 0.9222 0.9357 0.9474 0.9573 0.9656 0.9726 0.9783 0.9830 0.9868 0.9898 0.9922 0.9941 0.9956 0.9967 0.9976 0.9982 0.9987

0.5120 0.5517 0.5910 0.6293 0.6664 0.7019 0.7357 0.7673 0.7967 0.8238 0.8485 0.8708 0.8907 0.9082 0.9236 0.9370 0.9484 0.9582 0.9664 0.9732 0.9788 0.9834 0.9871 0.9901 0.9925 0.9943 0.9957 0.9968 0.9977 0.9983 0.9988

0.04

0.05

0.06

0.07

0.08

0.09

0.5160 0.5557 0.5948 0.6331 0.6700 0.7054 0.7389 0.7704 0.7995 0.8264 0.8508 0.8729 0.8925 0.9099 0.9251 0.9382 0.9495 0.9591 0.9671 0.9738 0.9793 0.9838 0.9875 0.9904 0.9927 0.9945 0.9959 0.9969 0.9977 0.9984 0.9988

0.5199 0.5596 0.5987 0.6368 0.6736 0.7088 0.7422 0.7734 0.8023 0.8289 0.8531 0.8749 0.8944 0.9115 0.9265 0.9394 0.9505 0.9599 0.9678 0.9744 0.9798 0.9842 0.9878 0.9906 0.9929 0.9946 0.9960 0.9970 0.9978 0.9984 0.9989

0.5239 0.5636 0.6026 0.6406 0.6772 0.7123 0.7454 0.7764 0.8051 0.8315 0.8554 0.8770 0.8962 0.9131 0.9279 0.9406 0.9515 0.9608 0.9686 0.9750 0.9803 0.9846 0.9881 0.9909 0.9931 0.9948 0.9961 0.9971 0.9979 0.9985 0.9989

0.5279 0.5675 0.6064 0.6443 0.6808 0.7157 0.7486 0.7794 0.8078 0.8340 0.8577 0.8790 0.8980 0.9147 0.9292 0.9418 0.9525 0.9616 0.9693 0.9756 0.9808 0.9850 0.9884 0.9911 0.9932 0.9949 0.9962 0.9972 0.9979 0.9985 0.9989

0.5319 0.5714 0.6103 0.6480 0.6844 0.7190 0.7517 0.7823 0.8106 0.8365 0.8599 0.8810 0.8997 0.9162 0.9306 0.9429 0.9535 0.9625 0.9699 0.9761 0.9812 0.9854 0.9887 0.9913 0.9934 0.9951 0.9963 0.9973 0.9980 0.9986 0.9990

0.5359 0.5753 0.6141 0.6517 0.6879 0.7224 0.7549 0.7852 0.8133 0.8389 0.8621 0.8830 0.9015 0.9177 0.9319 0.9441 0.9545 0.9633 0.9706 0.9767 0.9817 0.9857 0.9890 0.9916 0.9936 0.9952 0.9964 0.9974 0.9981 0.9986 0.9990

Source: This table was generated using the SAS1 function PROBNORM.

742

Example: P(t(30) ≤ 1.697) = 0.95 P(t(30) > 1.697) = 0.05

–4 –3 –2 –1

0 t

1

2

3

4

Ta b l e 2

Percentiles of the t-distribution

df

tð0:90,dfÞ

tð0:95,dfÞ

tð0:975,df Þ

tð0:99,dfÞ

tð0:995,dfÞ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 50 1

3.078 1.886 1.638 1.533 1.476 1.440 1.415 1.397 1.383 1.372 1.363 1.356 1.350 1.345 1.341 1.337 1.333 1.330 1.328 1.325 1.323 1.321 1.319 1.318 1.316 1.315 1.314 1.313 1.311 1.310 1.309 1.309 1.308 1.307 1.306 1.306 1.305 1.304 1.304 1.303 1.299 1.282

6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 1.746 1.740 1.734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1.697 1.696 1.694 1.692 1.691 1.690 1.688 1.687 1.686 1.685 1.684 1.676 1.645

12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064 2.060 2.056 2.052 2.048 2.045 2.042 2.040 2.037 2.035 2.032 2.030 2.028 2.026 2.024 2.023 2.021 2.009 1.960

31.821 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764 2.718 2.681 2.650 2.624 2.602 2.583 2.567 2.552 2.539 2.528 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.457 2.453 2.449 2.445 2.441 2.438 2.434 2.431 2.429 2.426 2.423 2.403 2.326

63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 3.169 3.106 3.055 3.012 2.977 2.947 2.921 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.750 2.744 2.738 2.733 2.728 2.724 2.719 2.715 2.712 2.708 2.704 2.678 2.576

Source: This table was generated using the SAS1 function TINV

744

APPENDIX D

Example: P(χ(24) ≤ 9.488) ! 0.95 P(χ(24) > 9.488) ! 0.05

0

10

20

2

Ta b l e 3

Percentiles of the Chi-square Distribution

df

x2ð0.90;dfÞ

x2ð0:95;dfÞ

x2ð0:975;dfÞ

x2ð0:99;dfÞ

x2ð0:995;dfÞ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 35 40 50 60 70 80 90 100 110 120

2.706 4.605 6.251 7.779 9.236 10.645 12.017 13.362 14.684 15.987 17.275 18.549 19.812 21.064 22.307 23.542 24.769 25.989 27.204 28.412 29.615 30.813 32.007 33.196 34.382 35.563 36.741 37.916 39.087 40.256 46.059 51.805 63.167 74.397 85.527 96.578 107.565 118.498 129.385 140.233

3.841 5.991 7.815 9.488 11.070 12.592 14.067 15.507 16.919 18.307 19.675 21.026 22.362 23.685 24.996 26.296 27.587 28.869 30.144 31.410 32.671 33.924 35.172 36.415 37.652 38.885 40.113 41.337 42.557 43.773 49.802 55.758 67.505 79.082 90.531 101.879 113.145 124.342 135.480 146.567

5.024 7.378 9.348 11.143 12.833 14.449 16.013 17.535 19.023 20.483 21.920 23.337 24.736 26.119 27.488 28.845 30.191 31.526 32.852 34.170 35.479 36.781 38.076 39.364 40.646 41.923 43.195 44.461 45.722 46.979 53.203 59.342 71.420 83.298 95.023 106.629 118.136 129.561 140.917 152.211

6.635 9.210 11.345 13.277 15.086 16.812 18.475 20.090 21.666 23.209 24.725 26.217 27.688 29.141 30.578 32.000 33.409 34.805 36.191 37.566 38.932 40.289 41.638 42.980 44.314 45.642 46.963 48.278 49.588 50.892 57.342 63.691 76.154 88.379 100.425 112.329 124.116 135.807 147.414 158.950

7.879 10.597 12.838 14.860 16.750 18.548 20.278 21.955 23.589 25.188 26.757 28.300 29.819 31.319 32.801 34.267 35.718 37.156 38.582 39.997 41.401 42.796 44.181 45.559 46.928 48.290 49.645 50.993 52.336 53.672 60.275 66.766 79.490 91.952 104.215 116.321 128.299 140.169 151.948 163.648

Source: This table was generated using the SAS1 function CINV.

Example: P(F(4,30) ≤ 2.69) ! 0.95 P(F(4,30) > 2.69) ! 0.05

0

1

2

3 F

Ta b l e 4

4

5

6

95th Percentile for the F-distribution

v2 =v1

1

2

3

4

7

8

9

10

12

15

20

60

1

1

161.45

199.50

215.71

224.58

230.16

233.99

236.77

238.88

240.54

241.88

243.91

245.95

248.01

250.10

252.20

254.31

2

18.51

19.00

19.16

19.25

19.30

19.33

...