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The Rules of Summationåni¼ 1xi¼x 1 þx 2 þþxnåni¼ 1 a¼naåni¼ 1axi¼aån i¼ 1xiåni¼ 1ðxiþyiÞ¼ån i¼ 1xiþån i¼ 1yiåni¼ 1 ðaxiþbyiÞ¼aån i¼ 1 xiþbån i¼ 1 yiåni¼ 1ðaþbxiÞ¼naþbån i¼ 1xix¼å n i¼ 1 xi n ¼x 1 þx 2 þþxn n åni¼ 1ðxixÞ¼ 0å2 i¼ 1 å3 j¼ 1 fðxi;yjÞ¼å2 i¼ 1 ½Šfðxi;y 1 Þþfðxi;y 2 Þþfðxi;y 3 Þ ¼fð...

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Expectations, Variances & Covariances

The Rules of Summation n

covðX; YÞ ¼ E½ðX E½X ÞðY E½Y Þ

å xi ¼ x1 þ x2 þ    þ xn

i¼1 n

å a ¼ na

¼ å å ½x  EðXÞ½ y  EðYÞ f ðx; yÞ x y

i¼1 n

covðX;Y Þ r ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi varðX ÞvarðY Þ

n

å axi ¼ a å xi i¼1

i¼1 n

n

n

E(c1 X þ c2 Y ) ¼ c1E(X ) þ c 2 E(Y ) E(X þ Y ) ¼ E(X ) þ E(Y )

å ðxi þ yi Þ ¼ å xi þ å yi i¼1

i¼1 n

i¼1

n

n

i¼1

i¼1

2

å ðaxi þ byi Þ ¼ a å xi þ b å yi

i¼1 n

å ða þ bxi Þ ¼ na þ b å xi

i¼1

xi

å

x1 þ x2 þ    þ xn ¼ n

x ¼ i¼1n n

varðaX þ bY þ cZÞ ¼ a2 varðX Þ

þ b2 varðYÞ þ c2 varðZ Þ

å ðxi  xÞ ¼ 0

i¼1 2

2

3

å å f ðxi ; yj Þ ¼ å ½f 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 å xi f ðxi Þ ¼ å x f ðxÞ x

i¼1

2

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

i¼1

n

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 )

n

E ½gðX Þ ¼ å gðxÞ f ðxÞ x

E ½g 1 ðXÞ þ g2 ðX Þ ¼ å½g1ðxÞ þ g2 ðxÞ f ðxÞ x

¼ å g1ðxÞ f ðxÞ þ å g2 ðxÞ f ðxÞ x

Xm  Nð0; 1Þ s If X  N(m, s ) and a is a constant, then   am PðX  aÞ 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

¼ E½ g1 ðX Þ þ E½g2 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(X 2)  [E(X )]2 var(a þ cX ) ¼ E[(a þ cX )  E(a þ cX )]2 ¼ c2var(X ) Marginal and Conditional Distributions f ðxÞ ¼ å f ðx; yÞ

for each value X can take

f ðyÞ ¼ å f ðx; yÞ

for each value Y can take

y

x

f ðx; yÞ f ðxjyÞ ¼ P½X 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

Normal Probabilities

f ðx; yÞ f ðxÞ f ðyÞ ¼ ¼ f ðxÞ f ðyÞ f ðyÞ

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

SR2 SR3 SR4 SR5 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 b2 x 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 e j is cov(ei , ej ) ¼ cov(yi , 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)

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 h¼ ¼ ¼  percentage change in x Dx=x Dx y h¼

DEðyÞ=EðyÞ DEðyÞ x x ¼  ¼ b2  EðyÞ Dx=x Dx EðyÞ

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

xi  x

Sðxi  xÞ2

Swi ¼ 0;

Swi xi ¼ 1;

Swi2 ¼ 1=Sðxi  xÞ2

Properties of the Least Squares Estimators " # s2 Sxi2 varðb2 Þ ¼ varðb1 Þ ¼ s2 Sðxi  xÞ2 NSðxi  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 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 b2 .

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 hypothesis test is smaller than the chosen value of a, then the test procedure leads to rejection of the null hypothesis. Prediction y0 ¼ b1 þ b2 x0 þ e0 ; y^0 ¼ b1 þ b2 x0 ; f ¼ y^0  y0 " # q ffiffiffiffiffiffiffiffiffiffiffiffiffi ðx0  xÞ2 1 2 b b þ ; seð f Þ ¼ varð fÞ varð f Þ ¼ s ^ 1þ 2 N Sðxi  xÞ

A (1  a)  100% confidence interval, or prediction interval, for y0 ^ y0  tc seð f Þ Goodness of Fit Sðyi  yÞ2 ¼ Sðy^i  yÞ2 þ S^ei2 SST ¼ SSR þ SSE SSE SSR ¼ 1 R2 ¼ ¼ ðcorrðy; ^yÞÞ2 SST SST

If we make the normality assumption, assumption Log-Linear Model SR6, about the error term, then the least squares estilnðyÞ ¼ b1 þ b2 x þ e; b mators are normally distributed. lnð yÞ ¼ b1 þ b2 x ! ! s2 s2 å x2i 100  b2  % change in y given a one-unit change in x: ; b2  N b2 ; b1  N b1 ; NSðxi  xÞ2 Sðxi  xÞ2 ^ yn ¼ expðb1 þ b2 xÞ Estimated Error Variance

s ^2 ¼

S^e2i N2

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

t-distribution

If assumptions SR1–SR6 of the simple linear regression model hold, then t¼

bk  bk  tðN2Þ ; k ¼ 1; 2 seðbk Þ

Interval Estimates

P[b2  t cse(b 2)  b2  b2 þ tc se(b2 )] 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 H 0 : b2 ¼ c is true, then t¼

b2  c  tðN2Þ seðb2 Þ

^ yc ¼ expðb1 þ b2 xÞexpð^ s2 =2Þ

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

yn ÞÞ Generalized goodness-of-fit measure Rg2¼ ðcorrðy;^

2

Assumptions of the Multiple Regression Model MR1 y i ¼ b1 þ b 2 xi2 þ    þ bK xiK þ e i MR2 E(y i) ¼ b 1 þ b2 xi2 þ    þ bK xiK , E(e i ) ¼ 0. 2 MR3 var(y i) ¼ var(e i) ¼ s MR4 cov(yi , yj ) ¼ cov(ei, e j ) 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 , . . . , b K minimize Sðb1 , b2 , . . . , b KÞ ¼ åðyi  b1  b 2 xi2     b K xiKÞ2

Estimated Error Variance and Estimator Standard Errors s ^2 ¼

å ^ei2 NK

seðbk Þ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b varðbk Þ

Hypothesis Tests and Interval Estimates for Single Parameters bk  bk 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 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi varðb2 Þ þ c2b seðb2 þ cb3 Þ ¼ b varðb3 Þ þ 2c  b covðb2 ; b3 Þ t¼

When H0 is true

Joint F-tests

To test J joint hypotheses,

H0 : b2 ¼ 0; b3 ¼ 0; : : : ; bK ¼ 0 H 1 : at least one of the bk is nonzero ðSST  SSEÞ=ðK  1Þ SSE=ðN  KÞ

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

y^i ¼ b1 þ b2 xi2 þ b3 xi3

H0 : g1 ¼ 0 yi ¼ b1 þ b2 xi2 þ b3 xi3 þ g1 ^yi 2þ ei ; H0 : g1 ¼ g2 ¼ 0 yi ¼ b1 þ b2 xi2 þ b3 xi3 þ g1y^2i þ g2y^i3 þ ei ; Model Selection AIC ¼ ln(SSE=N) þ 2K=N SC ¼ ln(SSE=N) þ K ln(N)=N Collinearity and Omitted Variables yi ¼ b1 þ b2 xi2 þ b3 xi3 þ ei s2 varðb2 Þ ¼ 2 ð1  r23 Þ å ðxi2  x2 Þ2 When x3 is omitted; biasðb2Þ ¼ Eðb2 Þ  b2 ¼ b3 Heteroskedasticity 2 var(y i) ¼ var(ei) ¼ si General variance function

b covðx2 ; x3 Þ b varðx2 Þ

s2i ¼ expða1 þ a2 zi2 þ    þ aS ziSÞ

Breusch-Pagan and White Tests for H 0: a2 ¼ a3 ¼    ¼ aS ¼ 0 When H0 is true

x2 ¼ N  R2  x2ðS1Þ

Goldfeld-Quandt test for H0 : s2M ¼ s2R versus H1 : sM2 6¼ sR2 sR2 F¼s ^ 2M =^ varðei Þ ¼ s2i

 FðNM KM ;NR KR Þ When H0 is true Transformed model for ¼ s2 x i pffiffiffiffi p ffiffiffiffi pffiffiffiffi p ffiffiffiffi yi = xi ¼ b1 ð1= xi Þþ b2ð xi = xiÞ þ ei = xi Estimating the variance function

lnð^e2i Þ ¼ lnðsi 2Þ þ vi ¼ a1 þ a2 zi2 þ    þ aS ziS þ vi

Grouped data varðei Þ ¼ si2 ¼

(

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 å ðyt  yÞðytk  yÞ= å 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 H0 :r ¼ 0 with t-test

et ¼ g1 þ g2 xt þ r^et1 þv^t ^

Test using LM ¼ T  R2

y t ¼ b1 þ b2 x t þ e t

AR(1) error

et ¼ ret1 þ vt

Nonlinear least squares estimation

ðSSER  SSEU Þ=J SSEU =ðN  KÞ To test the overall significance of the model the null and alternative hypotheses and F statistic are F¼

F¼

Regression with Stationary Time Series Variables

2 sM i ¼ 1; 2; . . . ; NM s2R i ¼ 1; 2; . . . ; NR

Transformed model for feasible generalized least squares  .p ffiffiffiffiffi  .p ffiffiffiffiffi . pffiffiffiffiffi .p ffiffiffiffiffi s ^ i ¼ b1 1 s ^ i þ b2 x i s ^ i þ ei s ^i yi

yt ¼ b1 ð1  rÞ þ b2 xt þ ryt1  b2 rxt1 þ vt ARDL(p, q) model yt ¼ d þ d0 xt þ dl xt1 þ    þ dq xtq þ ul yt1 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

Exponential smoothing y^t 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

 ðb0 þ b1 L þ b2 L2 þ   Þ

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 2 (with constant but no trend): Dyt ¼ a þ gyt1 þ vt Dickey-Fuller Test 3 (with constant and with trend): Dyt ¼ a þ gyt1 þ lt þ vt Augmented Dickey-Fuller Tests: m

Dyt ¼ a þ gyt1 þ å as 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, eis ) ¼ cts Fixed effects model b1i not random yit ¼ b1i þ b2 x2it þ b3 x3it þ eit 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 Ts2u þ se2 a ¼ 1  se

Hausman test

t ¼ ðbFE;k  bRE;k Þ

h i1=2 b b varðb FE;k Þ varðbRE;k Þ...