Econometrics-cheat-sheet - 2 All the formulas you need PDF

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General formulas used in econometrics for your every need. Use it for exams or for solving home works....

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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 Sx2i 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 b1 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 þ ; seð f Þ ¼ b varð f Þ 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 SR6, about the error term, then the least squares esti- Log-Linear Model lnðyÞ ¼ b1 þ b2 x þ e; b mators are normally distributed. lnð yÞ ¼ b1 þ b2 x ! ! s2 s2 å xi2 100  b2  % change in y given a one-unit change in x: ; b2  N b2 ; b1  N b1 ; Sðxi  xÞ2 NSðxi  xÞ2 ^ yn ¼ expðb1 þ b2 xÞ Estimated Error Variance s ^2 ¼

S^e2i N2

Estimator Standard Errors q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 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)  b 2  b 2 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 H0 : 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

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

2

Assumptions of the Multiple Regression Model MR1 yi ¼ b1 þ b2 x i2 þ    þ bK xiK þ e i MR2 E(y i) ¼ b 1 þ b 2xi2 þ    þ bK xiK , E(e i ) ¼ 0. 2 MR3 var(y i) ¼ var(e i) ¼ s MR4 cov(yi , y )j ¼ cov(e i , 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 , . . . , b K minimize Sðb1 , b 2, . . . , bK Þ ¼ åðyi  b1  b2 xi2      bK 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 c2 b seðb2 þ cb3 Þ ¼ b varðb3 Þ þ 2c  b covðb2 ; b3 Þ

When H0 is true

t¼

Joint F-tests

To test J joint hypotheses,

H0 : b2 ¼ 0; b3 ¼ 0; : : : ; bK ¼ 0 H1 : at least one of the bk is nonzero ðSST  SSEÞ=ðK  1Þ F¼ SSE=ðN  KÞ 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 g2 ^yi3 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 ð1  r223 Þ å ðxi2  x2 Þ2 When x3 is omitted; biasðb2 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 ziS Þ

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

x2 ¼ N  R2  x2ðS1Þ

2 Goldfeld-Quandt test for H0 : sM ¼ s2R versus H1 : s2M 6¼ sR2

s2R F¼s ^ 2M =^ varðei Þ ¼ si2

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

(

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 T rk  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 þ v^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

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

Regression with Stationary Time Series Variables

sM2 i ¼ 1; 2; . . . ; NM s2R i ¼ 1; 2; . . . ; NR

Transformed model for feasible generalized least squares  .p ffiffiffiffiffi  .pffiffiffiffiffi . 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 2 q d0 þ d1 L þ d2 L þ    þ dq L 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 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 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...