¹Y 1 ¹ Y 2 p s. 2 1 =n 1 + s 2 2=n 2. ¹X X n i. X i u i. i=1 ( ^Y i ¹ Y i ) 2 + P n

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1 Review Sheets for Stock ad Watso Hypothesis testig p-value: probability of drawig a statistic at least as adverse to the ull as the value actually computed with your data, assumig that the ull hypothesis is true. sigificace level ( ): a pre-specified probability of icorrectly rejectig the ull whe the ull is true; also called margial sigificace level. t-statistic Differece-i-meas test (two-sided): t = ¹Y ¹ Y 2 p s 2 = + s 2 2= 2 = ¹Y ¹ Y 2 SE( ¹ Y ¹ Y 2 ) ; cofidece iterval ( ¹Y ¹Y 2 )+z =2 SE( ¹Y ¹Y 2 ), where z =2 is the critical value, is the sigificace level. If the pre-specified sigificace level is 5%, you reject the ull hypothesis if jtj > :96, equivaletly, you reject if p 0:05. Itroductio to Liear Regressio (Simple Regressio). The coditioal distributio of u give X has mea zero, that is, E(ujX = x) =0. 2. (X i ;Y i ), i =,,, are i.i.d. (idepedetly ad idetically distributed; that is, (X i ;Y i ) are distributed idepedetly of (X j ;Y j ) for differet observatios i ad j. 3. X ad u have (fiite) four momets, that is: E(X 4 ) < ad E(u 4 ) <. Simple Regressio Y i = 0 + X i + u i ; Least Squares Estimate: miimize (Y i 0 X i ) 2 i= ^ = (X i X)(Y ¹ i Y ¹ P ) i= (X i X) ¹ i= = 2 + (X i X)u ¹ i i= (X i X) ¹, 2 ^ 0 = ¹ Y ^ ¹ X = 0 + P ³ i= ¹X i= X i ¹X 2 i= X2 i X i u i = 0 + P ³ i= i= Large sample property (whe sample size is large): ¾ 2^ = var[(x i ¹ x )u i ], ¾ 2^ 0 = var(h i u i ) [var(x i )] 2 [E(H, H i 2)]2 i = ¹ i E(Xi 2)X i ³ ¹X i= X i ¹X i= X i Squared residuals: Y i = ^Y i +^u i, i= (Y i ¹ Y ) 2 = i= ( ^Y i ¹ Y i ) 2 + i= (Y i ^Y i ) 2 R 2 : R 2 = ESS TSS = X i u i X i 2 i= ( ^Y i ¹ Y ) 2 i= (Y i ¹ Y ) 2, 0 R2 ; (simple regressio) R 2 = cov2 (X; Y ) var(x)var(y ) = ½2 XY Homoskedastic: if var(ujx = x) is costat; otherwise, heteroskedastic (always use heteroskedasticity-robust stadard errors)

2 Multiple Regressio omitted variable bias: bias i the OLS estimator occurs as a result of a omitted factor. If a omitted factor Z is a determiat of Y (that is, it is cotaied i u) ad correlated with X, the ½ Xu 6=0 ad the OLS estimator ^ is biased. multiple regressio: Y i = 0 + X i + + kx ki + u i ;i=;:::;; OLS for multiple regressio: i= [Y i (b 0 + b X i + + b k X ki )] 2 ;. The coditioal distributio of u give the X s has mea zero, that is,e(ujx = x ;:::;X k = x k )=0. 2. (X i,,x ki,y i ), i =,,, are i.i.d. 3. X ;:::;X k, ad u have four momets:e[xi] 4 <,, E[Xki 4 ] <, E[u4 i ] <. 4. There is o perfect multicolliearity. Iterpretatio of the coefficiets: suppose k=2, chage X by X but hold X 2 costat: before chage: Y = b 0 + b X + b 2 X 2; after the chage:y + Y = b 0 + b (X + X ) + b 2 X 2 the differece: Y = b X The F-statistic: tests all parts of a joit hypothesis at oce (special case, two coefficiets) F = μ t 2 + t 2 2 2^½ t ;t 2 t t 2 2 ^½ 2 t ;t 2 (special case, homoskedasticity-oly F-statistic) (Rurestricted 2 F = R2 restricted )=q ( Rurestricted 2 )=( k urestricted )» F q; k (large )» Â 2 q=q = F q; q = the umber of restrictios uder the ull k urestricted = the umber of regressors i the urestricted regressio. μ R 2 bar: R ¹ SSR 2 =, the portio the regressors explai the variatio i Y. k TSS Model Choice: o simple recipe for decidig which variables belog i a regressio; oe approach is to specify a base model the explore the sesitivity of the key estimate(s) i other alteratives. Noliear Regressio Fuctios Noliear Populatio Regressio Fuctio: Y i = f(x i ;X 2i ;:::;X ki ) + u i ;i=;:::;.e(u i jx i ;X 2i ;:::;X ki ) =0 (same); f is the coditioal expectatio of Y give the X s. 2. (X i ;:::;X ki ;Y i ) are i.i.d. (same). 3. eough momets exist (same idea; the precise statemet depeds o specific f). 4. No perfect multicolliearity (same idea; the precise statemet depeds o the specific f). Example - Polyomials Regressio: Y i = 0 + X i + 2Xi rxi r + u i Example 2 - Logarithmic fuctios of Y ad/or X

3 Case regressio fuctio Iterpretatio liear-log Y i = 0 + l(x i ) + u i a % icrease i X is associated with a 0.0 chage i Y. log-liear l(y i ) = 0 + X i + u i a chage i X by oe uit is associated with a 00 % chage i Y log-log l(y i ) = 0 + l(x i ) + u i a % chage i X is associated with a % chage i Y has the iterpretatio of a elasticity. Assessig Studies Based o Multiple Regressio Iteral validity: the statistical ifereces about causal effects are valid for the populatio beig studied. Threats to iteral validity: Threats Reaso Remedies Omitted variable bias Omit a variable that is both (i) a determiat of Y ad (ii) correlated with at least oe icluded regressor. If measurable, iclude as a regressor i multiple regressio; use pael data If ot measurable, use IV a radomized cotrolled experimet. Wrog fuctioal form fuctioal form is icorrect Cotiuous depedet variable: use oliear specificatios i X Discrete (example: biary) depedet variable: probit or logit, etc Errors-i-variables bias Sample selectio bias Simultaeous causality etry errors i admiistrative data Recollectio errors i surveys Ambiguous questios problems Itetioally false respose problems with surveys a selectio process (i) iflueces the availability of data (ii) is related to the depedet variable. Simultaeous causality bias i Obtai better data. Develop a specific model cross-checkig usig admiistrative records ad the discrepacies are aalyzed ad modeled. IV Collect the sample i a way that avoids sample selectio. Radomized cotrolled experimet. Costruct a model of the sample selectio problem ad estimate that model

4 bias equatios (a) Causal effect of X o Y: Y i = 0 + X i + u i (b) Causal effect of Y o X: X i = 0 + Y i + v i Radomized cotrolled experimet. Develop ad estimate a complete model of both directios of causality. Use IV regressio to estimate the causal effect of iterest. Exteral validity: the statistical ifereces ca be geeralized from the populatio ad settig studied to other populatios ad settigs, where the settig refers to the legal, policy, ad physical eviromet ad related saliet features. Regressio with Pael Data A pael dataset cotais observatios o multiple etities (idividuals), where each etity is observed at two or more poits i time. Pael data is also called logitudial data. Balaced pael: o missig observatios; ubalaced pael: some etities (states) are ot observed for some time periods (years). Fixed Effects Assumptios:.E(u it jx i ;:::;X it ; i ) =0: 2.(X i ;:::;X it ;Y i ;:::;Y it ), i =,,, are i.i.d. draws from their joit distributio. 3. (X it ;u it ) have fiite fourth momets. 4. There is o perfect multicolliearity (multiple X s) 5. corr(u it ;u is jx it ;X is ; i ) =0for t 6= s State Fixed Effects oly Y it = 0 + X it + 2Z i + u i Time Fixed Effects oly Y it = 0 + X it + 3S t + u it - biary regressor [by OLS] Y it = 0 + X it + 2 D 2i + + D i + u i Fixed effects Y it = X it + i + u i [by Etity-demeaed OLS] Y it = X it + i + u i, i is called a state fixed effect T- Biary regressor [by OLS] Y it = 0 + X it + 2B 2t + + T B Tt + u it Fixed effects

5 Y it = X it + ¹ t + u it, ¹ t is called a time fixed effect Both Time ad State Fixed Effects Y it = 0 + X it + 2Z i + 3S t + u it Biary regressor [by OLS] Y it = 0 + X it + 2 D 2i + + D i + ± 2 B 2t + + ± T B Tt + u it State ad time effects : [by State- ad year-demeaed OLS] Y it = X it + i + ¹ t + u it Regressio with a Biary Depedet Variable Model liear probability model (Predicted probabilities ca be <0 or >!) Y i = 0 + X i + u i Estimatio Probit Pr(Y =jx) = ( 0 + X) Noliear least squares, MLE Logit Pr(Y =jx) = Noliear least squares, MLE +e ( 0+ X) P Noliear least squares (NLS): mi b0 ;b [Y i F (b 0 + b X i )] 2, F ca be probit or logit; but NLS i= does t give ad explicit solutio, ad must be solved umerically usig the computer. Measures of fit:. fractio correctly predicted = fractio of Y s for which predicted probability is >50% (if Y i =) or is <50% (if Y i =0). 2. pseudo-r 2 : the improvemet i the value of the log likelihood, relative to havig o X s. OLS Istrumetal Variables Regressio Istrumetal variables regressio ca elimiate bias from () omitted variable bias; (2) simultaeous causality; (3) errors-i-variables bias. A edogeous variable is oe that is correlated with u; A exogeous variable is oe that is ucorrelated with u. Two coditios for a valid istrumet: Y i = 0 + X i + u i () Istrumet relevace: corr(z i ;X i ) 6= 0; (2) Istrumet exogeeity: corr(z i ;u i ) =0 Two Stage Least Squares (TSLS) For Simple Regressio () Rregress X o Z usig OLS X i = ¼ 0 + ¼ Z i + v i, compute the predicted values of X i, ^X i (2) Regress Y o ^X i usig OLS: Y i = 0 + ^Xi + u i

6 ^ TSLS = i= ( ^X i ¹^X i )(Y i ¹Y ) ( ^X i ¹^Xi ) 2 = i= ^¼ (Z i ¹Z)(Y i ¹Y ) i= ^¼2 (Z i ¹ Z) 2 ^¼ = i= ( ^X i ¹^Xi )(Z i ¹ Z) (Z i ¹ Z) 2, ^¼ = i= ( ^X i ¹^Xi )(Z i ¹ Z) i= (Z i ¹ Z) 2 Cosistecy of the TSLS estimator: ^ TSLS = S YZ S XZ p! cov(y;z) cov(x; Z) = Iferece usig TSLS: [assumes that the istrumets are valid] ¾ 2^ TSLS = var[(z i ¹ Z )u i ] [cov(z i ;X i )] 2, ^ TSLS a» N 0;¾ 2^ TSLS [for large samples] The stadard error: ^u i = Y i ^ TSLS 0 ^ TSLS X i ; ot ^X i! The geeral IV regressio model: Y i = b 0 + b X i + ::: + b k X ki + b k+ W i + ::: + b k+r W ri + u i W i ;:::;W ri are the icluded exogeous variables or icluded exogeous regressors (ucorrelated with u i ); Z i ;:::;Z mi are the m istrumetal variables (the excluded exogeous variables). Idetificatio: a parameter is said to be idetified if differet values of the parameter would produce differet distributios of the data. The coefficiets,, k are said to be () exactly idetified if m = k; (2) overidetified if m > k; (3) uderidetified if m < k. Two Stage Least Squares for Multiple Regressio. Regress X ji o all the exogeous regressors: regress X o W,,W r,z,,z m by OLS, do this for all j k; the compute ^Xi ;:::; ^X ki, i 2. Regress Y o ^X i ;:::; ^X ki, W,, W r by OLS. Assumptios:. E(u i W i,,w ri ) = 0 2. (Y i,x i,,x ki,w i,,w ri,z i,,z mi ) are i.i.d. 3. The X s, W s, Z s, ad Y have ozero, fiite 4 th momets 4. The W s are ot perfectly multicolliear 5. The istrumets (Z i,,z mi ) satisfy the coditios for a valid set of istrumets. Check for weak IV: Rule-of-thumb: If the first stage F-statistic is less tha 0, the the set of istrumets is weak. Overidetifyig Test (J test) () estimate the equatio usig TSLS ad all m istrumets; compute the predicted values ^Y i, usig the X i (ot ^X i ); (2) Compute the residuals ^u i ; (3) Regress ^u i agaist Z i,,z mi, W i,,w ri; (4)

7 Compute the F-statistic testig the hypothesis that the coefficiets o Z i,,z mi are all zero; (5) Compute the J-statistic as J = mf» Â 2 (m k). Experimets ad Quasi-Experimets experimet desiged ad implemeted cosciously by huma researchers. quasi-experimet has a source of radomizatio that is ot part of a coscious radomized or atural experimet treatmet ad cotrol desig. Program evaluatio the field of statistics aimed at evaluatig the effect of a program or policy Threats to Iteral Validity. Failure to radomize ( imperfect radomizatio) 2. Failure to follow treatmet protocol (or partial compliace ) 3. Experimetal effects Threats to Exteral Validity. Norepresetative sample 2. Norepresetative treatmet 3. Geeral equilibrium effects 4. Treatmet v. eligibility effects Model differeces Y X OLS differeces- i-differeces Y = Y after Y before X OLS adjusts for iitial differeces betwee treatmet ad cotrol groups differeces with add l Y X,W, OLS cotrols for additioal subject characteristics W regressors,w differeces-i-differeces Y = Y after X,W, OLS adjusts for group differeces + cotrols for with add l regressors Y before,w subject char s W Istrumetal variables Y X TSLS Z = iitial radom assigmet; elimiates bias from partial compliace Reasos for icludig additioal subject characteristics: () Efficiecy; (2) Check for radomizatio; (3) adjust for coditioal radomizatio. A cotrol variable W is a variable that results i X satisfyig the coditioal mea idepedece coditio: E(u X,W) = E(u W) Heterogeeous Causal Effects: causal (or treatmet) effect varies across idividuals.