Econometrics formula sheet, Cheat Sheet of Economics

Download Econometrics formula sheet and more Cheat Sheet Economics in PDF only on Docsity! Econometrics Cheat Sheet Data & Causality Basics about data types and causality.Types of data Experimental Observational Data Data from collected randomized passively experiment Time Cross-sectional series Single Multiple unit, units, multiple one point points in in time time Longitudinal (or Panel)Multiple time periods units followed over multiple Experimental data • Correlation ==> Causality • Very rare in Socia! Sciences Statistics basics We examine a random sample of data to learn about the population Random sample Parameter (0) Estimator of 0 Estimate of 0 Sampling distribution Bias of estimator W Efficiency Consistency Representative of population Some number describing population Rule assigning value of 0 to sample e.g. Sample average, Y = 1J I:�1 Y; What the estimator spits out for a particular sample ( 0) Distribution of estimates across ali possible samples E(W)- 0 W efficient if Var(W) < Var(W) W consistent if 0 ➔ 0 as N ➔ oo Hypothesis testing The way we answer yes/no questions about our population using a sample of data. e.g. "Does increasing public school spending increase student achievement?" null hypothesis (Ho) alt. hypothesis (Ha) significance leve! (a) test statistic (T) criticai value (c) p-value Typically, Ho : 0 = O Typically, Ho : 0 t O Tolerance for making Type I error; (e.g. 10%, 5%, or 1%) Some function of the sample of data Value of T such that reject Ho if ITJ > c; c depends on a; c depends on if 1- or 2-sided test Largest a at which fai! to reject Ho ; reject Ho if p < a Simple Regression Model Regression is useful because we can estimate a ceteris paribus relationship between some variable x and our outcome y Y = f3o + f31x + u We want to estimate /Ji, which gives us the effect of x on y. OLS formulas To estimate /Jo and /J1 , we make two assumptions: l. E(u) = O 2. E(ulx)=E(u) for all x When these hold, we get the following formulas: /Jo = y - /J1x /J1 = � (y, x) Var (x) fitted va.lues (y;) 1/ì = /Jo + /J1 x; resid uals (il;) u; = y; - y; Tota! Sum of Squares SST = I:�1 (y; -y)2 Expl. Sum of Squares SSE = I:;1 (y; -y)2 Resid. Sum of Squares SSR = I:i=l uf R-squared (R2 ) R2 = ii:; "frac. of var. in y explained by x" Algebraic properties of OLS estimates I:�1 u; = O (mean & sum of residuals is zero) I:�1 x;u; = O (zero covariance bet. x and resids.) The OLS line (SRF) always passes through (x, y) SSE + SSR = SST O '.S R2 '.S 1 lnterpretation and functional form Our model is restricted to be linear in parameters But not linear in x Other functional forms can give more realistic model Model DV RHS Interpretation of /31 Level-level y X 6.y = /316.x Level-log y log(x) 6.y = (/31/100) [1%6.x] Log-level log(y) X %6.y = (100/31) 6.x Log-log log(y) log(x) %6.y = /31 %.6.x Quadrati e y X+ X2 .6.y = (/31 + 2/32x) .6.x Note: DV dependent variable; RHS right hand side Multiple Regression Model Multiple regression is more useful than simple regression because we can more plausibly estimate ceteris paribus relationships (i.e. E (ulx) = E (u) is more plausible) Y = f3o + /31x1 + · · · + /3kXk + U /J1, ... , /Jk : partial effect of each of the x's on y /Jo = y -/J1x1 - · · · - /JkXk fJ _ � (y, residualized Xj ) J - Var (residualized Xj ) where "residua.lized x/' means the residuals from OLS regression of Xj on ali other x's (i.e. x1 , . .. , Xj-l, x 1+1, . .. Xk) Gauss-Markov Assumptions l. y is a linear function of the /3's 2. y and x's are randomly sampled from population 3. No perfect multicollinearity 4. E(ulx1 , . .. , xk ) = E(u) = O (Unconfoundedness) 5. Var (ulx1 , .. . , xk) = Var (u) = a2 (Homoskedasticity) When (1)-(4) hold: OLS is unbiased; i.e. E(/Jj ) = /3j When (1)-(5) hold: OLS is Best Linear Unbiased Estima.tor Variance of u (a.k.a. "error variance") a- 2 = SSR N-I<-1 1 � .2 = -N-- I<- - 1 � U; t=l Variance and Standard Error of /J j where • a2 Var(/3j) = SSTj (l-Ry)' j = 1, 2, .. . , k N SSTj = (N - l)Var(xj ) = L(Xij -Xj ) i=l RJ = R2 from a regression of Xj on ali other x's Standard deviation: JVar Standard error: � . I a- 2 se(/3j) = , SST1(1-R7)'j = l, .. . , k Classica! Linear Model ( CLM) Add a 6th assumption to Gauss-Markov: 6. u is distributed N (o, a2 ) Need this to know what the distribution of /Jj is Otherwise, can't conduct hypothesis tests about the f3's Testing Hypotheses about the /3's Under A (1)-(6), can test hypotheses about the (3's t-test for simple hypotheses To test a simple hypothesis like Ho : /3j = O Ha: /3j t O use a t-test: fJ - o t = _J __ se (/Ji) where O is the null hypothesized value. Reject Ho if p < a or if ltl > e (See: Hypothesis testing)