Econometrics Cheat Sheet, Cheat Sheet of Econometrics and Mathematical Economics

Download Econometrics Cheat Sheet and more Cheat Sheet Econometrics and Mathematical Economics in PDF only on Docsity! Econometrics Cheat Sheet University of Oklahoma Data & Causality Basics about data types and causality. Types of data Experimental Observational Data from randomized experiment Data collected passively Multiple units, one point in time ‘ional Cross-sec ‘Time series Single unit, multiple points in time Longitudinal (or Panel)Multiple units followed over multiple time periods Experimental data Correlation =} Causality © Very rare in Social Sciences Statistics basics We examine a random sample of data to learn about the population Random sample Parameter (8) Estimator of @ Representative of population Some number describing population Rule assigning value of @ to sample e.g. Sample average, Y = 3 ON What the estimator spits out: for a particular sample (0) Distribution of estimates across all possible samples Yi Estimate of @ Sampling distribution Bias of estimator WE (W) — 0 Efficiency W efficient if Var(W) < Var(W) Consistency W consistent if > @ as N > 00 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 level (a) Typically, Ho : @ = 0 Typically, Ho : 0 #0 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 |T'| > ¢ c depends on a; c depends on if 1- or 2-sided test test statisti al value (c) p-value Largest @ at which fail to reject Ho; Ho ifp<a Or, A (1)-(5) combined with asymptotic properties Simple Regression Model Regi eris paribus relationship between some variable x and our outcome y ession is useful because we can estimate a cet y=fo+ Pictu We want to estimate 81, which gives us the effect of x on y. OLS formulas To estimate and 1, we make two assumptions: L E(u)=0 2. E (ule) = E(u) for all x When these hold, we get the following formulas: Bo = 9- AE Var (a) fitted values (gi) residuals (é;) ‘Total Sum of Squares R-squared (R?) R? = 335; “frac. of var. in y explained by «” Algebraic properties of OLS estimates x a; = 0 (mean & sum of residuals is zero) ji; = 0 (zero covariance bet. « and resids.) ‘The OLS line (SRF) always passes through (¥,7) SSE +SSR= O<R?<1 Interpretation 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 61 Levellevel r Ay = fAr Levellog yy log(z) Ay = (81/100) [1% Ac] Log-level_ log(y) x %Ay ~ (10081) Ae Log-log —log(y)-—log(x) %Ay & Ai %Ax Quadratic -y ta? Ay= (6, +2Bor) Ae 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 (ul) = E(u) is more plausible) y= So + Biti +o++ + Bete tu B1,..., x: partial effect of each of the 2’s on y — BrEe ed 3) Bo =9- AF — ; Cov (y, residual = Var (residualized x;) where “residualized j” means the residuals from OLS regression of 2; on all other e's (ie. ©1,..-42j—150j41,---2h) Gauss-Markov Assumptions 1. y is a linear function of the 6’s 2. y and 2’s are randomly sampled from population 3. No perfect multicollinearity 4. E(ulri,...,2%) = E(u) 5. Var (ulai,...,2%) = Var (u) When (1)-(4) hold: OLS is unbiased; i.e. E(3;) = 8; When (1)-(5) hold: OLS is Best Linear Unbiased Estimator Variance of u (a.k.a. “error variance”) 0 (Unconfoundedness) o? (Homoskedasticity ) Variance and Standard Error of 3; 2 a Var = eo FH hee hk ar(s) SST; — R3) where SST; = (N ~1)Var(aj) = So (aj — F)) = R5 = R? from a regression of x on all other x’s VVar VVar Standard deviation: Standard error: se(3;) i BH Classical Linear Model (CLM) Add a 6th assumption to Gauss-Markov: 6. wis distributed N (0,0?) Need this to know what the distribution of 8; is Otherwise, need asymptotics to do hypothesis testing of A’s Testing Hypotheses about the (’s Under A (1)-(6),! can test hypotheses about the 8’s t-test for simple hypotheses To test a simple hypothesis like use a t-test: where 0 is the null hypothesize Reject Ho if p < «or if |t| > c (See: Hypothesis testing) F-test for joint hypotheses Can’t use a t-test for j int hypotheses, e.g.: Ho : 83 = 0, Ba =0, Bs =0 Ha : 63 #0 OR Bs #0 OR Bs 40 Instead, use F' statistic: (SSRr — SSRur)/(dj SSRur /dfur =dfur) _ (SSRy— SSRur)/q SSRur/(N —k— 1) where SSRy = SSR of restricted model (if Ho true) SSRuy = SSR of unrestricted model (if Ho false) q = # of equalities in Ho N~k~—1=Deg. Freedom of unrestricted model t Ho if p < aor if F > c (See: Hypothesis testing) Note: F > 0, always Qualitative data Can use qualitative data in our model Must create a dummy variable e.g. “Yes” represented by 1 and “No” by 0 dummy variable trap: Perfect collinearity that happens when too many dummy variables are included in the model y = Bo + Bihappy + B2not-happy + u ‘The above equation suffers from the dummy variable trap because units can only be “happy” or “not happy,” so luding both would result in perfect collinearity with the Interpretation of dummy variables Interpretation of dummy variable coefficients is always relative to the excluded category (e.g. not_-happy): y = Bo + Bihappy + Beage +u By: avg, y for those who are happy compared to those who are unhappy, holding fixed age Interaction terms interaction term: When two ’s are multiplied together y = Bo + Bihappy + Boage + Bshappy x age + u Bs: difference in age slope for those who are happy compared to those who are unhappy Linear Probability Model (LPM) When y is a dummy variable, e.g. happy = Bo + Brage + Beincome +u 8's are interpreted as change in probability: APr(y = 1) = fide By definition, homoskedasti Time Series (TS) data Observe one unit over many time periods ‘ity is violated in the LPM @ e.g. US quarterly GDP, 3-month T-bill rate, ete. « New G-M assumption: no serial correlation in u¢ « Remove random sampling assumption (makes no sense) Two focuses of TS data 1. Causality (e.g. T taxes = | GDP growth) 2. Forecasting (e.g. AAPL stock price next quartel Requirements for TS data To properly use ‘TS data for causal inf / forecasting, need data free of the following elements: ‘Trends: Seasonality: Non-stationarity: y always t or | every period y always } or | at regular intervals y has a unit root; ie. not stable Otherwise, R? and f;’s are misleading AR(1) and Unit Root Processes AR(1) model (Auto Regressive of order 1): ye = pyr + ut Stable if |p| < 1; Unit Root if |p| > 1 “Non-stationary,” “Unit Root,” “Integrated” are all synonymous Correcting for Non-stationarity Easiest way is to take a first differen First differen Test for unit root: Hp of ADF test: Use Ay Augmented Dicke; y has a unit root ye — yer instead of ye Fuller (ADF) test TS Forecasting A good forecast minimizes forecasting error fi: min E (e,a|le) = E [vera = #9)? [Ne] where J; is the information set RMSE measures forecast performance (on future dat: Root Mean Squared Error = Model with lowest RMSE is best forecast Can choose fy in many ways Basic way: Jr41 from linear model ¢ ARIMA, ARMA-GARCH are cutting-edge models Granger causality 2 Granger causes y if, after controlling for past values of y, past values of z help forecast ys CLM violations Heteroskedasticity ¢ Test: Breusch-Pagan or White tests (Ho : homosk.) © If Ho rejected, SEs, t-, and F-stats are invalid e Instead use heterosk.-robust SEs and t- and F-stats Serial correlation © Test: Breusch-Godfrey test (Ho : no serial corr.) e If Ho rejected, SEs, t-, and F-stats are invalid e Instead use HAC SEs and t- and F-stats e HAC: “Heterosk. and Autocorrelation Consistent” Measurement error « Measurement error in © can be a violation of A4 Attenuation bias: 8; biased towards 0 Omitted Variable Bias When an important 2 is excluded: omitted variable bias Bias depends on two forces: 1. Partial effect of 2 on y (i.e. 82) 2. Corre ition between x2 and x1 Which direction does the bias go? Corr(a1, 22) >0 Corr(«1,22) <0 Bo >0 Positive Bias Negative Bias Bo <0 Negative Bias Positive Bias Note: “Positive bias” means {; is too big; “Negative bias” means {1 is too small