Regressions with a dummy variable as the dependent variable y 0 1x1 e y can be a binary number either 0 or 1 Use D to denote dummy dependent variable Predicting D Estimate of a conditional expected value average value of D given values of X ex E D X 9 Implies that there is 9 outcomes of D 1 and 1 outcome of D 0 Estimates the probability that D 1 given the values of X This model is called a Linear Probability Model Change in probability of D 1 change in x Forcing this regression to a straight line can cause impossible probabilities you must use non linear regression to avoid this Called a Logit Regression change in log odds D 1 change in X change in odds D 1 change in X Odds ratio eB odds D 1 x 1 odds D 1 x As x increases by one unit the odds will change by a factor of the odds ratio Panel Data Longitudal Data time Data that contains multiple observations of the same individuals across different periods of Common types include Before and After studies multiple period tracking studies Variables have two subscripts Y it and Xit subscript i represents the individual subscript t represents a specific time There are different types of variables Type 1 Variables that differ across individuals but are constant in time i e the time index is irrelevant ex gender race paental education Type 2 Variables that change across time but the same for individuals i e the individual index is irrelevant Type 3 Variables that change over time and are different for each individual i e Employment marital status Pooled Regression A regression where we ignore that the data is in a panel E e Xi does not equal 0 This violates the CLRM assumption number 2 Yit 0 1xit eit eit may be composed of errors from all three types of variables ai error term from T1 variable bt error term from T2 variable cit error term from T3 variable Check first difference regression change in Yit 0 1 change in xit e Challenges to first differences The data needs observations for all individuals at all times You must have variation in the change in Xit or the Standard Error of will be very large Fixed effects regression Assign a unique Dummy variable to every individual Include these dummy variables in your regression Yit 0 1xit F1D1 F2D2 F3D3 bt cit Fixed effects regression has n intercepts You only create n 1 dummy variables One individual does not get a dummy variable Once you include fixed effects in your regression all other T1 variables must be dropped Distributed Lag Model DLM Yt o 0Xt 1xt 1 2Xt 2 pXt p et A model that past values affect current values Every lag you add reduces one observation values should taper off as lag length increases should be smooth 1 10 Problems with the DLM Values of lag x s are likely to be highly colinear large S E for your estimators s Due to large S E the tapering of s may not be smooth Each lag uses up one observation and we lose another degree of freedom Solution to the Problems with DLM i e every lag decreases observation by 2 Utilize or use the first lag of y to replace all lags of X This is called a dynamic model any model where lag values of y are used as a independent variable Dynamic Model Yt o 0Xt Yt 1 et A Dynamic Model has a dependent variable that changes over time Lag of dependent variable as individiual variable Independent variable that are notlinear combinations of the other independent variables If value of 1 We lose only one observation and 1 degree of freedom because we are only using a single lag 2 decreases and there is smooth tapering Completely avoids all three problems of DLM Don t directly put lag X s into model no multicollinearity Long run multiplier t M 0 t 0 t 1 1 if 1 t 0 M 0 1 1 Auto regressive Model AR q Yt o 1Yt 1 2Yt 2 qYt q et Auto regressive Distributed Lag Model ADL q p Yt o 1Yt 1 2Yt 2 qYt q 1Xt 1 2Xt 2 qXt q ADL q p include lags of dependent variable and lags of other independent variables Bayesian Criterion Information If you have T observations and K regressors BIC ln SSR T k ln T T You want to pick the combination of q and p based on minimizing BIC Granger Causality Stationarity F test that all coefficients for x variable 0 it says that past values of x have no effect on y if reject than x granger causes y A variable is stationary if its main properties do not change over time mean variance correlation between variable and its own lags is only dependent on number of lags If two variables are non stationary they may be correlated for non causal reasons this is called a spurious relationship Testing for mean stationary Run AR 1 Yt 0 Yt 1 e If 1 If 1 then Y will move towards a long run variable y has a unit root y is not mean stationary If both variables come as non stationary Are they both cointegrated cointegrated non stationary in the same way Dickey Fuller Test Change in Yt 0 Yt 1 1 et Regress change in Yt on Yt 1 If t stat critical value If they re not cointegrated cointegrated change in Y on change in x calculate t statistic in usual way and compare to critical value take the first differences of you variables and run regression of change in Y and change in x are likely to be stationary No spurious regression results Problems with this approach variables become noisy Hard to study because it doesn t follow economic theory Stationarity cont If the variables are cointegrated regression of y on x will not be spurious Regress y on X estimate the residuals e Test whether the e s are stationary If yes If no non stationary means it has a unit root y and x are cointegrated y and x are not cointegrated