Fixed Effects and Random Effects Models What do they do? Fixed effects and random effects models work to remove omitted variable bias by measuring change within a group. By measuring within a group (across time) you control for a number of potential omitted variables unique to the group. Difference between Fixed Effects and Random Effects: Perhaps the most fundamental difference between fixed and random effect is of inference. A fixed-effects analysis can only support inference about the group of measurements (subjects, etc.) you actually have - the actual subject pool you looked at. A random-effects analysis, by contrast, allows you to infer something about the population from which you drew the sample. If the effect size in each subject relative to the variance between your subjects is large enough, you can guess (given a large enough sample size) that your population exhibits that effect. Bottom line – If you use fixed effects on a random sample, you cannot make inferences outside your data set. Random effects assume a normal distribution, so y ou can make inferences to a larger population. Assumptions - Fixed Effects assumes that the individual specific effect is correlated to the independent variable. - Random Effects assumes the individual specific effects are uncorre lated with the independent variables.
What type of data do you use?
Examples
Pooled Data: Cross sectional AND time series data. Observations from different cross-sectional entities for different time periods. Example: State populations (cross-sectional) from 1971-2000 (time-series).
Food Desert Article (Panel Data): CARDIA surveys of the same people over 15 years asking the same questions at intervals. As CARDIA surveys were conducted, grocery/fast food data were taken at the same time
-OR-
Panel Data: Cross sectional AND time series data. Observations from the same cross-sectional entities (countries, people, firms) over time. More powerful than general pooled data.
Internet on Inflation Article (Panel Data): Cross-country data from 207 countries from 1991 through 2000 Variables included are Internet users, population, CPI inflation rate and unemployment rate from World Bank (2002) indicators and the West Texas Intermediate price from the International Monetary Fund (2002)
Fixed and Random Effects Regression:
in time t . Y it it is the dependent variable observed for individual i in X it it is the time-variant regressor Z i i is the time-invariant regressor; observed and can not be estimated directly by the fixed effect model mo del but can be estimated by the r andom effect model αi is the unobserved individual effect uit is the error term
Fixed Effects Model What the model does: 1. Assigns every cross-sectional entity a dummy variable. a. 0 = this is NOT the cross-sectional entity (Food De sert: not an observation about you) b. 1 = this IS the cross-sectional entity (Food Desert: an observation about you) 2. Use OLS to model the regression, LEAVING OUT THE CONSTANT TERM a. Leave out the constant term to avoid the dummy variable trap! 3. The betas on the dummy variables are different intercepts for each cross-sectional entity (Food Desert: each person’s individual intercept) Why would you use a Fixed Effects Model? Controlling for unobserved heterogeneity when heterogeneity is constant over time and correlated with independent variables. When there are ce rtain non-random characteristics you don’t want ending up in your error term. o Food Deserts: the “person-ness” of each respondent of the CARDIA survey that doesn’t change from time period to time period (tastes and prefere nces). Smaller standard errors (more powerful) Stata Commands: xtreg y x1 x2 x3, fe (variable you are controlling for) (reported intercept is the average value of the fixed effects, individual intercepts are the coefficients on the cross-sectional entity’s dummy variable) predict yhat Assumption: The fixed effect model assumes that no othe r factors are effecting changes in “data 1” over the period t 0 to t1 not including this will result in omitted variable bias.
Random Effects Model In Concept: Instead of thinking of each unit as having its own systematic baseline, w e think of each intercept as the result of a random deviation from some mean intercept. The intercept is a draw from some distribution for each unit, and it is independent of the error for a particular observation. Instead of trying to estimate N parameters as in fixed effects, we just need to estimate parameters describing the distribution from which each unit’s intercept is drawn. What the model does: 1. OLS regression is used on the entire pooled cross-section and time-series sample. 2. The error term observations from step 1 are used to estimate error variances and correlations between errors. 3. The estimates from step 2 are used to generate generalized least squares, giving estimates for the random effects model. 4. Some econometrics programs then use the results from step 3 to estimate how far each cross-sectional entity’s intercept is from the mean intercept, 0. Why would you use a Random Effects Model? If you have reason to believe t hat differences across entities have some influence on yo ur dependent variable then you should use random effects. A “group” effect is random if we can think of the levels we observe in that group to be samples from a larger population. Example: if collecting data from different medical centers, “center” might be thought of as random o Example: if surveying students on different campuses, “campus” may be a random effect o Advantages More degrees of freedom than fixed effects, because rather than estimating an intercept for virtually every cross-sectional unit, you estimate the parameters that descr ibe the distribution of the intercepts Can estimate coefficents for explanatory variables that are constant over time (like race or gender) Stata commands: xtreg y x1 x2 x3, re Add the option ‘robust’ to control for heteroskedasticity
How Do I Choose Fixed or Random? Which to pick? Durbin-Wu Test OR Hausman Specification Test Random Effects Assumption - If the Random Effects assumption holds (individual effects NOT correlated to independent variables) pick the Random Effects model because it will be a more efficient model. Graph Source: Dougherty
Resources Dougherty: Introduction to Econometrics 4e. “Chapter 14, Introduction to Panel Data Models.” Oxford University Press.
Available at http://www.oup.com/uk/orc/bin/9780199280964/dougherty_chap14.pdf Mindhive. A Community Portal for MIT Brain Research. “Random and Fixed Effects FAQ.” Available at
http://mindhive.mit.edu/node/92 Studenmund, A. H. Econometrics PBAF 528. Pearson Custom. “Chapter 14, Experimental and Panel Data.” Pgs. 486-496. Taylor, Jonathan. Statistics 203: Introduction to Regression and Analysis of Variance. “Fixed vs. Random Effects.” Available at http://www-stat.stanford.edu/~jtaylo/courses/stats203/notes/fixed+random.pdf Torres-Reyna, Oscar. “Panel Data Analysis, Fixed & Random Effects (using Stata 10.x) (ver. 4.1) .” Princeton University. Available at http://dss.princeton.edu/training/Panel101.pdf Yaffee, Robert. “A Primer for P anel Data Analysis.” New York University Information Technology Services. Fall 2003.
Available at http://www.nyu.edu/its/pubs/connect/fall03/yaffee_primer.html For more information on how to calculate a Hausman or Durbin-Wu Test in Stata see: http://www.stata.com/support/faqs/stat/endogeneity.htm
