Fixed and Random Effects Models: Vartanian, SW 683

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1 : Vartanian, SW 683 Fixed and random effects models See: When you have repeated observations per individual this is a problem and an advantage: o the observations are not independent (and their error component will likely be correlated) o we can use the repetition to get better parameter estimates If we pooled the observations and used e.g., OLS we would have biased estimates If we fit fixed-effect or random-effect models which take account of the repetition we can control for fixed or random individual differences. In the econometrics literature these models are called `cross-sectional time-series' models, because we have time-series of observations at individual rather than aggregate level. If we have a small number of individuals, we can simply fit a dummy for each individual: y = α + γ + β x + ε it i it t This can be considered a `fixed-effects' model because the regression line is raised or lowered by a fixed amount for each individual. Note that we may have 20 observations per individual and we can therefore use a dummy variable for each individual. We can also use a family dummy variable if we are examining siblings one dummy per family. If there are many individuals this cannot be done directly, but there are mathematically equivalent models which achieve the same effect This model is appropriate where we consider each individual to have a fixed effect shifting the y it up or down We may prefer to consider the individual differences as random disturbances drawn from some specified distribution: Here, the error term has two components: V i and ε t. ε t is the traditional error term, unique to each observation. V i is an error term representing the extent to which the intercept of the i th observation (not the i th individual) differs from the overall intercept. (Kennedy, 3 rd edition). Note that each individual or family will have a number of observations. Random effects models are considered to be more efficient than OLS models. Levy and Duncan (2001), in their study of family income using siblings, state the following: We compared the results of the OLS and fixed effects specifications with the ones obtained from the more efficient random effects model. This estimator is a weighted matrix average of the within (FE) and the between estimators, but is consistent only under the assumption that the family effect is uncorrelated with the regressors. Coefficients and standard errors from the random effects models were very similar to their OLS counterparts, which led us to present only the OLS results.. This has the advantage of using fewer degrees of freedom, and that individual differences are considered random rather than fixed and estimable. If effects for groups are not fixed, they are not estimable they are random.

2 It has the disadvantage of requiring no correlation between the regressors (the x it s) and the V i : there are tests for this assumption (Hausman test). o Example 1: Neighborhood characteristics and school grades are analyzed at each point of a child s life, with a new observation for each year of the child s life. Wages are then regressed on neighborhood conditions and school grades. Ability or IQ or some such measure are left out of the model because they are not measured in the data. IQ or ability of parents may also be left out of the model. Child IQ is likely to be correlated with school grades and parental ability/iq may be correlated with neighborhood conditions. These correlations will create bias in a random effects model but will not create bias in a fixed effect model. The reason ability will not bias fixed effect models is because fixed effect models difference out these effects for individuals (assuming that ability/iq is a permanent feature of an individual). Reference: William Greene, Econometric Analysis, Maxwell Macmillan 1991, Ch 16 section 4. Peter Kennedy, A Guide to Econometrics, 5 th edition, Example 2: We are examining siblings using two neighborhood indices under both fixed effects and random effects models. Is it appropriate to run a random effects model? We can use a Hausman test to determine if the individual error term is correlated with the included variables. Note that in a fixed effect model, those factors that do not change or do not differ among siblings cannot be used as independent variables. This is because fixed effect models difference out these factors. If every group in the sample has a value of 0 for a particular variable, you cannot use it in a regression analysis. Random effects models do not difference out these variable values within groups (they are assumed to be random effects) and therefore can use variables that cannot be used in a fixed effect model.. xtreg logefmns Nbhd1 Nbhd2 if caunt>1, fe Fixed-effects (within) regression Number of obs = 3652 R-sq: within = Obs per group: min = 2 between = avg = 2.9 overall = max = 10 А(2б2386) 5ю71 сщкк(г_шб Чи) = 0ю4562 Зкщи Ю А = 0ю0034 logefmns Coef. Std. Err. t P> t [95% Conf. Interval] Nbhd Nbhd _cons sigma_u sigma_e rho (fraction of variance due to u_i) F test that all u_i=0: F(1263, 2386) = 2.25 Prob > F = Random Effects Model: logefmns Coef. Std. Err. z P> z [95% Conf. Interval]. est store fixed. xtreg logefmns p1 p2 dropout hsg scoll afam othrace schild bigcity urbany city3y suby if caunt>1, re

3 Random-effects GLS regression Number of obs = 3652 R-sq: within = Obs per group: min = 2 between = avg = 2.9 overall = max = 10 Random effects u_i ~ Gaussian Wald chi2(2) = corr(u_i, X) = 0 (assumed) Prob > chi2 = logefmns Coef. Std. Err. z P> z [95% Conf. Interval] Nbhd Nbhd _cons sigma_u sigma_e rho (fraction of variance due to u_i). hausman fixed ---- Coefficients ---- (b) (B) (b-b) sqrt(diag(v_b-v_b)) fixed. Difference S.E. Nbhd Nbhd b = consistent under Ho and Ha; obtained from xtreg B = inconsistent under Ha, efficient under Ho; obtained from xtreg Test: Ho: difference in coefficients not systematic chi2(2) = (b-b)'[(v_b-v_b)^(-1)](b-b) = Prob>chi2 = This chi-square value indicates that a random effects model is not appropriate. Use the fixed effects model. ********************************************************************************

4 I ran this as an OLS model to compare the results with the Random Effects model.. reg logefmns NBHD1 NBHD2 if caunt>1 Source SS df MS Number of obs = F( 2, 3649) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = logefmns Coef. Std. Err. t P> t [95% Conf. Interval] Nbhd Nbhd _cons As you can see, the effects of the OLS models are very similar to the random effects model. This is often the case. I ran the random effects model using additional independent variables because these effects are not differenced out in random effects models, then ran a Hausman test to determine if a random effects model was appropriate. Random-effects GLS regression Number of obs = 3652 R-sq: within = Obs per group: min = 2 between = avg = 2.9 overall = max = 10 Random effects u_i ~ Gaussian Wald chi2(12) = corr(u_i, X) = 0 (assumed) Prob > chi2 = Nbhd Nbhd dropout hsg scoll afam othrace schild bigcity urbany city3y suby _cons sigma_u sigma_e rho (fraction of variance due to u_i) Level of education of the parents, schild (living in the south as a child), and area of residence differ little among siblings. We therefore do not use these variables in a fixed effect model. We can specifically see their effects in a random effects model.

5 hausman fixed ---- Coefficients ---- (b) (B) (b-b) sqrt(diag(v_b-v_b)) fixed. Difference S.E. Nbhd Nbhd b = consistent under Ho and Ha; obtained from xtreg B = inconsistent under Ha, efficient under Ho; obtained from xtreg Test: Ho: difference in coefficients not systematic chi2(2) = (b-b)'[(v_b-v_b)^(-1)](b-b) = Prob>chi2 = This Hausman test (and the chi-square value) indicates that we will reject the null hypothesis for the random effects model and go with the fixed effect model.