When should you use fixed effects estimation rather than random effects estimation, or vice versa? FIXED EFFECTS OR RANDOM EFFECTS? 1 NLSY 1980–1996 Dependent.

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When should you use fixed effects estimation rather than random effects estimation, or vice versa? FIXED EFFECTS OR RANDOM EFFECTS? 1 NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Random effects Married –0.134– (0.007)(0.012)(0.010) Soon-to-be – –0.075 married(0.009)(0.010)(0.008)(0.009)(0.007) Single–––0.106 – –0.134 (0.012)(0.010) R DWH test – – – n 20,343 20,343 20,343 20,343 20,343

In principle random effects is more attractive because observed characteristics that remain constant for each individual are retained in the regression model. In fixed effects estimation, they have to be dropped. FIXED EFFECTS OR RANDOM EFFECTS? 2 NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Random effects Married –0.134– (0.007)(0.012)(0.010) Soon-to-be – –0.075 married(0.009)(0.010)(0.008)(0.009)(0.007) Single–––0.106 – –0.134 (0.012)(0.010) R DWH test – – – n 20,343 20,343 20,343 20,343 20,343

Also with random effects estimation we do not lose n degrees of freedom, as is the case with fixed effects. FIXED EFFECTS OR RANDOM EFFECTS? 3 NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Random effects Married –0.134– (0.007)(0.012)(0.010) Soon-to-be – –0.075 married(0.009)(0.010)(0.008)(0.009)(0.007) Single–––0.106 – –0.134 (0.012)(0.010) R DWH test – – – n 20,343 20,343 20,343 20,343 20,343

However if either of the preconditions for using random effects is violated, we should use fixed effects instead. FIXED EFFECTS OR RANDOM EFFECTS? 4 NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Random effects Married –0.134– (0.007)(0.012)(0.010) Soon-to-be – –0.075 married(0.009)(0.010)(0.008)(0.009)(0.007) Single–––0.106 – –0.134 (0.012)(0.010) R DWH test – – – n 20,343 20,343 20,343 20,343 20,343

One precondition is that the observations can be described as being drawn randomly from a given population. This is a reasonable assumption in the case of the NLSY because it was designed to be a random sample. FIXED EFFECTS OR RANDOM EFFECTS? 5 NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Random effects Married –0.134– (0.007)(0.012)(0.010) Soon-to-be – –0.075 married(0.009)(0.010)(0.008)(0.009)(0.007) Single–––0.106 – –0.134 (0.012)(0.010) R DWH test – – – n 20,343 20,343 20,343 20,343 20,343

By contrast, it would not be a reasonable assumption if the units of observation in the panel data set were countries and the sample consisted of those countries that are members of the Organization for Economic Cooperation and Development (OECD). FIXED EFFECTS OR RANDOM EFFECTS? 6 NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Random effects Married –0.134– (0.007)(0.012)(0.010) Soon-to-be – –0.075 married(0.009)(0.010)(0.008)(0.009)(0.007) Single–––0.106 – –0.134 (0.012)(0.010) R DWH test – – – n 20,343 20,343 20,343 20,343 20,343

These countries certainly cannot be considered to represent a random sample of the 200- odd sovereign states in the world. FIXED EFFECTS OR RANDOM EFFECTS? 7 NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Random effects Married –0.134– (0.007)(0.012)(0.010) Soon-to-be – –0.075 married(0.009)(0.010)(0.008)(0.009)(0.007) Single–––0.106 – –0.134 (0.012)(0.010) R DWH test – – – n 20,343 20,343 20,343 20,343 20,343

The other precondition is that the unobserved effect be distributed independently of the X j variables. How can we tell if this is the case? FIXED EFFECTS OR RANDOM EFFECTS? 8 NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Random effects Married –0.134– (0.007)(0.012)(0.010) Soon-to-be – –0.075 married(0.009)(0.010)(0.008)(0.009)(0.007) Single–––0.106 – –0.134 (0.012)(0.010) R DWH test – – – n 20,343 20,343 20,343 20,343 20,343

The standard procedure is yet another implementation of the Durbin–Wu–Hausman test used to help us choose between OLS and IV estimation in models where there is suspected measurement error or simultaneous equations endogeneity. FIXED EFFECTS OR RANDOM EFFECTS? 9 NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Random effects Married –0.134– (0.007)(0.012)(0.010) Soon-to-be – –0.075 married(0.009)(0.010)(0.008)(0.009)(0.007) Single–––0.106 – –0.134 (0.012)(0.010) R DWH test – – – n 20,343 20,343 20,343 20,343 20,343

The null hypothesis is that the  i are distributed independently of the X j. FIXED EFFECTS OR RANDOM EFFECTS? 10 NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Random effects Married –0.134– (0.007)(0.012)(0.010) Soon-to-be – –0.075 married(0.009)(0.010)(0.008)(0.009)(0.007) Single–––0.106 – –0.134 (0.012)(0.010) R DWH test – – – n 20,343 20,343 20,343 20,343 20,343

If this is correct, both random effects and fixed effects are consistent, but fixed effects will be inefficient because, looking at it in its LSDV form, it involves estimating an unnecessary set of dummy variable coefficients. FIXED EFFECTS OR RANDOM EFFECTS? 11 NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Random effects Married –0.134– (0.007)(0.012)(0.010) Soon-to-be – –0.075 married(0.009)(0.010)(0.008)(0.009)(0.007) Single–––0.106 – –0.134 (0.012)(0.010) R DWH test – – – n 20,343 20,343 20,343 20,343 20,343

If the null hypothesis is false, the random effects estimates will be subject to unobserved heterogeneity bias and will therefore differ systematically from the fixed effects estimates. FIXED EFFECTS OR RANDOM EFFECTS? 12 NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Random effects Married –0.134– (0.007)(0.012)(0.010) Soon-to-be – –0.075 married(0.009)(0.010)(0.008)(0.009)(0.007) Single–––0.106 – –0.134 (0.012)(0.010) R DWH test – – – n 20,343 20,343 20,343 20,343 20,343

As in its other applications, the DWH test determines whether the estimates of the coefficients, taken as a group, are significantly different in the two regressions. FIXED EFFECTS OR RANDOM EFFECTS? 13 NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Random effects Married –0.134– (0.007)(0.012)(0.010) Soon-to-be – –0.075 married(0.009)(0.010)(0.008)(0.009)(0.007) Single–––0.106 – –0.134 (0.012)(0.010) R DWH test – – – n 20,343 20,343 20,343 20,343 20,343

If any variables are dropped in the fixed effects regression, they are excluded from the test. Under the null hypothesis the test statistic has a chi-squared distribution. FIXED EFFECTS OR RANDOM EFFECTS? 14 NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Random effects Married –0.134– (0.007)(0.012)(0.010) Soon-to-be – –0.075 married(0.009)(0.010)(0.008)(0.009)(0.007) Single–––0.106 – –0.134 (0.012)(0.010) R DWH test – – – n 20,343 20,343 20,343 20,343 20,343

In principle this should have degrees of freedom equal to the number of slope coefficients being compared, but for technical reasons that require matrix algebra for an explanation, the actual number may be lower. FIXED EFFECTS OR RANDOM EFFECTS? 15 NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Random effects Married –0.134– (0.007)(0.012)(0.010) Soon-to-be – –0.075 married(0.009)(0.010)(0.008)(0.009)(0.007) Single–––0.106 – –0.134 (0.012)(0.010) R DWH test – – – n 20,343 20,343 20,343 20,343 20,343

A regression application that implements the test, such as Stata, should determine the actual number of degrees of freedom. FIXED EFFECTS OR RANDOM EFFECTS? 16 NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Random effects Married –0.134– (0.007)(0.012)(0.010) Soon-to-be – –0.075 married(0.009)(0.010)(0.008)(0.009)(0.007) Single–––0.106 – –0.134 (0.012)(0.010) R DWH test – – – n 20,343 20,343 20,343 20,343 20,343

We will perform the test for the marriage-effect-on-earnings example. The fixed effects estimates, using the within-groups approach, of the coefficients of married men and soon- to-be married men are and 0.045, respectively. FIXED EFFECTS OR RANDOM EFFECTS? 17 NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Random effects Married –0.134– (0.007)(0.012)(0.010) Soon-to-be – –0.075 married(0.009)(0.010)(0.008)(0.009)(0.007) Single–––0.106 – –0.134 (0.012)(0.010) R DWH test – – – n 20,343 20,343 20,343 20,343 20,343

The corresponding random effects estimates are considerably higher, and 0.060, inviting the suspicion that they may be inflated by unobserved heterogeneity. FIXED EFFECTS OR RANDOM EFFECTS? 18 NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Random effects Married –0.134– (0.007)(0.012)(0.010) Soon-to-be – –0.075 married(0.009)(0.010)(0.008)(0.009)(0.007) Single–––0.106 – –0.134 (0.012)(0.010) R DWH test – – – n 20,343 20,343 20,343 20,343 20,343

The DWH test involves the comparison of 13 coefficients (those of MARRIED, SOONMARR, and 11 controls). The test statistic is FIXED EFFECTS OR RANDOM EFFECTS? 19 NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Random effects Married –0.134– (0.007)(0.012)(0.010) Soon-to-be – –0.075 married(0.009)(0.010)(0.008)(0.009)(0.007) Single–––0.106 – –0.134 (0.012)(0.010) R DWH test – – – n 20,343 20,343 20,343 20,343 20,343

With 13 degrees of freedom the critical value of chi-squared at the 0.1 percent level is 34.5, so we definitely conclude that we should be using fixed effects estimation. FIXED EFFECTS OR RANDOM EFFECTS? 20 NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Random effects Married –0.134– (0.007)(0.012)(0.010) Soon-to-be – –0.075 married(0.009)(0.010)(0.008)(0.009)(0.007) Single–––0.106 – –0.134 (0.012)(0.010) R DWH test – – – n 20,343 20,343 20,343 20,343 20,343

Suppose that the DWH test indicates that we can use random effects rather than fixed effects. FIXED EFFECTS OR RANDOM EFFECTS? 21 Random effects estimation

We should then consider whether there are any unobserved effects at all. It is just possible that the model has been so well specified that the disturbance term u consists of only the purely random component  it and there is no individual-specific  i term. FIXED EFFECTS OR RANDOM EFFECTS? 22 Random effects estimation

In this situation we should use pooled OLS, with two advantages. There is a gain in efficiency because we are not attempting to allow for non-existent within-groups autocorrelation. FIXED EFFECTS OR RANDOM EFFECTS? 23 Random effects estimation OLS

In addition we will be able to take advantage of the finite-sample properties of OLS, instead of having to rely on the asymptotic properties of random effects. FIXED EFFECTS OR RANDOM EFFECTS? 24 Random effects estimation OLS

Various tests have been developed to detect the presence of random effects. FIXED EFFECTS OR RANDOM EFFECTS? 25 Random effects estimation OLS

The most common, implemented in some regression applications, is the Breusch–Pagan Lagrange multiplier test, the test statistic having a chi-squared distribution with one degree of freedom under the null hypothesis of no random effects. FIXED EFFECTS OR RANDOM EFFECTS? 26 Random effects estimation OLS

In the case of the marriage effect example the statistic is very high indeed, 20,007, but in this case it is meaningless because are not able to use random effects estimation. FIXED EFFECTS OR RANDOM EFFECTS? 27 Random effects estimation OLS

A note on the random effects and fixed effects terminology. It is generally agreed that random effects/fixed effects terminology can be misleading, but that it is too late to change it now. FIXED EFFECTS OR RANDOM EFFECTS? 28 Random effects estimation OLS

It is natural to think that random effects estimation should be used when the unobserved effect can be characterized as being drawn randomly from a given population and that fixed effects should be used when the unobserved effect is non-random. FIXED EFFECTS OR RANDOM EFFECTS? 29 Random effects estimation OLS

The second part of that statement is correct. However the first part is correct only if the unobserved effect is distributed independently of the X j variables. If it is not, fixed effects should be used instead to avoid the problem of unobserved heterogeneity bias. FIXED EFFECTS OR RANDOM EFFECTS? 30 Random effects estimation OLS

The diagram summarizes the decision-making procedure. Make sure that you understand the reasons for making the choices. FIXED EFFECTS OR RANDOM EFFECTS? 31 Can the observations be described as being a random sample from a given population? Use fixed effectsPerform both fixed effects and random effects regressions. Does a DWH test indicate significant differences in the coefficients? Provisionally choose random effects. Does a test indicate the presence of random effects? Use fixed effects Use random effects Use pooled OLS YesNo Yes No YesNo

Copyright Christopher Dougherty These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 14.3 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre Individuals studying econometrics on their own and who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics or the University of London International Programmes distance learning course 20 Elements of Econometrics