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Christopher Dougherty EC220 - Introduction to econometrics (chapter 14) Slideshow: fixed effects regressions: LSDV method Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 14). [Teaching Resource] © 2012 The Author This version available at: Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms.

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In the third version of the fixed effects approach, known as the least squares dummy variable (LSDV) method, the unobserved effect is brought explicitly into the model. Fixed effects estimation (least squares dummy variable method) FIXED EFFECTS REGRESSIONS: LSDV METHOD 1

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If we define a set of dummy variables A i, where A i is equal to 1 in the case of an observation relating to individual i and 0 otherwise, the model can be rewritten as shown. Fixed effects estimation (least squares dummy variable method) 2 FIXED EFFECTS REGRESSIONS: LSDV METHOD

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Formally, the unobserved effect is now being treated as the coefficient of the individual- specific dummy variable, the i A i term representing a fixed effect on the dependent variable Y i for individual i (this accounts for the name given to the fixed effects approach). Fixed effects estimation (least squares dummy variable method) 3 FIXED EFFECTS REGRESSIONS: LSDV METHOD

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Having re-specified the model in this way, it can be fitted using OLS. Fixed effects estimation (least squares dummy variable method) 4 FIXED EFFECTS REGRESSIONS: LSDV METHOD

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Note that if we include a dummy variable for every individual in the sample as well as an intercept, we will fall into the dummy variable trap. Fixed effects estimation (least squares dummy variable method) 5 FIXED EFFECTS REGRESSIONS: LSDV METHOD

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To avoid this, we can define one individual to be the reference category, so that 1 is its intercept, and then treat the i as the shifts in the intercept for the other individuals. Fixed effects estimation (least squares dummy variable method) 6 FIXED EFFECTS REGRESSIONS: LSDV METHOD

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However, the choice of reference category is often arbitrary and accordingly the interpretation of the i not particularly illuminating. Fixed effects estimation (least squares dummy variable method) 7 FIXED EFFECTS REGRESSIONS: LSDV METHOD

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Fixed effects estimation (least squares dummy variable method) Alternatively, we can drop the 1 intercept and define dummy variables for all of the individuals, as has been done here. The i now become the intercepts for each of the individuals. 8 FIXED EFFECTS REGRESSIONS: LSDV METHOD

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Fixed effects estimation (least squares dummy variable method) Note that, in common with the first two versions of the fixed effects approach, the LSDV method requires panel data. 9 FIXED EFFECTS REGRESSIONS: LSDV METHOD

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Fixed effects estimation (least squares dummy variable method) With cross-sectional data, one would be defining a dummy variable for every observation, exhausting the degrees of freedom. The dummy variables on their own would give a perfect but meaningless fit. 10 FIXED EFFECTS REGRESSIONS: LSDV METHOD

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If there are a large number of individuals, using the LSDV method directly is not a practical proposition, given the need for a large number of dummy variables. Fixed effects estimation (least squares dummy variable method) 11 FIXED EFFECTS REGRESSIONS: LSDV METHOD

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However, it can be shown mathematically that the approach is equivalent to the within- groups method and therefore yields precisely the same estimates. Fixed effects estimation (least squares dummy variable method) Equivalent to within-groups method: 12 FIXED EFFECTS REGRESSIONS: LSDV METHOD

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Thus in practice we always use the within-groups method rather than the LSDV method. But it may be useful to know that the within-groups method is equivalent to modelling the fixed effects with dummy variables. Fixed effects estimation (least squares dummy variable method) Equivalent to within-groups method: 13 FIXED EFFECTS REGRESSIONS: LSDV METHOD

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The only apparent difference between the LSDV and within-groups methods is in the number of degrees of freedom. It is easy to see from the LSDV specification that there are nT – k – n degrees of freedom if the panel is balanced. Fixed effects estimation (least squares dummy variable method) Equivalent to within-groups method: 14 FIXED EFFECTS REGRESSIONS: LSDV METHOD

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In the within-groups approach, it seemed at first that there were nT – k. However n degrees of freedom are consumed in the manipulation that eliminate the i, so the number of degrees of freedom is really nT – k – n. Fixed effects estimation (least squares dummy variable method) Equivalent to within-groups method: 15 FIXED EFFECTS REGRESSIONS: LSDV METHOD

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To illustrate the use of a fixed effects model, we return to the example in Section 1 and use all the available data from 1980 to 1996, 20,343 observations in all. NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Married – (0.007)(0.012) Soon-to-be –0.061 married(0.009)(0.010)(0.008) Single–––0.106 (0.012) R n 20,343 20,343 20, FIXED EFFECTS REGRESSIONS: LSDV METHOD

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The table shows the extra hourly earnings of married men and of men who are single but married within the next four years. The omitted category in the first two columns is single men who are still single four years later. 17 FIXED EFFECTS REGRESSIONS: LSDV METHOD NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Married – (0.007)(0.012) Soon-to-be –0.061 married(0.009)(0.010)(0.008) Single–––0.106 (0.012) R n 20,343 20,343 20,343

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The controls (not shown) are the same as in the example in the first slideshow on panel data. 18 FIXED EFFECTS REGRESSIONS: LSDV METHOD NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Married – (0.007)(0.012) Soon-to-be –0.061 married(0.009)(0.010)(0.008) Single–––0.106 (0.012) R n 20,343 20,343 20,343

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The first column gives the estimates obtained by simply pooling the observations and using OLS with robust standard errors. The estimates are very similar to those in the wage equation for 1988 in the example in the first slideshow on panel data. 19 FIXED EFFECTS REGRESSIONS: LSDV METHOD NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Married – (0.007)(0.012) Soon-to-be –0.061 married(0.009)(0.010)(0.008) Single–––0.106 (0.012) R n 20,343 20,343 20,343

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The second column gives the fixed effects estimates, using the within-groups method, with single men as the reference category. The third gives the fixed effects estimates with married men as the reference category. 20 FIXED EFFECTS REGRESSIONS: LSDV METHOD NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Married – (0.007)(0.012) Soon-to-be –0.061 married(0.009)(0.010)(0.008) Single–––0.106 (0.012) R n 20,343 20,343 20,343

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The fixed effects estimates are considerably lower than the OLS estimates, suggesting that the OLS estimates were inflated by unobserved heterogeneity. Nevertheless the pattern is the same. 21 FIXED EFFECTS REGRESSIONS: LSDV METHOD NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Married – (0.007)(0.012) Soon-to-be –0.061 married(0.009)(0.010)(0.008) Single–––0.106 (0.012) R n 20,343 20,343 20,343

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Our findings confirm that married men earn more than single men. Part of the differential appears to be attributable to the characteristics of married men, since men who are soon-to- marry but still single also enjoy a significant earnings premium. 22 FIXED EFFECTS REGRESSIONS: LSDV METHOD NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Married – (0.007)(0.012) Soon-to-be –0.061 married(0.009)(0.010)(0.008) Single–––0.106 (0.012) R n 20,343 20,343 20,343

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However if we make married men the omitted category, as in the third column, we find that soon-to-be-married men earn significantly less than married men. Thus part of the marriage premium appears to be attributable to the effect of marriage itself. 23 FIXED EFFECTS REGRESSIONS: LSDV METHOD NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Married – (0.007)(0.012) Soon-to-be –0.061 married(0.009)(0.010)(0.008) Single–––0.106 (0.012) R n 20,343 20,343 20,343

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Hence both hypotheses relating to the marriage premium appear to be partly true. 24 FIXED EFFECTS REGRESSIONS: LSDV METHOD NLSY 1980–1996 Dependent variable logarithm of hourly earnings OLS Fixed effects Married – (0.007)(0.012) Soon-to-be –0.061 married(0.009)(0.010)(0.008) Single–––0.106 (0.012) R n 20,343 20,343 20,343

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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 14.2 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

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