christopher dougherty ec220 - introduction to econometrics (chapter 14) slideshow: fixed effects...

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: http://learningresources.lse.ac.uk/140/

Available in LSE Learning Resources Online: May 2012

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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)

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FIXED EFFECTS REGRESSIONS: LSDV METHOD

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If we define a set of dummy variables Ai, where Ai is equal to 1 in the case of an observation relating to individual i and 0 otherwise, the model can be rewritten as shown.


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Formally, the unobserved effect is now being treated as the coefficient of the individual-specific dummy variable, the iAi term representing a fixed effect on the dependent variable Yi for individual i (this accounts for the name given to the fixed effects approach).


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Having re-specified the model in this way, it can be fitted using OLS.


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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.


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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.


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However, the choice of reference category is often arbitrary and accordingly the interpretation of the i not particularly illuminating.


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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.

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Note that, in common with the first two versions of the fixed effects approach, the LSDV method requires panel data.

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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.

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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.


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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.


Equivalent to within-groups 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.



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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.



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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.



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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 1980–1996Dependent variable logarithm of hourly earnings

OLS Fixed effects

Married 0.184 0.106 –(0.007) (0.012)

Soon-to-be- 0.096 0.045 –0.061married (0.009) (0.010) (0.008)

Single – – –0.106(0.012)

R2 0.358 0.268 0.268

n 20,343 20,343 20,343

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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.

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OLS Fixed effects

Married 0.184 0.106 –(0.007) (0.012)


Single – – –0.106(0.012)

R2 0.358 0.268 0.268

n 20,343 20,343 20,343

The controls (not shown) are the same as in the example in the first slideshow on panel data.

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OLS Fixed effects

Married 0.184 0.106 –(0.007) (0.012)


Single – – –0.106(0.012)

R2 0.358 0.268 0.268

n 20,343 20,343 20,343

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.

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OLS Fixed effects

Married 0.184 0.106 –(0.007) (0.012)


Single – – –0.106(0.012)

R2 0.358 0.268 0.268

n 20,343 20,343 20,343

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.

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OLS Fixed effects

Married 0.184 0.106 –(0.007) (0.012)


Single – – –0.106(0.012)

R2 0.358 0.268 0.268

n 20,343 20,343 20,343

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.

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OLS Fixed effects

Married 0.184 0.106 –(0.007) (0.012)


Single – – –0.106(0.012)

R2 0.358 0.268 0.268

n 20,343 20,343 20,343

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.

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OLS Fixed effects

Married 0.184 0.106 –(0.007) (0.012)


Single – – –0.106(0.012)

R2 0.358 0.268 0.268

n 20,343 20,343 20,343

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.

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OLS Fixed effects

Married 0.184 0.106 –(0.007) (0.012)


Single – – –0.106(0.012)

R2 0.358 0.268 0.268

n 20,343 20,343 20,343

Hence both hypotheses relating to the marriage premium appear to be partly true.

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OLS Fixed effects

Married 0.184 0.106 –(0.007) (0.012)


Single – – –0.106(0.012)

R2 0.358 0.268 0.268

n 20,343 20,343 20,343

Copyright Christopher Dougherty 2011.

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

http://www.oup.com/uk/orc/bin/9780199567089/.

Individuals studying econometrics on their own and who feel that they might

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of Economics summer school course

EC212 Introduction to Econometrics

http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx

or the University of London International Programmes distance learning course

20 Elements of Econometrics

www.londoninternational.ac.uk/lse.

11.07.25

christopher dougherty ec220 - introduction to econometrics (chapter 14) slideshow: fixed effects...

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