Christopher Dougherty
EC220 - Introduction to econometrics (chapter 14)Slideshow: fixed effects regressions: LSDV method
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Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 14). [Teaching Resource]
© 2012 The Author
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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.
Fixed effects estimation (least squares dummy variable method)
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FIXED EFFECTS REGRESSIONS: LSDV METHOD
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).
Fixed effects estimation (least squares dummy variable method)
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FIXED EFFECTS REGRESSIONS: LSDV METHOD
Having re-specified the model in this way, it can be fitted using OLS.
Fixed effects estimation (least squares dummy variable method)
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FIXED EFFECTS REGRESSIONS: LSDV METHOD
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)
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FIXED EFFECTS REGRESSIONS: LSDV METHOD
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)
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FIXED EFFECTS REGRESSIONS: LSDV METHOD
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)
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FIXED EFFECTS REGRESSIONS: LSDV METHOD
Fixed effects estimation (least squares dummy variable method)
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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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FIXED EFFECTS REGRESSIONS: LSDV METHOD
Fixed effects estimation (least squares dummy variable method)
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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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FIXED EFFECTS REGRESSIONS: LSDV METHOD
Fixed effects estimation (least squares dummy variable method)
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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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FIXED EFFECTS REGRESSIONS: LSDV METHOD
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)
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FIXED EFFECTS REGRESSIONS: LSDV METHOD
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:
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FIXED EFFECTS REGRESSIONS: LSDV METHOD
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:
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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:
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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:
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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 1980–1996Dependent 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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FIXED EFFECTS REGRESSIONS: LSDV METHOD
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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FIXED EFFECTS REGRESSIONS: LSDV METHOD
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
The controls (not shown) are the same as in the example in the first slideshow on panel data.
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FIXED EFFECTS REGRESSIONS: LSDV METHOD
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
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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FIXED EFFECTS REGRESSIONS: LSDV METHOD
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
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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FIXED EFFECTS REGRESSIONS: LSDV METHOD
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
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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FIXED EFFECTS REGRESSIONS: LSDV METHOD
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
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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FIXED EFFECTS REGRESSIONS: LSDV METHOD
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
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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FIXED EFFECTS REGRESSIONS: LSDV METHOD
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
Hence both hypotheses relating to the marriage premium appear to be partly true.
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FIXED EFFECTS REGRESSIONS: LSDV METHOD
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
Copyright Christopher Dougherty 2011.
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Introduction to Econometrics, fourth edition 2011, Oxford University Press.
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11.07.25