In econometrics, fixed effects binary choice models are important tools for panel data analysis. Our package provides an approach suggested by Stammann, Heiss, and McFadden (2016) to estimate logit and probit panel data models of the following form:
\[ y_{it} = \mathbf{1}\left[\mathbf{x}_{it}\boldsymbol{\beta} + \alpha_{i} > \epsilon_{it}\right] \;, \]
where \(i = 1, \dots, N\) and \(t = 1, \dots, T_i\) denote different panel indices. In many applications, \(i\) represents individuals, firms or other cross-sectional units and \(t\) represents time in a longitudinal data set. But the setup is also useful for instance if \(i\) represents ZIP code areas and \(t\) is an index of individuals.
We are primarily interested in estimating the parameters \(\boldsymbol{\beta}\), but the model also includes individual fixed effects \(\alpha_{i}\). We assume \(E(\epsilon_{it} | \mathbf{X}_{i}, \alpha_{i}) = 0\) but do not make any assumptions about the marginal distribution of \(\alpha_{i}\) or its correlation with the regressors \(\mathbf{x}_{i1},\dots, \mathbf{x}_{iT_i}\).
The estimator implemented in this package is based on maximum likelihood estimation (ML) of both \(\boldsymbol{\beta}\) and \(\alpha_{1}, \dots, \alpha_{N}\). It actually is equivalent to a generalized linear model (glm()) for binomial data where the set of regressors is extended by a dummy variable for each individual. The main difference is that bife() applies a pseudo-demeaning algorithm proposed by Stammann, Heiss, and McFadden (2016) to concentrate out the fixed effects from the optimization problem.1 Its computational costs are lower by orders of magnitude if \(N\) is reasonably large.
It is well known that as \(N \rightarrow \infty\), the ML estimator is not consistent. This “incidental parameters problem” can be severe if \(T\) is small. To tackle this problem, we provide an analytical bias correction for the structural parameters \(\boldsymbol{\beta}\) and the average partial effects derived by Fernández-Val (2009).2 Thus this package is well suited to analyse big micro-data where \(N\) and/or \(T\) are large.
In the following we utilize an example from labor economics to demonstrate the capabilities of bife(). More precisely, we use a balanced micro panel data set from the Panel Study of Income Dynamics to analyze the intertemporal labor force participation of 1,461 married women observed for nine years. A similar empirical illustration is used in Fernández-Val (2009) and is an adoption from Hyslop (1999).
Before we start, we briefly inspect the data set to get an idea about its structure and potential covariates.
## ID LFP KID1 KID2 KID3 INCH AGE TIME
## 1: 1 1 1 1 1 58807.81 26 1
## 2: 1 1 1 0 2 41741.87 27 2
## 3: 1 1 0 1 2 51320.73 28 3
## 4: 1 1 0 1 2 48958.58 29 4
## 5: 1 1 0 1 2 53634.62 30 5
## 6: 1 1 0 0 3 50983.13 31 6ID and TIME are individual and time-specific identifiers, LFP is an indicator equal to one if a woman is in labor force, KID1 - KID3 are the number of children in a certain age group, INCH is the annual income of the husband, and AGE is the age of the woman.
First, we use a specification similar to Fernández-Val (2009) and estimate a static model of women’s labor supply where we control for unobserved individual heterogeneity (so called individual fixed effects).
library(bife)
stat <- bife(LFP ~ KID1 + KID2 + KID3 + log(INCH) + AGE + I(AGE^2) | ID, psid, "probit")
summary(stat)## binomial - probit link
##
## LFP ~ KID1 + KID2 + KID3 + log(INCH) + AGE + I(AGE^2) | ID
##
## Estimates:
## Estimate Std. error z value Pr(> |z|)
## KID1 -0.7144667 0.0562414 -12.704 < 2e-16 ***
## KID2 -0.4114554 0.0515524 -7.981 1.45e-15 ***
## KID3 -0.1298776 0.0415477 -3.126 0.00177 **
## log(INCH) -0.2417657 0.0541720 -4.463 8.08e-06 ***
## AGE 0.2319724 0.0375351 6.180 6.40e-10 ***
## I(AGE^2) -0.0028846 0.0004989 -5.781 7.41e-09 ***
## ---
## Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
##
## residual deviance= 6058.88,
## null deviance= 8152.05,
## nT= 5976, N= 664
##
## ( 7173 observation(s) deleted due to perfect classification )
##
## Number of Fisher Scoring Iterations: 6
##
## Average individual fixed effect= -1.121As glm(), the summary statistic of the model provides detailed information about the coefficients and some information about the model fit (residual deviance and null deviance). Furthermore, we report statistics that are specific to fixed effects models. More precisely, we learn that only 5,976 observations out of 13,149 contribute to the idenfication of the structural parameters. This is indicated by the message that 7,173 observations are deleted due to perfect classification. With respect to binary choice models those are observations that are related to women who never change their labor force participation status during the nine years observed. Thus those women were either always employed or unemployed. Overall the estimation results are based on 664 women observed for nine years.
Because coefficients itself are not very meaningful, researchers are usually interested in so called partial effects (also known as marginal or ceteris paribus effects). A commonly used statistic is the average partial effect. bife offers a post-estimation routine to estimate average partial effects and their corresponding standard errors.
## Estimates:
## Estimate Std. error z value Pr(> |z|)
## KID1 -9.278e-02 8.034e-03 -11.549 < 2e-16 ***
## KID2 -5.343e-02 7.228e-03 -7.393 1.44e-13 ***
## KID3 -1.687e-02 6.009e-03 -2.807 0.005 **
## log(INCH) -3.140e-02 7.515e-03 -4.178 2.95e-05 ***
## AGE 3.012e-02 5.306e-03 5.677 1.37e-08 ***
## I(AGE^2) -3.746e-04 7.071e-05 -5.298 1.17e-07 ***
## ---
## Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1A widespread reason that prevents the use of non-linear fixed effects models in practice is the so-called incidental parameter bias problem (IPP) first mentioned by Neyman and Scott (1948). Fortunately, for classical panel data sets, like in this example, there already exist several asymptotic bias corrections tackling the IPP (see Fernández-Val and Weidner (2018) for an overview). Our package provides a post-estimation routine that applies the analytical bias correction derived by Fernández-Val (2009).
## binomial - probit link
##
## LFP ~ KID1 + KID2 + KID3 + log(INCH) + AGE + I(AGE^2) | ID
##
## Estimates:
## Estimate Std. error z value Pr(> |z|)
## KID1 -0.6308839 0.0555073 -11.366 < 2e-16 ***
## KID2 -0.3635269 0.0511325 -7.110 1.16e-12 ***
## KID3 -0.1149869 0.0413488 -2.781 0.00542 **
## log(INCH) -0.2139549 0.0536613 -3.987 6.69e-05 ***
## AGE 0.2052708 0.0373054 5.502 3.75e-08 ***
## I(AGE^2) -0.0025520 0.0004962 -5.143 2.70e-07 ***
## ---
## Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
##
## residual deviance= 6062.8,
## null deviance= 8152.05,
## nT= 5976, N= 664
##
## ( 7173 observation(s) deleted due to perfect classification )
##
## Number of Fisher Scoring Iterations: 6 ## Estimates:
## Estimate Std. error z value Pr(> |z|)
## KID1 -9.127e-02 7.830e-03 -11.657 < 2e-16 ***
## KID2 -5.259e-02 7.146e-03 -7.359 1.85e-13 ***
## KID3 -1.664e-02 5.962e-03 -2.790 0.00526 **
## log(INCH) -3.095e-02 7.406e-03 -4.180 2.92e-05 ***
## AGE 2.970e-02 5.274e-03 5.632 1.79e-08 ***
## I(AGE^2) -3.692e-04 7.031e-05 -5.251 1.51e-07 ***
## ---
## Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Whereas analytical bias corrections for static models get more and more attention in applied work, it is not well known that they can also be used for dynamic models with fixed effects.
Before we can adjust our static to a dynamic specification, we first have to generate a lagged dependent variable.
Contrary to the bias correction for the static models, we need to additionally provide a bandwidth parameter (L) that is required for the estimation of spectral densities (see Hahn and Kuersteiner (2011)). Fernández-Val and Weidner (2018) suggest to do a sensitivity analysis and try different values for L but not larger than four.
dyn <- bife(LFP ~ LLFP + KID1 + KID2 + KID3 + log(INCH) + AGE + I(AGE^2) | ID, psid, "probit")
dyn_bc <- bias_corr(dyn, L = 1L)
summary(dyn_bc)## binomial - probit link
##
## LFP ~ LLFP + KID1 + KID2 + KID3 + log(INCH) + AGE + I(AGE^2) |
## ID
##
## Estimates:
## Estimate Std. error z value Pr(> |z|)
## LLFP 1.0025625 0.0473066 21.193 < 2e-16 ***
## KID1 -0.4741275 0.0679073 -6.982 2.91e-12 ***
## KID2 -0.1958365 0.0625921 -3.129 0.001755 **
## KID3 -0.0754042 0.0505110 -1.493 0.135482
## log(INCH) -0.1946970 0.0621143 -3.134 0.001722 **
## AGE 0.2009569 0.0477728 4.207 2.59e-05 ***
## I(AGE^2) -0.0024142 0.0006293 -3.836 0.000125 ***
## ---
## Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
##
## residual deviance= 4822.99,
## null deviance= 6549.14,
## nT= 4792, N= 599
##
## ( 1461 observation(s) deleted due to missingness )
## ( 6896 observation(s) deleted due to perfect classification )
##
## Number of Fisher Scoring Iterations: 6
##
## Average individual fixed effect= -1.939## Estimates:
## Estimate Std. error z value Pr(> |z|)
## LLFP 0.1537523 0.0072038 21.343 < 2e-16 ***
## KID1 -0.0617373 0.0078862 -7.828 4.94e-15 ***
## KID2 -0.0255003 0.0072685 -3.508 0.000451 ***
## KID3 -0.0098186 0.0058864 -1.668 0.095314 .
## log(INCH) -0.0253520 0.0070041 -3.620 0.000295 ***
## AGE 0.0261671 0.0054330 4.816 1.46e-06 ***
## I(AGE^2) -0.0003144 0.0000714 -4.403 1.07e-05 ***
## ---
## Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Chamberlain, Gary. 1980. “Analysis of Covariance with Qualitative Data.” The Review of Economic Studies 47 (1): 225–38.
Fernández-Val, Iván. 2009. “Fixed Effects Estimation of Structural Parameters and Marginal Effects in Panel Probit Models.” Journal of Econometrics 150 (1): 71–85.
Fernández-Val, Iván, and Martin Weidner. 2018. “Fixed Effects Estimation of Large-T Panel Data Models.” Annual Review of Economics 10 (1): 109–38.
Greene, William. 2004. “The Behaviour of the Maximum Likelihood Estimator of Limited Dependent Variable Models in the Presence of Fixed Effects.” Econometrics Journal 7 (1): 98–119.
Hahn, Jinyong, and Guido Kuersteiner. 2011. “Bias Reduction for Dynamic Nonlinear Panel Models with Fixed Effects.” Econometric Theory 27 (6): 1152–91.
Hahn, Jinyong, and Whitney Newey. 2004. “Jackknife and Analytical Bias Reduction for Nonlinear Panel Models.” Econometrica 72 (4): 1295–1319.
Hyslop, Dean R. 1999. “State Dependence, Serial Correlation and Heterogeneity in Intertemporal Labor Force Participation of Married Women.” Econometrica 67 (6): 1255–94.
Neyman, Jerzy, and Elizabeth L. Scott. 1948. “Consistent Estimates Based on Partially Consistent Observations.” Econometrica 16 (1): 1–32.
Stammann, Amrei, Florian Heiss, and Daniel McFadden. 2016. “Estimating Fixed Effects Logit Models with Large Panel Data.”