The rise and spread of female labour force participation at the US county level

ABSTRACT In 2011, Fogli and Veldkamp adopted a time–space recursive spatial econometric model to investigate whether the female labour force participation rate varies with past participation rates in their own and in contiguous US counties, based on decennial data over the period 1940–2000, but their results are problematic. The applied estimators are different from the provided descriptions, the predicted contributions of the control variables are not in line with expectations, and the coefficients of the temporal and spatiotemporal lags of the dependent variable indicate instability. This replication study demonstrates that a dynamic spatial econometric model with common factors yields more convincing results in favour of the postulated model.


INTRODUCTION
The significant rise of female labour force participation (LFP) in the United States in the previous century has attracted the attention of many scholars. According to Fogli and Veldkamp (2011) (hereinafter FV), many explanations focus on changes in wages, less discrimination, the introduction of household appliances, the less physical nature of jobs, the increased ability to control fertility, the decline in childcare costs and medical innovations. In addition to these common factors, FV provide another economic-theoretical explanation that relies on information transmitted from one woman to another located nearby, which can change their perceptions of the cost of maternal employment on children. Furthermore, FV expect a delay in the decision-making process since the transmission of this information takes time.
To empirically test for this, FV estimate a spatial econometric model based on decennial data over the period 1940-2000 at the county level in which the female LFP rate varies with past participation rates in the own county and in contiguous counties. Korniotis (2010) interprets the coefficients of these two variables as measures of the relative strength of internal and external habit persistence. Fogli and Veldkamp (2011, tab. II) report the main estimation results of the proposed model estimated by ordinary least squares (OLS), the Anderson and Hsiao (1982) instrumental variables (IV) estimator, and the Arellano and Bond (1991) generalized method of moments (GMM) estimator. However, the descriptions of these estimators are different from the reported results. Furthermore, the GMM estimation results, their preferred estimator, suffer from instability because the coefficients of the temporal and spatiotemporal lags of the dependent variable add to a value greater than 1. This implies that any change in one of the explanatory variables or any shock in the error term will blow up the female LFP rate in the long term. Another issue is that the predicted effects of the variables seem implausible.
The remainder of the paper is structured as follows. Section 2 first describes the model specified, the data and the estimation results implied by FV, representing our replication of their study in a narrow sense. We find that the observed problems cannot be solved when applying the correct estimators. Section 3 reports and discusses the results when the space-time recursive model is replaced by an advanced dynamic spatial econometric model with common factors and estimated by quasi-maximum likelihood (QML) (Shi & Lee, 2017), representing our replication in a wider sense, which produces more convincing results. Finally, section 4 concludes.

MODEL SPECIFICATION, DATA AND ESTIMATION ISSUES
FV postulate the following econometric model: where LFP it is the female LFP rate of county i at time t, LFP i(t−1) is the female LFP rate lagged one decade in time, L i(t−1) is the average female LFP rate lagged one decade in time observed in contiguous counties, x kit is a control variable with coefficient w k , a i represents a county fixed effect, g t is a time fixed effect, and 1 it is an error term. 1 b is the coefficient of interest because it measures information diffusion between counties. To estimate this model, FV collected data for 3074 US counties for 1940US counties for , 1950US counties for , 1960US counties for , 1970US counties for , 1980US counties for , 1990US counties for and 2000. The control variables are urban population (%), rural farm population (%), education (school years completed by the population aged 25 years and over), density (persons per square mile) and manufacturing wages (deflated and divided by 1000). Fogli and Veldkamp (2011, appx) report descriptive statistics for each of these variables, decade by decade. 2 Table 1 reports them for 1940 and 2000, as well as the increase in each of the variables over the period 1940-2000. The standard deviations indicate substantial heterogeneity in almost all variables, in both 1940 and 2000.
From a spatial econometric perspective, this is a challenging dataset. First, the dependent variable has increased significantly over the observation period, from 18.49% in 1940 to 54.69% in 2000, enabling stationarity problems. Second, the wage rate is characterised by  Fogli and Veldkamp (2011, appx, sect. S2) describe their panel data estimation procedure of the OLS, IV and GMM estimators. However, there are several differences between this description and their Stata code. FV claim to control for county fixed effects when applying OLS, but they report the results when controlling for random effects. Next, they state to adopt the Anderson and Hsiao (1982) IV estimator to instrument the second lag of the dependent variable (LFP i(t−2) ) and spatially lagged dependent variable ( L i(t−2) ) for the lagged difference of the dependent variable (LFP i(t−1) − LFP i(t−2) ) and spatially lagged dependent variable ( L i(t−1) − L i(t−2) ), respectively. However, their Stata code shows that they used There are also inconsistencies and ambiguities regarding the description of the Arellano and Bond (1991) GMM estimator and the reported results. First, FV claim to use three lags, However, their Stata code shows that they used LFP i(t−3) , LFP i(t−4) and LFP i(t−5) , and L i(t−3) , L i(t−4) and L i(t−5) . Second, FV treat wages as an endogenous regressor and instrument it by the same variables as the (space-) time lagged dependent variable, while this is not mentioned in the text, nor is there any explanation as to why wages would be endogenous. Third, FV include time dummies for 1970, 1980, 1990 and 2000, while it is possible to also include the time dummy for 1960. Moreover, they state using the entire time series of all the exogenous regressors x it as instruments. This description is ambiguous because one could interpret it as if all available lags and leads of x it are used, while FV 'only' use x it . To address the issue of instrument proliferation and the associated loss of observations, FV could refer to Roodman (2009), who discusses the option to collapse the instruments, but they do not. Table 2 shows the results when correcting for these differences. We present the OLS results with county fixed effects, the Anderson and Hsiao IV estimator, and we, just as FV, use the instrument x it when estimating the model by GMM. Furthermore, we include the full set of time dummies and treat wages as exogenous when applying GMM. As an extension, we also compute the contribution of each control variable to the increase in the participation rate of 36.2 percentage points over the period 1940-2000 by: where W BC is the row-standardized binary contiguity matrix used to construct L i(t−1) and i N is an N × 1 vector of 1s. The first term in this multiplication measures the marginal effect of each control variable (LeSage & Pace, 2009), and the second the extent to which the change of the kth control variable contributed to the change of the female LFP over the observation period (Table 1). In addition to the Nickell bias affecting the OLS results, we see that the signs of the contributions of the control variables are not in line with expectations. Education and wages have a negative rather than a positive effect, and the negative sign of education in particular is relatively large. Similar outcomes are obtained for the IV estimator. Although the results of this model are stable and not subject to the Nickell bias, FV point out that its estimates are inefficient because unlike GMM it does not take into account all possible moment restrictions. For both estimators, the sum of all contributions is also relatively small (11.54% and 18.88%). Importantly, the total contribution of all model components, the explanatory variables, the county and time fixed effects and the error term, sums to 100%. Since the county fixed effects are constant over time and the expected value of the error term is 0, the difference between 100% and the contribution of the explanatory variables is explained by the time period fixed effects. They act as a proxy for the Table 2. Adjusted estimation results of Fogli and Veldkamp (2011) and the contribution to labour force participation (LFP) of the control variables.
Ordinary least squares (OLS) Contribution

Instrumental variables (IV) Contribution
Generalized method of moments (GMM) Contribution

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Ioanna Tziolas and J. Paul Elhorst SPATIAL ECONOMIC ANALYSIS common factors, listed in section 1, covering the explanations offered by previous studies cited by FV.
Finally, the GMM estimation results are unstable as the sum of the coefficients of the time lagged and space-time lagged dependent variable exceeds 1. Besides, the Sargan test of overidentifying restrictions rejects this specification. The sign and magnitude of the predicted contributions are also implausible. The effects of urban population, farm population and education seem to be blown up, and the total contribution of the control variables is −144.93%, which would imply that the time dummies contribute more than 100% to the increase in female LFP. Overall, the results are not convincing. 4

A DYNAMIC SPATIAL ECONOMETRIC MODEL WITH COMMON FACTORS
A more fundamental solution is to adopt the following advanced spatial econometric model: The model is presented in vector form, such that each component denotes an N × 1 vector of individual observations across all counties at time t, and extends the time-space recursive spatial econometric model in four respects. The first involves the addition of the contemporaneous female LFP rate observed in contiguous counties ( L t ). Obviously, this regressor is not in line with the economic-theoretical model developed by FV. Nevertheless, the transmission of information may take place over a shorter time horizon than over a decade. It is not for nothing that the female LFP rates observed in 1940,1960,1980 and 2000 appear to be spatially clustered in Fogli and Veldkamp (2011, fig . 2). By also estimating the spatial autoregressive parameter t of L t , we can test whether there is any empirical evidence in favour of this hypothesis. Conversely, if the hypothesis of a generation-long decay of FV holds, then the coefficient t will not be statistically different from 0. A major objection to the model in equation (1) is that each time dummy g t has the same homogeneous impact on all counties in each decade. Fogli and Veldkamp (2011, pp. 1105-1107 explain that it is not likely that changes in the common factors generated local differences in the speed of transition. It is more likely that a simultaneous switch occurred from a low-to a high-participation outcome. However, this does not imply that this switch has been the same for all counties. In line with the heterogeneity (standard errors) identified in Table 1, some counties may have responded to these developments to a greater or lesser extent than others. To test for this, we replace the county and time fixed effects with two common factors. The common factors f rt denote principal components and the parameters Γ represent factor loadings of these principal components. This is the second and simplest extension since these principal components boil down to country and time period fixed effects when f 1t = (1, . . . , 1) ′ and f 2t = (g 1 , . . . , g T ) ′ , and the following two restrictions are imposed on the parameters G ′ 1 = (a 1 , . . . , a N ) and G ′ 2 = (1, . . . , 1). One common factor would be no generalization, while three or more common factors is more than strictly necessary.
The third extension concerns the inclusion of spatial lags in the explanatory variables, denoted by x, to test whether the female LFP rate is also affected by the control variables observed in neighbouring counties. This extension adds the parameter u k to the formula by which the predicted contribution of each control variable is determined and increases the flexibility with which this contribution can be determined (Elhorst, 2010): ,1940, −2000, /DLFP 1940, −2000 Note that the parameter t of L t is also added to this expression. The fourth and final extension is that we allow the error term to be spatially correlated to control for any additional spatial patterns among variables that have been omitted from the model potentially.
We use the quasi-maximum likelihood (QML) of Shi and Lee (2017) to estimate the parameters of this model. Shi and Lee (2017, theorem 1) show that this estimator is asymptotically normal and properly centred if N is large and T small. 5 They also developed a Matlab routine called SFactors 6 that can deal with spatial weight matrices when N is relatively large.
The first set of columns in Table 3 reports the estimation results when wages and spatial lags of the control variables are included, and is based on 9414 observations. This number is slightly smaller than in the last two columns of Table 2, as Shi and Lee's computer code requires the spatial panel to be balanced. This is because the log-likelihood function in comparison with a time-space recursive model is extended with the Jacobian term ln|I N tW BC | (Shi & Lee, 2017, eq. 3). In line with the average number of contiguous counties of 5.99, the spatial weight matrix is constructed based on the six nearest-neighbours principle for those counties that remain part of the sample.
The results provide convincing empirical evidence in favour of the economic-theoretical model proposed by FV. The internal habit persistence parameter is significant but small (0.153, t-value = 12.47), which implies that the female LFP rate in the own county has only a limited effect one decade later, while the external habit persistence parameter measuring information diffusion is significant and larger (0.504, t-value = 19.69). This contrasts with the OLS and IV results of Table 2 in which the internal habit persistence parameter was larger and the external habit persistence parameter substantially smaller. Finally, the parameter of the contemporaneous female LFP rate in other counties is small and insignificant (-0.075, t-value = -0.99). This is in line with the time-space recursive spatial econometric model proposed by FV, who set this parameter equal to 0 a priori. Importantly, the sum of these three parameters amounts to 0.581, which is smaller than 1 and thus points to a stable model despite the fact the female LFP rate increased over the observation period. In addition, we find evidence of remaining spatial patterns among the error terms representing variables that have been omitted from the model. The spatial autocorrelation coefficient is positive and significant (0.595, t-value = 8.19).
The control variables contribute to the rise of the female participation rate over the period 1940-2000 to a modest degree. The largest contribution of 54.19% is made by the fall in farm population. Yet, none of the individual contributions of the other control variables has the right sign. Especially the negative contribution of wages stands out. One reason is that wages and education appear to be highly correlated; their mutual correlation coefficient amounts to 0.71. To avoid biases in the parameter estimates due to multicollinearity, one of them is better removed from the model. We decided to remove wages for the following four reasons. First, when wages are excluded the number of observations is much larger: 18,396 versus 9414. Second, wages are measured by those in manufacturing; this measurement is not representative of females since most of them found a job in the services sector. Bowen and Finegan (1969) extensively documented this for the United States a long time ago. Third, it provides the opportunity to continue using the original binary contiguity matrix used by FV instead of the modified six-nearest neighbour matrix. Finally, Fogli andVeldkamp (2011, p.1131) mention that controlling for wages does not substantially alter their results and for this reason may be excluded. 7 The results when wages are excluded are reported in the second set of columns in Table 3. The results do not change much, except for the parameter of the female LFP rate in other counties at the same moment in time, which becomes negative and significant (−0.340, t-value = −3.84), and the spatial autocorrelation coefficient, which increases in magnitude substantially Table 3. Results of the dynamic spatial econometric models with common factors (CF).

Variable
Wages included x variables Contribution Wages excluded x variables Contribution Wages and x variables excluded Contribution  The rise and spread of female labour force participation at the US county level (0.830, t-value = 9.24). It seems as if these two coefficients blow each other up, which is a problem that more often occurs when all potential spatial lags are included in the model (Elhorst, 2010). To investigate this we finally also removed the spatial lags of the control variables in the last set of columns in Table 3, conforming FV who also did not include these spatial lags. According to this model, the internal habit persistence parameter and the spatial autocorrelation coefficient return to normal values, similar to those obtained in the first set of columns of Table 3, while the positive and weakly significant parameter of the contemporaneous spatial lag (0.087, t-value = 1.68) suggests that the time horizon of one decade might indeed be somewhat too long. The contributions of the control variables in the last two specifications turn out to have the right sign and contribute to the rise of the female participation rate over the period 1940-2000 to a modest degree; 24.81% when spatial lags of the control variables are included and 31.86% when they are not. The largest contribution is again made by the fall in farm population; it is the only control variable whose contribution really matters in terms of both significance and magnitude. Bowen and Finegan (1969) and Schultz (1990) provide an explanation for this. They showed that the shift in the composition of production, out of agriculture and into manufacturing and especially services in the previous century, was associated with expanded opportunities for women's employment relative to men's, particularly as wage earners. According to Jaumotte (2003), the possibility of part-time work, especially in the service sector, also was important because this option permitted women to combine work outside the household with their domestic activities within it. The finding that the individual contributions of the control variables in the last set of columns of Table 3 sum to 31.86% also implies that common factors explain the remaining 68.14%. These common factors reflect countrywide shocks that affect the benefits and costs of employment for women, including wages, less discrimination, the introduction of household appliances, the less physical nature of jobs, the increased ability to control fertility, the decline in childcare costs and medical innovations.
Finally, it should be emphasisedthat the columns of Table 3 differ in terms of the number of observations and spatial weight matrix. However, since they all point in a different direction than the results in Table 2, we conclude that the additional spatial lags and common factors are decisive. Fogli and Veldkamp (2011) claimed to have found empirical evidence in favour of their economic-theoretical model by estimating a time-space recursive model. However, the applied estimators are different from the descriptions provided by them. Moreover, the validity of their results may be questioned due to stationarity problems and because the predicted contributions of the control variable are unlikely. The high correlation coefficient between wages and education is also problematic.

CONCLUSIONS
We have shown that the dynamic spatial panel data model with common factors developed by Shi and Lee (2017) provides more convincing empirical evidence in favour of their economictheoretical model: the space-time lagged dependent variable is more important in explaining female LFP than the contemporaneous spatially lagged dependent variable, the estimates of this model are stable and the effects of the control variables more plausible.
For data, routines and output are made available as supplemental material at spatial-panels.com.

ACKNOWLEDGMENTS
This replication paper is based on an assignment of the research master course "Cross-sectional dependence and spatial econometrics" at the University of Groningen. The authors are grateful for the comments of two anonymous reviewers and one co-editor of this journal.