WORKING PAPER SERIES Jointly Modeling Male and Female Labor Participation and Unemployment

The COVID-19 pandemic resulted in the most abrupt changes in U.S. labor force participation and unemployment since the Second World War, and with different conse-quences for men and women. This paper models the U.S. labor market to help interpret the pandemic’s effects. After replicating and extending Emerson’s (2011) model of the labor market, we formulate a joint model of male and female unemployment and labor force participation rates for 1980–2019 and use it to forecast into the pandemic to understand the pandemic’s labor-market consequences. Gender-specific differences were particularly large at the pandemic’s outset; lower labor force participation persists.


Introduction
The COVID-19 pandemic resulted in the largest change in U.S. labor force participation and unemployment since the Second World War. As the economy recovers from the pandemicinduced recession, macroeconomic policy has focused on understanding changes in the labor market. Fed Chair Powell noted in his November 2021 post-FOMC press conference that the Federal Reserve's mandate of maximum employment has become much harder to assess, as "people are staying out of the labor market because of caretaking, [and] because of fear of COVID to a significant extent." This highlights the importance of examining gender-specific unemployment and labor force participation rates.
Previous research shows that, while labor force participation and unemployment rates move together over time, there are important differences between men's and women's roles in the the labor force; see Emerson (2011) and Kleykamp and Wan (2014). Furthermore, while labor-market dynamics were stable in past decades, they appear to have changed in the pandemic-induced recession, which has been called a "she-cession", in which job losses and burdens initially fell disproportionally on women; see Bluedorn et al. (2021).
To examine these and other issues, this paper formulates a joint model of male and female labor force participation and unemployment rates in the United States. We start by re-examining and extending the approach by Emerson (2011). We then focus on the four decades since 1980 and formulate a joint model of male and female labor force participation and unemployment rates to account for the interactions between those variables. The data reveal three long-run cointegrating relationships: a gap between male and female labor force participation rates (LFPRs), and stationary unemployment rates (URs) for both males and females. The female LFPR and UR adjust to close the gap in the LFPRs. The male LFPR and UR only respond to the male UR, whereas the female UR responds to the LFPR gap as well as to both the male and female URs.
While both male and female labor force participation rates declined and unemployment rates rose during the pandemic, women were much more adversely affected at the outset of the pandemic. More recently, the female LFPR and UR have recovered-relative to the male LFPR and UR-such that the gap between male and female LFPRs and the gap between 2 male and female URs are now similar to those pre-pandemic.
The rest of the paper is organized as follows. Section 2 gives background information on the historical trends and reviews the literature. Section 3 describes the data, Section 4 replicates the results in Emerson (2011), and Section 5 provides a visual analysis of the data to help inform model improvements. Section 6 presents the joint model analysis, while Section 7 uses the model to forecast the labor market over the pandemic. Section 8 concludes.    to 60.2 percent (April 2020), the largest two-month decline since the Second World War.
The pre-pandemic decline in LFPR from January 2000 to December 2019 has proved challenging to explain. Despite improvement in the unemployment rate and increases in non-farm payroll jobs, the LFPR did not recover to its January 2000 level. As demographics are generally correlated with participation, some research has explored this linkage to explain declining participation. Using demographic data, Hornstein et al. (2018) find that the labor market was at full employment in 2018 with an LFPR of 62.8 percent, and they predict declines in LFPR over the next decade. However, the LFPR grew until 2020.
Over the last two decades, the share of the population aged 55 and older has increased steadily. That sub-population may be important for understanding changes in the total LFPR. Jacobs and Piyapromdee (2016) investigate granular demographic effects to explain the high rates of partial retirement (50%) and reverse retirement (35%) of older Americans.
They find that a "work burnout-recovery process" helps to account for reverse retirement that cannot otherwise be explained by health and wealth shocks. Many competing theories have been ascribed to changes in the participation and unemployment rates. To understand these fluctuations, some empirical studies utilize disaggregated data, while others use aggregated data; see Hotchkiss and Rios-Avila (2013), Murphy and Topel (1997), and Emerson (2011) inter alia. In the first approach, household characteristics are utilized over shorter time spans. In the second approach, more parsimonious models are explored over longer horizons.
1 Fujita (2014) finds that, between 2000Q1 to 2013Q4, 65 percent of the decline is attributed to both retirements and disability, and that leaving the labor market is not determined by the business cycle. Technological progress and trade competition may also impact the participation rate; see Karabarbounis and Neiman (2014), Plant et al. (2017), and Abraham and Kearney (2020). Alternatively, Krueger (2017) notes that roughly half "prime age men who are not in the labor force take pain medication on any given day" and that opioid prescriptions are linked to lower participation. For other drivers of these dynamics, see Aaronson et al. (2012)   4 Replicating and extending Emerson (2011) We start by replicating the analysis in Emerson (2011), who models labor force participation and unemployment rates by gender. For each gender and for the total (male + female), we formulate a bivariate vector autoregressive (VAR) model with p lags: where y i,t = (LF P R i,t : U R i,t ) ′ is a bivariate vector of the labor force participation rate LF P R i,t and the unemployment rate U R i,t at time t for the ith aggregate i ∈ {T otal, M ale, F emale}, µ i is an unrestricted intercept, A i,j is an unrestricted matrix of coefficients, and ϵ i,t is a vector of residuals. Equation (1) can be rewritten in vector equilibrium-correction form: where Π i = p j=1 A i,j − I and Γ i,j = − p s=j+1 A i,s . The system in equation (2) is said to be cointegrated if Π i has a reduced rank r (r > 0), in which case Π i satisfies restrictions such that Π i = α i β ′ i for vectors α i and β i , and β ′ i y i,t is stationary. Under cointegration, r is the number of reduced-rank cointegrating relationships, the columns of β i are the cointegrating vectors, and the coefficients in α i are the adjustment parameters to the long-run cointegrating relationships. For the bivariate VAR in (1), cointegration implies reduced rank such that 0 < r < 2, i.e., r = 1: a single cointegrating relationship. Section 6 considers a 4-variate VAR, for which cointegration implies 0 < r < 4: i.e., one, two, or three cointegrating relationships; see Section 6.2.
We first test for the proper lag length of the general model in equation (1). Based on the joint evidence from F-tests and AIC and BIC information criteria, we choose 12 lags, in line with Emerson's analysis. We then test for the existence of a cointegrating relationship between unemployment and labor force participation rates with the trace statistic developed by Johansen (1988Johansen ( , 1991, treating the intercept as unrestricted. Cointegration rank deficiency, where the cointegration rank of the underlying data generation process is smaller than the rank used in the statistical analysis, is likely not a concern, given the large sample size and unambiguous rank tests; see Bernstein and Nielsen (2019).
The results in Table 2 indicate cointegration between the labor force participation rate and the unemployment rate and closely match those results reported in Emerson (2011).
Imposing a single cointegrating vector and normalizing on the labor force participation rate, the unrestricted estimate of the coefficient on the unemployment rate in the cointegrating relation indicates a long-run discouraged worker effect for males: an increase in the unemployment rate lowers the labor force participation rate. 3 For females and total workers, though, the estimates imply an added worker effect in the long run: an increase in the unemployment rate increases the labor force participation rate. Most estimated adjustment parameters are statistically significant, and they are in line with Emerson (2011); and each of the systems is equilibrium-correcting. The results in Table 2 are very similar to those in Emerson (2011). However, all three models in Table 2 appear misspecified. The system-based tests at the bottom of Table 2 test for evidence of autocorrelation, autoregressive conditional heteroskedasticity, normality, and heteroskedasticity in the vector of residuals; see Doornik (1996), Doornik and Hansen empirical instance; see also Fieller (1954).
(2008), and Kelejian (1982). In each model, the tests strongly reject the hypothesis of no model misspecification. Thus, we should be cautious in interpreting the model results.
Extending the model helps illuminate the empirical results. Specifically, we allow a trend in the cointegrating relationship, and we aim to capture business-cycle variation with an unrestricted dummy variable for NBER recessions. Including the trend generalizes the model to allow for the growth rates to be non-zero in the long-run, which may be important empirically over such a long sample. The recession dummy may capture business-cycle turning points, which can matter for modeling and forecasting and which can reduce model misspecification; see Kleykamp and Wan (2014), Ericsson (2017), and Lunsford (2021).   Table 2 normalizes (without loss of generality) on the unemployment rate rather than the labor force participation rate. Unlike the results in Table 2, statistical significance in Table 3 is ambiguous: although the labor force participation rate and the linear trend are jointly statistically significant, individually they are not. Thus, the unemployment rate could either be stationary around a linear trend or cointegrated with the labor force participation rate. The interpretation of discouraged / encouraged worker effects is also ambiguous in this formulation. However, adjustment parameters remain significant and indicate that the systems are equilibrium-correcting. Model specification tests still indicate misspecification.

Graphical Analysis
Before extending Emerson's analysis further, we return to the data and plot them. Panels A and B of Figure   First, male and female labor force participation rates converge over the sample. Second, the relationship between male and female unemployment rates shifts abruptly around 1980, and that shift appears permanent. This phenomena was analyzed in detail by Albanesi and Şahin (2018) and is explained by the convergence in LFPRs and differences in industry composition.
Third, the business cycle has clear and persistent effects, particularly on unemployment but also on labor force participation; see Hobijn and Şahin (2021)   To address these issues, we make the following changes to the modeling framework. First, we account for interaction between male and female labor force participation and unemployment rates by modeling them jointly. Second, we focus on the sample 1980-2019, allowing us to formulate a model on a relatively long sample (40 years), during which the relationships between male and female LFPRs and URs appear relatively stable. It also provides a baseline against which to assess how the labor market evolved during the pandemic. We leave modeling the entire post-war sample for future research. Finally, we include 12 unrestricted lags of the NBER recession dummy to better account for persistent effects of recessions over time. This approach is very much in the spirit of White (1990), Hendry and Mizon (1993), and Ericsson et al. (1998b), who formalize modeling as a progressive research strategy.

A Joint Model of Male and Female LFPRs and URs
This section develops a joint model of male and female labor force participation and unemployment rates. Section 6.1 performs additional bivariate analyses of male and female unemployment rates and of male and female labor force participation rates. Section 6.2 uses these findings to inform restrictions on the cointegrating vectors of a joint four-variable model of the male and female unemployment and labor force participation rates.

Bivariate Model Analysis
To help facilitate interpretation of the joint analysis, we start with bivariate analyses, looking at the relationship between male and female labor force participation rates and the relationship between male and female unemployment rates. In particular, we hypothesize that there are one-to-one long-run relationships between male and female LFPRs and between male and female URs. This framework is motivated by Figure 2, and it contrasts with Emerson (2011), who models the bivariate relationship between LFPR and UR for a given gender. Table 4 presents the results from this analysis. For the labor force participation rates, we find strong evidence of a single cointegrating vector and do not reject the restrictions imposing a one-to-one long-run relationship between male and female LFPRs. Unlike in the sample analyzed in Table 3, For the unemployment rates, the analysis in Table 4 finds two cointegrating relationships, implying that the unemployment rates are stationary after accounting for recessions.
However, if we impose a single cointegrating relationship, a one-to-one relation between male and female URs (i.e., a UR gap) is not rejected. The trend is significant in that estimated long-run relationship, indicating a small but steady long-run decline in the equilibrium UR gap. In this formulation, only the male UR responds significantly to the UR gap, while the female UR is weakly exogenous. The adjustment rate is rapid: about 10 percent per month.  Table 4. Table 5 presents the trace and max statistics and related results for the joint model of male and female LFPRs and URs. These results show consistent evidence of three cointegrating relationships. Identification of these cointegrating relationships draws on the bivariate cointegration results in Section 6.1. After some manipulation, we are able to clearly interpret the cointegrating relationships of the model and how variables in the system respond to them.
The first cointegrating relationship in Table 5 is the gap between the male and female LFPRs, as found in the first bivariate system in Table 4. The trend is insignificant, and only the female LFPR and the female UR respond directly to disequilbrium in this relationship. The estimated speed of adjustment for the female LFPR is faster than in the bivariate analysis, with a monthly adjustment of approximately 17 percent at an annual rate. However, estimating the actual multi-period adjustment is more complicated because the female UR responds to the LFPR gap and the female LFPR responds to the third cointegrating relationship (female UR).
The second and third cointegrating relations correspond to stationary male and female URs, where their stationarity is evident after controlling for NBER recessions and the recessions' lagged effects; see Table 4. Thus, the second cointegrating relation is the (stationary) male UR itself, to which the male LFPR and UR and the female UR respond. The third cointegrating relation is the (stationary) female UR, to which the female LFPR and the female UR respond. Table 5 shows dynamic evidence of discouraged worker effects. The male LFPR decreases in response to an increase in the male UR, and the female LFPR decreases in response to an increase in the female UR. However, the net effect of these dynamics is less obvious for females: the female LFPR also increases in response to an increase in the LFPR gap, thereby capturing the trending decline in the LFPR gap; and the female UR responds to all three cointegrating relations.
Jointly, the male LFPR and the male UR are weakly exogenous for cointegrating vectors involving the LFPR gap and the female UR. That is, the male LFPR and the male UR only Restricted Adjustment Parameters: respond to the male UR in the long-run, but not to the LFPR gap or the female UR. We can test whether the male LFPR and UR are strongly exogenous to the LFPR gap and the female UR (that is, both weakly exogenous and Granger non-causal) by jointly testing the weak exogeneity restrictions and whether the lags of the changes in female LFPR and UR can be excluded from the equations for male LFPR and UR; see Engle et al. (1983) andEricsson et al. (1998a). This hypothesis is strongly rejected: χ 2 (48) = 78.7, which has a probability of less than 1% under the null hypothesis. This rejection occurs primarily because changes in the male LFPR respond significantly to lagged changes in the female UR.
The joint system of equations appears to be well-specified, with most of the model specification tests failing to reject the null hypothesis of a correctly specified model. However, some autocorrelation remains in the residuals, possibly driven by the male UR equation.
The model could be simplified further by testing for and restricting lag structure. However, since we already see evidence of auto-correlated residuals and do not have a clear guide on how to reduce the model, we choose to not simplify the model further.
The robustness of the model can be assessed using the automatic model selection algorithm "Autometrics". Here, we perform a robustness check by selecting over lag structure and also over possible outliers and step shifts by saturating the model with impulse and step indicators; see Hendry and Krolzig (2005), Doornik (2009), and Hendry and Doornik (2014). 4 To protect against spurious results, we choose a "target gauge" of 0.1% such that, under the null hypothesis in which none of the indicators matter, we expect on average to include roughly 2 irrelevant indicators by chance; see Hendry (1999), Castle et al. (2011), andNielsen (2016). 5 We do not detect any relevant outliers or step shifts.
This provides further evidence that the model is reasonably well-specified in sample.
7 Forecasting the Labor Market During the Pandemic   that, for both men and women, the initial recession caused a sharp decline in the LFPRs and the EPOPs and a spike in the URs. The United States saw a dramatic jump in unemployment rates, unlike the United Kingdom, which had a job furlough program; see Castle et al. (2021). More than a year after the official end of the U.S. recession, the effects on the labor market continue to be pronounced. Unemployment rates have recently fallen to within two standard deviations of the model's forecasts. However, the LFPRs and EPOPs continue to be well below their pre-pandemic levels. This is especially striking because the forecasts were generated, conditional on the NBER 2020 recession dates. That highlights just how different the pandemic recession was, relative to previous recessions.
Next, we examine how the pandemic differentially affected men and women in the labor market by looking at how the forecasted gaps in LPFR, UR, and EPOP compare to the actual outcomes. Figure 4 plots the results from this exercise. The first row in Figure 4 shows the forecast and actual values for the UR gap, the LFPR gap, and the EPOP gap.

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The second row in Figure 4 presents the one-month changes in those variables and forecasts.  Unlike previous recessions in this sample, the unemployment gap became large and negative (Panel A). By April 2020, the female UR was nearly 3 percentage points higher than the male UR. During the initial months of the pandemic recession, the female UR increased to unprecedented levels, as service sector industries-which employ a larger share of womenwere disproportionately affected. As the pandemic progressed, the LFPR gap first fell and then increased rapidly (Panel B), with the increase driven by women leaving the labor force, in part due to limited child-care options; see Lim and Zabek (2021). Since then, both the LFPR and UR gaps have returned to pre-pandemic levels. This indicates that, while women were more severely affected at the outset of the pandemic relative to previous recessions, their relative participation in the labor market has recovered and is now in line with historical trends. Seasonal patterns also appear to have altered, making the observed data more volatile because seasonal adjustment failed to fully capture that change.

Conclusions
This paper jointly models male and female labor force participation rates and unemployment rates. We examine and extend the approach by Emerson (2011) and then focus on the last four decades of data to jointly model and forecast male and female LFPRs and URs to account for the relationships between those variables. There are three long-run relationships, corresponding to the LFPR gap and to stationarity in both male and female unemployment rates. The female LFPR and UR respond to the LFPR gap. In the long run, the male LFPR and UR only respond to the male UR, whereas the female UR responds to both the male and female URs as well as the LFPR gap.
During the pandemic, both male and female labor force participation rates declined and unemployment rates rose. However, women were much more adversely affected than men at the outset of the pandemic recession and immediately thereafter. Even so, the gap between male and female labor force participation rates has narrowed more recently, and the gap between them is now around its pre-pandemic level.
There are several possible avenues of future research. First, extending the model over the full post-war sample may allow additional insights on longer-term trends, such as (potentially) a much stronger convergence in the LFPR gap prior to 1980 and a different equilibrium relationship between male and female unemployment rates. 6 The full post-war sample may also benefit from an I(2) cointegration analysis, noting that the LFPR gap is itself strongly trending, and not just the male and female LFPRs; see Juselius (2006). Second, different recessions could have different effects on the labor market. 7 Also, additional information such as the number of COVID-19 cases or the stringency index developed by Hale et al.
(2021) could help better capture the pandemic period. 8 Finally, the model could be reformulated in terms of the underlying number of employees, the labor force, and population.
This could allow the model to address other interesting questions that cannot be answered 6 The Current Population Survey, from which the unemployment rate and labor force participation rate are derived, explicitly excludes the military population. This may have implications for analyses of the labor market in samples that include the Korean and Vietnam Wars. 7 Ahn and Hamilton (2021) argue that measurement bias of the participation rate has increased since the Great Recession, indicating the need for step indicators. Likewise, during COVID-19, the Bureau of Labor Statistics noted that the reported unemployment rate has been underestimated. 8 The index is available from Guidotti and Ardia (2020) as a simple average of nine response sub-indices.

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in the model's current form: e.g., are unemployed women more likely than men to leave the labor force than to remain unemployed? Such modeling could also help resolve changes in seasonality. This suggests a fruitful research agenda going forward.