# Jointly Modeling Male and Female Labor Participation and Unemployment

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## Abstract

**:**

## 1. Introduction

## 2. Background and Recent Literature

## 3. Data

## 4. Replicating and Extending Emerson

## 5. Graphical Analysis

## 6. A Joint Model of Male and Female LFPRs and URs

#### 6.1. Bivariate Model Analysis

#### 6.2. Joint Model Analysis

## 7. Forecasting the Labor Market during the Pandemic

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

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 of “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), Van Zandweghe (2012), Hotchkiss and Rios-Avila (2013), Bullard (2014), Aaronson et al. (2014), Perez-Arce et al. (2018), and Seshadri (2018). |

2 | Seasonal adjustment may confound dynamics, so additional analysis of the NSA data may prove fruitful; see Ericsson et al. (1993). |

3 | The implied t-statistics need to be interpreted with some caution, given that cointegrating vector parameters are ratios of the ${\Pi}_{i}$ parameters and so may be poorly estimated; see Staiger et al. (1997). However, likelihood ratio tests, which are invariant to these transformations, generally give similar inferences in this empirical instance; see also Fieller (1954). |

4 | This approach—general-to-specific (Gets) modeling—is also referred to as the “LSE approach to econometrics” or “Hendry’s methodology”. Hoover and Perez (1999) first automated this procedure using data from Lovell (1983) and demonstrated the ability of Gets to successfully recover the data generation process (DGP). |

5 | For more details on indicator saturation methods, see Doornik and Hendry (2013), Hendry and Doornik (2014), Castle et al. (2015), Pretis et al. (2016), and Ericsson (2017). |

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 the 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. It was introduced to provide a systematic way to track the stringency of government responses to COVID-19 across countries and time. |

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**Figure 1.**Total labor force participation rate (LFPR) and unemployment rate (UR). Note: shaded bars are NBER recessions.

**Figure 3.**Forecasts and outcomes of the pandemic labor market by gender. Notes: DX represents the change in X. Shaded bars are NBER recessions. Fan charts represent two standard deviation ex-ante forecast uncertainty.

**Figure 4.**Forecasts and outcomes of the pandemic labor market gender gaps. Note: See the Notes for Figure 3.

Variable | Unit | Description | FRED Code | Mean | Std.Dev. |
---|---|---|---|---|---|

LFPR | % | Labor Force Participation Rate: Total | CIVPART | 62.86 | 2.97 |

UR | % | Unemployment Rate: Total | UNRATE | 5.77 | 1.70 |

LFPR${}_{M}$ | % | Labor Force Participation Rate: Male | LNS11300001 | 77.18 | 5.18 |

UR${}_{M}$ | % | Unemployment Rate: Male | LNS14000001 | 5.63 | 1.88 |

LFPR${}_{F}$ | % | Labor Force Participation Rate: Female | LNS11300002 | 49.86 | 9.39 |

UR${}_{F}$ | % | Unemployment Rate: Female | LNS14000002 | 6.03 | 1.55 |

NBER | {0,1} | NBER-based Recession Dummy | USREC | 0.14 | 0.35 |

(0 = expansion, 1 = recession) |

Total | Male | Female | |
---|---|---|---|

Trace Statistics: | |||

${H}_{0}:$ r = 0 | 17.20 * [0.026] | 19.19 * [0.012] | 29.4 ** [0.000] |

${H}_{0}:$ r ≤ 1 | 0.84 [0.359] | 1.68 [0.195] | 2.86 [0.091] |

Cointegrating Vector: | |||

${\beta}_{LFPR}$ | 1 | 1 | 1 |

${\beta}_{UR}$ | −4.89 $\left(1.18\right)$ | 4.47 $\left(0.98\right)$ | −14.78 $\left(3.00\right)$ |

Intercept | −35.1 | −103 | 40.4 |

Adjustment Parameters: | |||

${\alpha}_{LFPR}$ | −0.002 $\left(0.001\right)$ | −0.0013 $\left(0.0012\right)$ | −0.0019 $\left(0.0004\right)$ |

${\alpha}_{UR}$ | 0.003 $\left(0.001\right)$ | −0.0056 $\left(0.0014\right)$ | 0.0010 $\left(0.0005\right)$ |

Model Specification Tests: | |||

AR 1-7: F(28,1390)= | 1.92 ** [0.002] | 1.79 ** [0.007] | 2.18 ** [0.000] |

ARCH 1-7: F(28,1424)= | 8.01 ** [0.000] | 10.4 ** [0.000] | 5.87 ** [0.000] |

Normality: ${\chi}^{2}\left(4\right)$= | 345 ** [0.000] | 582 ** [0.000] | 164 ** [0.000] |

Hetero: F(138,2054)= | 2.89 ** [0.000] | 3.27 ** [0.000] | 2.32 ** [0.000] |

Total | Male | Female | |
---|---|---|---|

Trace Statistics: | |||

${H}_{0}:$ r = 0 | 21.89 [0.146] | 28.61 * [0.021] | 28.77 * [0.019] |

${H}_{0}:$ r ≤ 1 | 0.53 [0.999] | 2.45 [0.921] | 1.02 [0.994] |

Cointegrating Vector: | |||

${\beta}_{LFPR}$ | −0.087 $\left(0.287\right)$ | 0.287 $\left(0.303\right)$ | −0.012 $\left(0.139\right)$ |

${\beta}_{UR}$ | 1 | 1 | 1 |

Trend | −0.003 $\left(0.004\right)$ | −0.001 $\left(0.006\right)$ | −0.006 $\left(0.006\right)$ |

Intercept | 1.02 | −27.8 | −4.18 |

Adjustment Parameters: | |||

${\alpha}_{LFPR}$ | 0.011 $\left(0.005\right)$ | −0.007 $\left(0.005\right)$ | 0.026 $\left(0.006\right)$ |

${\alpha}_{UR}$ | −0.016 $\left(0.005\right)$ | −0.026 $\left(0.005\right)$ | −0.013 $\left(0.006\right)$ |

Model Specification Tests: | |||

AR 1-7: F(28,1388)= | 1.66 * [0.017] | 1.58 * [0.028] | 2.61 ** [0.000] |

ARCH 1-7: F(28,1424)= | 8.03 ** [0.000] | 8.71 ** [0.000] | 5.57 ** [0.000] |

Normality: ${\chi}^{2}\left(4\right)$= | 232 ** [0.000] | 383 ** [0.000] | 172 ** [0.000] |

Hetero: F(141,2050)= | 2.73 ** [0.000] | 2.87 ** [0.000] | 2.29 ** [0.000] |

LFPR | UR | |
---|---|---|

Trace Statistics: | ||

${H}_{0}:$ r = 0 | 26.10 * [0.045] | 49.95 ** [0.000] |

${H}_{0}:$ r ≤ 1 | 1.71 [0.971] | 14.02 * [0.026] |

Restricted Cointegrating Vector: | ||

$Male$ | 1 | 1 |

$Female$ | −1 | −1 |

Trend | 0 | −0.0016 $\left(0.0005\right)$ |

Restricted Adjustment Parameters: | ||

${\alpha}_{M}$ | 0 | −0.102 $\left(0.018\right)$ |

${\alpha}_{F}$ | 0.011 $\left(0.002\right)$ | 0 |

Test of Over-identifying Restrictions: | ${\chi}^{2}\left(3\right)=0.49\left[0.921\right]$ | ${\chi}^{2}\left(2\right)=3.01\left[0.222\right]$ |

Model Specification Tests: | ||

AR 1-7: F(28,856)= | 1.32 [0.128] | 1.24 [0.182] |

ARCH 1-7: F(28,916)= | 0.91 [0.601] | 1.18 [0.238] |

Normality: ${\chi}^{2}\left(4\right)$= | 3.97 [0.410] | 3.54 [0.472] |

Hetero: F(177,1254)= | 1.02 [0.429] | 1.20 [0.050] |

Trace Statistics | Max Statistics | ||
---|---|---|---|

${H}_{0}:$ r = 0 | 109.10 ** [0.000] | 48.45 ** [0.000] | |

${H}_{0}:$ r ≤ 1 | 60.64 ** [0.000] | 32.13 ** [0.007] | |

${H}_{0}:$ r ≤ 2 | 29.52 * [0.015] | 19.45 * [0.046] | |

${H}_{0}:$ r ≤ 3 | 10.06 [0.126] | 10.06 [0.126] | |

Restricted Cointegrating Vectors: | |||

LFPR Gap | Male UR | Female UR | |

$LFP{R}_{M}$ | 1 | 0 | 0 |

$LFP{R}_{F}$ | −1 | 0 | 0 |

$U{R}_{M}$ | 0 | 1 | 0 |

$U{R}_{F}$ | 0 | 0 | 1 |

Trend | 0 | 0 | 0 |

Restricted Adjustment Parameters: | |||

${\alpha}_{LFP{R}_{M}}$ | 0 | −0.021 $\left(0.006\right)$ | 0 |

${\alpha}_{LFP{R}_{F}}$ | 0.014 $\left(0.003\right)$ | 0 | −0.019 $\left(0.008\right)$ |

${\alpha}_{U{R}_{M}}$ | 0 | −0.029 $\left(0.006\right)$ | 0 |

${\alpha}_{U{R}_{F}}$ | 0.012 $\left(0.004\right)$ | 0.095 $\left(0.023\right)$ | −0.136 $\left(0.030\right)$ |

Test of Over-identifying Restrictions: | ${\chi}^{2}\left(11\right)=$15.45 [0.163] | ||

Model Specification Tests: | |||

AR 1-7: | F(112,1551) = 1.31 * [0.019] | ||

ARCH 1-7: | F(112,1758) = 0.90 [0.755] | ||

Normality: | ${\chi}^{2}\left(8\right)=$ 7.59 [0.475] | ||

Hetero: | F(460,1446) = 1.07 [0.191] |

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**MDPI and ACS Style**

Bernstein, D.H.; Martinez, A.B.
Jointly Modeling Male and Female Labor Participation and Unemployment. *Econometrics* **2021**, *9*, 46.
https://doi.org/10.3390/econometrics9040046

**AMA Style**

Bernstein DH, Martinez AB.
Jointly Modeling Male and Female Labor Participation and Unemployment. *Econometrics*. 2021; 9(4):46.
https://doi.org/10.3390/econometrics9040046

**Chicago/Turabian Style**

Bernstein, David H., and Andrew B. Martinez.
2021. "Jointly Modeling Male and Female Labor Participation and Unemployment" *Econometrics* 9, no. 4: 46.
https://doi.org/10.3390/econometrics9040046