Table 7 Estimated hazard ratios for the piecewise exponential model of the apprenticeship non-completion hazard for different starting ages and years

\(W/W^{(l)}\)

0.023\(***\)

(0.002)

0.021\(***\)

(0.002)

0.029\(***\)

(0.004)

0.024\(***\)

(0.002)

0.034\(***\)

(0.003)

0.027\(***\)

(0.003)

\(ln \text { }W^{(m)}\)

0.456\(***\)

(0.028)

0.448\(***\)

(0.028)

0.469\(***\)

(0.044)

0.484\(***\)

(0.040)

0.381\(***\)

(0.029)

0.692\(***\)

(0.067)

Time

(ref: (547–1000] days)

   (0–60]

0.985

(0.027)

0.983

(0.027)

0.699\(***\)

(0.034)

1.121\(**\)

(0.038)

1.596\(***\)

(0.058)

0.517\(***\)

(0.026)

   (60–121]

1.536\(***\)

(0.038)

1.541\(***\)

(0.039)

1.014

(0.044)

1.810\(***\)

(0.058)

2.550\(***\)

(0.086)

0.811\(***\)

(0.035)

   (121–212]

1.296\(***\)

(0.031)

1.298\(***\)

(0.032)

0.874\(**\)

(0.036)

1.536\(***\)

(0.048)

2.041\(***\)

(0.069)

1.786\(***\)

(0.031)

   (212–425]

0.933\(**\)

(0.020)

0.944\(**\)

(0.021)

0.772\(***\)

(0.026)

0.996

(0.029)

1.297\(***\)

(0.042)

0.737\(***\)

(0.023)

   (425–547]

0.983

(0.026)

0.985

(0.026)

0.794\(***\)

(0.033)

1.096\(**\)

(0.038)

1.409\(***\)

(0.055)

0.764\(***\)

(0.029)

Female

0.864\(***\)

(0.023)

0.859\(***\)

(0.023)

0.767\(***\)

(0.038)

0.963

(0.032)

0.852\(***\)

(0.029)

0.947

(0.039)

High school diploma

(ref: lowest track)

   None

0.907\(***\)

(0.027)

0.909\(***\)

(0.023)

0.841\(***\)

(0.032)

0.930\(*\)

(0.031)

0.933\(*\)

(0.032)

0.870\(***\)

(0.032)

   Intermediate track

0.733\(***\)

(0.013)

0.738\(***\)

(0.013)

0.749\(***\)

(0.022)

0.733\(***\)

(0.016)

0.744\(***\)

(0.017)

0.739\(***\)

(0.020)

   Academic track

0.590\(***\)

(0.016)

0.595\(***\)

(0.016)

0.724\(***\)

(0.033)

0.555\(***\)

(0.019)

0.556\(***\)

(0.019)

0.694\(***\)

(0.030)

Start. age

2.023\(***\)

(0.116)

2.627\(***\)

(0.207)

2.000\(***\)

(0.191)

1.854\(***\)

(0.133)

1.907\(***\)

(0.139)

1.972\(***\)

(0.182)

Start. age sqr.

0.984\(***\)

(0.001)

0.977\(***\)

(0.002)

0.984\(***\)

(0.002)

0.986\(***\)

(0.002)

0.985\(***\)

(0.002)

0.985\(***\)

(0.002)

Start. year

1.187\(***\)

(0.005)

1.189\(***\)

(0.005)

1.512\(***\)

(0.015)

1.121\(***\)

(0.008)

1.100\(***\)

(0.006)

1.303\(***\)

(0.010)

Foreign

1.106\(***\)

(0.031)

1.119\(***\)

(0.032)

1.071

(0.055)

1.121\(**\)

(0.038)

1.090\(*\)

(0.039)

1.080

(0.050)

Est. size

(ref: 1–49 employees)

   50–249

0.859\(***\)

(0.005)

0.862\(***\)

(0.018)

0.859\(***\)

(0.029)

0.855\(***\)

(0.023)

0.934\(***\)

(0.025)

0.776\(***\)

(0.026)

   250+

0.798\(***\)

(0.023)

0.800\(***\)

(0.023)

0.807\(***\)

(0.037)

0.793\(***\)

(0.030)

0.853\(***\)

(0.032)

0.735\(***\)

(0.033)

Est. age

0.991\(***\)

(0.001)

0.991\(***\)

(0.001)

0.991\(***\)

(0.001)

0.990\(***\)

(0.001)

0.990\(***\)

(0.001)

0.991\(***\)

(0.001)

East Germany

0.975

(0.030)

0.972

(0.030)

0.696\(***\)

(0.043)

1.071

(0.042)

0.998

(0.039)

0.958

(0.046)

Industry

(ref: Retail/wholesale/gastronomy)

   Farming/fishing/forestry

1.039

(0.079)

1.047

(0.080)

0.907

(0.112)

1.117

(0.106)

0.938

(0.089)

1.185

(0.146)

   Earths/food/textiles

0.719\(***\)

(0.028)

0.714\(***\)

(0.028)

0.767\(***\)

(0.047)

0.687\(***\)

(0.035)

0.757\(***\)

(0.038)

0.707\(***\)

(0.045)

   Chemical/oil/metal

0.581\(***\)

(0.023)

0.576\(***\)

(0.023)

0.551\(***\)

(0.036)

0.582\(***\)

(0.029)

0.576\(***\)

(0.030)

0.581\(***\)

(0.035)

   Electronics/vehicles/misc. prod.

0.617\(***\)

(0.030)

0.612\(***\)

(0.030)

0.595\(***\)

(0.048)

0.609\(***\)

(0.037)

0.625\(***\)

(0.041)

0.628\(***\)

(0.047)

   Construction/water, energy supply

0.815\(***\)

(0.032)

0.822\(***\)

(0.033)

0.828\(**\)

(0.053)

0.775\(***\)

(0.039)

0.836\(**\)

(0.044)

0.765\(***\)

(0.047)

   Logistics/Postal/Banking

0.961

(0.043)

0.972

(0.044)

0.972

(0.072)

0.941

(0.052)

0.909

(0.054)

1.045

(0.070)

   Public/private services

0.843\(***\)

(0.028)

0.844\(***\)

(0.029)

0.792\(***\)

(0.044)

0.868\(**\)

(0.037)

0.830\(***\)

(0.036)

0.859\(**\)

(0.045)

   Health/education

1.309\(***\)

(0.038)

1.297\(***\)

(0.038)

1.097

(0.053)

1.487\(***\)

(0.053)

1.220\(***\)

(0.046)

1.398\(***\)

(0.063)

   NGOs/culture/sports

1.229\(***\)

(0.044)

1.217\(***\)

(0.044)

1.164\(*\)

(0.070)

1.331\(***\)

(0.059)

1.088

(0.051)

1.455\(***\)

(0.081)

SDR

0.728

(0.123)

0.720

(0.125)

0.013\(***\)

(0.007)

1.173

(0.218)

0.941

(0.176)

1.772\(*\)

(0.476)

MES

1.286\(***\)

(0.058)

1.285\(***\)

(0.059)

1.283\(**\)

(0.098)

1.324\(***\)

(0.074)

1.332\(***\)

(0.075)

1.228\(***\)

(0.091)

\(u^{(l)}/u^{(m)}\)

0.770\(***\)

(0.013)

0.767\(***\)

(0.013)

0.502\(***\)

(0.021)

0.824\(***\)

(0.017)

0.841\(***\)

(0.018)

0.835\(***\)

(0.023)

\(u^{(h)}/u^{(m)}\)

1.438\(***\)

(0.087)

1.447\(***\)

(0.089)

1.331\(**\)

(0.134)

1.111

(0.098)

1.558\(***\)

(0.121)

1.098

(0.105)

Constant

0.000\(***\)

(0.000)

0.000\(***\)

(0.000)

0.000\(***\)

(0.000)

0.000\(***\)

(0.000)

0.000\(***\)

(0.000)

0.000\(***\)

(0.000)

\(\theta\)

0.115

0.120

0.205

0.098

0.083

0.171

LR-test of \(\theta\)

1,509

1,500

873

743

506

775

Log pseudolikelihood

− 171,485

− 167,376

− 66,548

− 104,393

− 94,778

− 74,157

Wald test

10,821

10,821

4,300

6,186

5,840

4,915

AIC

343,038

334,821

133,164

208,854

189,625

148,381

Observations

641,921

631,465

296,318

345,603

253,124

388,797

Apprenticeships

94,223

92,544

40,881

53,342

66,623

27,600

Non-completion

19,697

19,211

7,412

12,285

11,584

8,113