Letting $\mathbf{M^{I}}=\mathbf{M^{I}_{m}+M^{I}_{s}}$ denote the matrix that contains the aggregate occupational transition flows across two interview dates under independent interviewing, it follows that $\mathbf{M^{I}_{s}}=\mathbf{M^{I}-M^{I}_{m}}=\mathbf{\Gamma^{\prime} M_{s} \Gamma}$. By virtue of the symmetry of $\mathbf{M_{s}}$ and assumption A2, $\mathbf{M_{s} \Gamma=\Gamma^{\prime} {M_{s}}^{'}=\Gamma^{'} M_{s}}$. Substituting back yields $\mathbf{M^{I}_{s}}=\mathbf{M_{s}\Gamma \Gamma}$. Next note that $\mathbf{M^{I}_{s}=M_{s}T^{I}_{s}}$, where $\mathbf{T^{I}_{s}}$ is the occupational transition probability matrix of the employer/activity stayers in this population observed under independent interviewing. Substitution yields $\mathbf{M_{s} T^{I}_{s}}=\mathbf{M_{s} \Gamma \Gamma}$. Multiply both sides by $\mathbf{M_{s}^{-1}}$, which exists as long as all the diagonal elements of $\mathbf{M_{s}}$ are non-zero, yields the key relationship we exploit to estimate $\mathbf{\Gamma}$,

\begin{equation}\label{eq:est_gamma_2}
\mathbf{T^{I}_{s}}=\mathbf{\Gamma \Gamma}.
\end{equation}

To use this equation we first need to show that it implies a unique solution for $\mathbf{\Gamma}$. Towards this result, we now establish that $\mathbf{\Gamma}$ and $\mathbf{T^{I}_{s}}$ are diagonalizable. For the latter it is useful to interpret the coding error process described above as a Markov chain such that $\mathbf{\Gamma}$ is the one-step probability matrix associated with this process.

Lemma A.1: Assumptions A2 and A3 imply that $\mathbf{\Gamma}$ and $\mathbf{T^{I}_{s}}$ are diagonalizable.

In general one cannot guarantee the uniqueness, or even existence, of a transition matrix that is the $(nth)$ root of another transition matrix. Here, however, existence is obtained by construction: $\mathbf{T_s}$ is constructed from $\mathbf{\Gamma}$, and in reverse, we can find its roots. The next result shows that $\mathbf{T_s}$ has a unique root satisfying assumptions A2 and A3.

Proposition A.1: $\mathbf{\Gamma}$ is the unique solution to $\mathbf{T^{I}_{s}=\Gamma \ \Gamma}$ that satisfies assumptions A2 and A3. It is given by $\mathbf{P \Lambda^{0.5} P^{-1}}$, where $\mathbf{\Lambda}$ is the diagonal matrix with eigenvalues of $\mathbf{T^{I}_{s}}$, $0<\lambda_i\leq 1$, and $\mathbf{P}$ is the orthogonal matrix with the associated (normalized) eigenvectors.

The above results imply that under A2 and A3, $\mathbf{\Gamma}$ is uniquely identified from the transition matrix of true occupational stayers under independent interviewing, $\mathbf{T^{I}_{s}}$.

§.§.§ A.2 Estimation of $\mathbf{\Gamma}$

The next lemma provides an intermediate step towards estimating $\mathbf{\Gamma}$. For this purpose let $PDT(.)$ denote the space of transition matrices that are similar, in the matrix sense, to positive definite matrices.

Lemma A.2: The function $f:PDT(\mathbb{R}^{O\times O})\to PDT(\mathbb{R}^{O\times O})$ given by $f(\mathbf{T})=\mathbf{T^{0.5}}$ exists and is continuous with $f(\mathbf{T^{I}_{s}})=\mathbf{\Gamma}$ in the spectral matrix norm.

Let $\mathbf{\hat{T}^{I}_{s}}$ denote the sample estimate of $\mathbf{T^{I}_{s}}$ and let $\mathbf{\hat{\Gamma}}$ be estimated by the root $\mathbf{(\hat{T}^{I}_{s})}^{0.5} \in PDT(\mathbb{R}^{O \times O})$ such that $\mathbf{\hat{\Gamma}}=\mathbf{(\hat{T}^{I}_{s})^{0.5}}=\mathbf{\hat{P} \hat{\Lambda}^{0.5} \hat{P}^{-1}}$, where $\mathbf{\hat{\Lambda}}$ is the diagonal matrix with eigenvalues of $\mathbf{\hat{T}^{I}_{s}}$, $0<\hat{\lambda_i}^{0.5}\leq 1$ and $\mathbf{\hat{P}}$ the orthogonal matrix with the associated (normalized) eigenvectors. We then have the following result.

Proposition A.2: $\mathbf{\Gamma}$ is consistently estimated from $\mathbf{(\hat{T}^{I}_{s})}^{0.5} \in PDT(\mathbb{R}^{O \times O})$ such that $\mathbf{\hat{\Gamma}}=\mathbf{(\hat{T}^{I}_{s})^{0.5}}=\mathbf{\hat{P} \hat{\Lambda}^{0.5} \hat{P}^{-1}}$. That is, $plim_{n \to \infty} \mathbf{\hat{\Gamma}}=\mathbf{\Gamma}$.

To identify and estimate $\mathbf{\Gamma}$ in the SIPP it is not sufficient to directly compare the aggregate occupational transition flows under independent interviewing with the aggregate occupational transition flows under dependent interviewing. Let $\mathbf{M^{D}}=\mathbf{M^{I}_{m}+M^{D}_{s}}$ denote the matrix that contains the aggregate occupational transition flows across two interview dates under dependent interviewing for employer/activity stayers and under independent interviewing for employer/activity movers. Subtracting $\mathbf{M^{I}}=\mathbf{M^{I}_{m}+M^{I}_{s}}$ from $\mathbf{M^{D}}$ yields $\mathbf{M^{D}_{s}-M^{I}_{s}}=\mathbf{M_{s} -\Gamma' M_{s} \Gamma}$. Given the symmetry assumed in A2, the latter expression has $0.5n(n-1)$ exogenous variables on the LHS and $0.5n(n+1)$ unknowns (endogenous variables) on the RHS, leaving $\mathbf{\Gamma}$ (and $\mathbf{M_s}$) unidentified.

In addition to $\mathbf{M^{D}-M^{I}}=\mathbf{M_{s} -\Gamma' M_{s} \Gamma}$ one can use $\mathbf{M^D}=\mathbf{\Gamma^{'} M_{m} \Gamma + M_{s}}$, which contains the remainder information. When $\mathbf{M_{m}}$ has mass on its diagonal, however, this additional system of equations has $n^{2}$ exogenous variables on the LHS and $n^{2}$ unknowns (arising from $\mathbf{M_m}$) on the RHS. This implies that with the $n$ unknowns remaining from $\mathbf{M^{D}-M^{I}}=\mathbf{M_{s} -\Gamma' M_{s} \Gamma}$, one is still unable to identify $\mathbf{\Gamma}$ and $\mathbf{M_s}$.

Corollary A.1:
If $\mathbf{M_{m}}$ has mass on its diagonal, $\mathbf{\Gamma}$ cannot be identified from $\mathbf{M^I}$ and $\mathbf{M^D}$ alone.

The intuition behind this result is that by comparing aggregate occupational transition flows under dependent and independent interviewing, it is unclear how many workers are `responsible' for the change in occupational mobility between $\mathbf{M^D}$ and $\mathbf{M^I}$. Only when the diagonal of $\mathbf{M_{m}}$ contains exclusively zeros, identification could be resolved and one can recover $\mathbf{M_s}$, $\mathbf{\Gamma}$ and $\mathbf{M_m}$ as the number of equations equals the number of unknowns.[However, in the SIPP this case is empirically unreasonable as it requires that all employer/activity changers be true occupational movers.] An implication of the above corollary is that interrupted time-series analysis that is based on the difference in occupational mobility at the time of a switch from independent to dependent interviewing, does not identify the precise extent of the average coding error, but provides a downwards biased estimate.

To identify $\mathbf{\Gamma}$, however, Proposition A.2 implies that one can use the observed occupational transition flows of a sample of true occupational stayers that are subject to two rounds of independent interviewing. Some of these workers will appear as occupational stayers and some of them as occupational movers. Ideally, such a sample of workers should be isolated directly from the 1985 panel. Unfortunately, the questions on whether the individual changed activity or employer were only introduced in the 1986 panel, as a part of the switch to dependent interviewing. As a result, the 1985 panel by itself does not provide sufficient information to separate employer/activity stayers from employer/activity movers. Instead we use 1986 panel to estimate $\mathbf{\hat{M}^I_{m}}$. We can infer $\mathbf{M^I_{s}}$ indirectly by subtracting the observed occupational transition flow matrix $\mathbf{\hat{M}^I_m}$ in the 1986 panel from the observed occupational transition flow matrix $\mathbf{\hat{M}^I}$ in the 1985 panel. This is possible as the 1986 panel refers to the same underlying population as the 1985 panel and separates the employer/activity changers, who are independently interviewed.

Corollary A.2:
$\mathbf{\hat{\Gamma}}$ is consistently estimated from $\mathbf{\hat{T}_s^I}$ when the latter is estimated from $\mathbf{\hat{M}^{I}}-\mathbf{\hat{M}^{I}_{m}}$

This result is important to implement our approach. It follows as the population proportions underlying each cell of $\mathbf{\hat{M}_{s}}$, the sample estimate of $\mathbf{M_{s}}$, are consistently estimated. In turn, the latter follows from the standard central limit theorem for estimating proportions, which applies to $\mathbf{\hat{M}^{I}}$, $\mathbf{\hat{M}^{I}_{m}}$ and its difference. Proposition A.2 then implies that $\mathbf{\hat{\Gamma}}$ is consistently estimated.

§ QUANTITATIVE ANALYSIS

This Appendix is divided into three parts that complement Sections 4 and 5 of the paper. The first part provides further details of the full model calibration done in Section 4. The second part presents the calibration results from the “excess mobility model”, where we analyse its ability to reproduce the long-run and cyclical patterns of several labor market variables.

§.§ Full Model: Gross and Net Mobility

In the main text we show that the calibrated version of the full model is able to replicate well all the targeted long-run occupational mobility, job separation, job finding and unemployment patterns of the US labor market. It does so by generating within each task-based category periods of search, rest and reallocation unemployment as $A$, $p_{o}$ and $z$ evolve.

Here we expand on the analysis presented in Sections 4 and 5 along three dimensions. First, we further show the model's implied unemployment durations by presenting (i) the job finding rates as a function of duration, (ii) the (incomplete) unemployment duration distribution and (iii) the mobility-duration profile decomposed by excess and net mobility. Second, we provide further details of the differences between occupational categories with respect to their relative cyclical unemployment responses, and the cyclical inflow and net flow responses that are used to estimate occupation-specific cyclical differences in the model. Third, we present the full correlation tables describing the cyclical performance of the model using the 5Q-MA smoothed and quarterly HP-filtered measures. We also discuss the cyclicality of an alternative unemployment measure that includes entrants; show the cyclicality of the unemployment, job finding and job separation rates by age groups; and present the decomposition of search, rest and reallocation unemployment episodes for a given value of $A$ in a comparable way to the one derived for the excess mobility model discussed in Section 5.1 of the main text.

Unemployment duration moments Figure <ref> shows the aggregate and age-specific unemployment hazard functions, comparing the model to the data.
The observed duration dependence patterns are captured very well, where the young exhibit stronger negative duration duration dependence than the prime-aged. In our sample (and in the calibration) the degree of duration dependence in the unemployment hazard is relatively weak as we have tried to minimise the presence of unemployed workers who were in temporary layoff and/or returned to their previous employers (see Supplementary Appendix B.4 for a further discussion of this issue).

Hazard Functions. Data and Model Comparison

1.0![All Workers]

[Young Workers]

[Prime-aged Workers]

Figure <ref> shows the aggregate and age-specific hazard functions separately for occupational movers and stayers. The model captures these well, where we find a stronger degree of negative duration dependence among occupational stayers than occupational movers, particularly among young workers, as in the data.

Occupational Movers/Stayers Hazard Functions. Data and Model Comparison

1![All Workers]
[Young Workers]
[Prime-aged Workers]

The unemployment duration distribution is also well matched. Table <ref> shows it reproduces very well both the proportion of short and long durations spells across the distribution. This fit is achieved when pooling together all workers and when separating them between young and prime-aged workers. Crucially, the fit of the duration distribution is not implied by targeting the empirical unemployment survival functions. The reported duration distribution is constructed by averaging duration distributions across quarters, while the survival functions are derived from pooling all observations. For example, the observed long-term unemployment in the pooled survival functions occur mainly in recessions and these observations get down-weighted when averaging across quarters (instead of counting each observation equally). Indeed, we show below that matching the survival functions does not imply also matching the duration distribution.

Incomplete Unemployment Duration Distribution Behavior (1-18 months)

3cAll workers     3cYoung workers     3cPrime-aged workers

(lr)2-4 (lr)5-7 (lr)8-10
Unemp.      Full     Excess     Data      Full     Excess       Data           Full     Excess      Data

Duration       Model      Model            Model     Model         Model     Model

(lr)2-4 (lr)5-7 (lr)8-10
1-2 m      0.43     0.42          0.43      0.53     0.52     0.47     0.41     0.39        0.41

1-4 m      0.65     0.64      0.67      0.76     0.75       0.71     0.63     0.61       0.65

5-8 m      0.20     0.21      0.20      0.16     0.17      0.19      0.21     0.22      0.21

9-12 m     0.09     0.09      0.08      0.05     0.05     0.07       0.10    0.10     0.09

13-18m      0.06     0.06      0.05     0.03     0.03       0.03     0.07     0.07      0.06

Supplementary Appendix B (Figures 9 and 10) shows that both excess and net mobility increase with unemployment duration and that the former is the main driver of the overall increase in the mobility-duration profile. Figure <ref> (below) depicts the model's equivalent decomposition using task-based categories and without considering the “management” occupation. In the model both excess and net mobility increase with unemployment duration. Given the countercyclicality of net mobility, the latter occurs as net mobility is more prominent in recessions where workers' unemployment durations are typically longer. As in the data, excess mobility is the main driver of the mobility-duration profile.

Task-based occupational categories over the cycle

The cyclical productivity loadings $\epsilon_o$ are the only four cyclical parameters that explicitly differ across task-based categories $o \in \{NRC, RC, NRM, RM \}$. Together with the elasticity of the cross-occupation search, $\nu$, these parameters shape the differential cyclical response of each category $o$ along three dimensions, summarised by 12 moments in the main text. (i) The cyclical response of net mobility for each task-based category, (ii) the cyclical change in the proportion of occupational movers that choose an occupation category $o$, and (iii) the strength of each category's unemployment durations response relative to the economy-wide average response to the aggregate unemployment rate.

Task-based Unemployment Duration Elasticities

1cNRC     1cRC     1cNRM     1cRM

$\varepsilon^{Data }_{UD_{o},u}$     0.410     0.384     0.284     0.418

(s.e.)     (0.068)     (0.050)     (0.045)     (0.053)

$\varepsilon^{Model}_{UD_{o},u}$     0.390     0.413     0.342     0.423
$\varepsilon^{Data}_{UD_{o},u}/\varepsilon^{Data}_{UD_{avg},u}$ (targeted)     1.096     1.026     0.759     1.119

(s.e.)     (0.183)     (0.132)     (0.119)     (0.141)

$\varepsilon^{Model}_{UD_{o},u}/\varepsilon^{Model}_{UD_{avg},u}$    0.988     1.055     0.890     1.072

We highlight that in (iii) we target the unemployment duration elasticities for each task-based category relative to the economy-wide elasticity. We do this as we want to leave untargeted the amplification of aggregate unemployment. In particular, as a first step to derive these elasticities in the SIPP we regress for each task-based category $o$ the log unemployment durations of workers who lost their job in $o$ on the log (aggregate) unemployment rate and a linear trend. Let $\varepsilon_{UD_{o},u}$ for $o \in \{NRC, RC, NRM, RM \}$ denote the resulting unemployment duration elasticities with respect to aggregate unemployment. The first row of Table <ref> presents these elasticities and compares them to the simulated ones in the calibration. These elasticities show that NRM occupations have a more muted cyclical response than RM occupations. This differential response is also statistically significant: a Wald test on equality of the two corresponding coefficients has an associated p-value of 0.02. In the second step, we normalize each elasticity by the (occupation size-weighted) average of all four elasticities. The resulting normalized elasticities are the ones we target in the model. The last two rows of Table <ref> shows these ratios, showing that the model fits the data well. In particular, it shows that RM occupations are the most cyclically sensitive in terms of unemployment durations; while NRM occupations are the least cyclically sensitive. These elasticities are in line with the data. Below we show that the model is also successful in generating the untargeted aggregate unemployment amplification.

As shown in the main text, the model is consistent with the cyclical changes in net mobility as well as the cyclical changes in the inflows for each task-based category. This occurs as differences in $\epsilon_o$ translate into cyclically changing incentives for workers to leave an occupation in category $o$ and, depending on $\nu$, to sample $z$-productivities from another occupation in category $o'$. Figure <ref> displays the relationship between these two set of moments. For each task-based category, it shows the relationship between the cyclical changes in net mobility on the x-axis (“Net mobility $o$, Expansions - Net mobility $o$, Recessions”) and the cyclical changes in inflows as a proportion of all occupational movers on the y-axis ($\Delta_{exp - rec}$ (inflow $o / \text{all flows}$)). RM occupations have the strongest cyclical response of net outflows, increasing in recessions, as well as the strongest response in the inflow proportion, also larger in recessions. In contrast, NRM occupations are the ones which experience the largest increase in net inflows in recessions and the largest increase in inflows as destination category.

Task-Based Occupational Mobility

1![Decomposition of the Mobility-Duration Profile]

[Cyclical Shifts across Occupation Categories in Netflows vs Inflows]

As Figure <ref> and Table <ref> show, the model captures well the co-movement along dimensions (i), (ii) and (iii) discussed above. In particular, with only one set of parameters indexed by task-based occupation category, $\epsilon_{o}$, the model reproduces well the average co-movement of the cyclical inflow shift with the cyclical changes in net flows. Figure <ref> shows that when comparing the fitted regression lines in the data and the model both display very similar slopes, where changing the net flow by 1% goes together with an inflow response of more than 2%.

Economy-wide cyclical outcomes

In terms of the cyclical properties of the unemployment, vacancies, job finding and separation rates, Table <ref> show the full set of correlations for the model and data. The model's aggregate time series arise from the distributions of employed and unemployed workers across all labor markets, combined with agents' decisions. The top panel compares the data and model using centered 5Q-MA time series of quarterly data. The cyclical components of the (log) of these time series are obtained by using an HP filter with parameter 1600. It shows that the model matches the data very well. The bottom panel compares the data and model without using this smoothing procedure. The version now yields vacancy and job separation rates that are much less persistent than their data counterparts. This happens because we have used a relative coarse grid for the simulated productivity process, as making the productivity grid finer will make the computational time of the calibration unmanageable. This implies that the discreteness of the $z^s$ and $z^r$ cutoffs functions (relative to the productivity grid) makes the vacancy and job separation rates change value too often. Using a centered 5Q-MA on quarterly data alleviates this feature without further compromising on computation time. Note, however, that this comes at the cost of reducing the volatility of the vacancy rate (and labor market tightness) in the model from 0.07 to 0.05 (0.21 to 0.17), while in the data they remain stable. Similarly on the data side, the job finding rate, measured in a consistent way with the model while taking into account censoring in the SIPP, is somewhat noisy at quarterly frequency. Smoothing this time-series using the 5Q-MA helps diminish this noise.

Full Model: Logged and HP-filtered Business Cycle Statistics

13|l|Smoothed data: centred 5Q MA time series of quarterly data
6|c||Data (1983-2014)     6c|Full Model
$u$     $v$     $\theta$     $s$     $f$     $outpw$     $u$     $v$     $\theta$     $s$     $f$      $outpw$
$\sigma$      0.15     0.11	    0.25	    0.10     	0.09    	0.01     0.14	    0.04    	0.17	    0.07	    0.09	    0.01

$\rho_{t-1}$     0.98    	0.99    0.99    	0.94    	0.93    	0.92     0.93    	0.91    	0.92    	0.88    	0.93    	0.88
12|c|Correlation Matrix
$u$     1.00    	-0.95    	-0.99    	0.83    	-0.86    	-0.50      1.00    	-0.66    	-0.97    	0.79    	-0.89    	-0.94

$v$          1.00     0.98     	-0.79    	0.81     	0.61     	    1.00     	0.80     	-0.83     	0.90     	0.81

$\theta$               1.00    	-0.82     	0.85     	0.55               1.00    	-0.84     	0.96    	0.97

$s$                       1.00     -0.71     	-0.43                   1.00      -0.87     	-0.91

$f$                               1.00      0.40                       1.00       0.94

$outpw$                             1.00                          1.00
13|l|Un-smoothed data
$u$     $v$     $\theta$     $s$     $f$     $outpw$    $u$     $v$     $\theta$     $s$     $f$      $outpw$
$\sigma$      0.16     	0.11    	0.26    	0.12     	0.13     	0.01     0.16     	0.07    	0.20     	0.11     	0.12    	0.01

$\rho_{t-1}$     0.85    	0.96    	0.94    	0.58    	0.33    	0.75     0.88    	0.54     	0.83     	0.39    	0.78    	0.76
12|c|Correlation Matrix
$u$     1.00    	-0.86     	-0.98    	0.69    	-0.65    	-0.38    1.00    	-0.54     	-0.96     	0.52     	-0.80    	-0.89

$v$          1.00     0.95     	-0.75     	0.59    	0.50         1.00     0.75    	-0.74    	0.85    	0.78

$\theta$               1.00     -0.74     	0.65    	0.44               1.00     -0.65     0.90    	0.95

$s$                       1.00     -0.57    	-0.32                  1.00      -0.79    	-0.79

$f$                               1.00      0.26                       1.00       0.91

$outpw$                             1.00                          1.00

As argued in main text (and in Supplementary Appendix B.7) we consider the unemployment rate of those workers who are unemployed between jobs $(EUE)$, so that the occupational mobility of these workers can be straightforwardly measured. The resulting $EUE$ unemployment rate $(EUE/(EUE+E))$, under the definitions and restrictions we explained in the main text, is significantly lower than the BLS at 3.5% (vs 6.3%), but drives much of its changes. In particular, for every one percentage point change in the BLS unemployment rate, we find that about 0.75 percentage points originate from the response of the $EUE$ unemployment rate. This means that the relative cyclical response of the $EUE$ unemployment rate is much stronger than the relative response of the BLS unemployment rate. Indeed, the volatility of the HP-filtered logged quarterly $EUE$ unemployment rate is 0.16 while the corresponding BLS unemployment measure (which includes inflows from non-participation) over the same period is 0.11. For the 5Q-MA smoothed time series, the difference is from 0.15 $(EUE)$ to 0.10 (BLS). The above also means that the focus on $EUE$ unemployment raises the bar further to achieve sufficient amplification. Nevertheless, Table <ref> shows that our model performs well.

In the model we also can calculate a measure of unemployment that includes unemployment following first entry into the labor market. Relative to the BLS measure, this measure still excludes unemployment associated with workers who re-enter the labor market during their working life or who subsequently leave the labor force but not before spending time in unemployment. Including entrants in unemployment raises the average total unemployment rate to 5.2% in the model, exhibiting a lower volatility of 0.12 (5Q-MA smoothed). The latter arises as with this unemployment measure, roughly 60% of the way from the $EUE$ to BLS unemployment measures, the volatility gets closer to that of the BLS measure. Cross-correlation and autocorrelation statistics of this alternative unemployment measure are very similar to the $EUE$ unemployment measure.

The ability of the model to replicate the cyclical behavior of many labor market variables is down to the coexistence of episodes of search, rest and reallocation unemployment during workers' jobless spells. Figure <ref> shows that when aggregating across all occupations the distribution of these types of unemployment episodes across values of $A$ is very similar to the one generated by the excess mobility model depicted in the main text. That is, search unemployment episodes are the most common when the economy moves from mild recessions up to strong expansions. It is only as recessions get stronger that rest unemployment episodes become more common.

The middle and right panels of Figure <ref> shows that among young and prime-aged workers the calibration generates similar search and rest unemployment dynamics over the business cycle. This yields high and similar cyclical volatilities for the unemployment, job finding and separation rates across age groups. In particular, the $u$ volatilities for the young and the prime-aged are 0.139 and 0.141, the volatilities of $f$ for young and prime-aged workers are 0.099 and 0.096; and the volatilities of $s$ are 0.059 for young workers and 0.063 for prime-aged workers. We return to this point in the next section when presenting the calibration details of the excess mobility model.

Unemployment Decomposition - Full model

1![All Workers]
[Young Workers]
[Prime-aged Workers]

§.§ Excess Mobility and Cyclical Unemployment

To show the importance of idiosyncratic occupation-worker ($z$) productivity shocks in allowing the full model to replicate the cyclical behavior of many labor market variables, we re-estimate the model by shutting down occupation-wide heterogeneity (level and business cycle loadings), effectively setting $p_{o,t}=1$ at all $t$. In this case, a worker's productivity at time $t$ in an occupation $o$ is completely described by aggregate productivity $A$, worker-occupation match productivity $z$ and occupation-specific human capital $x$. Workers do not (and do not want to) prefer a new occupation over another before knowing their $z$. Note that although we label this model as the “excess mobility model”, it can easily be made consistent with the observed average net flows by imposing an exogenous transition matrix that governs the probabilities with which a worker in occupation $o$ observes a $z$ in a different occupation $o'$.[With a cyclically varying exogenous transition matrix, we would also be able to match the observed cyclical net flows.] This is in contrast to our full model, where occupational productivities $p_o, p_{o'}$ differ and change relative to one another over the cycle and in response workers change the direction of their cross-occupation search. As such the full model can be considered as the “endogenous net mobility” model, while the excess mobility model as the “exogenous net mobility” model.

§.§.§ Benchmark Excess Mobility Model

This version of the model corresponds to the excess mobility model in the main text. Except for occupation-wide productivity differences and a cross-occupation search decisions, everything else remains as described in Section 3 of the main text. We use the same functional forms as done to calibrate the full model in Section 4 of the main text. This implies that to capture economic choices and gross mobility outcomes, we now have a set of 14 parameters to recover, where $\left[c, \rho_z, \sigma_z, \underline{z}_{norm} \right]$ governs occupational mobility due to idiosyncratic reasons (excess mobility); $\left[x^2, x^3, \gamma _{d},\delta_{L}, \delta_{H} \right]$ governs differences in occupational human capital; and the remainder parameters $\left[ k, b, \eta, \rho_A, \sigma_A \right]$ are shared with standard DMP calibrations. We jointly calibrate these parameters by matching the moments reported in Table <ref> and Figures <ref>-<ref>.

Targeted Moments, Excess Mobility Model

Moment          Model     Data     Moment          Model         Data
(lr)1-3 (lr)4-6
Agg. output per worker mean     1.005     1.000     Rel. separation rate young/prime-aged     2.134     1.994

Agg. output per worker persistence, $\rho_{outpw}$     0.761     0.753      Rel. separation rate recent hire/all      5.230     4.944

Agg. output per worker st. dev., $\sigma_{outpw}$     0.0094     0.0094       Prob (unemp. within 3 yr for empl.)     0.148      0.122

Mean unemployment     0.0355     0.0355     Empirical elasticity matching function     0.522     0.500

Average u. duration movers/stayers     1.211     1.139     5-year OLS return to occ. tenure     0.150     0.154

Repeat mobility: occ. stay after stay     0.600     0.634       10-year OLS return to occ. tenure     0.230      0.232

Occ. mobility young/prime-aged     1.162     1.163

Occ. mobility-duration profiles     2cFig 5a,b,c     U. survival profiles     2cFig 5d,e

Excess Mobility Model: Data and Model Comparison
[Mobility-Duration Profile (All)]

[Mobility-Duration Young-Prime]

[Cyclical Mobility-Duration Profile]

1.0![Survival Profile (All)]

[Survival Profile Young-Prime]

[Survival Profile Movers-Stayers]

The excess mobility model matches very well the targeted occupational mobility moments as well as the job finding and job separation moments. The fit is comparable with the one of the full model. This model also matches well the untargeted moments pertaining to workers' gross occupational mobility and job finding hazards discussed in the previous section. The fit of other untargeted moments is not shown here to save space, but available upon request. The estimated parameter values in this calibration are also very similar to the ones obtained in the full model. These are $c=7.549$, $k=125.733$, $b=0.843$, $\eta=0.241$, $\delta_{L}=0.0034$, $\delta_{H}=0.0004$, $\underline{z}_{corr}=0.349$, $\rho_{A}=0.998$, $\sigma_{A}=0.00198$, $\rho_{z}=0.998$, $\sigma_{z}=0.00707$, $x^{2}=1.181$, $x^{3}=1.474$ and $\gamma_{h}=0.0039$.

As shown in the main text, the excess mobility calibration is also able to fit a wide range of cyclical features of the labor market. The left panel of Table <ref> (below) shows the time series properties of the unemployment, vacancy, job finding and job separation rates and labor market tightness as well as the full set of correlations between them, obtained from the excess mobility calibration. Here we find that the cyclical implications of the excess mobility model are very similar to that of the full model, as shown and discussed in the main text.

Unemployment Decomposition - Excess Mobility Model

1![All workers]
[Young Workers]
[Prime-aged Workers]

(Un)Employment Distributions and Aggregate Productivity by Age Groups

0.75![Young Unemployed]
[Young Employed]

0.75![Prime-Aged Unemployed]
[Prime-Aged Employed]

Age patterns Figure <ref> shows that the occurrence of search, rest and reallocation unemployment episodes not only happens when pooling all workers together but also for each age group, as in the full model. Figure <ref> shows these age group dynamics more clearly by depicting the distribution of unemployed and employed workers among young and prime-aged workers. It shows that during recessions unemployment among young workers is concentrated slightly above $z^s(.,x^{1})$ and between $z^s(.,x^{1})$ and $z^r(.,x^{1})$. During expansions, however, unemployment is located above the $z^s(.,x^{1})$ cutoff. In the case of prime-aged workers, the concentration of unemployment during recessions and expansions occurs mostly above $z^s(.,x^{3})$ and between $z^s(.,x^{3})$ and $z^r(.,x^{3})$, but also between the $z^s(.,x^{1})$ and $z^r(.,x^{1})$ cutoffs. This difference implies that during expansion episodes of rest unemployment are still prevalent among prime-aged workers, while for young workers these episodes basically disappear (as shown in Figure <ref>) and are consistent with a lower occupational mobility rate among prime-aged workers.

As in the full model, the excess mobility calibration obtains a similar cyclicality for the unemployment, job finding and separations rates across age-groups. In both models this occurs because the estimated $z$-productivity process places enough workers on the $z^s$ cutoffs across the respective human capital levels. Figure <ref> shows that, as is the case for low human capital workers, many high human capital workers enjoy high $z$-productivities, but for a number of them their $z$-productivities have drifted down, positioning themselves close to $z^s(.,x^3)$. Some of these high human capital workers will subsequently leave the occupation, but over time the stock of workers close to $z^s(.,x^3)$ will be replenished by those workers who currently have high $z$-productivities but will suffer bad $z$-realizations in the near future. As $z^s(.,x^3)<z^s(.,x^1)$ the average level of separations is lower for high human capital workers, but this nevertheless does not preclude the similarity in the aforementioned cyclical responsiveness.

Given that it is clear the excess mobility model is able to replicate on its own many critical features of the full model, in what follows we use it to perform two key exercises. The first one highlights the effect of human capital depreciation in attenuating the cyclical properties of the above labor market variables and motivates our use of the cyclical shift of the mobility-duration profile as a target. The second exercise investigates the quantitative implications of our model when considering that a worker's varying job finding prospects (due to the stochastic nature of the $z$-productivity process) during a jobless spell can be linked to observed transitions between the states of unemployment and non-participation (or marginally attached to the labor force) as defined in the SIPP. For this exercise we recompute all of the relevant empirical targets using non-employment spells that contain a mix of periods of unemployment and non-participation. To save space we refer the reader to our working paper Carrillo-Tudela and Visschers (2020) for the list of targeted moments, the fit and the estimated parameter values under both exercises.

§.§.§ The Importance of Human Capital Depreciation

To estimate the full and the excess mobility models we used the mobility-duration profiles at different durations during recessions and expansions. These patterns informed us about the rate of occupational human capital depreciation during spells of unemployment. In the main text, we argued that these profiles were crucial in helping us identify the depreciation parameter, $\gamma_{h}$. The reason for the latter is that a model which did not incorporate human capital depreciation will generate very similar long-run moments as a model which did incorporate depreciation, but generate different cyclical predictions. To show this, we now discuss the result of the excess mobility model without human capital depreciation. To estimate the latter we target the same long-run moments as in the calibration described above, but do not target the cyclical behaviour of the mobility-duration profile.

We find that the fit is very good, similar to the models which incorporates human capital depreciation. It also does well in matching the same untargeted long-run moments described above. The estimated parameter values are also similar. Further, this calibration finds that periods of search, rest and reallocation unemployment can arise during a worker's jobless spell across all levels of occupational human capital.

The Cyclical Mobility-Duration Profile in Relation to Human Capital Depreciation

0.80![Model with HC Depreciation]
[Model without HC Depreciation]

Figure <ref> shows the first key difference between the excess mobility model with and without occupational human capital depreciation. We plot the mobility-duration profile in times of expansions and recessions (low and high unemployment, respectively), where Figure <ref> shows the mobility-duration profiles from the model with human capital depreciation and Figure <ref> shows the ones for the model without human capital depreciation. The latter finds that the lack of human capital depreciation implies the model completely misses the profile at nearly all durations during recessions, particularly at low durations. Is precisely this lack of fit that motivated us to add the cyclical patterns of the mobility-duration profile as targets in order to help identify the rate of human capital depreciation.

Excess Mobility Models: HP-filtered Business Cycle Statistics

6|c||Excess Mobility Model with HC dep.      6c|Excess Mobility Model with No HC dep.
$u$     $v$     $\theta$     $s$     $f$     $outpw$     $u$     $v$     $\theta$     $s$     $f$     $outpw$
$\sigma$       0.14     0.05     0.17     0.06      0.10    0.01     0.20     0.06     0.25     0.10     0.14    0.01

$\rho_{t-1}$     0.95     0.91     0.94     0.89     0.94     0.94     0.94     0.87     0.94     0.88     0.93     0.94
6|c||Correlation Matrix     6c|Correlation Matrix
$u$     1.00     -0.72     -0.98     0.80     -0.89      -0.95      1.00     -0.62     -0.97     0.78     -0.89     -0.92

$v$          1.00      0.84     -0.81     0.94     0.85          1.00      0.76     -0.61     0.77     0.72

$\theta$               1.00     -0.84     0.96     0.97                  1.00     -0.78     0.94     0.93

$s$                       1.00     -0.87    -0.90                       1.00     -0.81     -0.84

$f$                               1.00      0.95                                1.00      0.89

$outpw$                             1.00                             1.00
13p0.85Note: Each model's aggregate time series arise from the distributions of employed and unemployed workers across all labor markets, combined with agents' decisions. Times series are centered 5Q-MA series of quarterly data to smooth out the discreteness in the relatively flat cutoffs (relative to the grid). The cyclical components of the logged time series are obtained by using an HP filter with parameter 1600.

Table <ref> shows the second key difference. The model without depreciation generates a larger amount of cyclical volatility in the aggregate unemployment, job finding and job separation rates in relation to the model with human capital depreciation. Relative to the data, Table <ref> shows an overshooting in the volatilities of the unemployment and job finding rates. This occurs as the area of inaction between $z^r$ and $z^s$ becomes too cyclical. Without human capital depreciation there is a sharp drop in the proportion of rest unemployment and a sharper rise in the proportions of search and reallocation unemployment as the economy improves. Human capital depreciation attenuates these effects, improving the model's fit.

§.§.§ The Unemployed and Marginally Attached

Previous calibrations build the analysis based on the interpretation that, although a worker who is currently in rest unemployment cannot find a job, he would want to search for jobs (as opposed to stay idle at home) because he still faces a positive expected job finding probability in the near future. Episodes of rest unemployment, however, could conceptually be extended to incorporate marginally attached workers. To investigate the latter we expand our analysis to capture more broadly the occupational mobility decisions of the unemployed and marginally attached in shaping the cyclicality of aggregate unemployment.

We do this by re-estimating the excess mobility model, recomputing the targets using non-employment spells in which workers transition between unemployment and non-participation as labelled in the SIPP. We consider non-employment spells with at least one period of unemployment, which we label `NUN' spells. To avoid maternity and related issues in non-participation we restrict the focus to men. We show that when considering NUN spells the model still reproduces the observed cyclical amplification in the non-employment rate.[Here we also focus on spells of at least one month and workers who say that they are “without a job”, mirroring these sample restrictions for unemployment spells.]

The fit is once again very good in this case even though the survival probability in NUN spells shifts up significantly at longer durations, compared to the corresponding patterns for unemployment spells, both for all workers and across age groups. The estimated parameter values are also broadly similar to the ones in the previous versions of the excess mobility model, changing in expected directions. The $z$ process is now somewhat more volatile, but the higher reallocation cost implies that the area of inaction between $z^s$ and $z^r$ is (in relative terms) also larger. The latter leaves more scope for workers to get “trapped” for longer periods in rest unemployment episodes, thus creating an increase in the survival functions across all, young and prime-aged workers as observed in the data.

NUN Spells: HP-filtered Business Cycle Statistics

6|c||Data (1983-2014) - NUN spells      6c|Excess Mobility Model - NUN spells
$u$     $v$     $\theta$     $s$     $f$     $outpw$     $u$     $v$     $\theta$     $s$     $f$     $outpw$
$\sigma$       0.11    	0.11	    0.21    0.10    0.08    0.01     0.09    	0.03	    0.11    	0.04    	0.07    	0.01

$\rho_{t-1}$     0.97    0.99	    0.94    	0.99	    0.94    	0.93	    0.95    0.85	    0.94    	0.87	    0.93	    0.94
6|c||Correlation Matrix     6c|Correlation Matrix
$u$     1.00    -0.92	    -0.98	    0.79    	-0.96	    -0.51      1.00     -0.51    	-0.96	    0.65    	-0.81    	-0.89

$v$          1.00      0.98    	-0.79    	0.94	    0.61          1.00      0.72    	-0.50    	0.75	    0.65

$\theta$               1.00     -0.80    	0.96	    0.57                 1.00     -0.68    	0.89	    0.92

$s$                       1.00     -0.87    	-0.43                       1.00     -0.64	    -0.80

$f$                               1.00      0.46                               1.00      0.86

$outpw$                             1.00                             1.00
13p0.85See description under Table <ref> for details.

Table <ref> shows the main takeaway of this exercise. The model remains able to generate cyclical movements of the non-employment, job finding and job separation rates as well as a relatively strong Beveridge curve. In particular, the cyclical volatilities of the non-employment and job finding rates are the same as in the data. As in the previous estimations, here we also find that the reason for the amplification of the non-employment rate is that the model generates period of search, rest and reallocation unemployment, whose relative importance changes over the cycle.

Including marginally attached workers in our analysis increases the overall importance of rest unemployment in normal times. This is consistent with the fact that in these times the non-employment rate is higher and the associated job finding rate lower, compared to our benchmark unemployment and job finding rates measures. Further, this version of the model still needs to accommodate short-term outflows as before and does so mostly through search unemployment episodes. As a result, the proportion of rest unemployment decreases at a slower rate with $A$. Even at the highest aggregate productivity levels rest unemployment is very prevalent, representing about 40% of all episodes during a non-employment spell, with a large role for prime-aged workers. Overall we find that a version of the excess mobility model that considers NUN spells exhibits a higher non-employment rate but a lower cyclicality than in our benchmark model. This is consistent with the data, where we observe a lower cyclicality among the non-employment than among the unemployment.

§.§ The Importance of Occupational Mobility

To demonstrate the importance of including the $z^r$ cutoff to simultaneously replicate the cyclical behaviour of the unemployment duration distribution and the aggregate unemployment rate, we re-estimate the model by shutting down occupational mobility. This is done by exogenously setting $c$ to a prohibiting level. We present two calibrations. Model I targets the same moments as in the full model with the exception of those pertaining to occupational mobility.[To make the estimation of Model I as comparable as possible with the previous ones, we continue targeting the returns to occupational mobility to inform the human capital levels, $x^{2}$ and $x^{3}$. Under no occupational mobility, human capital could also be interpreted as general and not occupation specific, depending on the aim of the exercise. In this sense it would be more appropriate to target the returns to general experience. However, a comparison between the OLS returns to general experience and the OLS returns to occupational human capital estimated by Kambourov and Manovskii (2009) from the PSID (see their Table 3 comparing columns 1 and 3 or 6 and 8), suggests that this bias should be moderate. Using their estimates, the 5 year return to general experience is about 0.19, while the 10 years returns is about 0.38.] Model 2 is chosen to achieve a higher cyclical volatility in the aggregate unemployment rate but is more permissive of deviations from the targets.

Model I To save space our working paper, Carrillo-Tudela and Visschers (2020), shows the table of targeted moments and the fit as well as the estimated parameter values. The fit is largely comparable along nearly all corresponding dimensions to the full and excess mobility models. The estimated parameters are largely sensible. Two key features stand out: (i) there is now a higher role for search frictions as the model estimates a higher value $k=195.58$; and (ii) the $z$ process is now less persistent $\rho_{z}=0.9923$ and exhibits a much larger variance $\sigma_{z}=0.0300$ in the stationary distribution, generating a perhaps too large $Mm$ ratio of 2.28.[In this context the $z$-productivity process can be interpreted as an idiosyncratic productivity shock affecting a worker's overall productivity, rather than a worker's idiosyncratic productivity within an occupation.] Table <ref> under “No Occupational Mobility - Model I”, however, shows that this model cannot generate enough cyclical volatility on all the relevant labor market variables.

Models Without Reallocation: HP-filtered Business Cycle Statistics

6|c||No Occupational Mobility - Model I      6c|No Occupational Mobility - Model II
$u$     $v$     $\theta$     $s$     $f$     $outpw$     $u$     $v$     $\theta$     $s$     $f$     $outpw$

$\sigma$     0.04       0.02       0.06       0.03       0.03       0.01      0.10     0.03      0.12     0.07     0.05     0.01

$\rho_{t-1}$     0.94       0.85       0.93       0.83       0.87       0.94     0.94     0.85     0.94     0.89      0.89     0.94
6|c||Correlation Matrix     6c|Correlation Matrix
$u$     1.00      -0.42      -0.93       0.71      -0.77      -0.86     1.000     -0.57       -0.98        0.83       -0.79       -0.96

$v$          1.000      0.72      -0.26       0.58       0.60.          1.00      0.73       -0.60       0.72        0.66

$\theta$               1.00     -0.65       0.82       0.90               1.00     -0.85      0.84        0.97

$s$                       1.00     -0.52       -0.71                       1.00     -0.85       -0.89

$f$                               1.00     0.74                               1.00      0.84

$outpw$                             1.00                             1.00
13p0.89See description under Table <ref> for details.

Incomplete Unemployment Duration Distribution Behavior

2c     10cPanel A: Incomplete Unemployment Distribution (1-18 months)    1c
4cAll workers     4cYoung workers     4cPrime-aged workers

(lr)2-5 (lr)6-9 (lr)10-13
Unemp.      Occ     No Occ.    No Occ.     Data      Occ.     No Occ.     No Occ.       Data           Occ.     No Occ.     No Occ.     Data

Duration       Model      Model I     Model II              Model     Model I     Model II             Model     Model I     Model II
(lr)2-5 (lr)6-9 (lr)10-13
1-2 m      0.43     0.38         0.36          0.43      0.53     0.45     0.36     0.47     0.41     0.35         0.36          0.41

1-4 m      0.65     0.58     0.56      0.67      0.76     0.67     0.55       0.71     0.63     0.56      0.56       0.65

5-8 m      0.20     0.22     0.22      0.20      0.16     0.19     0.22       0.19      0.21     0.23      0.22     0.21

9-12 m     0.09     0.11     0.12     0.08      0.05     0.08     0.12     0.07       0.10    0.12     0.12    0.09

13-18m      0.06     0.09     0.10     0.05     0.03     0.06     0.11      0.03     0.07     0.09     0.09     0.06
2c     10cPanel B: Cyclical Changes of the Incomplete Unemployment Distribution (1-18 months)    1c
2c     4cElasticity wrt $u$     4cHP-filtered Semi-elasticity wrt $u$     2c
(lr)4-7 (lr)8-11
Unemp.             Occ     No Occ.    No Occ.     Data     Occ     No Occ.    No Occ.     Data      2c

Duration             Model     Model I     Model II          Model     Model I     Model II       2c
(lr)4-7 (lr)8-11
1-2 m               -0.450    1c-0.328     1c-0.265     -0.464     -0.155     1c-0.122     1c-0.099     -0.169     2c

1-4 m             -0.321     1c-0.241     1c-0.186      -0.363     -0.168     1c-0.134     1c-0.107      -0.184     2c

5-8 m               0.414     1c0.148      1c0.119     0.320     0.067     1c0.046      1c0.040      0.076      2c

9-12 m              1.102     1c0.422      1c0.300      0.864     0.058     1c0.049       1c0.039      0.072      2c

$>$13 m                  1.817     1c0.752     1c0.504     1.375     0.044     1c0.039    1c0.029     0.043     2c

Table <ref> further shows that Model I is not able to reproduce the observed average quarterly unemployment duration distribution at short or long durations (Panel A), nor does it capture the cyclical behavior of this distribution (Panel B). While Model I and the occupational mobility models replicate the same unemployment survival functions, they generate different incomplete duration distributions. This occurs because the survival functions are computed pooling the entire sample, while the distribution of incomplete spells between 1 and 18 months is calculated for each quarter and then averaged across quarters. Model I generates about 50% more long-term unemployment (9-12 months) relative to the data and the discrepancy is even stronger, about 80%, at longer durations. That is, this model matches the unemployment survival functions by creating too dispersed unemployment durations within a typical quarter, in particular too many long spells, but its distribution then responds too little to the cycle.

Model II If one is willing to compromise on replicating the aggregate and age-group survival functions, however, the model without occupational mobility is able to generate larger cyclical volatilities. To show this we re-estimated the model by de-emphasising these survival functions. Once again we refer to Carrillo-Tudela and Visschers (2020) for the fit of this model and the estimated parameter values. There we show that the unemployment survival functions of young and prime-aged workers are no longer well matched. This model misses the distribution at longer durations for young workers and at shorter durations for prime-aged workers, such that age differences in job finding hazards have nearly disappeared. However, there is now a lower degree of search frictions, $k=102.019$ and the $z$ process is more persistent, $\rho_{z}=0.993$, and its overall dispersion is much lower than in Model I, $\sigma_{z}=0.0132$. It is these last properties that allow Model II to create more cyclical volatility as shown in the right panel of Table <ref>.

Panel A of Table <ref> - No Occ. Model II shows that this model still creates too much long-term unemployment (13-18 months) in the average quarter, where the proportion of long-term unemployed (among those with spells between 1-18 months) is missed by a large margin for all, young and prime-aged workers. Further, Panel B of Table <ref> shows that this feature is also reflected in a muted cyclical response of the unemployment duration distribution. Here we also find that Model II misses the semi-elasticity with respect to the unemployment rate by an average of about 40% across the duration distribution.[In these versions of the model the behavior of spells with durations beyond 18 months might also impact the overall unemployment rate more than empirically warranted, especially as persistence can create a “first-in last-out pattern” when entering a recession in the form of very long unemployment spells for those who lost their job early on in the recession. We focus on the distribution of spells up to 18 months because censoring issues in the SIPP restrict how accurately we can investigate the behavior of very long spells over the cycle.]

Discussion These calibrations show that without the $z^r$ cutoff versions of our model with no occupational mobility cannot resolve the tension between individual unemployment outcomes and aggregate unemployment volatility. This arises as without the possibility of occupational mobility the new area of inaction is defined by the set of $z \in[\underline{z},z^s]$, where $\underline{z}$ is the lowest value of $z$, and its cyclical response now solely depends on $\partial z^s / \partial A$.

In the case of Model I, a less persistent and much more volatile $z$ process creates enough heterogeneity in unemployment durations that allows it to match the empirical unemployment survival functions at the aggregate and across age groups. However, it also increases the heterogeneity in $z$-productivities relative to the cyclical range of $A$. This dampens the model's cyclical performance as it implies less responsive $z^s$ cutoffs relative to the workers' $z$ distribution, weakening the cyclical responses of job separations and the rate at which workers leave the area of inaction. Moreover, with a larger vacancy posting cost, Figure <ref>.a shows that search unemployment is now more prominent than rest unemployment at any point of the cycle. Larger search frictions imply larger surpluses and therefore further reducing the cyclical responsiveness of the model.

Unemployment Decomposition - Models Without Occupational Mobility

0.85![All Workers - Model I]

[All Workers - Model II]

The increased cyclical performance of Model II arises as its estimated $z$ process becomes more persistent and less volatile, creating more responsive $z^s$ cutoffs leading to stronger cyclical responses in job separations, as well as much more episodes of rest unemployment over all $A$. Figure <ref>.b shows that rest unemployment episodes are now more prominent than search unemployment episodes even during economic recoveries. It is only for the highest values of $A$ that search unemployment is more prominent, but only by a relatively small margin. With more responsive $z^s$ cutoffs, an aggregate shock can now move somewhat larger masses of workers from rest into search unemployment episodes creating more amplification. This result is in line with Chassamboulli (2013), who extends the DMP model by adding permanent productivity differences among workers and shows that this feature increases it cyclical performance relative to the data. Table <ref> demonstrates, however, that this comes at the cost of not being able to match the distribution of unemployment durations nor the dynamic behavior of this distribution over the cycle.[An extrapolation from the above discussion appears to suggest a model with permanent heterogeneity, where all moves in and out of rest unemployment would be because of aggregate productivity changes, cannot resolve the tension between cyclical performance and fitting the unemployment duration distribution moments. In principle our estimation allows and has evaluated in its procedure parameter tuples with a near-permanent $z$-productivity process (i.e. a persistence approximating $1$). However, such parameter tuples yield larger deviations from the individual-level unemployment outcomes we targeted compared to Model II.] Thus, in Model II the average unemployment rate responds more to the cycle as aggregate productivity takes on a more prominent role in shaping the amount of rest unemployment, but the individual unemployment outcomes that underlie these dynamics become counterfactual.

The possibility of occupational mobility would have given workers in rest unemployment episodes another margin through which they can escape the area of inaction and get re-employed faster. As a result, the individual-level unemployment duration dependence (given an aggregate state) is not only affected by $z^s$ but also by the distance to $z^r$. Further, since the $z^r$ cutoff is at different distances from the $z^s$ across the cycle, it creates a more cyclically sensitive area of inaction. That is, occupational mobility creates a more responsive area of inaction with respect to both worker heterogeneity and the business cycle that resolves the tensions discussed above.

[29] Carrillo-Tudela, Carlos and Ludo Visschers (2020): “Unemployment and Endogenous Separations over the Business Cycle”. IZA DP No. 13307, Germany.

[30] Chassamboulli, Andri (2013): “Labor-Market Volatility in a Matching Model with Worker Heterogeneity and Endogenous Separations”. Labour Economics, 24: 217-229.

[31] Kambourov, Gueorgui and Iourii Manovskii. (2009): “Occupational Specificity of Human Capital”. International Economic Review, 50(1): 63-115.

Supplementary Appendices

Supplementary Appendix A: Occupational Coding Error and its Correction

This appendix complements Section 2 of the paper and expands Online Appendix A. Here we quantify the extent to which coding errors affect the probability of an occupational change after a non-employment spell. This is done in two ways. (1) We exploit the change from independent to dependent interviewing that occurred across the 1985 and 1986 SIPP panels, which overlap between February 1986 to April 1987. The change of interviewing technique allows us to estimate a garbling matrix $\mathbf{\Gamma}$ whose elements denote the probability that a worker's true occupation $i$ gets miscoded as another occupation $j$. (2) We then take advantage of the retrospective coding exercise done to occupational codes in the PSID. Retrospective coding improved the assignment of the occupational codes obtained during the 1970s, but did not affect the codes obtained during later years (see also Kambourov and Manovskii, 2008). We use probabilistic models to estimate the average effect retrospective coding had on the probability of an occupational change.

Across these data sets we obtain a very similar conclusion. Our correction method implies that on average 82% of the observed wave-to-wave  occupational transitions after re-employment in the SIPP are genuine, were each wave covers a 4 months interval. In stark contrast, when pooling together workers who changed and did not change employers, we find that 40% of the observed changes among all workers are genuine. Similarly, retrospective coding in the PSID implies that 84% of the observed year-to-year occupational transitions after re-employment are genuine. When pooling together workers who changed and did not change employers, retrospective coding implies that only 44% are genuine. A key insight is that the same propensities to miscode occupations affect very differently the measured occupational change of employer movers and employer stayers. This is important as it implies one cannot use the error correction estimates obtained from pooled samples to adjust the occupational mobility statistics of those who changed employers through spells of unemployment.

We use the estimated $\mathbf{\Gamma}$-correction matrix to eliminate the impact of coding errors on the occupational mobility statistics of the unemployed obtained from the SIPP. We show that occupational miscoding increases observed excess mobility and reduces the importance of net mobility (see also Kambourov and Manovskii, 2013). It also makes occupational mobility appear less responsive to unemployment duration and the business cycle.
In Supplementary Appendix B we use the SIPP to evaluate robustness of the occupational mobility patterns derived from our $\Gamma$-correction method. To do so, we follow two alternative approaches of measuring occupational mobility: (i) simultaneously mobility of major occupational and major industrial groups at re-employment and (ii) self-reported duration of occupational tenure obtained from the topical modules. The first measure is considered less sensitive to miscoding as it typically requires errors to be made simultaneously along two dimensions. The second captures the worker's own perception of occupational mobility and is not based on occupational coding. We find that the occupational mobility patterns obtained through our classification error model applied to the SIPP, the retrospective coding in the PSID and these alternative measures provide a very consistent picture.

§ A CLASSIFICATION ERROR MODEL

In Section 2.1 of the main text we defined the matrix $\mathbf{\Gamma}$, where its elements are the probabilities that an occupation $i$ is miscoded as an occupation $j$, for all $i,j=1,2,...,O$. There is an important literature that investigates classification error models. Magnac and Visser (1999) identify two main branches. The first one uses assumptions on the measurement error process and auxiliary data on the error rates, where it is assumed that the error rates can be directly observe from the auxiliary data (see Abowd and Zellner, 1985, Poterba and Summers, 1986, and Magnac and Visser, 1999, among others). The second one does not rely on auxiliary data on error rates, but estimates parametric models in the presence of misclassified data, where these models are either reduced form statistical models (see Hausman et al. 1998, among others) or structural economic models (see Kean and Wolpin, 2001, Sullivan, 2009, Roys and Taber, 2017, among others).

Our classification error model builds on Abowd and Zellner (1985) and Poterba and Summer (1986) in that it uses a garbling matrix that captures the errors made in classifying workers. These authors investigate the misclassification of individuals' employment status in the CPS and use reported employment status at the original interview date, at the re-interviewing date (which occurred one week after the original interview) and the reconciliation information provided by the CPS to directly observe the garbling matrix. Their key assumption is that the reconcile information provides the true individuals' labor market status. In contrast, we do not have auxiliary information that allows us to directly recover the garbling matrix. Our challenge is to estimate the garbling matrix.

Our classification error model also relates to Sullivan (2009) in that both approaches provide estimates of an occupation garbling matrix. In contrast, our approach does not rely on the observed optimal choices of economic agents for identification. Rather, miscoding can be estimated using the occupational transitions alone, without requiring further information on wages which themselves can be subject to measurement error. Therefore, it is much simpler to implement relative to Sullivan's (2009) approach which relies on time costly simulation maximum likelihood methods that end up restricting the size of the estimated garbling matrix. This is important as it reduces the overall computational burden when estimating our economic model.

To identify and estimate $\mathbf{\Gamma}$ we make three assumptions:

(A1) Independent classification errors: conditional on the true occupation, the realization of an occupational code does not depend on workers' labor market histories, demographic characteristics or the time it occurred in our sample. This assumption is also present in Poterba and Summers (1986) and Abowd and Zellner (1985) and is consistent with independent interviewing. In the standard practise of independent interviewing, professional coders base their coding on the verbatim description of the reported work activities without taking into account the respondents' demographic characteristics or earlier work history.[For example, during the 1980s and 1990s independent occupational coding in the PSID was done without reference to respondents' characteristics or their work history. However, this information was used in the retrospective coding exercise done to the 1970s occupational codes.] Errors introduced by the respondents, however, could be correlated with their characteristics. Assumption $\emph{A1}$ implies that errors in the individuals' verbatim responses are fully captured by the nature of the job these individuals are performing and hence only depend on their true occupation. Another implication of assumption $\emph{A1}$ is that $\mathbf{\Gamma}$ is time-invariant. This is important as we will apply our correction method across all years in our sample and one could be concern whether the coding errors estimated in the 1980's are similar to those founds 20 years later. In Section 4 (below) we investigate further these implications.

(A2) “Detailed balance” in miscoding: $\mathbf{diag(c)\Gamma}$ is symmetric, where $\mathbf{c}$ is a $O$x$1$ vector that describes the distribution of workers across occupations and $\mathbf{diag(c)}$ is the diagonal matrix of $\mathbf{c}$. This assumption is known as “detailed balance”. It implies that the number of workers whose true occupation $i$ gets mistakenly coded as $j$ is the same as the number of workers whose true occupation $j$ gets mistakenly coded as $i$, such that the overall size of occupations do not change with coding error. This assumption allow us to invert $\Gamma$ and hence aids our identification arguments. Although undeniably strong, this assumption is a weaker version of the one proposed by Keane and Wolpin (2001) and subsequently used Roys and Taber (2017) to correct of occupation miscoding. We return to discuss A2 in Section 4.

(A3) Strict diagonal dominance: $\mathbf{\Gamma}$ is strictly diagonally dominant in that $\gamma_{ii}>0.5$ for all $i=1,2,...,O$. This assumption is also present in Hausman et al., (1998) and implies that it is more likely to correctly code a given occupation $i$ than to miscoded it. The converse would imply occupational mobility rates that are of a magnitude inconsistent with our data. Indeed, we derive an upper bound on code error directly from the data and find that A3 is verified in our data.

To estimate $\mathbf{\Gamma}$ we exploit the change of survey design between the 1985 and 1986 SIPP panels. Until the 1985 panel the SIPP used independent interviewing for all workers: in each wave all workers were asked to describe their job anew, without reference to answers given at an earlier date. Subsequently, a coder would consider that wave's verbatim descriptions and allocate occupational codes. This practise is known to generate occupational coding errors. In the 1986 panel, instead, the practise changed to one of dependent interviewing (see Lynn and Sala, 2007, Jäckle, 2008, and Jäckle and Eckman, 2020). Respondents were only asked “independently” to describe their occupation if they reported a change in employer or if they reported a change in their main activities without an employer change within the last 8 months. If respondents declared no change in employer and in their main activities, the occupational code assigned to the respondent in the previous wave is carried forward.

It is important to note that during February 1986 to April 1987, the 1985 and 1986 panels overlap, representing the same population under different survey designs. The identification theory we develop in the next section refers to this population. We then show how to consistently estimate $\mathbf{\Gamma}$ using the population samples.

§.§ Identification of $\mathbf{\Gamma}$

Consider the population represented by 1985/86 panels during the overlapping period and divide it into two groups of individuals across consecutive interviews by whether or not they changed employer or activity. Label those workers who stayed with their employers in both interviews and did not change activity as “employer/activity stayers”. By design this group only contains true occupational stayers. Similarly, label those workers who changed employers or changed activity within their employers as “employer/activity changers”. By design this group contains all true occupational movers and the set of true occupational stayers who changed employers.

Suppose that we were to subject the employer/activity stayers in this population to dependent interviewing as applied in the 1986 panel. Let $\mathbf{c_{s}}$ denote the $O$x$1$ vector that describes their true distribution across occupations and let $\mathbf{M_{s}}=\mathbf{diag(c_{s})}$. In what follows we will use the convention that the $(i,j)$'th element of an $M$ matrix indicates the flow from occupation $i$ to $j$. Let $\mathbf{c^{D}_{s}}$ denote the $O$x$1$ vector that describes their observed distribution across occupations under dependent interviewing and let $\mathbf{M^{D}_{s}}=\mathbf{diag(c^{D}_{s})}$. Note that $\mathbf{c^{D}_{s}}=\mathbf{\Gamma^{'} M_{s} \overrightarrow{\bf{1}}}$, where $\overrightarrow{\bf{1}}$ describes a vector of ones. $\mathbf{M_{s}}$ is pre-multiplied by $\mathbf{\Gamma^{'}}$ as true occupations would have been miscoded in the first of the two consecutive interviews. “Overall balance”, a weaker version of A2, implies that $\mathbf{c^{D}_{s}}=\mathbf{diag(c_{s})\Gamma \overrightarrow{\bf{1}}}=\mathbf{c_{s}}$ and hence $\mathbf{M^{D}_{s}=M_{s}}$.[Overall balance only requires that classification errors do not change the overall occupational distribution rather than the bilateral flows between occupations as also required by detailed balance in A2.]

Next suppose that instead we were to subject the employer/activity stayers in this population to independent interviewing as applied in the 1985 panel. Let $\mathbf{M^{I}_{s}}$ denote the matrix that contains these workers' observed occupational transition flows under independent interviewing. In this case $\mathbf{M^{I}_{s}}=\mathbf{\Gamma^{\prime} M_{s} \Gamma}$. Here $\mathbf{M_{s}}$ is pre-multiplied by $\mathbf{\Gamma^{'}}$ and post-multiplied by $\mathbf{\Gamma}$ to take into account that the observed occupations of origin and destination would be subject to coding error.

Let $\mathbf{M_{m}}$ denote the matrix that contains the true occupational transition flows of employer/activity changers in this population. The diagonal of $\mathbf{M_{m}}$ describes the distribution of true occupational stayers across occupations among employer/activity changers. The off-diagonal elements contain the flows of all true occupational movers. Under independent interviewing we observe $\mathbf{M^{I}_{m}}=\mathbf{\Gamma^{\prime} M_{m} \Gamma}$. Once again $\mathbf{M_{m}}$ is pre-multiplied by $\mathbf{\Gamma^{'}}$ and post-multiplied by $\mathbf{\Gamma}$ as the observed occupations of origin and destination would be subject to coding error.

Letting $\mathbf{M^{I}}=\mathbf{M^{I}_{m}+M^{I}_{s}}$ denote the matrix that contains the aggregate occupational transition flows across two interview dates under independent interviewing, it follows that $\mathbf{M^{I}_{s}}=\mathbf{M^{I}-M^{I}_{m}}=\mathbf{\Gamma^{\prime} M_{s} \Gamma}$. By virtue of the symmetry of $\mathbf{M_{s}}$ and “detailed balance” (A2), $\mathbf{M_{s} \Gamma=\Gamma^{\prime} {M_{s}}^{\prime}=\Gamma^{\prime} M_{s}}$. Substituting back yields $\mathbf{M^{I}_{s}}=\mathbf{M_{s}\Gamma \Gamma}$. Next note that $\mathbf{M^{I}_{s}=M_{s}T^{I}_{s}}$, where $\mathbf{T^{I}_{s}}$ is the occupational transition probability matrix of the employer/activity stayers in this population observed under independent interviewing. Substitution yields $\mathbf{M_{s} T^{I}_{s}}=\mathbf{M_{s} \Gamma \Gamma}$. Multiply both sides by $\mathbf{M_{s}^{-1}}$, which exists as long as all the diagonal elements of $\mathbf{M_{s}}$ are non-zero, yields the key relationship we exploit to estimate $\mathbf{\Gamma}$,

\begin{equation}\label{eq:est_gamma_2}
\mathbf{T^{I}_{s}}=\mathbf{\Gamma \Gamma}.
\end{equation}

To use this equation we first need to show that it implies a unique solution for $\mathbf{\Gamma}$. Towards this result, we now establish that $\mathbf{\Gamma}$ and $\mathbf{T^{I}_{s}}$ are diagonalizable. For the latter it is useful to interpret the coding error process described above as a Markov chain such that $\mathbf{\Gamma}$ is the one-step probability matrix associated with this process.

Lemma A.1: Assumptions A2 and A3 imply that $\mathbf{\Gamma}$ and $\mathbf{T^{I}_{s}}$ are diagonalizable.

First note that without loss of generality we can consider the one-step probability matrix $\mathbf{\Gamma}$ to be irreducible. To show this suppose that $\mathbf{\Gamma}$ was not irreducible, we can (without loss of generality) apply a permutation matrix to re-order occupations in $\mathbf{\Gamma}$ and create a block-diagonal $\mathbf{\Gamma'}$, where each block is irreducible and can be considered in isolation. Given A3, it follows directly that $\mathbf{\Gamma}$ is aperiodic. Further, assumption A2 implies that $\mathbf{c_{s}}$ is a stationary distribution of $\mathbf{\Gamma}$. The fundamental theorem of Markov chains then implies that $\mathbf{c_{s}}$ is the unique stationary distribution of $\mathbf{\Gamma}$. Assumption A2 also implies that the Markov chain characterised by $\mathbf{\Gamma}$ is reversible with respect to $\mathbf{c_s}$. This means that $\mathbf{\Gamma}$ is similar (in matrix sense) to a symmetric matrix $\mathbf{G}$ such that $\mathbf{G}=\mathbf{diag(\sqrt{\mathbf{c_s}})} \ \mathbf{\Gamma} \ \mathbf{diag(\sqrt{\mathbf{c_s}})^{-1}}$. By symmetry, $\mathbf{G}$ is orthogonally diagonalizable by $\mathbf{Q \Delta Q^{-1}}$, where the diagonal matrix $\mathbf{\Delta}$ contains the associated (real) eigenvalues and $\mathbf{Q}$ is the orthogonal matrix of associated (normalized) eigenvectors. It then follows that $\mathbf{\Gamma}$ is diagonalizable as well. Further, $\mathbf{T^{I}_{s}}= \mathbf{diag(\sqrt{\mathbf{c_s}})^{-1}\  G \ G \ diag(\sqrt{\mathbf{c_s}})}=\mathbf{diag(\sqrt{\mathbf{c_s}})^{-1}\ Q \ \Delta^2 \ Q^{-1} \ diag(\sqrt{\mathbf{c_s}})}$, and hence $\mathbf{T^{I}_{s}}$ is also orthogonally diagonalizable, with a root of $\mathbf{P \Lambda^{0.5} P^{-1}}$, where $\mathbf{\Lambda}$ is the diagonal matrix of eigenvalues of $\mathbf{T^{I}_{s}}$, and $\mathbf{P}$ the associated orthogonal matrix with eigenvectors of $\mathbf{T^{I}_{s}}$.

In general one cannot guarantee the uniqueness, or even existence, of a transition matrix that is the $(nth)$ root of another transition matrix (see Higham and Lin, 2011). Here, however, existence is obtained by construction: $\mathbf{T_s}$ is constructed from $\mathbf{\Gamma}$, and in reverse, we can find its roots. The next result shows that $\mathbf{T_s}$ has a unique root satisfying assumptions A2 and A3.

Proposition A.1: $\mathbf{\Gamma}$ is the unique solution to $\mathbf{T^{I}_{s}=\Gamma \ \Gamma}$ that satisfies assumptions A2 and A3. It is given by $\mathbf{P \Lambda^{0.5} P^{-1}}$, where $\mathbf{\Lambda}$ is the diagonal matrix with eigenvalues of $\mathbf{T^{I}_{s}}$, $0<\lambda_i\leq 1$, and $\mathbf{P}$ is the orthogonal matrix with the associated (normalized) eigenvectors.

Following from the proof of Lemma A.1, a root of $\mathbf{T^{I}_{s}}$ is given by $\mathbf{P \Lambda^{0.5} P^{-1}}$, where $\mathbf{\Lambda}$ is the diagonal matrix with eigenvalues of $\mathbf{T^{I}_{s}}$ and $\mathbf{P}$ is the orthogonal matrix with the associated (normalized) eigenvectors. Since A3 implies $\mathbf{\Gamma}$ is strictly diagonally dominant, it follows that the determinant of all its leading principal minors are positive. Moreover, under the similarity transform by pre-/post-multiplication with the diagonal matrices $\mathbf{diag(\sqrt{c_s}), diag(\sqrt{c_s})^{-1}}$, the determinant of all principals minors of the symmetric matrix $\mathbf{G}=\mathbf{diag(\sqrt{\mathbf{c_s}})} \ \mathbf{\Gamma} \ \mathbf{diag(\sqrt{\mathbf{c_s}})^{-1}}$ are positive as well. Hence $\mathbf{G}$ is a symmetric positive definite matrix (with all eigenvalues between $0$ and $1$, as has $\mathbf{\Gamma}$). It follows that $\mathbf{G \ G}=\mathbf{S}$ is also positive definite, and $\mathbf{T^{I}_{s}}=\mathbf{diag(\sqrt{\mathbf{c_s}})^{-1} \ S \ diag(\sqrt{\mathbf{c_s}})}$ is positive definite in the sense that $\mathbf{v'\ T^{I}_{s}\ v}>0$ for all $\mathbf{v}\neq \mathbf{0}$, while also all eigenvalues of $\mathbf{T^{I}_{s}}$ will be between $0$ and $1$.

To show the uniqueness of the root of $\mathbf{T^{I}_{s}}$ suppose (towards a contradiction) that there exists two different roots $\mathbf{\Gamma}$ and $\mathbf{\Upsilon}$ such that each are similar (in matrix sense), with the same transform involving $\mathbf{diag(\sqrt{c_s})}$, to different symmetric positive definite matrices $\mathbf{G}$ and $\mathbf{Y}$, where $\mathbf{G\ G}= \mathbf{S}$ and $\mathbf{Y\ Y} = \mathbf{S}$. Both $\mathbf{G}$ and $\mathbf{Y}$ are diagonalizable, and have the square roots of the eigenvalues of $\mathbf{S}$ on the diagonal. Given that the squares of the eigenvalues need to coincide with the eigenvalues of $\mathbf{S}$ and assumptions A2 and A3 imply that all eigenvalues must be between $0$ and $1$, without loss of generality we can consider both diagonalizations to have the same diagonal matrix $\mathbf{\Delta}$, where $\mathbf{\Delta}$ is the diagonal matrix of eigenvalues of $\mathbf{T^{I}_{s}}$ and these eigenvalues are ordered using a permutation-similarity transform with the appropriate permutation matrices. Let $\mathbf{G}=\mathbf{H \Delta  H^{-1}}$ and $\mathbf{Y}=\mathbf{K \Delta K^{-1}}$. Then, it follows that $\mathbf{K^{-1}H \Delta^2 H^{-1} K = \Delta^2}$ and since $\mathbf{K^{-1} H}$ and $\mathbf{\Delta^2}$ commute, implies that $\mathbf{K^{-1} H}$ is a block-diagonal matrix with the size of the blocks corresponding to the multiplicity of squared eigenvalues. Again, since all eigenvalues of $\mathbf{\Delta}$ are positive, this equals the multiplicity of the eigenvalues $\delta_i$ itself. But then it must be true that $\mathbf{K^{-1} H \Delta H^{-1} K = \Delta}$. Then, $\mathbf{G}=\mathbf{H\ \Delta\  H^{-1}}=\mathbf{K K^{-1} H\ \Delta\  H^{-1} K K^{-1}}=\mathbf{K\ \Delta\  K^{-1}}=\mathbf{Y}$ which leads to a contradiction.

The above results imply that under assumptions A2 and A3, $\mathbf{\Gamma}$ is uniquely identified from the transition matrix of true occupational stayers under independent interviewing, $\mathbf{T^{I}_{s}}$.

§.§ Estimation of $\mathbf{\Gamma}$

The next lemma provides an intermediate step towards estimating $\mathbf{\Gamma}$. For this purpose let $PDT(.)$ denote the space of transition matrices that are similar, in the matrix sense, to positive definite matrices.

Lemma A.2: The function $f:PDT(\mathbb{R}^{O\times O})\to PDT(\mathbb{R}^{O\times O})$ given by $f(\mathbf{T})=\mathbf{T^{0.5}}$ exists and is continuous with $f(\mathbf{T^{I}_{s}})=\mathbf{\Gamma}$ in the spectral matrix norm.

Existence follows from Lemma A.1 and Proposition A.1. To establish continuity of the mapping, we follow Horn and Johnson (1990). Let $\mathbf{T_{1}}$ and $\mathbf{T_{2}}$ be any two transition matrices in $PDT$ and let $\mathbf{U_{1}}$ and $\mathbf{U_{2}}$ be two symmetric positive definite matrices constructed as $\mathbf{U_{1}}=\mathbf{diag(\sqrt{\mathbf{c_1}})\ T_{1} \ diag(\sqrt{\mathbf{c_1}})^{-1}}$ and $\mathbf{U_{2}}=\mathbf{diag(\sqrt{\mathbf{c_2}}) \ T_{2} \ diag(\sqrt{\mathbf{c_2}})^{-1}}$, where $\mathbf{c_{1}}$ and $\mathbf{c_{2}}$ are the unique stationary distributions associated with $\mathbf{T_{1}}$ and $\mathbf{T_{2}}$, respectively. We want to show that if $\mathbf{U_{1}}\to \mathbf{U_{2}}$, then $\mathbf{U_{1}^{0.5}}\to \mathbf{U_{2}^{0.5}}$. First, note that $\| \mathbf{U_{1}-U_{2}} \|_2=\| \mathbf{U_{1}^{0.5}(U_{1}^{0.5}-U_{2}^{0.5})}+\mathbf{(U_{1}^{0.5}-U_{2}^{0.5})U_{2}^{0.5}} \|_2\geq | \mathbf{x'U_{1}^{0.5}(U_{1}^{0.5}-U_{2}^{0.5})x}+\mathbf{x'(U_{1}^{0.5}-U_{2}^{0.5})U_{2}^{0.5}x }|$, where $\mathbf{x}$ is any normalised vector. Assumptions A2 and A3 imply $\mathbf{U_{1}^{0.5}-U_{2}^{0.5}}$ exists and is a symmetric matrix. Let $|\lambda|=\rho(\mathbf{U_{1}^{0.5}-U_{2}^{0.5}})$ be the absolute value of the largest eigenvalue of $\mathbf{U_{1}^{0.5}-U_{2}^{0.5}}$ and let $\mathbf{z}$ be the normalized eigenvector associated with $\lambda$. Note that $\|\mathbf{U_{1}-U_{2}}\|_2=|\lambda|$ and $(\mathbf{U_{1}^{0.5}-U_{2}^{0.5}})\mathbf{z}=\lambda\mathbf{z}$. Then $| \mathbf{z'U_{1}^{0.5}(U_{1}^{0.5}-U_{2}^{0.5})z}+\mathbf{z'(U_{1}^{0.5}-U_{2}^{0.5})U_{2}^{0.5}z }|\geq |\lambda| \ |\lambda_{min}^{0.5}(\mathbf{U_{1}})+\lambda_{min}^{0.5}(\mathbf{U_{2}})| = \|\mathbf{U_{1}^{0.5}-U_{2}^{0.5}}\|_2 \ (\lambda_{min}^{0.5}(\mathbf{U_{1}})+\lambda_{min}^{0.5}(\mathbf{U_{2}})) $, where $\lambda_{min}(\mathbf{U_{1}})$ denotes the smallest eigenvalue of $\mathbf{U_{2}}$, which is positive by virtue of assumptions A2 and A3. Then choose a $\delta=\varepsilon \lambda_{min}^{0.5}(\mathbf{U_{1}})$. It follows that if $\|\mathbf{U_{1}-U_{2}}\|_2<\delta$, then $\|\mathbf{U_{1}^{0.5}-U_{2}^{0.5}}\|_2 $ $\times$ $\ \frac{(\lambda_{min}^{0.5}(\mathbf{U_{1}})+\lambda_{min}^{0.5}(\mathbf{U_{2}}))}{\lambda_{min}^{0.5}(\mathbf{U_{1}})}$ $<\varepsilon$,  and therefore $\|\mathbf{U_{1}^{0.5}-U_{2}^{0.5}}\|_2<\varepsilon$, which establishes the desired continuity. From the fact that $\mathbf{U_{1} }\to \mathbf{U_{2}}$ implies $\mathbf{U_{1}^{0.5}}\to \mathbf{U_{2}^{0.5}}$, it then also follows that $f(\mathbf{T})$ is continuous.

Let $\mathbf{\hat{T}^{I}_{s}}$ denote the sample estimate of $\mathbf{T^{I}_{s}}$ and let $\mathbf{\hat{\Gamma}}$ be estimated by the root $\mathbf{(\hat{T}^{I}_{s})}^{0.5} \in PDT(\mathbb{R}^{O \times O})$ such that $\mathbf{\hat{\Gamma}}=\mathbf{(\hat{T}^{I}_{s})^{0.5}}=\mathbf{\hat{P} \hat{\Lambda}^{0.5} \hat{P}^{-1}}$, where $\mathbf{\hat{\Lambda}}$ is the diagonal matrix with eigenvalues of $\mathbf{\hat{T}^{I}_{s}}$, $0<\hat{\lambda_i}^{0.5}\leq 1$ and $\mathbf{\hat{P}}$ the orthogonal matrix with the associated (normalized) eigenvectors. We then have the following result.

Proposition A.2: $\mathbf{\Gamma}$ is consistently estimated from $\mathbf{(\hat{T}^{I}_{s})}^{0.5} \in PDT(\mathbb{R}^{O \times O})$ such that $\mathbf{\hat{\Gamma}}=\mathbf{(\hat{T}^{I}_{s})^{0.5}}=\mathbf{\hat{P} \hat{\Lambda}^{0.5} \hat{P}^{-1}}$. That is, $plim_{n \to \infty} \mathbf{\hat{\Gamma}}=\mathbf{\Gamma}$.

From Lemma A.1 and Proposition A.1 it follows that if we know $\mathbf{T^{I}_{s}}$, then we can find the unique $\mathbf{\Gamma}$ that underlies it, constructing it from the eigenvalues and eigenvectors of $\mathbf{T^{I}_{s}}$. To estimate $\mathbf{T^{I}_{s}}$ one can use the sample proportion $\mathbf{\hat{M}_{ij}}/\sum_{k=1}^O \mathbf{\hat{M}_{ik}}$ and note that this converges in probability to $\mathbf{(T^{I}_{s})_{ij}}$ (see Anderson and Goodman, 1957; Billingsley 1961, thm 1.1-3.) for all occupations, given assumptions A2 and A3. Hence, $plim_{n\to \infty} \mathbf{\hat{T}^{I}_{s}} = \mathbf{T^{I}_{s}}$. Then, we derive $\mathbf{\hat{\Gamma}}$ from $\mathbf{\hat{T^{I}_{s}}}$ according to $\mathbf{\hat{P}\hat{\Lambda}^{0.5}\hat{P}^{-1}}$, per Proposition A.1. By continuity of the mapping in Lemma A.2, it follows that $plim_{n\to \infty} \mathbf{\hat{\Gamma}} = \mathbf{\Gamma}$, and our estimator is consistent.

Note that to identify and estimate $\mathbf{\Gamma}$ in the SIPP it is not sufficient to directly compare the aggregate occupational transition flows under independent interviewing with the aggregate occupational transition flows under dependent interviewing. To show this let $\mathbf{M^{D}}=\mathbf{M^{I}_{m}+M^{D}_{s}}$ denote the matrix that contains the aggregate occupational transition flows across two interview dates under dependent interviewing for employer/activity stayers and under independent interviewing for employer/activity movers. Subtracting $\mathbf{M^{I}}=\mathbf{M^{I}_{m}+M^{I}_{s}}$ from $\mathbf{M^{D}}$ yields $\mathbf{M^{D}_{s}-M^{I}_{s}}=\mathbf{M_{s} -\Gamma' M_{s} \Gamma}$. Given the symmetry assumed in A2, the latter expression has $0.5n(n-1)$ exogenous variables on the LHS and $0.5n(n+1)$ unknowns (endogenous variables) on the RHS, leaving $\mathbf{\Gamma}$ (and $\mathbf{M_s}$) unidentified.

In addition to $\mathbf{M^{D}-M^{I}}=\mathbf{M_{s} -\Gamma' M_{s} \Gamma}$ one can use $\mathbf{M^D}=\mathbf{\Gamma^{'} M_{m} \Gamma + M_{s}}$, which contains the remainder information. When $\mathbf{M_{m}}$ has mass on its diagonal, however, this additional system of equations has $n^{2}$ exogenous variables on the LHS and $n^{2}$ unknowns (arising from $\mathbf{M_m}$) on the RHS. This implies that with the $n$ unknowns remaining from $\mathbf{M^{D}-M^{I}}=\mathbf{M_{s} -\Gamma' M_{s} \Gamma}$, one is still unable to identify $\mathbf{\Gamma}$ and $\mathbf{M_s}$.

Corollary A.1:
If $\mathbf{M_{m}}$ has mass on its diagonal, $\mathbf{\Gamma}$ cannot be identified from $\mathbf{M^I}$ and $\mathbf{M^D}$ alone.

The intuition behind this result is that by comparing aggregate occupational transition flows under dependent and independent interviewing, it is unclear how many workers are `responsible' for the change in occupational mobility between $\mathbf{M^D}$ and $\mathbf{M^I}$. Only when the diagonal of $\mathbf{M_{m}}$ contains exclusively zeros, identification could be resolved and one can recover $\mathbf{M_s}$, $\mathbf{\Gamma}$ and $\mathbf{M_m}$ as the number of equations equals the number of unknowns.[However, in the SIPP this case is empirically unreasonable as it requires that all employer/activity changers be true occupational movers.] An implication of the above corollary is that interrupted time-series analysis that is based on the difference in occupational mobility at the time of a switch from independent to dependent interviewing, does not identify the precise extent of the average coding error, but provides a downwards biased estimate.

To identify $\mathbf{\Gamma}$, however, Proposition A.2 implies that one can use the observed occupational transition flows of a sample of true occupational stayers that are subject to two rounds of independent interviewing. Some of these workers will appear as occupational stayers and some of them as occupational movers. Ideally, such a sample of workers should be isolated directly from the 1985 panel. Unfortunately, the questions on whether the individual changed activity or employer were only introduced in the 1986 panel, as a part of the switch to dependent interviewing. As a result, the 1985 panel by itself does not provide sufficient information to separate employer/activity stayers from employer/activity movers. Instead we use 1986 panel to estimate $\mathbf{\hat{M}^I_{m}}$. We can infer $\mathbf{M^I_{s}}$ indirectly by subtracting the observed occupational transition flow matrix $\mathbf{\hat{M}^I_m}$ in the 1986 panel from the observed occupational transition flow matrix $\mathbf{\hat{M}^I}$ in the 1985 panel. This is possible as the 1986 panel refers to the same underlying population as the 1985 panel and separates the employer/activity changers, who are independently interviewed.

Corollary A.2:
$\mathbf{\hat{\Gamma}}$ is consistently estimated from $\mathbf{\hat{T}_s^I}$ when the latter is estimated from $\mathbf{\hat{M}^{I}}-\mathbf{\hat{M}^{I}_{m}}$

This result is important to implement our approach. It follows as the population proportions underlying each cell of $\mathbf{\hat{M}_{s}}$, the sample estimate of $\mathbf{M_{s}}$, are consistently estimated. In turn, the latter follows from the standard central limit theorem for estimating proportions, which applies to $\mathbf{\hat{M}^{I}}$, $\mathbf{\hat{M}^{I}_{m}}$ and its difference. Proposition A.2 then implies that $\mathbf{\hat{\Gamma}}$ is consistently estimated.

§.§ Implementation

To implement our correction method we take the overlapping period of the 1985/86 panels. To increase the sample size we also use observations from the 1987 panel for the period between February 1987 and April 1987.[To avoid seasonality effects we re-weight all observations such that each observation in a given month has the same weight as another observation in any other month.] This panel has an identical setup to that of the 1986 panel (dependent interviewing and other relevant aspects) and is likewise representative of the population during the period of study.

Interviews throughout the SIPP are conducted every four months and collect information pertaining to the last four months, where these four months are considered to be a wave. We compare the reported occupational code of a worker in a given interview with the reported occupational code of that worker in the subsequent interview. An observation is therefore a pair of occupational codes, a reported `source' and (potentially identical) `destination' occupation. To keep comparability across interviews as clean as possible, and to focus on measuring occupations in the primary job, we only consider those workers who throughout the two waves stayed in full-time employment and who reported having only one employer at any point in time. These restrictions imply that in the estimation of $\mathbf{\Gamma}$ we do not include non-temporary laid off workers who experienced a short unemployment episodes and returned to their same jobs and employers. We also restrict attention to those workers who do not have imputed occupations, were not enrolled in school and were between 19-66 years old. These restrictions yield 28,302 wave/individual observations for the 1985 panel, 27,801 wave/individual observations for the 1986 panel and 5,922 wave/individual observations for the 1987 panel.

Demographic characteristics - February 1986 to April 1987

SIPP 1985     SIPP 1986     SIPP 1987     p-value

(no difference)

less than high school        14.72       15.27       14.55       0.386

high school grad        38.10       37.36       36.80       0.361

some college        24.51       24.94       24.71       0.546

college degree        22.67       22.43       23.95       0.746

Age category

19-24     12.29	    12.62	    12.90     0.458

25-29      16.72	    16.15     	 16.36     0.357

30-34     15.84	    15.40	    16.00    	0.512

35-39     15.02     15.30    	13.99    	0.806

40-44      11.65	     11.50	    11.80    	0.804

45-49     9.10    	8.87     9.22    	0.667

50-54     7.72	    7.90    	8.25	     0.600

55-59     7.07    	7.31	     7.01     0.600

60-64      4.20    	 4.64      4.05    	0.221

white        86.29      86.57     86.25    	0.729

black        10.62      10.81     10.73    	0.782

american indian, eskimo         0.49	    0.62     	0.42     0.401

asian or pacific islander         2.60    	2.01     2.60    	0.090

men        54.20       55.27       53.93       0.112

married       65.51    	66.15     	64.54      0.550

living in metro area        76.33      76.24    	75.97       0.885
5p0.8Workers aged 19-66, not enrolled in school, in two adjacent waves, measured in the first month of the current wave, with employment in one firm only in the previous wave, and employment in one (but possibly different) employer only in the wave that follows, without any self-employment, with un-imputed occupations reported in both waves. Person weights are used to scale observations per month within panel group (1985 versus 1986+1987).

Tables <ref> and <ref> show the demographic and occupational characteristics (based on the major occupations of the 1990 SOC), respectively, of the samples across the three panels. In the last column of each table we test characteristic-by-characteristic whether the proportion of workers with a given characteristic in the 1985 sample is statistically indistinguishable from the proportion of workers with the same characteristic in the pooled 1986/87 sample. Across all the characteristics analysed, we cannot reject at a 5% level that the proportions in the 1985 sample and the corresponding proportions in the 1986-87 sample are the same. With the exception of the proportion of Asian Americans and the proportion of workers whose source occupation is management, similarity cannot be rejected even at a 10% level. Although not shown here, we also cannot rejected at a 10% level that the proportion of workers across source and destination industries are the same when comparing the 1985 and 1986/87 samples. This analysis thus confirm that the observations used for our exercise are taken from the same underlying population.[To further rule out any meaningful impact from the observed differences in occupational distributions, we re-calculated all statistics after re-balancing the weights on the source/destination occupations to create identical occupational distributions. This exercise yields minimal effects on our statistics. For example, the observed occupational mobility changes by 0.01 percentage point at most. The reason for this small change is because there is the large proportion of the implied true stayers in the sample, which means that $\mathbf{\hat{T}^I_s}$ is not very sensitive to proportional changes in $\mathbf{\hat{M}_m^I}$.]

Distribution of workers across occupations - February 1986 to April 1987

5cDistribution across source occupations (occupation code) (%)
SIPP 1985     SIPP 1986     SIPP 1987     p-value (no difference)
managing occupations       12.31     13.11      13.99    	0.064

professional speciality       13.40     12.92     13.18     	0.380

technicians and related support        3.85     3.89     4.08    	0.829

sales occ.        9.74     9.98     9.86    	0.599

admin support       18.67    	 18.17    	18.26    	0.366

services       10.96    	11.66     11.19     0.166

farming/fish/logging        1.09     1.08     1.03     0.940

mechanics and repairers        4.87    	4.47     4.69     0.231

construction and extractive        3.47     3.65     3.60    	 0.503

precision production        4.01    	 4.19     3.75    	0.643

machine operators/assemblers        9.27      8.89     8.63    	0.355

transportation and materials moving        4.63    	 4.34     4.31     0.340

laborers        3.73    	3.66     3.42    	0.714

5cDistribution across destination occupations (occupation code) (%)
managing occupations       12.58     13.27     14.18    	0.103

professional speciality       13.31     12.90     13.10    	0.451

technicians and related support        3.82     3.92      4.01    	0.693

sales occ.        9.76     9.89     9.77    	0.797

admin support       18.53    	 18.20     18.20     0.546

services       10.96     11.57     11.19    	0.229

farming/fish/logging       1.08    	1.07     1.03    	0.924

mechanics and repairers        4.87    	4.46     4.73    	0.222

construction and extractive        3.59     3.60     3.56    	0.954

precision production        4.01      4.26     3.77    	0.480

machine operators/assemblers       9.24    	8.93     8.77    	0.455

transportation and materials moving       4.62    	4.31     4.33    	0.310

laborers        3.62    	3.62     3.36    	0.906

5p0.8Workers aged 19-66, not enrolled in school, in two adjacent waves, measured in the first month of the current wave, with employment in one firm only in the previous wave, and employment in one (but possibly different) firm only in the wave that follows, without any self-employment, with un-imputed occupations reported in both waves. Person weights are used to scale observations per month within panel group (1985 versus 1986+1987).

We estimate the occupational flows of employer/activity stayers by $\mathbf{\hat{M}^{I,85}}-\mathbf{\hat{M}^{I, 86/87}_m}=\mathbf{\hat{M}^{I}_s}$, where $x\!=\!85$ ($x\!=86/87$) in $\mathbf{\hat{M}_i^{I,x}}$ refers to the 1985 sample (1986/87 sample). In the 1986/87 sample, where we can observe employer/activity changers directly, we find that in 2.31% of (weighted) observations workers changed employers and in 4.65% workers report an activity change within their employers. This implies that more than 93% of the 1985 sample should be made up of employer/activity stayers.

A potential concern from using this survey design change could be the reliability in the 1986 implementation of dependent interviewing and its comparison with the data collected in the overlapping period of the 1985 panel. For example, any trail/learning period in the implementation of dependent interviewing in the 1986 panel could affect our results. To evaluate whether any trial/learning period in the implementation of dependent interviewing affected the occupational mobility rates we compare the occupational mobility rates obtained from the 1986 panel to those obtained from subsequent ones. Assuming improvements made after 1986 affected the measurement of occupational mobility, we should observe a meaningful change in these rates when using subsequent panels. Figure <ref> (see below for a detail explanation of the graph) suggests that any trail/learning period in the implementation of dependent interviewing did not have a major effect on average occupational mobility.[This is consistent with Hill (1994), who instead points to a potential higher level of attrition in the “older” 1985 panel relative to the “newer” 1986 panel as a concern. He argues that using the “1986 final panel weights, taken from the 1985 and 1986 Full Panel Longitudinal Research Files”, while not perfect, seem to be the best available solution to tackle this problem. We follow the same practice in our analysis.]

§ RESULTS AND DISCUSSION

Table 3 shows the occupational transition matrix $\mathbf{\hat{T}^{I}_{s}}$ for true occupational stayers derived from $\mathbf{\hat{M}^{I}_s}$ using the 1985 sample based on the major occupations of the 1990 SOC. The $(i,j)$'th element of $\mathbf{\hat{T}^{I}_{s}}$ indicates the transition probability from occupation $i$ to $j$. This matrix implies that 18.46% of true occupational stayers get classified as occupational movers.[This value lies within the expected bounds. To construct an upper bound consider the 1985 sample and assume that all observed occupational transitions are spurious. In this case we can expect that at most 19.71% of the observations would be miscoded. To construct a lower bound consider the 1986/87 sample and calculate the number of observations in which an occupational move is reported among the employer/activity changers. These observations account for 2.60% of all observations in the 1986/87 sample. Assuming that the effect of miscoding is to generate a net increase in the number of occupational changes, we can expect that at least 19.71%-2.60%$=$17.11% of the observations would be miscoded.] Although not shown here, a similar conclusion arises when we calculate the same matrix based on the 2000 SOC, which we use for our results in the
main text.

Observed occupational transition matrix of true occupational stayers, SOC 1990, $\mathbf{\hat{T}^{I}_{s}}$, (%)

OCCUPATIONS     (1)      (2)      (3)      (4)      (5)      (6)      (7)      (8)      (9)      (10)      (11)      (12)      (13)
(1) Managing Occ.      75.7     3.7     1.0     5.1     8.8     1.5     0.1     0.7     0.7     1.5     0.5     0.4     0.3

(2) Professional Spec.     3.3     87.9     2.9     0.4     2.2     1.5     0.1     0.5     0.2     0.6     0.2     0.1     0.1

(3) Technicians       3.1     10.1     69.5     0.5     5.9     3.8     0.1     2.7     0.6     1.3     2.1     0.2     0.3

(4) Sales Occ.     6.5     0.5     0.2     83.3     3.8     1.5     0.1     0.7     0.2     0.5     0.2     0.8     1.8

(5) Admin. Support       5.8     1.6     1.2     2.0     84.5     1.0     0.1     0.3     0.1     0.5     0.7     0.6     1.6

(6) Services       1.7     1.8     1.3     1.3     1.8     87.2     0.2     1.3     0.7     0.5     0.6     0.6     0.9

(7) Farm/Fish/Logging     1.3     1.7     0.5     1.3     1.1     2.0     84.7     0.6     0.6     0.2     1.0     3.1     2.0

(8) Mechanics     1.8     1.3     2.1     1.3     1.1     2.9     0.1     79.2     2.1     2.6     3.0     0.8     1.6

(9) Construction      2.4     0.7     0.6     0.5     0.4     2.2     0.2     3.0     77.7     1.7     1.9     1.2     7.6

(10) Precision Prod.     4.5     2.0     1.2     1.1     2.4     1.4     0.0     3.2     1.5     69.1     11.1     0.2     2.3

(11) Mach. Operators      0.7     0.3     0.9     0.2     1.4     0.7     0.1     1.6     0.7     4.8     84.1     0.7     3.8

(12) Transport     1.1     0.4     0.2     1.6     2.2     1.4     0.7     0.8     0.9     0.2     1.5     85.1     3.9

(13) Laborers      1.0     0.4     0.3     4.6     8.2     2.7     0.6     2.1     7.3     2.5     9.7     5.0     55.7

§.§ The estimate of $\mathbf{\Gamma}$

Following Proposition A.1 we can then recover the garbling matrix $\mathbf{\Gamma}$ by using $\mathbf{\hat{T}^{I}_{s}}$ and equation (<ref>). Table 4 shows the estimated $\mathbf{\hat{\Gamma}}$ based on the major occupations of the 1990 SOC, while Table <ref> shows the estimated $\mathbf{\hat{\Gamma}}$ based on the major occupations of the 2000 SOC. These estimates imply that on average the incorrect occupational code is assigned in around 10% of the cases. Since a spurious transition is likely to be created when either the source or destination occupation is miscoded, the probability of observing a spurious transition for a true occupational stayer is nearly twice as large. Our methodology then suggests that coding error is indeed substantial under independent interviewing. Its magnitude is of similar order as found in other studies analysing the extent of errors in occupational coding (see Campanelli et al., 1997, Sullivan, 2009, Roys and Taber, 2017, and vom Lehn et al., 2021).

Estimate of the garbling matrix, SOC 1990, $\mathbf{\hat{\Gamma}}$, (%)

OCCUPATIONS     (1)      (2)      (3)      (4)      (5)      (6)      (7)      (8)      (9)      (10)      (11)      (12)      (13)
(1) Managing Occ.       86.8     2.0     0.5     2.8     4.9     0.8     0.1     0.4     0.4     0.8     0.2     0.2     0.1

(2) Professional Spec.       1.8     93.7     1.6     0.2     1.1     0.8     0.1     0.2     0.1     0.3     0.1     0.1     0.0

(3) Technicians           1.7     5.6     83.3     0.2     3.3     2.1     0.1     1.5     0.3     0.7     1.1     0.1     0.1

(4) Sales Occ.         3.6     0.2     0.1     91.2     1.9     0.8     0.1     0.3     0.1     0.2     0.0     0.4     1.0

(5) Admin. Support       3.2     0.8     0.7     1.0     91.8     0.5     0.0     0.1     0.0     0.3     0.3     0.3     1.0

(6) Services             0.9     0.9     0.7     0.7     0.9     93.3     0.1     0.7     0.4     0.3     0.3     0.3     0.5

(7) Farm/Fish/Logging       0.7     0.9     0.3     0.7     0.5     1.0     92.0     0.3     0.3     0.1     0.5     1.7     1.1

(8) Mechanics      1.0     0.6     1.2     0.7     0.5     1.5     0.1     88.9     1.2     1.5     1.6     0.4     0.9

(9) Construction     1.3     0.3     0.3     0.2     0.0     1.1     0.1     1.6     88.0     0.9     0.8     0.6     4.7

(10) Precision Prod.      2.6     1.0     0.7     0.6     1.2     0.7     0.0     1.8     0.8     83.0     6.3     0.1     1.3

(11) Mach. Operators      0.3     0.1     0.5     0.0     0.7     0.4     0.1     0.8     0.3     2.7     91.5     0.4     2.3

(12) Transport      0.5     0.2     0.1     0.8     1.1     0.7     0.4     0.4     0.4     0.0     0.7     92.2     2.3

(13) Laborers       0.3     0.1     0.1     2.7     4.8     1.5     0.3     1.2     4.5     1.4     5.7     3.0     74.3

Estimate of the garbling matrix, SOC 2000, $\mathbf{\hat{\Gamma}}$, (%)

OCCUPATIONS     (11)      (13)      (15)      (17)      (19)      (21)      (23)      (25)      (27)      (29)      (31)      (33)      (35)      (37)     (39)     (41)     (43)     (45)     (47)     (49)     (51)     (53)
(11) Management Occ.              84.2       2.3       0.3       0.8       0.3       0.3       0.1      0.3      0.1      0.3      0.1      0.1      0.4      0.1       0.4       3.3       3.5       0.5       0.7        0.5       1.2       0.3

(13) Business & Finance Oper.      4.7       82.8        0.9       0.5       0.2       0.2       0.2       0.3      0.0       0.0      0.0      0.1      0.0      0.0      0.0      1.6      7.6      0.0      0.1      0.1      0.5      0.1

(15) Computer & Math. Occ.     2.6        3.7       85.7       1.6      0.4      0.0      0.0     0.4      0.5      0.0     0.1      0.1      0.0      0.0      0.0      0.0           4.4      0.0      0.1       0.7      0.2       0.2

(17) Architect & Eng. Occ.      2.2       0.7      0.5       87.0      1.3      0.0      0.0     0.0      1.2      0.3      0.0      0.0     0.0     0.1      0.0     0.1              1.1      0.0     0.9      2.4      2.2      0.1

(19) Life, Phys, and Soc. Sci. Occ.      2.4      0.6      0.4      3.3      82.9      0.0      0.2      1.1      0.4      2.2      0.5      0.0     0.0    0.5      0.0             0.3      1.7      0.7      0.5      1.1     1.0      0.3

(21) Comm & Soc. Service Occ.      2.8      0.7      0.0     0.0     0.0      89.8      0.0     1.0      0.4      0.5      0.7      0.3      0.2      0.1      0.4              0.0      2.6      0.0     0.2      0.1      0.1      0.3

(23) Legal      1.3      1.6      0.0     0.0     0.5      0.0     93.0      0.0      0.0      0.0     0.0      0.5      0.0      0.0      0.1      0.1      3.5      0.0      0.0         0.0      0.0      0.0

(25) Educ., Training & Library       0.5      0.3      0.1       0.0      0.3      0.2      0.0     97.3       0.1      0.2      0.1      0.0     0.0     0.0      0.3       0.1             0.6      0.0      0.0     0.0     0.0     0.0

(27) Arts, Dsgn, Ent., Sports & Media       1.0      0.2      0.6      3.8        0.5       0.4       0.0      0.3      89.2      0.0     0.0     0.0     0.0     0.0              0.0      0.7      1.3      0.0     0.2      0.2      1.7       0.4

(29) Healthcare Pract. & Tech. Occ.       0.7      0.0       0.0     0.2      0.7      0.1      0.0      0.3      0.0     92.7      3.3        0.0     0.1      0.1        0.1       0.1      1.0      0.0     0.0     0.1      0.2      0.1

(31) Healthcare Support     0.5       0.0       0.0     0.0      0.3       0.5      0.0      0.2      0.0     6.8       88.5      0.0      0.6      0.4       0.3      0.0          1.6      0.0     0.0     0.1       0.3      0.0

(33) Protective Service      0.7      0.3      0.1      0.0      0.0      0.2      0.2      0.1      0.1      0.1      0.0       95.0      0.5       0.1      0.2      0.1      1.8              0.3      0.0     0.1      0.4      0.5

(35) Food Prep/Serving & Rel.      0.9      0.0      0.0     0.0      0.0      0.0      0.0      0.0     0.0       0.1      0.3      0.2       95.3       0.5      0.0      1.5      0.4      0.0      0.0      0.0     0.4       0.2

(37) Building/grounds Clean. & Maint.       0.3      0.0      0.0      0.1      0.2      0.1      0.0      0.0      0.0      0.1      0.2      0.0     0.6      90.3              0.1       0.4      0.2      0.3      1.7      2.5      1.1      1.8

(39) Personal Care & Service Occ.      3.3      0.0      0.0      0.0     0.0     0.4      0.1      1.6      0.0     0.5       0.6       0.4      0.0      0.3      91.4              0.9       0.6      0.0      0.0     0.1      0.0     0.2

(41) Sales & Rel. Occ.      3.0      0.7      0.0      0.0     0.0      0.0     0.0      0.0      0.1      0.0     0.0       0.0      0.5      0.1      0.1      91.7       1.7              0.1      0.1      0.4      0.3      1.3

(43) Office & Admin. Support     1.7       1.8       0.3      0.2      0.1      0.2      0.1      0.2       0.1       0.2       0.2      0.2      0.1      0.0     0.0     0.9              91.6       0.0      0.0     0.1      0.6      1.3

(45) Farm, Fish. & Forestry     4.5      0.2      0.0      0.0      1.0      0.1      0.0      0.1      0.0     0.0     0.0     0.5      0.1      0.9      0.1      0.7      0.3              87.5       0.1      0.3      0.8        3.4

(47) Construction & Extraction     1.3      0.1       0.0      0.6      0.1       0.0      0.0      0.0     0.0     0.0     0.0     0.0     0.0     1.1      0.0     0.2      0.1              0.0     90.8      1.6       1.9      2.1

(49) Install., Maint. & Repair Occ.        0.8      0.1      0.1      1.4       0.2      0.0      0.0     0.0      0.0     0.1      0.0      0.0     0.0     1.5      0.0     0.6              0.4       0.1       1.4      89.0      3.0       1.1

(51) Production Occ.      0.8       0.2      0.0     0.5       0.1       0.0      0.0     0.0     0.1      0.1      0.0     0.0     0.1      0.3      0.0      0.2       0.8              0.0      0.6       1.1       93.1       1.9

(53) Transportation & Mater. Moving     0.3      0.1      0.0     0.1      0.0       0.0      0.0      0.0      0.1      0.1       0.0       0.1      0.1      0.7      0.0       1.5              2.9      0.4       1.2       0.7       3.4       88.2

Two additional messages come out of Tables 4 and <ref>. (i) Different occupations have very different propensities to be assigned a wrong code. For example, when using the 1990 SOC we find that individuals whose true occupation is “laborers” have a 74% probability of being coded correctly, while individuals whose true occupation is “professional speciality” have a 94% probability of being coded correctly. (ii) Given a true occupation, some coding mistakes are much more likely than others. For example, workers whose true occupation is “laborers” have a much larger probability to be miscoded as “machine operators” (5.7%), “construction” (4.5%) or “admin. support” (4.8%) than as “managers” (0.3%) or “professionals” (0.1%). Our methodology enable us to take these differences into account by correcting observed occupational flows by source-destination occupation pair. This provides cleaner net mobility estimates, where the identity of the origin and destination occupation matters.

§.§ The effect of $\mathbf{\hat{\Gamma}}$ on the gross occupational mobility rate
Figure <ref> depicts the effect of $\mathbf{\hat{\Gamma}}$ when computing wave-to-wave occupational mobility rates using the major occupational groups of the 2000 SOC. For this exercise we augment the 1985/86/87 samples with workers from the 1984 and 1988 panels, which satisfy the same sample restrictions as before. Figure <ref> depicts the average wave-to-wave occupational mobility rate obtained during each year. For the year 1986 we only use the observations that cover the February to December period obtained from the 1986 sample. We label these observations “1986s” as they are the ones we use in our original sample to estimate $\mathbf{\hat{\Gamma}}$. In the case of the year 1987, we present the average occupational mobility rate obtained for the January to April period (labelled “1987s”) separately from the average occupational mobility rate obtained from the remaining months (labelled “1987r”). The two vertical lines mark the time period in which dependent and independent interviewing overlap.

Correcting occupational mobility rates for workers employed in two subsequent waves

The short-dashed line with hollow circular markers depicts the observed occupational mobility rates obtained from pooling together employer/activity stayers and changers using the 1984/85 samples, where all respondents were subject to independent interviewing. This pooled sample yields occupational mobility rates that lie between 19.6%-20% and average 19.7% between the vertical lines. Under independent interviewing, $\mathbf{M_m}$ is garbled both at the source and destination occupations and hence is observed as $\mathbf{\Gamma'} \mathbf{M_m }\mathbf{\Gamma}$. Pre- and post-multiplying the latter by the respective inverses $\mathbf{(\Gamma')^{-1}}, \mathbf{\Gamma^{-1}}$ recovers $\mathbf{(\Gamma')^{-1}\Gamma' M_m \Gamma \Gamma^{-1}}=\mathbf{M_m}$. Applying this procedure to the 1984/85 samples yields the long-dashed line with hollow triangular markers. The result is a drop in the occupational mobility rate to about 1.6%.

Next consider the 1986/88 samples. Here independent interviewing is only applied to employer/activity changers. Therefore, the observed overall occupational mobility rate for these samples is based on $\mathbf{M^{D,86/87}_s} + \mathbf{M^{I, 86/87}_m}$. The matrix $\mathbf{M^{D,86/87}_s}$ does not contribute to any occupational transitions while matrix $\mathbf{M^{I, 86/87}_m}$ does. This difference implies a much lower observed aggregate mobility rate. The latter is depicted by the short-dashed line with filled circular markers in Figure <ref> and averages 2.6%. Correcting the occupational flows of the 1986-88 samples using $\mathbf{\hat{M}^{D,86/87}_s} + \mathbf{(\hat{\Gamma}')^{-1}}\mathbf{\hat{M}^{I, 86/87}_m}\mathbf{\hat{\Gamma}^{-1}}$ yields the series depicted by the long-dashed line with solid triangular markers. The result is a drop in the occupational mobility rate to about 1.7%, which is very close to the $\mathbf{\Gamma}$-corrected occupational mobility rate from the 1984/85 samples. Indeed, in Figure <ref> the blue (independent-interviewing) and red (dependent-interviewing) long-dashed lines of the $\mathbf{\Gamma}$-corrected measures nearly coincide between the two vertical lines.[Note that in the estimation of $\mathbf{\Gamma}$ we used $\mathbf{\hat{M}^{I,86/87}_m}$, but we did not impose the additional restriction on $\mathbf{(\hat{\Gamma}')^{-1}}\mathbf{\hat{M}^{I,86/87}_m }\mathbf{\hat{\Gamma}^{-1}}$ to equal $\mathbf{(\hat{\Gamma}')^{-1}}\mathbf{\hat{M}^{I,85}}\mathbf{\hat{\Gamma}^{-1}}$.]

The fact that we obtain very similar corrected mobility rates after using the same underlying $\mathbf{\Gamma}$-correction matrix in two different survey designs, suggests that our methodology captures the extent of coding error quite well. Our methodology also seems to work well in other dimensions. We find that the occupational mobility rates in the years before 1986 are adjusted downwards to numbers that are very similar to the numbers obtain during the 1986s-1987s window. For example, the $\mathbf{\Gamma}$-corrected occupational mobility in 1985 is 1.63%, while during the 1986s-1987s window we obtain 1.62%. For the years after the 1986s-1987s window, we find that the $\mathbf{\Gamma}$-corrected occupational mobility rate is within 0.02% of the one obtained during this window. Further, Figure <ref> shows that any changes in the level of the $\mathbf{\Gamma}$-corrected occupational mobility rate series appear to track changes in the uncorrected series. This also suggests that our correction method does not seem to introduce additional randomness into the occupational mobility process.

We apply our $\mathbf{\Gamma}$-correction method to those who changed employers with an intervening spell of unemployment or non-employment. We then compute the host of statistics shown in the main text and Supplementary Appendix B. We show that occupational miscoding increases observed gross mobility. In the raw data we compute an occupational mobility rate at re-employment of 53.1% based on the 2000 SOC. After applying our $\mathbf{\Gamma}$-correction, we obtain an occupational mobility rate at re-employment of 44.4%. We also find that coding errors makes occupational mobility appear less responsive to unemployment duration. Since in short unemployment spells true occupational staying is more common, miscoding creates relatively more spurious mobility and therefore our method corrects more the short spells, leading to a steeper relationship between occupational mobility and unemployment duration (see Table 1 and Figure 1 in Supplementary Appendix B). Miscoding also reduces the degree of procyclicality of gross mobility. This arises as coding errors will generate more spurious mobility in times were there are more true stayers (see Table 6 and Figure 11 in Supplementary Appendix B).

In contrast, we find that miscoding reduces the contribution of net occupational mobility (see also Kambourov and Manovskii, 2013). The average net mobility rate (as defined in the main text) increases from 3.6% (uncorrected) to 4.2% (corrected), a nearly 15% increase. To understand why this arises, consider the true net mobility transition flow matrix, $\mathbf{M_{net}}$, and note that this matrix does not have mass on its diagonal. Under independent interviewing coding errors imply that the true net mobility matrix would be observed as $\mathbf{M^{I}_{net}}=\mathbf{\Gamma'M_{net}\Gamma}$, which could have mass on the diagonal and hence biasing downwards net mobility flows. That is, coding errors mistakenly convert some true mobility flows into occupational stays, while miscoding for stayers is completely symmetric with respect to origin and destination occupations, and therefore should not give rise to spurious net mobility.

§.§ The differential impact of coding error on employer/activity stayers and movers
Note that our $\mathbf{\Gamma}$-corrected occupational mobility rate for those workers who changed employers through a spell of unemployment is 16.4% lower than the raw one (44.4% vs 53.1%, 2000 SOC). This adjustment is much smaller in relative terms than the one suggested by Kambourov and Manovskii (2008). They argue that on average around 50% of all year-to-year observed occupational mobility in the raw PSID data is due to coding error. Indeed when constructing the year-to-year occupational mobility rate for our pooled sample of employer/activity stayers and movers in the SIPP, we find that only 39.3% of observed occupational changes are genuine (an uncorrected rate of 26.7% vs a $\mathbf{\Gamma}$-corrected rate of 10.5%).

These findings are not mutually inconsistent. The key to this difference is that the relative importance of coding error varies greatly with the true propensity of an occupational change among employer stayers and among employer changers. True occupational changes are more likely to be accompanied by changes in employers (for examples of this argument see Hill, 1994, Moscarini and Thomsson, 2007, and Kambourov and Manovskii, 2009) and, vice versa, employer changes are more likely to be accompanied by occupational changes. As such, the relative adjustment that Kambourov and Manovskii (2008) find does not automatically carry over to different subsets of workers or flows measured at a different frequency.

As an example, consider an individual who is observed moving from “managers” to “laborers”. Suppose “managers” was this individual's true source occupation, but the observed destination occupation was the result of coding error. If this individual is a true occupational stayer, the observed transition will wrongly tell us that the individual stopped being a manager and will generate a false occupational move. Instead, if this individual is a true occupational mover, the observed transition, although wrongly coded, will still capture the fact that the individual stopped being a manager and hence will capture a true occupational move. Given that true occupational changes are more likely to occur along side employer changes, there will be more workers among employer changers (relative to employer stayers) whose categorization as an observed occupational mover will not change after using the $\mathbf{\Gamma}$-correction. This implies that the measured occupational mobility rate of employer changers would have a relative smaller adjustment than the measured occupational mobility rate of employer stayers. Kambourov and Manovskii (2008) pooled together employer changers and stayers. Since the latter group represent the vast majority of workers in their sample (as well as in our sample), the relative adjustment proposed by these authors is naturally much larger.

Consider the following iterative back-of-the-envelope approximation to understand why there must be a larger proportion of true occupational movers among those who changed employers than among those who did not change employers. Recall that the $\mathbf{\Gamma}$-correction method implies that true occupational stayers will be coded as movers in about 20% of the times. This happens irrespectively of whether the worker changed employers or not. If we were to suppose that all of the unemployed who regain employment were true occupational stayers, the difference between their observed mobility rate (53%) and coding error (20%) would immediately imply that among the unemployed there must be true occupational movers and these true movers would represent at least 33% of the unemployed. This result then shrinks the population of occupational stayers (the “population at risk”) among the unemployed to at most 67%, which (proceeding iteratively) implies the maximum extent of spurious flows produced under the same miscoding propensity is reduced to $0.2 \times 67\%$. In turn, the latter implies an updated lower bound on the percentage of true occupational movers among the unemployed of 39.6%. Proceeding iteratively, one arrives to a lower bound on the percentage of true occupational movers among the unemployed of 41.25%, which is close to the gross mobility we obtained after applying the $\mathbf{\Gamma}$-correction. A similar procedure but applied to employer stayers shows a much lower `true' mobility rate among this group.

Inferred Coding Error Probabilities and Observed vs. Underlying Occupational Mobility

Classification     1c$\mathbb{P}(\tilde{M} |S)$      1c      $\mathbb{P}(\tilde{M}|U)$     1c       $\mathbb{P}(M|U)$     1c    $\frac{\mathbb{P}(M|U)}{\mathbb{P}(\tilde{M}|U)} - 1$      1c$\mathbb{P}(\tilde{o}\neq o | U)$
2000 SOC (22 cat)     0.178           0.531            0.444          -0.164     0.095

1990 SOC (13 cat)     0.197          0.507           0.401           -0.209      0.105

1990 SOC (6 cat)     0.148           0.402           0.317          -0.213      0.077

NR/R Cognitive, NR/R Manual  (4 cat)     0.110           0.332            0.263           -0.208     0.058

Cognitve, R Manual / NR Manual (3 cat)     0.083           0.273            0.218           -0.199     0.043

Major industry groups (15 cat)     0.101            0.523           0.477           -0.088      0.055

6p1Sample: unemployed between 1983-2013, in 1984-2008 SIPP panels, subject to conditions explained in data construction appendix (most importantly: unimputed occupations (resp. industries), with restrictions to avoid right and left censoring issues.) $\mathbb{P}(\tilde{M} | S)$: probability that the wrong code is assigned to a true stayer; $\mathbb{P}(\tilde{o}\neq o | U)$: probability that the wrong code is assigned to an unemployed worker; $\mathbb{P}(\tilde{M})$: observed occupational mobility among the unemployed; $\mathbb{P}(M)$: inferred underlying true mobility (proportion of unemployed). NR/R refers to routine vs. non-routine. Further details on the classifications are explained in the data construction appendix.

Table <ref> shows that the differential impact of coding error on employer changers and stayers is present when considering several alternative occupational classifications as well as mobility across industries. In all these cases we use wave-to-wave mobility rates. The first column, $\mathbb{P}(\tilde{M} |S)$, presents the probability that a true employer/activity stayer in the 1985 sample is assigned the wrong occupational code and hence is observed as an occupational mover $\tilde{M}$. This probability is 17.8% when using the 2000 SOC. This implies that under independent interviewing we will observe an occupational mobility rate of 17.8% among the employer/activity stayers. This rate increases slightly when using the 1990 SOC, 19.7%, and remains high even when we aggregate occupations into six categories, 14.8%.[The six groups are: (1) managers/professional speciality; (2) tech support/admin support/sales; (3) services; (4) farm/forest/fisheries; (5) precision production/craft/repair; and (6) operators, fabricators and laborers. These correspond to the summary occupational group of the 1990 SOC.] Aggregating occupations into four tasked based categories (routine vs. non-routine and manual vs. cognitive) only brings down the observed mobility rate of true occupational stayers to 11%. As many other studies, we also find the probability that a true stayer is observed as a mover is lower when considering industries instead of occupations.

The second and third columns show the observed, $\mathbb{P}(\tilde{M}|U)$, and $\mathbf{\Gamma}$-corrected, $\mathbb{P}(M|U)$, occupational mobility rate of those workers who changed employers through unemployment. These are obtained using the probability that an unemployed worker in the 1984-2008 SIPP panels is assigned the wrong occupational code at re-employment, $\mathbb{P}(\tilde{o}\neq o | U)$. We observe that across all classifications the (relative) difference between the observed and the $\mathbf{\Gamma}$-corrected occupational mobility rates of the unemployed, $\mathbb{P}(\tilde{M}|U) - \mathbb{P}(M|U)$, is about half the size of $\mathbb{P}(\tilde{M} |S)$. The fourth column shows this difference in relative terms. It is clear that across all classifications the same coding error generates a much larger difference (in absolute and relative terms) between the observed and $\mathbf{\Gamma}$-corrected occupational mobility rates of employer stayers than among employer changers.

§ MEASURING CODING ERROR IN THE PSID