Ashenfelter & Card (1985): Using the Longitudinal Structure of Earnings to Estimate the Effect of Training Programs

An explanation with a simulation

1 The problem

Ashenfelter and Card (1985) ask how to estimate the effect of a job-training program on earnings when participants are not randomly assigned. Their data are longitudinal: they have annual earnings histories for trainees and for a comparison group that are drawn from the general population.

Two features of participant earnings make the problem hard:

1. Selection on permanent differences. Trainees tend to have lower long-run earnings than the average person.
2. The “Ashenfelter dip.” Trainees’ earnings fall sharply in the year(s) just before they enter training. People enrol when they have had a bad spell, so their earnings are unusually low right before the program.

If the transitory component of earnings is serially correlated (a temporary earnings shock today tends to carry over into tomorrow), then a low draw in the selection year (the worker happens to receive an unusually negative temporary earnings shock in that year) predicts low earnings in neighbouring years too. Any estimator that uses the selection year (or its neighbours) as the “before” period will mix up recovery from a bad shock with the treatment effect.

What actually happened in the real data

As previously stated, the evaluation of the training program in the paper does not correspond to a randomized experiment. Nobody assigned people to the program by lottery, so the authors had to work with observational data.

• The participants were people who entered a US federal job-training program (CETA) in 1976. Their annual earnings histories come from Social Security records covering several years before and after training.
• The authors created a comparison group made by a sample of the general population from a household survey (the CPS), linked to the same kind of Social Security earnings records.
• This setting is not ideal: people who enrol are not a random slice of the population. Their earnings are lower on average, and they fall sharply just before entry (this was coined as the Ashenfelter dip). A simple before-after or standard DiD comparison therefore mixes up the program’s effect with selection and rebound. The paper’s estimators are built to deal with this issue.

2 The model

Suppose the earnings process is

\[ Y_{it} = \alpha_i + d_t + \beta D_{it} + \epsilon_{it}, \]

where

• \(\alpha_i\) is a permanent individual component,
• \(d_t\) is an economy-wide year effect,
• \(D_{it}\) indicates that person \(i\) has completed training by year \(t\), and \(\beta\) is the effect of interest,
• \(\epsilon_{it}\) is a transitory component following an AR(1) process, \(\epsilon_{it} = \rho\,\epsilon_{i,t-1} + u_{it}\).

Participation depends on earnings in a selection year \(T-k\), so it depends on \(\alpha_i\) and on \(\epsilon_{i,T-k}\). Under the AR(1) process,

\[ E[\epsilon_{it} \mid \epsilon_{i,T-k}] = \rho^{|t-(T-k)|}\,\epsilon_{i,T-k}, \]

so among participants (who were selected for having low \(\epsilon_{i,T-k}\)) the expected transitory component is negative around the selection year and gradually returns to zero.

More explanations to understand why selecting on a bad year is a problem

Suppose the transitory component follows an AR(1) process, \(\varepsilon_{it}=\rho\,\varepsilon_{i,t-1}+u_{it}\) with \(\rho>0\): a temporary shock fades gradually instead of disappearing after one year.

Take a worker who enters training in 1976 because 1975 was unusually bad (illness, cut hours, a struggling employer), say \(\varepsilon_{i,1975}=-1000\). With \(\rho=0.7\), the expected shock is \(-700\) in 1974 and 1976, and \(-490\) in 1973 and 1977.

Details on how to obtain those values

• Forward. The innovation has mean zero, so \(E[\varepsilon_{i,1976}\mid\varepsilon_{i,1975}]=\rho\,\varepsilon_{i,1975}=-700\), and \(E[\varepsilon_{i,1977}\mid\varepsilon_{i,1975}]=\rho^2\varepsilon_{i,1975}=-490\).
• Backward. Stationarity gives every year the same variance \(\sigma_\varepsilon^2=\sigma_u^2/(1-\rho^2)\), and a covariance that depends only on the distance between years, \(\mathrm{Cov}(\varepsilon_{it},\varepsilon_{i,t-k})=\rho^k\sigma_\varepsilon^2\). The slope of the prediction is covariance over variance, so \[ E[\varepsilon_{i,1974}\mid\varepsilon_{i,1975}]=\frac{\rho\,\sigma_\varepsilon^2}{\sigma_\varepsilon^2}\,\varepsilon_{i,1975}=-700 . \] Note that this is a statement about likely histories, not causation: a very bad 1975 is more likely to follow a somewhat bad 1974 than a normal one.

In both directions, \[ E[\varepsilon_{it}\mid\varepsilon_{i,1975}]=\rho^{|t-1975|}\,\varepsilon_{i,1975}. \]

Two problems arise:

• First, selection is on a low draw. Participants enter precisely when they are in a trough (the Ashenfelter dip).
• Second, recovery looks like a treatment effect. Participants’ earnings would have rebounded anyway, so a naive before-after comparison credits the program with the natural recovery.

The authors suggest a symmetric fix: 1973 and 1977 are equally far from 1975 and carry the same expected shock (\(-490\)), so comparing them cancels it.

3 Three estimators

Let us consider three stages, each showing a specific estimator.

3.1 Stage 1: naive before-after

The naive estimator compares participants’ average earnings after the program with their average earnings before it. It needs no comparison group, but it attributes to the program everything that changed over time for participants, including aggregate growth and the rebound from the pre-program dip.

\[ \hat\beta_{\text{naive}} = E(Y_{\text{post}} - Y_{\text{pre}} \mid D=1). \]

This ignores aggregate growth (\(d_t\)) and, if “pre” is the dip year, attributes the rebound from the dip to the program.

3.2 Stage 2: conventional difference-in-differences

The conventional estimator subtracts from the participants’ before-after change the same change for a comparison group. This removes common time effects and permanent differences between the groups. However, if the base year is close to the selection year, the participants’ rebound from the bad draw remains in the estimate.

\[ \hat\beta_{\text{DiD}} = E(Y_{\text{post}} - Y_{\text{pre}} \mid D=1) - E(Y_{\text{post}} - Y_{\text{pre}} \mid D=0). \]

This removes common time effects and permanent differences \(\alpha_i\). But with a serially correlated \(\epsilon_{it}\) and a base year close to the selection year, the trainees’ rebound is still in the estimate. The paper shows that one can compute a different DiD estimate for each possible pre-program base year, and these estimates differ.

3.3 Stage 3: symmetric difference-in-differences

The symmetric estimator also uses a comparison group, but it chooses the pre-program year so that it is as far before the selection year as the post-program year is after it. With selection in period \(T-k\), this gives

\[ \boxed{ \hat\beta_{AC} = \left[E(Y_{T+1} - Y_{T-2k-1} \mid D=1)\right] - \left[E(Y_{T+1} - Y_{T-2k-1} \mid D=0)\right] } \]

The selection year \(T-k\) is the midpoint of \([T-2k-1,\,T+1]\), so both endpoints are \(k+1\) years away from it. Under stationarity, the selection shock then affects both years equally and cancels out of the change: \(E[\epsilon_{T+1}-\epsilon_{T-2k-1}\mid\epsilon_{T-k}]=(\rho^{k+1}-\rho^{k+1})\,\epsilon_{T-k}=0.\)

The paper also discusses a scaling factor \(1/(1-p)\), where \(p\) is the fraction of the comparison population that participates. It is needed when the comparison group is the whole population, trainees included. See the caveat in Section 6.4.

This is not the paper’s final estimator

Ashenfelter and Card go further. They estimate a components-of-variance model on the comparison group, use a model of participation to predict what trainees’ earnings histories would have looked like, and take the gap between predicted and actual post-training earnings as the program effect. The symmetric DiD is an intermediate step. Simulating the full prediction-based estimator is the natural extension (see Section 9).

4 Simulation

We simulate 5,000 individuals over 1970-1978. Selection is on 1975 earnings, so \(T-k = 1975\) with \(T = 1976\) and \(k = 1\). Training takes effect from 1977. The true effect is $500 per year.

library(dplyr)
library(tidyr)
library(ggplot2)
library(broom)
library(purrr)

set.seed(1985)

4.1 Parameters and individual characteristics

N     <- 5000
years <- 1970:1978

beta      <- 500   # true annual treatment effect ($)
rho       <- 0.70  # persistence of transitory shocks
sigma_eps <- 500   # sd of innovations

individuals <- tibble(
  id      = 1:N,
  alpha_i = rnorm(N, mean = 0, sd = 1200),   # permanent component
  age     = round(rnorm(N, mean = 31, sd = 6)),
  educ    = pmax(8, pmin(18, round(rnorm(N, mean = 12, sd = 2))))
)

4.2 Panel, year effects and AR(1) shocks

data <- individuals |>
  crossing(year = years) |>
  arrange(id, year)

# Economy-wide earnings growth
year_effects <- tibble(
  year = years,
  d_t  = c(0, 100, 250, 450, 650, 700, 800, 950, 1100)
)

data <- data |> left_join(year_effects, by = "year")

# AR(1) transitory component, started from its stationary distribution
data <- data |>
  group_by(id) |>
  mutate(u_it = rnorm(n(), mean = 0, sd = sigma_eps)) |>
  mutate(
    eps_it = {
      eps <- numeric(n())
      eps[1] <- u_it[1] / sqrt(1 - rho^2)
      for (j in 2:n()) eps[j] <- rho * eps[j - 1] + u_it[j]
      eps
    }
  ) |>
  ungroup()

4.3 Potential earnings without treatment

data <- data |>
  mutate(
    base_earnings = 3500 + 250 * (educ - 12) + 20 * (age - 31),
    earnings_0    = pmax(0, base_earnings + alpha_i + d_t + eps_it)
  )

4.4 Selection on 1975 earnings

Lower 1975 earnings raise the probability of participating. Treatment status is a person-level constant.

selection_data <- data |>
  filter(year == 1975) |>
  select(id, earnings_1975 = earnings_0)

data <- data |>
  left_join(selection_data, by = "id") |>
  mutate(p_treat = plogis(-0.5 - 0.0015 * (earnings_1975 - 3500))) |>
  group_by(id) |>
  mutate(treated = rbinom(1, size = 1, prob = first(p_treat))) |>
  ungroup()

data |> distinct(id, treated) |> count(treated)

# A tibble: 2 × 2
  treated     n
    <int> <int>
1       0  3521
2       1  1479

4.5 The dip and the treatment effect

The dip is an additional $600 drop in 1975 for participants. The treatment adds $500 from 1977 on.

data <- data |>
  mutate(
    ash_dip        = if_else(year == 1975 & treated == 1, -600, 0),
    earnings_0_dip = pmax(0, earnings_0 + ash_dip),
    treatment_effect = if_else(treated == 1 & year >= 1977, beta, 0),
    earnings       = earnings_0_dip + treatment_effect
  )

5 Describing the data

Our simulated dataset is a panel dataset where each row gives the earnings of an individual in a given year, as well as their characteristics

data

# A tibble: 45,000 × 17
      id alpha_i   age  educ  year   d_t  u_it eps_it base_earnings earnings_0
   <int>   <dbl> <dbl> <dbl> <int> <dbl> <dbl>  <dbl>         <dbl>      <dbl>
 1     1    508.    30    13  1970     0 -626. -876.           3730      3361.
 2     1    508.    30    13  1971   100  630.   17.0          3730      4355.
 3     1    508.    30    13  1972   250 -746. -734.           3730      3754.
 4     1    508.    30    13  1973   450  605.   91.4          3730      4779.
 5     1    508.    30    13  1974   650  358.  422.           3730      5310.
 6     1    508.    30    13  1975   700 -440. -144.           3730      4793.
 7     1    508.    30    13  1976   800 -828. -929.           3730      4109.
 8     1    508.    30    13  1977   950  521. -129.           3730      5059.
 9     1    508.    30    13  1978  1100 -290. -380.           3730      4957.
10     2  -1336.    27    16  1970     0 -149. -209.           4420      2875.
# ℹ 44,990 more rows
# ℹ 7 more variables: earnings_1975 <dbl>, p_treat <dbl>, treated <int>,
#   ash_dip <dbl>, earnings_0_dip <dbl>, treatment_effect <dbl>, earnings <dbl>

mean_earnings <- data |>
  group_by(year, treated) |>
  summarise(mean_earnings = mean(earnings), .groups = "drop")

mean_earnings |>
  pivot_wider(names_from = "treated", values_from = "mean_earnings",
              names_prefix = "treated_")

# A tibble: 9 × 3
   year treated_0 treated_1
  <int>     <dbl>     <dbl>
1  1970     3936.     2381.
2  1971     4036.     2464.
3  1972     4190.     2627.
4  1973     4399.     2761.
5  1974     4623.     2869.
6  1975     4720.     2215.
7  1976     4791.     2983.
8  1977     4929.     3704.
9  1978     5061.     3919.

ggplot(mean_earnings,
       aes(x = year, y = mean_earnings, group = treated,
           linetype = factor(treated))) +
  geom_line(linewidth = 1) +
  geom_point() +
  geom_vline(xintercept = 1976, linetype = "dashed") +
  labs(x = "Year", y = "Mean annual earnings",
       linetype = "Program participant",
       title = "Simulated earnings histories") +
  theme_minimal()

Figure 1: Mean earnings of participants and non-participants. Participants start lower, dip in 1975, then recover.

The dashed line marks the start of training. Participants’ earnings fall before it and rebound afterwards, even without any treatment effect. That rebound is what the estimators must not mistake for a program effect.

6 Estimators in practice

6.1 Naive before-after

\(\hat\beta_{\text{naive}} = E[Y_{i,1978} - Y_{i,1975} \mid D_i = 1]\)

naive <- data |>
  filter(year %in% c(1975, 1978)) |>
  select(id, treated, year, earnings) |>
  pivot_wider(names_from = year, values_from = earnings,
              names_prefix = "earnings_") |>
  filter(treated == 1) |>
  summarise(naive_effect = mean(earnings_1978 - earnings_1975))

naive

# A tibble: 1 × 1
  naive_effect
         <dbl>
1        1703.

Because 1975 is the dip year, this is much larger than the true $500.

6.2 Conventional DiD (base 1975, post 1978)

We simply compute, for each treated, the change in earnings between 1975 and 1978 and then calculate the average over each group (treated and untreated).

did_1975 <- data |>
  filter(year %in% c(1975, 1978)) |>
  select(id, treated, year, earnings) |>
  pivot_wider(names_from = year, values_from = earnings,
              names_prefix = "earnings_") |>
  mutate(change = earnings_1978 - earnings_1975) |>
  group_by(treated) |>
  summarise(mean_change = mean(change), .groups = "drop")

A glimpse at the dataset before computing the average

data |>
  filter(year %in% c(1975, 1978)) |>
  select(id, treated, year, earnings) |>
  pivot_wider(names_from = year, values_from = earnings,
              names_prefix = "earnings_") |>
  mutate(change = earnings_1978 - earnings_1975) |>
  group_by(treated)

# A tibble: 5,000 × 5
# Groups:   treated [2]
      id treated earnings_1975 earnings_1978 change
   <int>   <int>         <dbl>         <dbl>  <dbl>
 1     1       0         4793.         4957.  164. 
 2     2       0         4059.         3941. -118. 
 3     3       0         3775.         4053.  278. 
 4     4       1         1944.         4163. 2218. 
 5     5       0         5155.         5073.  -81.1
 6     6       0         3174.         3679.  505. 
 7     7       0         5523.         6609. 1086. 
 8     8       0         4312.         4372.   59.7
 9     9       0         3552.         3405. -147. 
10    10       0         3143.         3403.  260. 
# ℹ 4,990 more rows

did_1975

# A tibble: 2 × 2
  treated mean_change
    <int>       <dbl>
1       0        341.
2       1       1703.

did_estimate <- did_1975 |>
  summarise(DID = mean_change[treated == 1] - mean_change[treated == 0])

did_estimate

# A tibble: 1 × 1
    DID
  <dbl>
1 1362.

This is better than the naive estimate: it removes aggregate growth. But it still uses the selection/dip year as its base, so it is still biased upward.

6.3 Symmetric DiD (1973 vs 1977)

With \(T = 1976\) and \(k = 1\): post period \(T+1 = 1977\), pre period \(T-2k-1 = 1973\). Both are two years from the selection year 1975.

symmetric_did <- data |>
  filter(year %in% c(1973, 1977)) |>
  select(id, treated, year, earnings) |>
  pivot_wider(names_from = year, values_from = earnings,
              names_prefix = "earnings_") |>
  mutate(change = earnings_1977 - earnings_1973) |>
  group_by(treated) |>
  summarise(mean_change = mean(change), .groups = "drop")

A glimpse at the dataset before computing the average

data |>
  filter(year %in% c(1973, 1977)) |>
  select(id, treated, year, earnings) |>
  pivot_wider(names_from = year, values_from = earnings,
              names_prefix = "earnings_") |>
  mutate(change = earnings_1977 - earnings_1973) |>
  group_by(treated)

# A tibble: 5,000 × 5
# Groups:   treated [2]
      id treated earnings_1973 earnings_1977 change
   <int>   <int>         <dbl>         <dbl>  <dbl>
 1     1       0         4779.         5059.  280. 
 2     2       0         2523.         3541. 1018. 
 3     3       0         3927.         4011.   83.5
 4     4       1         2360.         3449. 1089. 
 5     5       0         4359.         5186.  827. 
 6     6       0         2482.         3667. 1185. 
 7     7       0         4715.         6765. 2049. 
 8     8       0         3858.         4571.  714. 
 9     9       0         1988.         3340. 1352. 
10    10       0         3677.         3291. -386. 
# ℹ 4,990 more rows

symmetric_did

# A tibble: 2 × 2
  treated mean_change
    <int>       <dbl>
1       0        530.
2       1        943.

symmetric_estimate <- symmetric_did |>
  summarise(symmetric_DID = mean_change[treated == 1] - mean_change[treated == 0])

symmetric_estimate

# A tibble: 1 × 1
  symmetric_DID
          <dbl>
1          413.

6.4 The \(1/(1-p)\) adjustment

The paper’s comparison group is a general population sample, which includes some trainees. Because those trainees pull the comparison group’s average toward the participants’, the estimated gap is too small. Ashenfelter and Card correct for this by dividing the estimate by \((1-p)\), where \(p\) is the share of participants in the population.

A numerical example

Take 100 people, 30 of whom join the program, so \(p = 0.3\). Suppose the change in earnings between 1973 and 1977 is \(+500\) for participants and \(0\) for non-participants. The gap we want to measure is \(500 - 0 = 500\).

If the comparison group is the whole population, its average change mixes both groups:

\[ 0.3 \times 500 + 0.7 \times 0 = 150 . \]

Comparing participants with this group gives \(500 - 150 = 350\), which is \((1-p) = 0.7\) times the true gap. Dividing by \((1-p)\) recovers it:

\[ \frac{350}{1-p} = \frac{350}{0.7} = 500 . \]

Not needed in our simulation

In our simulation, the controls are non-participants only, so the comparison group is not contaminated by trainees and the unadjusted symmetric DiD already targets the true effect.

Dividing it by \((1-p)\) would overshoot here. Hence, we will do so for illustrative purposes only.

The proportion of treated:

p <- data |> distinct(id, treated) |> summarise(p = mean(treated)) |> pull(p)
p

[1] 0.2958

Then the symmetric DiD with each comparison group:

change_1973_1977 <- data |>
  filter(year %in% c(1973, 1977)) |>
  select(id, treated, year, earnings) |>
  pivot_wider(names_from = year, values_from = earnings,
              names_prefix = "earnings_") |>
  mutate(change = earnings_1977 - earnings_1973)

avg_change <- change_1973_1977 |>
  summarise(
    participants     = mean(change[treated == 1]),
    non_participants = mean(change[treated == 0]),
    whole_population = mean(change)
  )

avg_change |>
  transmute(
    `Non-participants only`      = participants - non_participants,
    `Whole population`           = participants - whole_population,
    `Whole population, adjusted` = (participants - whole_population) / (1 - p)
  ) |>
  pivot_longer(everything(), names_to = "comparison_group", values_to = "estimate")

# A tibble: 3 × 2
  comparison_group           estimate
  <chr>                         <dbl>
1 Non-participants only          413.
2 Whole population               291.
3 Whole population, adjusted     413.

6.5 Regression versions

The same three estimators can be written as regressions. The relevant coefficient is on post (naive) or on treated:post (DiD variants).

# Naive
naive_reg_data <- data |>
  filter(year %in% c(1975, 1978), treated == 1) |>
  mutate(post = as.integer(year == 1978))
tidy(lm(earnings ~ post, data = naive_reg_data))

# A tibble: 2 × 5
  term        estimate std.error statistic   p.value
  <chr>          <dbl>     <dbl>     <dbl>     <dbl>
1 (Intercept)    2215.      29.7      74.6 0        
2 post           1703.      42.0      40.6 1.36e-286

# Conventional DiD
did_reg_data <- data |>
  filter(year %in% c(1975, 1978)) |>
  mutate(post = as.integer(year == 1978))
tidy(lm(earnings ~ treated + post + treated:post, data = did_reg_data))

# A tibble: 4 × 5
  term         estimate std.error statistic   p.value
  <chr>           <dbl>     <dbl>     <dbl>     <dbl>
1 (Intercept)     4720.      20.7     228.  0        
2 treated        -2505.      38.1     -65.7 0        
3 post             341.      29.3      11.6 4.15e- 31
4 treated:post    1362.      53.9      25.3 1.47e-136

# Symmetric DiD
symmetric_reg_data <- data |>
  filter(year %in% c(1973, 1977)) |>
  mutate(post = as.integer(year == 1977))
tidy(lm(earnings ~ treated + post + treated:post, data = symmetric_reg_data))

# A tibble: 4 × 5
  term         estimate std.error statistic  p.value
  <chr>           <dbl>     <dbl>     <dbl>    <dbl>
1 (Intercept)     4399.      21.0    210.   0       
2 treated        -1639.      38.6    -42.5  0       
3 post             530.      29.7     17.9  3.26e-70
4 treated:post     413.      54.6      7.57 4.03e-14

6.6 Comparing the estimators

comparison <- tibble(
  estimator = c("True treatment effect",
                "Naive before-after",
                "Conventional DiD: 1975-1978",
                "Symmetric DiD: 1973-1977"),
  estimate = c(beta,
               naive$naive_effect,
               did_estimate$DID,
               symmetric_estimate$symmetric_DID)
)

comparison

# A tibble: 4 × 2
  estimator                   estimate
  <chr>                          <dbl>
1 True treatment effect           500 
2 Naive before-after             1703.
3 Conventional DiD: 1975-1978    1362.
4 Symmetric DiD: 1973-1977        413.

Note that the true effect is $500 in each post year, and the 1977 and 1978 effects are both $500, so the symmetric estimate (post = 1977) and the conventional one (post = 1978) target the same number.

7 DiD using every possible pre-program base year

Instead of fixing 1975, compute

\[ \hat\beta_j = E(Y_{1978} - Y_j \mid D=1) - E(Y_{1978} - Y_j \mid D=0) \]

for each pre-treatment base year \(j\).

base_years <- 1970:1975

did_by_base_year <- base_years |>
  map_dfr(
    ~ data |>
      filter(year %in% c(.x, 1978)) |>
      select(id, treated, year, earnings) |>
      pivot_wider(names_from = year, values_from = earnings) |>
      mutate(change = `1978` - .data[[as.character(.x)]]) |>
      group_by(treated) |>
      summarise(mean_change = mean(change), .groups = "drop") |>
      summarise(
        base_year = .x,
        DID = mean_change[treated == 1] - mean_change[treated == 0]
      )
  )

did_by_base_year

# A tibble: 6 × 2
  base_year   DID
      <int> <dbl>
1      1970  413.
2      1971  429.
3      1972  420.
4      1973  496.
5      1974  612.
6      1975 1362.

ggplot(did_by_base_year, aes(x = base_year, y = DID)) +
  geom_line() +
  geom_point(size = 3) +
  geom_hline(yintercept = beta, linetype = "dashed") +
  labs(x = "Pre-program base year",
       y = "Estimated treatment effect",
       title = "Sensitivity of DiD to the choice of pre-program base year",
       subtitle = "Dashed line = true treatment effect in the simulation") +
  theme_minimal()

Figure 2: DiD estimate as a function of the pre-program base year. The estimate is most distorted near the selection year.

The pattern shows the core message: the answer depends on which “before” year you pick, and base years near the selection shock give the most misleading estimates. A base year is only “safe” if it sits at the symmetric distance from the selection year that matches the post year (here 1973 for a 1977 post period). With \(\rho = 0.7\), base years farther back (1970-1972) are also close to unbiased, because the shock has largely faded.

8 Takeaways

Estimator	Removes common trends	Removes permanent differences	Handles serially correlated selection shock
Naive before-after	no	(yes, within person)	no
Conventional DiD	yes	yes	no
Symmetric DiD	yes	yes	yes, under covariance stationarity

9 Next step: the full paper

To reproduce the paper’s final estimator in this simulation, one would:

1. Fit a components-of-variance model to the comparison group’s earnings histories (permanent component, year effects, serially correlated transitory component).
2. Model participation as a function of pre-program earnings.
3. Predict each trainee’s counterfactual post-program earnings from that model.
4. Estimate the program effect as actual minus predicted post-training earnings.

10 Reference

Ashenfelter, Orley, and David Card. 1985. “Using the Longitudinal Structure of Earnings to Estimate the Effect of Training Programs.” The Review of Economics and Statistics 67 (4): 648. https://doi.org/10.2307/1924810.