1.1 Can it be useful for you?

I recently presented a paper using the synthetic control method to understand the impact of a California policy. I received a lot of great feedback–even if the feedback made me understand how far I have to go as a presenter, as an economist, and with this paper in particular–and one of the comments was that my synthetic controls had somewhat poor pre-period fits. This is actually quite rare for synthetic controls; in fact, the issue most people point out is that (especially when we have long pre-periods), SCM has a tendency to overfit. Regardless, we got into a discussion of how to alleviate any concerns folks would have over my method. I have multiple related outputs (i.e. a factor model might find some strong common factors across the outcomes), and we discussed what would happen if we used a multiple-outcome synthetic control.

As I learn more about this method, I thought it would be best to write about it. This helps me, selfishly, because I have to feel somewhat confident before putting this on my website. I also thought it might be good to get an overview from someone less statistically nuanced. I might end up putting together a presentation on this for my graduate group brown bag and, if so, will post that here.

This post is meant to be a setup note on the difference between the standard synthetic control method in Abadie, Diamond, and Hainmueller (2010) and the newer multi-outcome extensions in Tian, Lee, and Panchenko (2026) and Sun, Ben-Michael, and Feller (2025). The goal is not to cover every identification result, but to write down what changes in the implementation. It would be good, prior to presenting this, to briefly go over the proofs in the texts to understand any changes made to our typical assumptions, though they do not change all that much.

2 Setup and Notation

Suppose we observe units over periods. Unit is WOLG, the treated unit. The donor units are indexed by . Treatment begins after period , so there are pre-treatment periods and post-treatment periods.

Let denote the untreated potential outcome for unit at time , and let denote the treated potential outcome. For the treated unit, we observe

For donor units, which are never treated, we observe for all . The missing object is the post-treatment untreated path for the treated unit:

Synthetic control estimates this missing path using a weighted average of donor units:

The weights live on the simplex

The intuition: we are just choosing the best weights to match the pre-treatment outcome while restricting the weights to be non-negative and sum to one. So if you are tired of reading up on difference-in-differences and worrying about negative weights, fear not.

3 Standard Synthetic Control

If we match only on the pre-treatment outcome path (something worth being careful about; check out this cherry-picking article), the standard synthetic control problem chooses donor weights that make the pre-treatment treated outcome look as close as possible to the weighted donor outcome path:

In matrix form, define

Then the same problem is

With the dimensions written underneath, something I tend to do since my econometrics class with George Evans who always paid particular attention to this, the implementation object is

Once we have , the period-specific treatment effect estimate is

The average post-treatment effect is

Then you can get into placebo inference, or there is some work on t-statistics for SCM I haven’t quite read up on too much.

4 Why Multiple Outcomes Help

Standard synthetic control can fit the pre-treatment path very tightly, especially when there are many donor units relative to the number of pre-treatment periods. That can be useful, but it also creates an overfitting concern: the weights can match noise in one outcome rather than the common structure that generates the untreated path. One of the first things we discuss in econometrics courses is that errors exist–we will never be able to model everything. Synthetic control sometimes tries to be an overzealous student in that regard.

The multiple-outcome idea is to use related outcomes to discipline the weight-selection problem. Instead of finding weights that only match one outcome, we estimate a common set of weights that performs well across several outcome series. In the language of the newer papers, this can help focus the fit on shared factors rather than outcome-specific noise.

5 Multi-Outcome SCM: Two Implementations

Now suppose there are outcomes. Let

denote outcome for unit in period , where . The main implementation change is that we still want one donor weight vector , but we use multiple outcomes to choose it.

There are two main ways that we can do this. The first is probably what you were initially considering when you found out people did multiple-outcome synthetic control: find the weights that minimize the distance of a whole host of outcomes. The second is an averaging method that, frankly, sounded odd to me at first but has some perks that I will go over.

6 Normalization

One last thing that matters more than I initially wanted it to: the outcomes need to be put on comparable footing. Otherwise, stable level differences across units or outcomes can end up doing a lot of work in the optimization problem. The math is not being mean; it is just doing exactly what we asked it to do.

The normalization used in these papers is pretty simple. They use demeaned outcomes, where each unit-outcome series is centered by its own pre-treatment mean:

So rather than matching on the raw level of each outcome, we are matching on movement relative to that unit’s own pre-period average. This is also why it is sometimes described as an intercept-shifted synthetic control. We let units differ by a stable level difference, but still ask the synthetic control to track the treated unit’s dynamics.

In practice, this means the concatenation or averaging step above is done with instead of . Then, when we construct the counterfactual for a specific outcome, we can add the treated unit’s pre-treatment mean back in:

The short version: before estimating the weights, subtract each unit’s pre-treatment average for each outcome. This is not a separate research design choice so much as a practical step that keeps the multiple outcomes from turning into a contest of levels.

7 Matrix Size Comparison

So all in all, if you understand the base SCM well, it is a fairly simple jump to the multi-outcome methods described here.

TL;DR:

Method	Outcomes Used For Weights	Treated Pre Matrix	Donor Pre Matrix	Weight Vector	Main Idea
Standard SCM					Match one treated outcome path
Multi-outcome concatenation					Match all outcome paths jointly
Multi-outcome average/index					Match an averaged/indexed outcome path

8 Conclusion

When I have a working paper version of what I’m doing right now, after I implement the feedback I got before, I’ll post it and you can see the results for yourself. For the most part, my results are stable to whether I run many separate single-outcome SCMs or the two different types of multi-outcome SCMs.

2 Review

As a quick refresher for what we’re doing here, I’ll write in words what the random inference procedure we need to do is, then show you how to implement it with the assumption you’ve created a panel for your synthetic control already. I’m using the notation from the synthdid package in this case.

We observe one treated unit and donor units over periods. Let denote the number of pre-treatment periods and the number of post-treatment periods.

Let:

• denote the outcome for the treated unit
  
• denote the outcome for donor unit

3 Implementing in synthdid

# Assumes you already have: setup <- panel.matrices(...)
# setup$Y is (N x T), rownames(setup$Y) are unit names, setup$T0 is pre-period length.
# This loops over EVERY unit, treats it as "treated" by moving it to the last row,
# runs sc_estimate, computes pre/post RMSE and MSPE ratios, and ranks California.

library(synthdid)

sc_placebo_inference <- function(setup, stat = c("mspe", "rmse")) {
  stat <- match.arg(stat)

  Y0 <- setup$Y
  T0 <- setup$T0
  TT <- ncol(Y0)
  units <- rownames(Y0)
  N <- nrow(Y0)

  out <- data.frame(
    unit = units,
    att = NA_real_,
    pre_mspe = NA_real_,
    post_mspe = NA_real_,
    pre_rmse = NA_real_,
    post_rmse = NA_real_,
    ratio_rmse = NA_real_,
    ratio_mspe = NA_real_
  )

  for (i in seq_len(N)) {
    # reorder Y so unit i is last (treated)
    ord <- c(setdiff(seq_len(N), i), i)
    Y <- Y0[ord, , drop = FALSE]

    run <- synthdid::sc_estimate(Y, N - 1, T0)
    out$att[i] <- as.numeric(run)

    # Use the full donor weight vector (not summary(run)$controls display output).
    w_full <- attr(run, "weights")$omega

    # synthetic path and gaps
    y_synth <- as.numeric(t(w_full) %*% Y[1:(N - 1), , drop = FALSE])
    y_tr <- as.numeric(Y[N, ])
    gap     <- y_tr - y_synth

    pre_mspe <- mean(gap[1:T0]^2, na.rm = TRUE)
    post_mspe <- mean(gap[(T0 + 1):TT]^2, na.rm = TRUE)
    pre_rmse <- sqrt(pre_mspe)
    post_rmse <- sqrt(post_mspe)

    out$pre_mspe[i] <- pre_mspe
    out$post_mspe[i] <- post_mspe
    out$pre_rmse[i]  <- pre_rmse
    out$post_rmse[i] <- post_rmse
    out$ratio_rmse[i] <- post_rmse / pre_rmse
    out$ratio_mspe[i] <- post_mspe / pre_mspe
  }

  ratio_vec <- if (stat == "mspe") out$ratio_mspe else out$ratio_rmse
  treated_unit <- rownames(setup$Y)[setup$N0 + 1]
  R_treated <- ratio_vec[out$unit == treated_unit]
  R_placebo <- ratio_vec[out$unit != treated_unit]
  p_val <- (1 + sum(R_placebo >= R_treated, na.rm = TRUE)) /
    (1 + sum(is.finite(R_placebo)))

  table_ranked <- out[order(ratio_vec, decreasing = TRUE), ]
  table_ranked$rank <- seq_len(nrow(table_ranked))
  rank_treated <- table_ranked$rank[table_ranked$unit == treated_unit]

  list(
    treated_unit = treated_unit,
    statistic = stat,
    R_treated = R_treated,
    rank_treated = rank_treated,
    p_val = p_val,
    table = table_ranked
  )
}

# Example:
setup <- panel.matrices(california_prop99)
res <- sc_placebo_inference(setup, stat = "mspe")
res[c("treated_unit", "statistic", "R_treated", "rank_treated", "p_val")]

$treated_unit
[1] "California"

$statistic
[1] "mspe"

$R_treated
[1] 154.94

$rank_treated
[1] 3

$p_val
[1] 0.07692308

require(tidysynth)

Loading required package: tidysynth

Warning: package 'tidysynth' was built under R version 4.4.3

data("smoking")

smoking_out <-
  smoking %>%

  synthetic_control(
    outcome = cigsale,
    unit    = state,
    time    = year,
    i_unit  = "California",
    i_time  = 1988,
    generate_placebos = TRUE
  ) %>%

  # Outcome-only matching:
  # use the pre-treatment outcome path (or a subset of pre-years) as predictors.
  # Here I include every pre-treatment year up through 1988.
  generate_predictor(
    time_window = 1970:1988,
    cigsale_pre = mean(cigsale, na.rm = T)
  ) %>%

  generate_weights(
    optimization_window = 1970:1988,
    margin_ipop = .02, sigf_ipop = 7, bound_ipop = 6
  ) %>%

  generate_control()

  head(smoking_out %>% grab_significance())

# A tibble: 6 × 8
  unit_name     type    pre_mspe post_mspe mspe_ratio  rank fishers_exact_pvalue
  <chr>         <chr>      <dbl>     <dbl>      <dbl> <int>                <dbl>
1 West Virginia Donor      13.6      351.        25.8     1               0.0256
2 Illinois      Donor       9.36     202.        21.6     2               0.0513
3 California    Treated    61.4     1004.        16.3     3               0.0769
4 Missouri      Donor      23.4      324.        13.8     4               0.103 
5 Indiana       Donor      34.8      453.        13.0     5               0.128 
6 South Dakota  Donor       6.92      70.2       10.1     6               0.154 
# ℹ 1 more variable: z_score <dbl>

2 Assumptions of the Model

We start off assuming that there exists a total cost function of quantity supplied and quality supplied .

We also assume there is a function of quantity demanded, as a function of price (given) and quality demanded

s.t. and

We assume that non-profit firms make zero profit, thus . Since and $ TR = P Q$ (abusing notation of quantity and quality before establishing equilibrium conditions, sorry), and further assuming that we can solve for an inverse demand function , we have the setup for the breakevn condition which will allow us to solve for quantity as a function of price.

In particular, we have:

which allows us to obtain equlibrium quality as a function of equilibrium quantity:

This sets up the maximization problem, where both quality and quantity are part of the DM problem, and we can solve for them both assuming we havethe quantity-quality tradeoff.

3 DM Problem

The hospital administrator choose and to maximizes the institutions utility. We subject this to the quantity-quality tradeoff which builds in the zero-profit/break-even condition already.

we can then get first order conditions

We can then combine these: substitute for from Equation 4 into Equation 5 to get that

Newhouse discusses some concavity conditions to ensure an interior solution, mostly some second-order conditions.

4 Implications

The optimality condition tells us that the MRS of quantity and quality is equal to the slope of the break-even condition. In economic terms, this condition means the administrator will expand quantity up to the point where the value of treating one more patient (in utility terms) is exactly offset by the reduction in quality that must occur to remain at break-even.

If the break-even locus is steep (large (h’(Q))), then each additional patient requires a substantial reduction in quality, so the hospital leans toward protecting quality. If the locus is relatively flat, the hospital can expand patient volume with only modest quality sacrifices, so output increases. This helps explain observed nonprofit hospital behavior: rather than maximizing profit or output alone, they balance prestige- and reputation-enhancing quality with the social goal of treating more patients.

Newhouse argued this framework accounts for investments in high-cost technology, the provision of uncompensated care, and other behaviors that appear puzzling under a pure profit-maximization model but are consistent with a utility-maximizing nonprofit decision maker.

I had Chat-GPT compare how this might look, the first image is basically just what you would typically think of in a utility maximization problem but with different labels, the

library(ggplot2)
library(dplyr)
library(tibble)

# ---- Parameters (tweak to taste) ----
k0    <- 8      # base quality intercept of break-even
b     <- 0.6    # linear cost/demand pressure on quality as Q increases
c     <- 0.05   # curvature of break-even; >0 makes the locus get steeper in Q
alpha <- 0.55   # weight on Q in utility U(K,Q) = alpha*ln(Q) + (1-alpha)*ln(K)

# ---- Functions ----
h     <- function(Q) k0 - b*Q - c*Q^2                 # break-even locus: K as fn of Q
hdash <- function(Q) -b - 2*c*Q                       # slope dK/dQ of break-even
Ufun  <- function(Q,K) alpha*log(Q) + (1-alpha)*log(K)

# MRS = -U_Q/U_K = -(alpha/Q)/((1-alpha)/K) = - (alpha/(1-alpha)) * (K/Q)
MRS   <- function(Q,K) - (alpha/(1-alpha)) * (K/Q)

# ---- Tangency (solve MRS = h'(Q) along K = h(Q)) ----
# Condition: - (alpha/(1-alpha)) * (h(Q)/Q) = hdash(Q)
# Rearranged root condition: h(Q) - ((1-alpha)/alpha) * Q * (b + 2*c*Q) = 0
root_fn <- function(Q) h(Q) - ((1-alpha)/alpha) * Q * (b + 2*c*Q)

# search interval where K>0 and Q>0; adjust if needed
Q_star <- uniroot(root_fn, interval = c(0.2, 8))$root
K_star <- h(Q_star)
U_star <- Ufun(Q_star, K_star)

# ---- Data for plotting ----
Q_grid <- seq(0.2, 8, length.out = 400)
df     <- tibble(
  Q = Q_grid,
  K = pmax(h(Q_grid), 0.05)   # keep K positive for log utility
) %>% filter(K > 0)

# Utility levels for indifference curves (one through the tangency, ± offsets)
u_levels <- c(U_star - 0.35, U_star, U_star + 0.35)

# Make a grid for contours
Q_seq <- seq(0.2, 8, length.out = 200)
K_seq <- seq(0.2, 8, length.out = 200)
grid  <- expand.grid(Q = Q_seq, K = K_seq) %>%
  mutate(U = Ufun(Q, K))

# ---- Optional: tangent line at (Q*,K*) to visualize tangency ----
slope_tan <- hdash(Q_star)                 # dK/dQ at the tangency
K_tan     <- function(Q) K_star + slope_tan*(Q - Q_star)
tan_df    <- tibble(Q = Q_grid, K = K_tan(Q_grid)) %>% filter(K > 0)

# ---- Plot ----
ggplot() +
  # Indifference curves (three levels)
  geom_contour(data = grid, aes(Q, K, z = U), breaks = u_levels, linewidth = 0.6) +
  # Break-even locus
  geom_line(data = df, aes(Q, K), linewidth = 1) +
  # Tangency point
  geom_point(aes(Q_star, K_star), size = 3) +
  # Tangent line at the optimum
  geom_line(data = tan_df, aes(Q, K), linewidth = 0.4, linetype = "longdash") +
  annotate("text", x = Q_star + 0.4, y = K_star + 0.4,
           label = "Tangency (Q*, K*)", hjust = 0, size = 3.5) +
  labs(x = "Quantity (Q)", y = "Quality (K)",
       caption = "Indifference curves: U(K,Q) = alpha ln Q + (1-alpha) ln K (solid: through optimum).\nBreak-even locus: K = k0 - b Q - c Q^2. Tangency shows MRS = slope of constraint.") +
  coord_cartesian(xlim = c(0, 8), ylim = c(0, 8)) +
  theme_minimal(base_size = 12)

library(ggplot2)
library(dplyr)
library(tibble)

# utility weight (same across panels)
alpha <- 0.55

# helper to build one scenario's data
build_scenario <- function(b, c, label, k0 = 8) {
  h     <- function(Q) k0 - b*Q - c*Q^2
  hdash <- function(Q) -b - 2*c*Q
  Ufun  <- function(Q,K) alpha*log(Q) + (1-alpha)*log(K)
  root_fn <- function(Q) h(Q) - ((1-alpha)/alpha) * Q * (b + 2*c*Q)
  Q_star  <- tryCatch(uniroot(root_fn, interval = c(0.2, 8))$root,
                      error = function(e) 3.5)
  K_star  <- h(Q_star)
  U_star  <- Ufun(Q_star, K_star)

  Q_grid <- seq(0.2, 8, length.out = 400)
  locus  <- tibble(Q = Q_grid, K = pmax(h(Q_grid), 0.05), scenario = label) %>% filter(K > 0)

  Q_seq <- seq(0.2, 8, length.out = 200)
  K_seq <- seq(0.2, 8, length.out = 200)
  grid  <- expand.grid(Q = Q_seq, K = K_seq) %>% mutate(U = Ufun(Q, K), scenario = label)

  slope_tan <- hdash(Q_star)
  K_tan     <- function(Q) K_star + slope_tan*(Q - Q_star)
  tan_df    <- tibble(Q = Q_grid, K = K_tan(Q_grid), scenario = label) %>% filter(K > 0)

  list(
    locus = locus,
    grid  = grid,
    ulevels = c(U_star - 0.35, U_star, U_star + 0.35),
    star  = tibble(Q = Q_star, K = K_star, scenario = label),
    tan   = tan_df
  )
}

flat   <- build_scenario(b = 0.4, c = 0.01, label = "Flat break-even locus")
steep  <- build_scenario(b = 1.2, c = 0.15, label = "Steep break-even locus")

locus   <- bind_rows(flat$locus, steep$locus)
grid    <- bind_rows(flat$grid, steep$grid)
stars   <- bind_rows(flat$star, steep$star)
tangent <- bind_rows(flat$tan, steep$tan)

# map of scenario -> its contour levels
ulevels_by <- list(
  "Flat break-even locus"  = flat$ulevels,
  "Steep break-even locus" = steep$ulevels
)

# Draw contours separately per scenario
ggplot() +
  geom_contour(
    data = grid %>% filter(scenario == "Flat break-even locus"),
    aes(Q, K, z = U),
    breaks = flat$ulevels, linewidth = 0.6
  ) +
  geom_contour(
    data = grid %>% filter(scenario == "Steep break-even locus"),
    aes(Q, K, z = U),
    breaks = steep$ulevels, linewidth = 0.6
  ) +
  geom_line(data = locus, aes(Q, K)) +
  geom_point(data = stars, aes(Q, K), size = 3) +
  geom_line(data = tangent, aes(Q, K), linewidth = 0.4, linetype = "longdash") +
  facet_wrap(~ scenario, nrow = 1) +
  labs(x = "Quantity (Q)", y = "Quality (K)") +
  coord_cartesian(xlim = c(0, 8), ylim = c(0, 8)) +
  theme_minimal(base_size = 12)

5 Extension and Relevance to Treatment Centers

The question for myself, now, is whether or not this truly applies to treatment centers? What comes to mind are the sort of luxury spa-esque TCs, very high quality with low output. Newhouse’s model does seem to predict these sorts of cases. A possible extension to review is Pauly & Redisch’s The Non-For-Profit Hospital as a Physicians’ Cooperative. This paper may be interesting to review in light of the many opioid prescribing papers in health econ literature.

It also brings to mind how we might measure quality. SAMHSA keeps track of service offerings for each TC. Would we consider TCs with a high level of quality those who offer tons of services, or is that a high-quantity TC with possibly low quality in each of these areas? Are TCs that have a high availability of MAT lower quality, since they treat so many individuals? This also ties into the idea of how we might measure treatment itself; in all of this work, I consistently combine different types of treatment, but the heterogeneity of TC characteristics is certainly important to consider.

Regardless, I’m thankful for Newhouse’s contribution — the quantity-quality tradeoff and the model that spurred much work — for allowing me to think of these questions.

Series Posts

Series Posts