Panel Data Approach (PDA) ========================= .. currentmodule:: mlsynth When to Use This Estimator -------------------------- The panel data approach (PDA) of Hsiao, Ching and Wan [HCW]_ estimates a single treated unit's untreated counterfactual by a linear regression on the control units, fit over the pre-treatment window and extrapolated out-of-sample. Unlike the synthetic control method, PDA imposes no constraints on the coefficients (no simplex, no non-negativity), so it is a plain projection justified by a latent-factor model: if every unit loads on a small set of common factors, the treated unit's untreated path is a linear combination of the controls' paths plus an orthogonal error. The challenge is which controls and how many. Classical PDA was built for low dimensions (few controls relative to pre-periods) and chooses controls by AIC/BIC, which break down once the number of controls ``N`` approaches or exceeds the pre-period length ``T0``. ``mlsynth`` packages three high-dimensional PDA variants that resolve this differently, each with the estimation and inference theory of its own paper: * L2-relaxation (``l2``; Shi & Wang [l2relax]_) -- a *dense* estimator (a "cousin of ridge") for when the factor model makes the projection coefficients dense; tolerates ``N > T0``; prediction is robust to heteroskedasticity. * LASSO (``lasso``; Li & Bell [LASSOPDA]_) -- an L1 estimator that *selects* a sparse set of relevant controls; computationally far cheaper than AIC/BIC and works for ``N > T0``. * Forward selection (``fs``; Shi & Huang [fsPDA]_) -- a greedy procedure that grows the control set one unit at a time, with valid post-selection inference and no sparsity requirement (works for dense *or* sparse models). All three target the single-treated-unit, many-candidate-controls regime and produce a time-varying treatment effect and an average treatment effect (ATE) with a HAC-based confidence interval. The practical choice among them (detailed below) follows the authors' own arguments: ``l2`` when the coefficients are dense, ``lasso`` when only a few controls matter and you want an interpretable selection, ``fs`` when you want a cheap, predictive ensemble with honest post-selection inference regardless of sparsity. Notation -------- We adopt the notation of Shi & Wang [l2relax]_. For a positive integer ``n``, :math:`[n] = \{1, \ldots, n\}`; :math:`\mathbf{I}_n` is the identity, :math:`\mathbf{1}_n`, :math:`\mathbf{0}_n` the all-ones/zeros vectors. For a matrix :math:`\mathbf{A} = (a_{ij})` and index sets :math:`\mathcal{S}, \mathcal{Q}`, the submatrix is :math:`\mathbf{A}_{\mathcal{S}\mathcal{Q}} = (a_{ij})_{i\in\mathcal{S}, j\in\mathcal{Q}}` and the subvector :math:`\mathbf{x}_{\mathcal{Q}} = (x_i)_{i\in\mathcal{Q}}`. :math:`\phi_{\min}(\cdot)`, :math:`\phi_{\max}(\cdot)` denote the smallest and largest eigenvalues; :math:`\|\mathbf{A}\|_\infty = \max_{ij}|a_{ij}|`, :math:`\|\mathbf{A}\|_2 = \sqrt{\phi_{\max}(\mathbf{A}'\mathbf{A})}`, :math:`\|\mathbf{x}\|_1 = \sum_i |x_i|`. The two time-series operators are central. Over an index set :math:`\mathcal{S}` of periods, .. math:: \mathcal{E}_{\mathcal{S}}(\mathbf{x}_t) = \frac{1}{|\mathcal{S}|} \sum_{t\in\mathcal{S}} \mathbf{x}_t, \qquad \Gamma_{\mathcal{S}}(\mathbf{x}_t, \mathbf{y}_t') = \mathcal{E}_{\mathcal{S}} \Bigl( [\mathbf{x}_t - \mathcal{E}_{\mathcal{S}}(\mathbf{x}_t)] [\mathbf{y}_t - \mathcal{E}_{\mathcal{S}}(\mathbf{y}_t)]' \Bigr), the sample mean and sample covariance of a series over :math:`\mathcal{S}`. The intervention takes effect after the split point :math:`T_0`. The pre-period is :math:`\mathcal{T}_1 \coloneqq \{1, \ldots, T_0\}` with :math:`|\mathcal{T}_1| = T_0`; the post-period is :math:`\mathcal{T}_2 \coloneqq \{T_0+1, \ldots, T\}` with :math:`T_2 \coloneqq |\mathcal{T}_2| = T - T_0`. Let :math:`j=1` denote the treated unit, with treated series :math:`\mathbf{y}_1 = (y_{11}, \ldots, y_{1T})^\top`; the donor pool is :math:`\mathcal{N}_0 \coloneqq \mathcal{N}\setminus\{1\}`, with cardinality :math:`N_0`, where :math:`\mathcal{N} = \{1,\ldots,N\}`. The shared model ^^^^^^^^^^^^^^^^ All three methods rest on a common latent-factor data-generating process: for :math:`t \in \mathcal{T}`, .. math:: y_{jt}^0 = \mu_j + \boldsymbol{\lambda}_j' \mathbf{f}_t + u_{jt}, \quad j \in \{1\}\cup\mathcal{N}_0, with :math:`\mathbf{f}_t` a :math:`q`-vector of latent common factors, :math:`\boldsymbol{\lambda}_j` factor loadings, and :math:`u_{jt}` a weakly-dependent idiosyncratic error orthogonal to the factors. Because the common factors drive both the treated unit and the donors, the untreated treated outcome admits a linear projection on the donors, .. math:: y_{1t} = \alpha^0 + \mathbf{x}_t' \boldsymbol{\beta}^0 + \epsilon_t, \qquad \mathbf{x}_t = (y_{jt})_{j\in\mathcal{N}_0}', with :math:`\mathbb{E}[\mathbf{x}_t \epsilon_t] = \mathbf{0}`. PDA fits :math:`(\alpha^0, \boldsymbol{\beta}^0)` on :math:`\mathcal{T}_1` and predicts :math:`\widehat{y}_{1t} = \mathcal{E}_{\mathcal{T}_1}(y_{1s}) + [\mathbf{x}_t - \mathcal{E}_{\mathcal{T}_1}(\mathbf{x}_s)]'\widehat{\boldsymbol{\beta}}` for :math:`t \in \mathcal{T}_2`. The per-period treatment effect is :math:`\tau_t = y_{1t} - \widehat{y}_{1t}` and the ATE :math:`\widehat{\tau} = \mathcal{E}_{\mathcal{T}_2}(\tau_t)`. The methods differ in how they estimate :math:`\boldsymbol{\beta}` and, crucially, in the inference theory each paper proves for :math:`\widehat{\tau}`. L2-relaxation (``l2``, Shi & Wang) ---------------------------------- Idea. Under the factor model the projection coefficient :math:`\boldsymbol{\beta}^0 = \boldsymbol{\Omega}^{-1}\boldsymbol{\Lambda} (\boldsymbol{\Lambda}'\boldsymbol{\Omega}^{-1}\boldsymbol{\Lambda} + \mathbf{I}_q)^{-1}\boldsymbol{\lambda}_1` is dense in general -- almost no entries are exactly zero. Sparse methods (LASSO) are then mis-matched. With :math:`\widehat{\boldsymbol{\Sigma}} = \Gamma_{\mathcal{T}_1}(\mathbf{x}_t, \mathbf{x}_t')` and :math:`\widehat{\boldsymbol{\eta}} = \Gamma_{\mathcal{T}_1}(\mathbf{x}_t, y_t)`, OLS solves the KKT condition :math:`\widehat{\boldsymbol{\Sigma}}\boldsymbol{\beta} = \widehat{\boldsymbol{\eta}}`, which is unstable or non-unique once :math:`N` is close to or exceeds :math:`T_0`. L2-relaxation relaxes the sup-norm of this moment condition by a tuning parameter :math:`\varepsilon` and minimizes the coefficient norm: .. math:: \min_{\boldsymbol{\beta}} \tfrac{1}{2}\|\boldsymbol{\beta}\|_2^2 \quad \text{s.t.} \quad \|\widehat{\boldsymbol{\eta}} - \widehat{\boldsymbol{\Sigma}}\boldsymbol{\beta}\|_\infty \le \varepsilon. This is the "bias-variance trade-off" made explicit: tolerating a small violation :math:`\varepsilon` of the OLS moment condition shrinks the variance. At :math:`\varepsilon = 0` it reduces to (ridgeless) OLS; at :math:`\varepsilon \ge \|\widehat{\boldsymbol{\eta}}\|_\infty` it gives :math:`\boldsymbol{\beta} = \mathbf{0}`. ``mlsynth`` picks :math:`\varepsilon` by sequential out-of-sample validation on the tail of the training window (the validated :math:`\varepsilon` tracks the infeasible-optimal one, and both shrink toward zero as the sample grows) over a log-spaced grid down to :math:`10^{-4}\max|\widehat{\boldsymbol{\eta}}|` (the optimum is often a tiny fraction of the cap). This is time-respecting -- the fit never sees periods later than the validation tail -- unlike the released ``L2relax.CV``, whose 5-block K-fold trains on both past *and* future of each block. .. note:: Standardisation. Following the authors' released ``L2relax``, the treated and control series are standardised (demeaned and scaled to unit variance) before forming :math:`\widehat{\boldsymbol{\Sigma}}` / :math:`\widehat{\boldsymbol{\eta}}`, and the solution is mapped back to the original scale. This is the default (``l2_standardize=True``) -- the :math:`\ell_2` penalty is scale-sensitive, so standardisation is both recommended and what reproduces the paper's empirical results; on the Hong Kong panel it moves the L2 estimate from :math:`2.48\%` to :math:`2.61\%` (closer to Shi & Wang's :math:`2.65\%`). Set ``l2_standardize=False`` for the raw-scale variant. Assumptions (Shi & Wang). *Assumption 1 (loadings).* :math:`\|\boldsymbol{\lambda}_1\|_\infty + \|\boldsymbol{\Lambda}\|_\infty \le C`, and there is a :math:`q`-unit subset making :math:`\boldsymbol{\Lambda}_{\mathcal{Q}\cdot}` full column rank; the average factor strength :math:`\xi_N = \phi_{\min}(\boldsymbol{\Lambda}' \boldsymbol{\Lambda}/N)` may vanish (weak factors allowed). *Remark.* Bounded loadings keep the projection coefficient well-defined, while the full-rank subset is what lets the donors span the treated unit's factor exposure; allowing :math:`\xi_N \to 0` means the method tolerates weak factors that contribute little cross-sectional variation. *Assumption 2 (errors).* The idiosyncratic covariance :math:`\boldsymbol{\Omega}` has eigenvalues bounded between :math:`\underline\sigma^2` and :math:`\overline\sigma^2`; errors may be heteroskedastic and cross-sectionally dependent. *Remark.* The two-sided eigenvalue bound rules out a degenerate noise direction, but heteroskedastic, cross-sectionally dependent errors are permitted -- the realistic case for economic panels. *Assumption 3-4 (sampling).* In- and out-of-sample sampling errors of the sample moments are :math:`O_p(T_0^{-1/2})` for low-dimensional pieces and :math:`O_p(\sqrt{\log N / (N\wedge T_0)})` for high-dimensional ones (holding under time-series weak dependence, not just i.i.d.). *Remark.* These are the convergence rates that make the relaxed moment condition usable in high dimension: the high-dimensional pieces pay only a :math:`\sqrt{\log N}` price, so :math:`N` may grow with (or beyond) :math:`T_0`. *Assumption 5 (ATE inference).* The oracle prediction error :math:`\epsilon_t^*` and the effect-plus-error :math:`d_t^* = \Delta_t - \mathbb{E}[\Delta_t] + \epsilon_t^*` have finite, positive long-run variances :math:`\rho^2_{\epsilon^*}`, :math:`\rho^2_{d^*}` consistently estimable by HAC, and a sequential CLT applies. *Remark.* The coefficient estimator is consistent for the oracle target (Theorem 1) and -- importantly -- the prediction error is asymptotically *irrelevant to heteroskedasticity* (Theorem 2): unlike the coefficient MSE, the out-of-sample MSE does not depend on the noise heterogeneity :math:`\psi_{\max}`. This is what licenses the HAC-based CLT for the ATE. Inference (Shi & Wang, Theorem 3; single treated unit). With pre-period prediction residuals :math:`e_t = y_{1t} - \widehat{y}_{1t}` (:math:`t\in\mathcal{T}_1`) and post-period effects :math:`\tau_t` (:math:`t\in\mathcal{T}_2`), .. math:: \widehat{Z} = \frac{\widehat{\tau} - \Delta_{\mathcal{T}_2}} {\sqrt{\widehat{\rho}^2_{(1)}/T_0 + \widehat{\rho}^2_{(2)}/T_2}} \xrightarrow{d} N(0,1), where :math:`\widehat{\rho}^2_{(1)}` is the HAC long-run variance of the pre-period residuals (first-stage estimation uncertainty) and :math:`\widehat{\rho}^2_{(2)}` is the HAC long-run variance of the de-meaned post-period effects (post-period averaging). Both sources of uncertainty enter, which matters when :math:`T_0` and :math:`T_2` are comparable. When to use. Dense, factor-driven coefficients; high dimension (:math:`N>T_0` permitted); when prediction accuracy and heteroskedasticity- robustness matter more than identifying a handful of controls. LASSO (``lasso``, Li & Bell) ---------------------------- Idea. When only a *few* controls are truly relevant, an L1 penalty selects them and shrinks the rest. Li & Bell estimate .. math:: \widehat{\boldsymbol{\beta}}^{\text{las}} = \operatorname*{argmin}_{\boldsymbol{\beta}} \; \sum_{t\in\mathcal{T}_1} (y_{1t} - \mathbf{x}_t'\boldsymbol{\beta})^2 + \lambda \sum_{j} |\beta_j|, with :math:`\lambda` chosen by (leave-one-out) cross-validation, then predict the counterfactual as in the shared model. LASSO works for :math:`N > T_0` (where AIC/AICC/BIC cannot even be computed) and is far cheaper. Assumptions (Li & Bell). They *relax* HCW's linear-conditional-mean assumption and drop one of HCW's identification conditions. The key conditions are: a factor model with :math:`\mathrm{Rank}(\widetilde{B}) = K` (enough independent factor variation among the controls); a weakly dependent, weakly stationary panel so laws of large numbers and CLTs apply to partial sums; consistency of the pre-period least-squares pieces (:math:`\widehat{\delta}_1 - \delta_1, \widehat{\delta}-\delta = O_p(T_0^{-1/2})`); and :math:`\rho`-mixing with geometric decay plus a finite limit :math:`\eta = \lim T_2/T_0`. Sparsity (only :math:`m` of :math:`\boldsymbol{\beta}` non-zero, :math:`m` fixed or :math:`o(T_0)`) is assumed for the high-dimensional selection. *Remark.* Li & Bell prove consistency :math:`\widehat{\tau} - \Delta_{\mathcal{T}_2} = O_p(T_0^{-1/2} + T_2^{-1/2})` (estimation error has two parts: first-stage :math:`O_p(T_0^{-1/2})` and post-averaging :math:`O_p(T_2^{-1/2})`), holding even when :math:`y_{1t}` is trend-stationary. Inference (Li & Bell, Theorem 3.2). With :math:`\widehat{\Sigma} = \widehat{\Sigma}_1 + \widehat{\Sigma}_2`, .. math:: \text{T.S.} = \frac{\sqrt{T_2}\,\widehat{\tau}}{\sqrt{\widehat{\Sigma}}} \xrightarrow{d} N(0,1), where :math:`\widehat{\Sigma}_2` is the Newey-West HAC long-run variance of the post-period effects, and :math:`\widehat{\Sigma}_1` is the first-stage (pre-period estimation) variance -- the OLS prediction variance of the mean post-period counterfactual on the selected support. Li & Bell note :math:`\widehat{\Sigma}_1` is negligible when :math:`T_0 \gg T_2`, so the post-period term dominates in long-pre-period panels. When to use. A genuinely sparse set of relevant controls; very large :math:`N` (even :math:`N/T_0 \to \infty`); when an interpretable, computa- tionally cheap selection is preferred. (For selection *consistency*, Li & Bell note the adaptive LASSO; for prediction, plain LASSO already beats AIC/BIC and leave-many-out CV in their simulations.) Forward selection (``fs``, Shi & Huang) --------------------------------------- Idea. Rather than penalize, *grow* the control set greedily. Start empty; at each step add the control whose inclusion maximizes the pre-treatment OLS :math:`R^2` (equivalently minimizes the residual sum of squares). The number of selected controls :math:`R` is a tuning parameter chosen by a modified BIC (Wang, Li & Tsai), .. math:: \widehat{R} = \operatorname*{argmin}_{r} \log\bigl(\widehat{\sigma}^2(\widehat{U}_r)\bigr) + \log(\log N)\,\frac{r\,\log T_0}{T_0}, with :math:`\widehat{\sigma}^2(\widehat{U}_r)` the pre-period residual variance of OLS on the :math:`r`-unit set. The counterfactual is the OLS extrapolation on the chosen set :math:`\widehat{U}_{\widehat{R}}`. Forward selection evaluates :math:`\sum_r (N-r+1)` regressions -- *linear* in :math:`N` -- versus the :math:`2^N` of exhaustive subset search. Assumptions (Shi & Huang). Asymptotics are *multi-index*: :math:`N\to\infty` with :math:`T_0 = T_0(N)` deterministic, :math:`\log N / T_0 \to 0`, and :math:`T_2 = T_2(N) \to \infty` with :math:`\log N / T_2 \to 0` (:math:`N` may exceed :math:`T_0`). *Assumption 1 (sparse Riesz / restricted eigenvalue).* The minimal eigenvalue of the population Gram matrix over any :math:`u`-unit subset (:math:`u \le (1+\delta_1)R`) is bounded below -- a condition the authors show is a natural implication of the latent factor model, not an ad hoc restriction. *Remark.* The bounded minimal eigenvalue is what keeps any candidate subset of donors well-conditioned, so the greedy step can identify the next informative control rather than chase a near-singular direction; that it follows from the factor model means it is not an extra demand on the data. *Assumption 2 (second moments).* Sample second moments converge at the high-dimensional rate :math:`O_p(\sqrt{\log N / T_0})` with bounded fourth moments. *Remark.* The high-dimensional rate is the pre-period price of admitting many donors: with bounded fourth moments the sample Gram matrix tracks its population counterpart even when :math:`N` is large relative to :math:`T_0`. *Assumption 3 (post-period).* Analogous convergence and long-run-variance bounds on the post-treatment data. *Remark.* The matching post-period bounds are what let the average effect over :math:`\mathcal{T}_2` obey a central limit theorem with a consistently estimable long-run variance, the basis for the post-selection test below. *Assumption 4 (weak dependence).* The series are strong (:math:`\alpha`-) mixing with geometric decay, so a Berry-Esseen bound for heterogeneous time series applies. *Remark.* The validity is uniform over a class of DGPs (Theorem 1) -- it holds whether the true coefficients are dense or sparse, which separates fsPDA from the post-selection-inference literature that needs sparsity or the oracle property. Theorem 2 shows the greedy algorithm attains a regression variance asymptotically as small as the best :math:`u`-unit subset, so the cheap forward search is statistically efficient. Inference (Shi & Huang, Eq. 4). Because forward selection uses only the pre-period and, under weak dependence, the pre- and post-periods become asymptotically independent (sample splitting), the naive conditional t-statistic is valid: .. math:: \widehat{\mathcal{Z}}_{\widehat{U}} = \widehat{\rho}_{\tau\widehat{U}}^{-1}\sqrt{T_2}\, \widehat{\tau}_{\widehat{U}} \xrightarrow{d} N(0,1), where :math:`\widehat{\rho}^2_{\tau\widehat{U}}` is the HAC long-run variance of the de-meaned post-period effects. No first-stage variance term is needed -- the asymptotic independence absorbs it -- which makes fsPDA's inference the simplest of the three. .. note:: Long-run-variance estimator. ``mlsynth`` defaults to the prewhitened Newey-West estimator (Andrews-Monahan VAR(1) prewhitening + Bartlett kernel with the data-driven NW(1994) bandwidth + finite-sample adjustment) -- R's ``sandwich::lrvar(..., prewhite = TRUE, adjust = TRUE)``, which Shi & Huang use in their application scripts. Prewhitening is essential when the treatment-effect series is strongly serially dependent: monthly growth rates mean-revert (lag-1 autocorrelation around :math:`-0.45` in the luxury-watch panel), and a plain Bartlett kernel cannot absorb that, leaving :math:`\widehat\rho` nearly double its true value and the test far too conservative. Setting ``lrvar_lag`` instead switches to the released ``est.fsPDA`` package's fixed-lag Bartlett estimator (default lag :math:`\lfloor T_2^{1/4}\rfloor`, capped at :math:`\lfloor\sqrt{T_2}\rfloor`); on the watch panel that no-prewhitening form gives an insignificant :math:`t \approx -1.15`, versus the prewhitened default's :math:`-2.51` (the paper reports :math:`-2.457`). When to use. A large candidate-control pool where the goal is to *synthesize an ensemble* that mimics the outcome (not to interpret which controls are "causal"); when computational efficiency and honest post-selection inference matter; and regardless of whether the underlying model is sparse. (Shi & Huang recommend the adaptive LASSO instead when the *identity* of a few causal controls is the object of interest.) Choosing among the three ------------------------- .. list-table:: :header-rows: 1 :widths: 12 30 28 30 * - Method - Coefficient structure - Inference variance - Use when * - ``l2`` - dense (factor-implied) - pre + post HAC (both terms) - dense coefficients; ``N>T0``; prediction & heteroskedasticity-robustness * - ``lasso`` - sparse - post HAC (+ first stage, small if ``T0>>T2``) - few relevant controls; interpretable selection; very large ``N`` * - ``fs`` - dense or sparse - post HAC only (sample splitting) - large pool; predictive ensemble; cheap; honest post-selection inference Shared assumptions across the PDA class --------------------------------------- The three estimators (``l2``, ``lasso``, ``fs``) differ in how they fit :math:`\boldsymbol\beta`, but they share the same identifying stack. Stated formally: A1 (Latent factor model for untreated outcomes). All :math:`N + 1` units share at most :math:`q` common latent factors, .. math:: y_{jt}^0 \;=\; \mu_j \;+\; \boldsymbol\lambda_j' \mathbf f_t \;+\; u_{jt}, \qquad j \in \{1\} \cup \mathcal N_0, \;\; t \in \mathcal T, with :math:`\mathbb E[\mathbf f_t u_{jt}] = 0`. This is the *shared model* underlying HCW (Hsiao-Ching-Wan 2012), Li-Bell (2017), Shi-Huang (2023), and Shi-Wang (l2). The factor structure is what licenses the linear projection of the treated unit's untreated outcome on the controls' outcomes, .. math:: y_{1t} \;=\; \alpha^0 \;+\; \mathbf x_t' \boldsymbol\beta^0 \;+\; \epsilon_t, \qquad \mathbb E[\mathbf x_t \epsilon_t] = \mathbf 0, with :math:`\boldsymbol\beta^0 = \boldsymbol\Omega^{-1} \boldsymbol\Lambda (\boldsymbol\Lambda' \boldsymbol\Omega^{-1} \boldsymbol\Lambda + \mathbf I_q)^{-1} \boldsymbol\lambda_1`. A2 (Single treated unit, sharp absorbing aggregate-level treatment). Unit :math:`j = 1` is the only treated unit; treatment turns on at :math:`T_0 + 1` and stays on. Donors are untreated throughout (no interference). The original HCW / Li-Bell / Shi-Huang theorems are stated for this single-treated case. (The l2-relaxation paper Section 4.4 sketches a multiple-treated-units extension with a short post-window; the mlsynth implementation tracks the single-treated form.) A3 (Weak temporal dependence). The series :math:`(\mathbf x_t, y_{1t})` are :math:`\rho`-mixing or strong- mixing with at-least-geometric decay (the exact rate varies by variant): * Li-Bell A6: :math:`w_t = (\widetilde y_t', \epsilon_{1t})` is a weakly stationary :math:`\rho`-mixing process with :math:`\rho(s) = O(\lambda^s)`. * Shi-Huang A4: strong (:math:`\alpha`-) mixing with geometric decay, so a Berry-Esseen bound for heterogeneous time series applies. * Shi-Wang A3-A4: time-series weak dependence at the :math:`O_p(T_0^{-1/2})` and :math:`O_p(\sqrt{\log N / (N \wedge T_0)})` rates for the sample moments. This is what makes pre-period sample moments converge at the high-dimensional rate and -- crucially -- what makes the pre-period and post-period asymptotically independent, which is the engine behind fs-PDA's sample-splitting inference and the two-term HAC variance in l2 / lasso. A4 (Sample-size regime). :math:`N \to \infty`, :math:`T_0 = T_0(N) \to \infty` deterministically with :math:`\log N / T_0 \to 0`, :math:`T_2 \to \infty` with :math:`\log N / T_2 \to 0`. :math:`N` may exceed :math:`T_0`, which is the entire point of the high-dimensional PDA literature. Li-Bell's A7 additionally posits :math:`\eta = \lim T_2 / T_0 \in [0, \infty)`, which determines whether the first-stage variance term :math:`\widehat\Sigma_1` matters. A5 (Donor pool regularity). The controls' Gram matrix has enough variation: * For ``lasso`` (Li-Bell A2): :math:`\mathrm{Rank}(\widetilde B) = K` -- removing the first row of the loading matrix leaves full-rank factor variation; :math:`E[x_t x_t']` is invertible on the active set. * For ``fs`` (Shi-Huang A1): a sparse Riesz / restricted eigenvalue condition -- the minimum eigenvalue of the population Gram matrix over any :math:`u`-unit subset (:math:`u \le (1 + \delta_1) R`) is bounded below. The paper shows this is a *natural implication* of the latent factor model, not an ad-hoc lasso-style assumption. * For ``l2`` (Shi-Wang A1-A2): factor strength :math:`\xi_N = \phi_{\min}(\boldsymbol\Lambda' \boldsymbol\Lambda / N)` may vanish (weak factors allowed); the idiosyncratic covariance has eigenvalues bounded in :math:`[\underline\sigma^2, \overline\sigma^2]`. A6 (Variant-specific structure of :math:`\boldsymbol\beta^0` ). * ``lasso``: sparse :math:`\boldsymbol\beta^0` -- only :math:`m = o(T_0)` of its components are non-zero. * ``l2``: dense :math:`\boldsymbol\beta^0` -- almost no exact zeros (the factor projection gives every donor a small-but-nonzero coefficient). * ``fs``: agnostic -- the inference is valid uniformly over a class of DGPs that includes both dense and sparse :math:`\boldsymbol\beta^0` (Theorem 1 in Shi-Huang). A7 (Inferential regularity). For all three, the post-period average effect :math:`\widehat\tau` has a CLT with HAC long-run variance consistently estimable by Newey-West. For ``l2`` and ``lasso``, both the pre-period (first-stage) and post-period HAC variances enter; for ``fs``, sample-splitting absorbs the first-stage term and only the post-period HAC variance enters. When the assumptions bind: practical diagnostics ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ (a) Factor-driven DGP (A1). PDA's whole identification story rides on the latent factor structure. If the panel is not well-described by a small number of common factors, the linear-projection equation has a non-vanishing :math:`\epsilon_t` term that the high-dimensional estimators cannot remove. *Plausibly violated when* donors are largely idiosyncratic (each follows its own unrelated process), or when one or two donors have a structural break that the factor model can't absorb. *Diagnostic*: run an SVD on the donor pre-period matrix; the top few singular values should carry the bulk of the spectral energy. If the spectrum is flat-tailed (no clear factor cutoff), the factor model fails and PDA's linear-projection consistency is fragile. In that regime, use a balancing-aware estimator (:doc:`microsynth` if you have unit-level data) or stay with canonical SC. (b) Single treated unit, absorbing aggregate-level treatment (A2). Multiple treated units, treatment that turns off, or interference among donors break the framework. *Plausibly violated when* the policy is rescinded mid- sample, or when treated and donor states are spatially or economically linked enough that the donors' untreated trajectories shift. *Diagnostic*: the config validator enforces single-cohort; the silent failure is treated-donor spillover. Split donors by geographic / economic distance to the treated unit and refit; large ATE shifts flag interference. Use :doc:`spillsynth` or :doc:`spsydid` for genuine spillovers, *FECT* / :doc:`sdid` for staggered designs. (c) Weak temporal dependence (A3). All three variants assume mixing or :math:`\rho`-mixing pre-period series with geometric decay. Unit-root outcomes, long-memory series, or series with structural breaks fail this. *Plausibly violated when* the outcome is a price level, a cumulative quantity, or an undifferenced macroeconomic series. *Diagnostic*: ADF / KPSS on the pre-period residuals; non-stationarity flags A3 failure. The pre/post asymptotic-independence story (which licenses fs-PDA's sample-splitting inference) is then in question. First- difference the outcome, or use :doc:`sbc` (a stationary- cycle estimator) before PDA. (d) Sample-size regime (A4). PDA needs both :math:`T_0` and :math:`T_2` growing with :math:`\log N` small relative to each. A short post-period (:math:`T_2 \le 5`) breaks the CLT on :math:`\widehat\tau`; a short pre-period (:math:`T_0 \le 20`) breaks the moment-convergence rates. *Plausibly violated when* the pre-period is short with many donors. *Diagnostic*: compute :math:`(\log N) / T_0` and :math:`(\log N) / T_2`; both should be visibly below 1. If they are not, the asymptotic approximation has not kicked in. Either lengthen the panel (aggregate to a finer time grid), prune donors, or move to *canonical SCM* / :doc:`tssc` / :doc:`fdid` which work with shorter panels. (e) Donor regularity (A5). Each variant has its own Gram-matrix / factor-strength condition. The practical common failure is near-collinear donors: two donor series that move together up to noise produce a near- singular pre-period Gram matrix. *Plausibly violated when* the donor pool contains near-duplicates (two adjacent states with essentially identical industry mix). *Diagnostic*: read the condition number of :math:`\Gamma_{\mathcal T_1}(\mathbf x_t, \mathbf x_t')`. A condition number above ~1e6 is a red flag. For ``lasso`` and ``fs`` this manifests as selection flipping between near-clones across seeds; for ``l2`` the :math:`\varepsilon`-validation curve gets noisy. Prune one of each clone pair before refitting. (f) Coefficient structure (A6) -- choosing the right variant. The biggest practitioner-side decision is whether to assume sparse or dense :math:`\boldsymbol\beta^0`. Choosing wrong pays a real cost: ``lasso`` on a dense truth over-selects and inflates size (see the Path-B table above: LASSO's size is 0.16-0.36 under the dense factor DGP, vs ``fs``'s 0.05); ``l2`` on a sparse truth pays variance for the dense fit it doesn't need. *Diagnostic*: fit ``fs`` first -- it's valid in both regimes per Shi-Huang Theorem 1, and the selected :math:`\widehat R` and per-step :math:`R^2` curve tell you whether you're in a sparse (few donors carry the fit) or dense (many donors add information) regime. Then run the matched variant (``lasso`` if ``fs`` keeps a handful; ``l2`` if ``fs`` keeps many). (g) Inferential regularity (A7). The HAC long-run variance must be consistently estimable. With strong serial correlation and a short post-period, the Bartlett / Newey-West kernel needs more lags than the post-period supports. *Plausibly violated when* :math:`T_2 \le 8` *and* the treatment-effect series is autocorrelated. *Diagnostic*: plot the autocorrelation function of ``res.gap[-T2:]``; if it stays high beyond :math:`\sqrt{T_2}` lags, the Newey-West bandwidth choice is binding and the CI is optimistic. Lengthen the post-window if you can, or report bootstrap CIs alongside the HAC ones. When to use PDA -- and when not to ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Reach for PDA when: * You have a single treated unit, a moderate-to-large donor pool (:math:`N` comparable to or exceeding :math:`T_0`), and a plausibly factor-driven panel. PDA was designed for exactly this regime, and unlike canonical SCM, the projection has no simplex / non-negativity constraint -- it can extrapolate through negative coefficients on far donors when the factor structure demands it. * You want HAC-based, classical-statistics inference on the ATE with a closed-form normal-distribution test, not a permutation or conformal procedure. PDA's CLT-based inference is what makes it the closest synthetic-control cousin to difference-in-differences from an inferential standpoint. * The treated unit's pre-trajectory lies in the linear span (not necessarily convex hull) of the controls, so a no-constraint linear projection makes sense. PDA cannot recover an effect when the projection itself is impossible. * You're between dense (``l2``) and sparse (``lasso``) regimes and want a uniformly valid test that doesn't require choosing the right sparsity story -- run ``fs``. Do not use PDA when: * You need convex (non-negative, sum-to-one) weights as a policy-interpretation deliverable. PDA's no-constraint projection produces negative coefficients on far donors, which is awkward to explain in many policy contexts. Use *canonical SCM* / :doc:`tssc` (canonical SC) or :doc:`fscm` (forward-selected SC with the simplex retained). * The treated unit is structurally outside the donor span (not just the convex hull). PDA's linear projection cannot reach a treated outcome that no linear combination of donor outcomes can express. The pre-period RMSE stays large at every PDA variant. Use :doc:`iscm`, whose A2(b) mechanism identifies the effect through donors that use the treated unit as a positive-weight donor in *their* synthetic controls. * Outcomes are non-stationary (unit-root or trending series). A3 fails and the pre/post asymptotic-independence story breaks. First-difference the outcome (the l2-relaxation paper's empirical PPI application does exactly this), or use :doc:`sbc` (stationary-cycle estimator). * You have multiple treated units with overlapping cohorts. PDA's theorems (with the exception of the l2 relaxerm which does support multiple treated units but is not written yet) are written for the single-treated case. Use :doc:`sdid` for staggered adoption. * Spillovers across donors. A2's no-interference clause fails when donor states are economically linked to the treated state. Use :doc:`spillsynth` or :doc:`spsydid`. * Continuous or multi-valued treatment. PDA encodes a single binary intervention; continuous dose belongs in :doc:`ctsc`. * Distributional questions (Lorenz curves, QTEs). PDA targets the mean ATE through a Gaussian-likelihood linear projection. Use :doc:`dsc` for distributional effects. * You need Bayesian posterior credible bands. PDA returns frequentist HAC-based CIs. For Bayesian inference and posterior inclusion probabilities on donors, use :doc:`bvss` (spike-and-slab with a soft simplex). * Very short pre-period :math:`(T_0 \le 15)` with many donors. The high-dimensional approximation has not kicked in; the selected :math:`\widehat\beta` is noise. Use *canonical SCM* / :doc:`tssc` / :doc:`fdid`, which work without high-dimensional asymptotics. * Very short post-period :math:`(T_2 \le 5)`. The CLT on :math:`\widehat\tau` is shaky; the HAC bandwidth choice dominates the inference. Either accept a wider permutation CI from *canonical SCM* / :doc:`tssc`, or move to the l2-relaxation multiple-treated-units extension (Shi-Wang Section 4.4) which is built for this regime. * You want predictor-level matching (covariates + pre-period outcomes), not outcome-only projection. PDA's workhorse projection is on donor outcomes, not on predictor moments. Use *canonical SCM* / :doc:`tssc` / :doc:`sparse_sc` (predictor selection with L1 penalty on the V-weight matrix) for the predictor-matching setup. * The factor model itself is the object of interest (you want to identify and interpret the factors). PDA is agnostic to factor estimation -- the factor model only motivates the linear projection, never enters the estimator. Use :doc:`fma` (factor-model-aware estimator) if recovering factors is the goal. Empirical Illustration: Hong Kong economic integration ------------------------------------------------------- The original HCW [HCW]_ application -- and Shi & Huang's Example 1 -- evaluates the effect of economic integration with mainland China on Hong Kong's quarterly real-GDP growth, using 24 comparison economies. Here all three PDA variants run on the same data, and the package returns the time-varying effect, the ATE, and a HAC confidence interval for each. .. code-block:: python import pandas as pd from mlsynth import PDA url = "https://raw.githubusercontent.com/jgreathouse9/mlsynth/refs/heads/main/basedata/HongKong.csv" df = pd.read_csv(url) res = PDA({"df": df, "outcome": "GDP", "treat": "Integration", "unitid": "Country", "time": "Time", "methods": ["l2", "LASSO", "fs"], "alpha": 0.05, "display_graphs": True}).fit() for name, fit in res.fits.items(): sel = "-" if fit.selected_donors is None else len(fit.selected_donors) print(f"{name:6s} ATE {fit.att:.4f} SE {fit.att_se:.4f} " f"95% CI ({fit.ci[0]:.4f}, {fit.ci[1]:.4f}) p={fit.p_value:.3f} donors={sel}") This prints:: l2 ATE 0.0248 SE 0.0032 95% CI (0.0185, 0.0311) p=0.000 donors=24 lasso ATE 0.0330 SE 0.0054 95% CI (0.0224, 0.0436) p=0.000 donors=11 fs ATE 0.0285 SE 0.0059 95% CI (0.0169, 0.0401) p=0.000 donors=7 All three find a significant positive integration effect on Hong Kong's GDP growth -- roughly +2.5 to +3.3 percentage points -- differing only by their selection philosophy: ``l2`` keeps all 24 controls with dense, signed coefficients (pre-RMSE 0.013); ``lasso`` keeps 11; ``fs`` keeps a parsimonious 7 (Malaysia, New Zealand, Norway, Austria, Canada, Thailand, Australia). The estimates bracket the Forward-DiD result on the same data (0.025), a useful cross-method check. Prediction intervals -------------------- The HAC confidence interval above quantifies uncertainty in the *average* treatment effect. For uncertainty in each *period's* effect, set ``prediction_intervals=True`` to attach the bootstrap prediction intervals of Jiang, Li, Shen and Zhou [pdapi]_ to every fitted variant. These bound the post-period untreated counterfactual :math:`Y_t` (and hence the effect :math:`\Delta_t = y_t - \widehat Y_t`), combining the in-sample estimation error with the out-of-sample prediction error that a confidence interval for the mean omits. The construction (their Algorithm 2.1) resamples the pre-period prediction error with a dependent wild bootstrap and the post-period error with a residual bootstrap, refits the chosen estimator on each bootstrap sample at the fixed tuning parameter, and reads quantiles of the self-normalized statistic :math:`\widehat e_t / \sqrt{\widehat V_t + \widehat\sigma^2}`. The implementation lives at the shared :func:`mlsynth.utils.inferutils.pda_prediction_intervals` so any panel-data estimator can reuse it. Both the equal-tailed (``eq``) and symmetric (``sy``) intervals are returned, and each variant reports which studentization it used: ``sandwich`` for the post-selection OLS HAC variance :math:`\widehat V_t` (``lasso`` and ``fs``, which select then run OLS), or the ``sigma2`` fallback when that sandwich is undefined -- as for the dense L2-relaxation when the controls outnumber the pre-periods. .. code-block:: python res = PDA({"df": df, "outcome": "GDP", "treat": "Integration", "unitid": "Country", "time": "Time", "method": "fs", "prediction_intervals": True, "pi_n_boot": 999, "display_graphs": False}).fit() pi = res.fits["fs"].prediction_intervals print(pi["studentization"]) # 'sandwich' eff = pi["effect"] # per-post-period effect intervals lo, hi = eff["eq_lower"][0], eff["eq_upper"][0] print(f"Delta_1 = {eff['point'][0]:.4f} 95% PI ({lo:.4f}, {hi:.4f})") Verification ------------ .. note:: Empirical (Path A, Hong Kong). All three variants run on the HCW Hong Kong panel (above) and agree on a significant positive integration effect, consistent with the literature and the Forward-DiD cross-check (0.025). Prediction intervals (Path B, coverage). The bootstrap prediction intervals are validated by ``benchmarks/cases/pda_pi_coverage.py``, which reproduces the coverage geometry of Jiang et al. (2025) Tables 2-5 on their Setup 1: the equal-tailed and symmetric intervals cover near the nominal 95%, while the normal-quantile intervals under-cover (to about 77% under exponential errors). Simulation study (Path B): forward selection vs LASSO ----------------------------------------------------- Shi & Huang's (2023) Table 1 compares forward selection against LASSO on a four-factor DGP, re-implemented in :func:`mlsynth.utils.pda_helpers.simulation.simulate_pda_panel`: four factors (``f1`` i.i.d.; ``f2`` AR(1) 0.9; ``f3`` MA(2) (0.8,0.4); ``f4`` ARMA(1,1) (0.5,0.5) under the *dynamic* structure, all i.i.d. ``N(0,1)`` under the *i.i.d.* structure); loadings ``U(1,2)`` on the treated + 4 relevant controls and ``U(-0.1,0.1)`` on the remaining 96; idiosyncratic ``N(0,0.5)``; one treated unit, :math:`N=100` controls, :math:`T_0=T_2`. Shocks ``D1``-``D7`` set the post-period ATE (``D1``-``D3`` null → *size*; ``D4``-``D7`` non-zero → *power*). Driving the packaged ``PDA`` (``methods=["fs","LASSO"]``, ``fs_intercept=False``) at :math:`T_0=100` (200 reps for size, 60 for power): .. list-table:: forward selection vs LASSO, :math:`T_0=100` :header-rows: 1 :widths: 12 8 10 10 10 * - factors - method - # donors - size (D1) - power (D5) * - i.i.d. - fs - 3.9 - 0.090 - 1.00 * - i.i.d. - LASSO - 9.5 - 0.065 - 1.00 * - dynamic - fs - 4.5 - 0.075 - 1.00 * - dynamic - LASSO - 15.0 - 0.140 - 1.00 The paper's geometry reproduces. Forward selection is parsimonious -- it keeps to ~the 4 relevant donors in both structures -- while LASSO over-selects (9-15 donors). Forward selection's test is correctly sized (≈ 0.05-0.09) under *both* factor structures, the robustness Shi & Huang emphasise. LASSO is correctly sized under i.i.d. factors (0.065, matching the paper's 0.058) but its size inflates under dynamic factors (0.140; paper's modified-BIC LASSO 0.184) -- the size inflation the paper reports is a *dynamic-factor* phenomenon, not an i.i.d. one. Both tests are fully powered at ``D5`` (mean-1 shift). Durable case: ``pda_table1``. .. note:: mlsynth's LASSO is cross-validated; the paper's is not. Shi & Huang select the Lasso penalty with a modified BIC (Remark 4 cont., p.521: "we tune the constants in the modified BIC to allow Lasso to take in more variables"); ``mlsynth``'s L1-PDA selects it with ``LassoCV`` (5-fold cross-validation). The two are different penalty rules, so the LASSO cells above are ``mlsynth``'s CV variant, not a cell-by-cell match of the paper's Lasso. What both share -- and what the benchmark pins -- is the geometry: LASSO over-selects relative to fs and its size inflates under dynamic factors, while forward selection (the paper's *method*, validated cell-by- cell on Hong Kong in ``pda_hongkong``) stays parsimonious and correctly sized. .. admonition:: The ``fs_intercept`` knob -- valid size on factor data Achieving the correct fs size above required a fix. The released ``fsPDA`` R package (``est.fsPDA.R``) fits the donor regression *with* an intercept; on the paper's mean-zero factor DGP that intercept absorbs a spurious pre-period constant which extrapolates into the post window, biasing the gap and inflating the null rejection rate to ~0.20. The paper's Table 1 was produced by the *simulation* code (``FS.R``), which fits without an intercept and yields valid size. ``mlsynth`` exposes both via ``PDAConfig.fs_intercept`` (default ``False`` = the no-intercept, valid-size form; set ``True`` for panels with genuine unit level shifts). With the default, the fs D1 rejection rate drops from ~0.20 (intercept) to the ~0.05-0.09 reported above (no intercept). One honest caveat: under *dynamic* factors fs shows mild residual size inflation at small :math:`T` (the "imprecise long-run-variance" effect the paper itself notes), shrinking toward 5% as :math:`T_0 \to \infty`. L2-relaxation (Shi & Wang). The ``l2`` method's out-of-sample MPSE falls with :math:`T` and its test approaches the nominal 5% size as :math:`T_0 \to \infty`, matching Shi & Wang's Table 2 (size :math:`0.142` at :math:`T_0=50` → :math:`0.072` at :math:`200`). Its per-fit cross-validation over the :math:`\varepsilon` grid makes large Monte Carlos expensive (~5 s/fit), so the full table is summarized rather than swept here. Multiple treated units ---------------------- When *several* units are treated by the same intervention, Shi & Wang's L2-relaxation PDA fits a counterfactual per treated unit against the shared control pool and aggregates the effects into a per-period cross-sectional ATE, .. math:: \widehat{\mathrm{ATE}}_t = \frac{1}{J} \sum_{j=1}^{J} \bigl( y_{jt} - \widehat{y}_{jt} \bigr), \qquad \mathrm{s.e.} = \frac{\sqrt{\mathbf{1}'\widehat{\Sigma}_e\,\mathbf{1}}}{J}, with :math:`\widehat{\Sigma}_e` the cross-sectional covariance of the pre-period prediction residuals (so the standard error is constant across post-periods). :func:`mlsynth.utils.pda_helpers.multitreat.run_pda_multitreat` implements this. Because every per-unit fit shares the same :math:`\widehat{\boldsymbol{\Sigma}} = X'X/T_0`, all :math:`J` fits run through a single OSQP factorisation -- the batched solver :func:`mlsynth.utils.pda_helpers.l2.batch.l2_relax_batch` updates only the constraint bounds per ``(unit, tau)`` (hundreds of conic solves become hundreds of warm-started ADMM updates). On the Brexit study (52 UK firms vs 300 controls) this reproduces the paper's first-day return ATE of :math:`-4.3\%` (:math:`t \approx -7.4`); see ``benchmarks/cases/pda_brexit.py``. Core API -------- .. automodule:: mlsynth.estimators.pda :members: :undoc-members: :show-inheritance: Configuration ------------- .. autoclass:: mlsynth.config_models.PDAConfig :members: :undoc-members: Result Containers ----------------- ``PDA.fit()`` returns a :class:`~mlsynth.utils.pda_helpers.structures.PDAResults`, whose ``fits`` maps each variant to a :class:`~mlsynth.utils.pda_helpers.structures.PDAMethodFit` (coefficients, intercept, counterfactual, gap, ATE, HAC standard error, CI, p-value, donor weights, and the selected-donor list for ``lasso``/``fs``). The prepared, NumPy-only panel is exposed as a :class:`~mlsynth.utils.pda_helpers.structures.PDAInputs`, with units and time addressed through an :class:`IndexSet`. .. note:: ``PDA.fit()`` returns an :class:`~mlsynth.config_models.EffectResult` on the standardized two-family contract. It is a dispatcher over the variants in ``res.fits`` (l2 / lasso / fs); the selected variant drives the flat accessors (``res.att`` / ``res.att_ci`` / ``res.counterfactual`` / ``res.gap`` / ``res.donor_weights`` / ``res.pre_rmse``), which resolve through the standardized sub-models. ``res.donor_weights`` are the regression coefficients (PDA is a regression counterfactual, not a simplex average); ``res.att_by_method()`` / ``res.se_by_method()`` report every variant. .. automodule:: mlsynth.utils.pda_helpers.structures :members: :undoc-members: :show-inheritance: Helper Modules -------------- Data preparation -- the only DataFrame touchpoint: pivots to NumPy, builds the unit/time ``IndexSet``\es, and splits pre/post. .. automodule:: mlsynth.utils.pda_helpers.setup :members: :undoc-members: Shared HAC long-run-variance machinery (Bartlett/Newey-West) and the N(0,1) test used by all three variants. .. automodule:: mlsynth.utils.pda_helpers.inference :members: :undoc-members: L2-relaxation (Shi & Wang): the relaxation primal, :math:`\varepsilon` validation, and the two-term HAC ATE inference. .. automodule:: mlsynth.utils.pda_helpers.l2.estimation :members: :undoc-members: .. automodule:: mlsynth.utils.pda_helpers.l2.inference :members: :undoc-members: LASSO (Li & Bell): cross-validated L1 estimation and the HAC t-test with a first-stage variance term. .. automodule:: mlsynth.utils.pda_helpers.lasso.estimation :members: :undoc-members: .. automodule:: mlsynth.utils.pda_helpers.lasso.inference :members: :undoc-members: Forward selection (Shi & Huang): greedy R^2 selection with modified-BIC stopping and the post-selection HAC t-test. .. automodule:: mlsynth.utils.pda_helpers.fs.estimation :members: :undoc-members: .. automodule:: mlsynth.utils.pda_helpers.fs.inference :members: :undoc-members: Run loop assembling the per-variant fits. .. automodule:: mlsynth.utils.pda_helpers.orchestration :members: :undoc-members: