Stats

Estimating remaining time to sustained recovery with repeated daily landmarks within stock drawdowns

Step-by-step statistics solution: Estimating remaining time to sustained recovery with repeated daily landmarks within stock drawdowns

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1. What the question is asking – plain‑language restatement

You have daily closing prices for a few Egyptian stocks.
During a draw‑down (the price is below the recent high) you would like, at each trading day, to predict the probability that the stock will “recover” within a short, medium, or long horizon:

horizon meaning in the problem
5 sessions recovery within the next 5 trading days
21 sessions recovery within the next 21 trading days
126 sessions recovery within the next 126 trading days

A recovery is defined (for now) as:

  1. The first closing price that exceeds the maximum of the previous 10 closes, and
  2. The next 5 closes all stay above that breakout level.

Those five “post‑breakout” closes are the outcome; they must never be used as predictors.

Because many days belong to the same draw‑down episode, the daily “landmark” observations are not independent.
You want to know:

  • How to build the risk set (i.e. which landmarks are eligible to be counted for each horizon) while respecting censoring (administrative end‑of‑study, unresolved prices, etc.)?
  • What validation unit (what should stay together when you split the data into training / test folds) is appropriate?
  • Whether a discrete‑time landmark survival model is suitable, and if so, how to handle the episode‑level dependence when you compare one predictor at a time (e.g. 3‑day return, volume‑to‑20‑day‑average, …).

The answer below walks through every step required to construct a proper dataset, fit a calibrated model, and evaluate it while accounting for the within‑episode correlation.

2. Step‑by‑step solution

2.1 Define the basic entities

Entity Symbol Description
Stock s   e.g. ABUK, FWRY
Draw‑down episode e (belonging to stock s)   a contiguous period that starts at the first day the price falls below the preceding 10‑day high and ends when a consistent recovery (the 5‑day post‑breakout rule) is observed or the episode is censored
Landmark day l (within episode e)   a trading day on which a prediction is made (i.e. every day while the episode is “open”)
Horizon h ∈ {5, 21, 126}   number of sessions ahead for which we want the recovery probability
Outcome Yl,h 0/1 1 if a consistent recovery occurs strictly within the next h sessions after landmark l; 0 otherwise (including censored cases)
Censoring indicator Cl 0/1 1 if the observation is right‑censored before we can see the outcome for the longest horizon (e.g. episode ends, data series ends, unresolved price)

2.2 Build the episode table

  1. Detect draw‑down start
    • For each day t compute max10_t = max(Close_{t‑9}, …, Close_t).
    • A draw‑down starts on the first day after a day where Close_t < max10_t.
  2. Detect a “consistent” recovery (the label)
    • Scan forward until a day b where Close_b > max10_b.
    • Check that Close_{b+1}, …, Close_{b+5} are all ≥ Close_b.
    • If yes, the episode ends on day b+5 (the last of the five confirming closes).
  3. Censoring
    • If the series ends before a consistent recovery is seen → right‑censor episode at the last observed day.
    • If any of the 5 confirming closes are missing or marked unresolved → treat the episode as censored at the first missing day.
  4. Store for each episode e:
    • stock, episode_id, start_date, end_date (or censoring flag), max10_at_start, etc.

2.3 Expand to landmark‑level rows

For every episode e with start date t₀ and (possibly censored) end date t_end:

for each day t = t₀, t₀+1, … , t_end:
        for each horizon h in {5,21,126}:
                if t + h ≤ t_end   (i.e. we can observe h days ahead)
                        Y_lh = 1 if a consistent recovery occurs on any day ≤ t+h
                        else Y_lh = 0
                else
                        C_l = 1   (right‑censored for horizon h)

Resulting long data set has one row per (episode, landmark day, horizon).
Columns include:

  • stock, episode_id, landmark_date (t)
  • horizon (h)
  • outcome (Y), censor (C)
  • Predictor values computed only from information available at t (e.g. 3‑day return ending on t, volume/t‑20‑day‑mean ending on t, etc.)

2.4 Construct the risk set for each horizon

In discrete‑time survival language the risk set at horizon h consists of all landmark rows where the observation is not censored for that horizon:

RiskSet_h = { (l, h) : C_lh = 0 }

Only those rows enter the likelihood for horizon h.
Rows censored before horizon h contribute to the likelihood for shorter horizons (if they are uncensored there) but not for longer ones.

2.5 Choose the validation unit

Because landmarks within the same episode share the same future price path, splitting the data at the daily‑landmark level would leak information (the training set could contain a landmark that is only a few days away from a landmark in the test set, and both share the same eventual recovery).

Recommendation:

  • Unit of validation = whole episode (i.e. all landmarks belonging to the same draw‑down).
  • When you create folds, keep every landmark of an episode together and never place two landmarks from the same episode into different folds.

2.5.1 Chronological (time‑series) folds

  • Sort episodes by their start date.
  • Divide the ordered list into K contiguous blocks (e.g. K = 5).
  • For fold k: train on episodes in blocks 1,…,k‑1; test on block k.
  • This respects the “future‑cannot‑inform‑past” principle and mimics a realistic forecasting situation.

2.6 Model: Discrete‑time landmark survival

For each horizon h you can fit a logistic (or complementary‑log‑log) model of the form

[ \Pr(Y_{lh}=1 \mid \mathbf X_{l}) = \text{logit}^{-1}\bigl(\beta_{0h} + \beta_{1h} X_{1l} + \dots + \beta_{ph} X_{pl}\bigr) ]

where

  • (\mathbf X_{l}) are the predictors measured at landmark l (they are the same for all horizons, but you may allow horizon‑specific coefficients).
  • The likelihood uses only rows in RiskSet_h.

Why discrete‑time?

  • Your horizons are measured in whole sessions, not in continuous calendar time.
  • The event (recovery) can only be observed at integer session boundaries.
  • The model directly yields calibrated probabilities for each horizon.

2.6.1 Handling episode‑level dependence

Two common ways:

Method How it works When to use
Cluster‑robust (sandwich) SE Fit the logistic model ignoring dependence, then compute variance‑covariance matrix that is robust to arbitrary correlation within each episode_id. If you only need inference (p‑values, CI) and the number of episodes is moderate‑large (>30).
Mixed‑effects (random intercept) model Add a random intercept (u_e \sim N(0,\sigma^2u)) for each episode *e*: (\text{logit}(p{leh}) = \beta_{0h}+u_e+\beta^{\top}\mathbf X_l). Fit by Laplace approximation or adaptive Gaussian quadrature. When you suspect that episodes have heterogeneous baseline hazards and you want the model to borrow strength across episodes. Also useful for out‑of‑sample prediction because the random effect can be set to its posterior mean (≈0) for a new episode.
Frailty survival model (continuous‑time analog) Same idea as random intercept but expressed in a proportional‑hazards framework; not needed here because we are already in discrete time. –

For predictor‑screening (one predictor at a time) the simplest approach is to fit a marginal logistic model with cluster‑robust SE. When you later combine predictors, you may switch to the mixed‑effects version to improve calibration.

2.7 Model‑assessment metrics (calibration & discrimination)

Because the goal is calibrated probabilities, use:

  • Brier score (scaled) for each horizon, computed on the held‑out episodes of each fold.
  • Calibration plot (decile‑wise observed vs. predicted recovery rates).
  • Time‑dependent AUC (c‑index for discrete time) if you also care about ranking.

All metrics must be aggregated episode‑wise (i.e., average over all landmark rows in the test episodes) to avoid double‑counting the same future recovery.

2.8 Putting it all together – workflow

  1. Pre‑process raw price/volume data → clean dates, handle corporate actions.
  2. Identify draw‑down episodes → start, end, censoring flag.
  3. Create landmark‑horizon rows → compute predictors using only data up to the landmark date.
  4. Assign risk‑set flags (C_lh).
  5. Split episodes into chronological folds (e.g., 5‑fold).

  6. For each fold:

    a. Training set = all rows from episodes in earlier folds.

    b. Fit a discrete‑time logistic model for each horizon (or a pooled model with horizon indicator).

    c. Obtain predictions on the test episodes (use fixed‑effects only; random intercept set to 0 if mixed model).

    d. Compute Brier score, calibration curves, AUC for each horizon.

  7. Aggregate metric averages across folds → final estimate of out‑of‑sample performance.

  8. Screen predictors: repeat steps 6‑7 for each candidate predictor (or small sets). Compare Brier score improvements, use a paired‑fold test (e.g., DeLong for AUC, or bootstrap for Brier) that respects the episode‑level clustering.

2.9 Summary of the “what to use” answer

Question Answer
Risk‑set construction Use all landmark‑horizon rows that are not censored for that horizon (C_lh = 0). Rows censored before horizon h are excluded from the likelihood for h but may be kept for shorter horizons.
Validation unit The draw‑down episode (all its daily landmarks) is the proper unit. Build chronological (time‑series) folds that keep whole episodes together.
Is a discrete‑time landmark survival model appropriate? Yes. The problem is naturally cast as predicting a binary event within a fixed number of

Original question: Estimating remaining time to sustained recovery with repeated daily landmarks within stock drawdowns on Cross Validated (Stats Stack Exchange), licensed CC BY-SA.

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