---
title: "Time-Transformed Test (STADF/GSTADF)"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Time-Transformed Test (STADF/GSTADF)}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE, warning = FALSE, message = FALSE,
  comment = "#>"
)
```

```{r load}
library(exuber)
```

## Why another test

`radf()`, the classic PSY GSADF test, assumes that the innovation variance is constant. Real financial series usually do not have constant volatility. Harvey, Leybourne, Sollis & Taylor (2016) show that when volatility changes over time, the standard critical values of `radf()` no longer control the size of the test. `radf_wb_cv()` addresses this in exuber with a wild bootstrap.

`radf_tt()` implements a different fix that needs no bootstrap, from Kurozumi, Skrobotov & Tsarev (2024, *Journal of Financial Econometrics*). Instead of resampling, it time-deforms the series using a nonparametric estimate of its variance profile, so that under the null the deformed series behaves like a random walk with constant volatility. The null distribution of the resulting statistic is then the same pivotal distribution as under homoskedasticity. Ordinary asymptotic critical values therefore apply, with no bootstrap and no resimulation for each dataset.

## Basic usage

`radf_tt()` targets non-stationary volatility, meaning a permanent shift or trend in the unconditional innovation variance, which is the setting of the simulations in Kurozumi, Skrobotov & Tsarev. It does not target stationary conditional heteroskedasticity such as GARCH. In that case the variance profile is asymptotically flat, the time deformation is close to the identity, and plain `radf()` already has the correct size. `sim_vol_break()` generates innovations with a permanent volatility shift, and the `e` argument of `sim_psy1()` passes them into the PSY bubble process. Here the innovation standard deviation triples half-way through the sample:

```{r radf-tt}
y <- sim_psy1(n = 200, seed = 1, e = sim_vol_break(199))
res <- radf_tt(y)
res
```

`radf_tt_cv()` returns the matching pivotal asymptotic critical values. The null distribution does not depend on the volatility path, so one call with a large `n` approximates the whole family of cases. `radf_wb_cv()`, in contrast, runs a separate bootstrap for each dataset:

```{r radf-tt-cv}
cv <- radf_tt_cv(n = 300, minw = 30, nrep = 1000, seed = 1)
cv$gsadf_cv
```

## What is estimated

`radf_tt()` works in three steps.

1. It estimates the time-varying AR(1) coefficient with a local kernel regression. From the truncated residuals it builds a monotone variance profile `eta_hat(s)`, with `s` in `[0, 1]`.
2. It inverts the profile and uses the inverse to resample and time-deform the series.
3. It computes a recursive sup-ADF statistic (GLS-demeaned, with no intercept) on the deformed series. This belongs to the same family of statistics as `radf()`, but it needs no fitted intercept, which matches the derivation in the paper.

You can adjust `kernel` (`"uniform"`, the choice of the paper, or `"gaussian"`) and `h`, the bandwidth. The default for `h` is a fixed plug-in value and not the full cross-validation search of the paper.

## Dating and plotting a detected bubble

`radf_tt()` returns the same `radf_obj` class as `radf()`. `radf_tt_cv()` also computes the full time-varying boundary that dating and plotting need, and not only the summary-level critical values. The usual pipeline therefore works unchanged:

```{r radf-tt-datestamp}
res <- radf_tt(y, minw = 20)
cv <- radf_tt_cv(n = 200, minw = 20)

datestamp(res, cv = cv)
autoplot(res, cv = cv)
```

`vignette("naming-and-analysis")` explains which other exuber functions work with this pipeline and which do not.