Intro to exuber

For our analysis we are going to use the datasets::EuStockMarkets dataset, which contains the daily closing prices of four major European stock indices: Germany DAX, Switzerland SMI, France CAC, and UK FTSE (see ?EuStockMarkets). The data are sampled in business time, i.e., weekends and holidays are omitted. In this particular exercise we want to focus on weekly observations. To do so we aggregate to a weekly frequency and reduce the number of observations from 1860 to 372.

stocks <- aggregate(EuStockMarkets, nfrequency = 52, mean)

Estimation

We estimate the above series using the recursive Augmented Dickey-Fuller test with 1 lag.

est_stocks <- radf(stocks, lag = 1)

Analysis

The summary will print the test statistic and the critical values for 10%, 5% and 1% significance level. For a plain recursive ADF (lag = 0) and up to 600 observations, the package ships pre-simulated critical values that summary()/diagnostics()/datestamp()/autoplot() use automatically when cv is omitted. With lag = 1 (as estimated above), critical values instead need to be simulated for this specific (n, lag) combination – radf_mc_cv() does that locally, and we pass the result via cv to every downstream call.

cv_stocks <- radf_mc_cv(NROW(stocks), lag = 1)

summary(est_stocks, cv = cv_stocks)
#> 
#> ── Summary (minw = 38, lag = 1) ─────────────────── Monte Carlo (nrep = 1000) ──
#> 
#> DAX :
#> # A tibble: 3 × 5
#>   stat  tstat   `90`    `95`  `99`
#>   <fct> <dbl>  <dbl>   <dbl> <dbl>
#> 1 adf    1.45 -0.440 -0.0429 0.537
#> 2 sadf   4.95  1.23   1.54   2.06 
#> 3 gsadf  5.18  2.13   2.38   3.00 
#> 
#> SMI :
#> # A tibble: 3 × 5
#>   stat  tstat   `90`    `95`  `99`
#>   <fct> <dbl>  <dbl>   <dbl> <dbl>
#> 1 adf    1.77 -0.440 -0.0429 0.537
#> 2 sadf   4.28  1.23   1.54   2.06 
#> 3 gsadf  4.49  2.13   2.38   3.00 
#> 
#> CAC :
#> # A tibble: 3 × 5
#>   stat  tstat   `90`    `95`  `99`
#>   <fct> <dbl>  <dbl>   <dbl> <dbl>
#> 1 adf   0.987 -0.440 -0.0429 0.537
#> 2 sadf  2.91   1.23   1.54   2.06 
#> 3 gsadf 2.97   2.13   2.38   3.00 
#> 
#> FTSE :
#> # A tibble: 3 × 5
#>   stat  tstat   `90`    `95`  `99`
#>   <fct> <dbl>  <dbl>   <dbl> <dbl>
#> 1 adf   0.194 -0.440 -0.0429 0.537
#> 2 sadf  2.56   1.23   1.54   2.06 
#> 3 gsadf 2.67   2.13   2.38   3.00

It seems that all stocks exhibit exuberant behaviour but we can also verify it using diagnostics(). This function is particularly useful when we deal a large number of series.

diagnostics(est_stocks, cv = cv_stocks)
#> 
#> ── Diagnostics (option = gsadf) ───────────────────────────────── Monte Carlo ──
#> 
#> DAX:      Rejects H0 at the 1% significance level
#> SMI:      Rejects H0 at the 1% significance level
#> CAC:      Rejects H0 at the 5% significance level
#> FTSE:     Rejects H0 at the 5% significance level

If we need to know the exact period of exuberance we can do so with the function datestamp(). datestamp() works in a similar manner with summary() and diagnostics().

# Minimum duration of an explosive period
rot = psy_ds(stocks) # log(n) ~ rule of thumb

dstamp_stocks <- datestamp(est_stocks, cv = cv_stocks, min_duration = rot)
dstamp_stocks
#> 
#> ── Datestamp (min_duration = 6) ───────────────────────────────── Monte Carlo ──
#> 
#> DAX :
#>        Start       Peak        End Duration   Signal Ongoing
#> 1 1997-02-10 1997-08-05 1997-11-04       38 positive   FALSE
#> 2 1998-01-27 1998-07-22 1998-08-19       29 positive   FALSE
#> 
#> SMI :
#>        Start       Peak        End Duration   Signal Ongoing
#> 1 1993-12-02 1994-02-03 1994-02-17       11 positive   FALSE
#> 2 1997-04-14 1997-07-15 1997-09-02       20 positive   FALSE
#> 3 1997-09-09 1997-10-07 1997-11-04        8 positive   FALSE
#> 4 1997-11-25 1998-04-07 1998-08-19       39 positive    TRUE
#> 
#> CAC :
#>        Start       Peak        End Duration   Signal Ongoing
#> 1 1997-07-08 1997-08-05 1997-08-19        6 positive   FALSE
#> 2 1998-03-10 1998-07-15 1998-08-12       22 positive   FALSE
#> 
#> FTSE :
#>        Start       Peak        End Duration   Signal Ongoing
#> 1 1997-07-08 1997-08-12 1997-09-02        8 positive   FALSE
#> 2 1997-09-23 1997-10-07 1997-11-04        6 positive   FALSE
#> 3 1998-02-10 1998-04-14 1998-06-24       19 positive   FALSE

We can extract the datestamp as a dummy variable 1 = Exuberance, 0 = No exuberance.

dummy <- attr(dstamp_stocks, "dummy")
tail(dummy)
#>     DAX SMI CAC FTSE
#> 367   1   1   1    1
#> 368   1   1   1    1
#> 369   1   1   1    1
#> 370   1   1   1    0
#> 371   1   1   0    0
#> 372   0   1   0    0

Plotting

The autoplot function returns a faceted ggplot2 object for all the series that reject the null hypothesis at 5% significance level.

autoplot(est_stocks, cv = cv_stocks)
#> Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
#> ℹ Please use `linewidth` instead.
#> ℹ The deprecated feature was likely used in the exuber package.
#>   Please report the issue at <https://github.com/kvasilopoulos/exuber/issues>.
#> This warning is displayed once per session.
#> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
#> generated.

Finally, we can plot just the periods the periods of exuberance. Plotting datestamp object is particularly useful when we have a lot of series, and we are interested to identify explosive patterns in all of them.

datestamp(est_stocks, cv = cv_stocks) %>%
  autoplot()