# 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. The package provides simulated critical values for up to 600 observations, so we use them by omitting the `cv` argument in the `summary` function.

    summary(est_stocks)
    #>
    #> ── Summary (minw = 38, lag = 1) ─────────────────── Monte Carlo (nrep = 2000) ──
    #>
    #> DAX :
    #> # A tibble: 3 × 5
    #>   stat  tstat   `90`    `95`  `99`
    #>   <fct> <dbl>  <dbl>   <dbl> <dbl>
    #> 1 adf    1.45 -0.437 -0.0900 0.511
    #> 2 sadf   4.95  1.14   1.42   2.04
    #> 3 gsadf  5.18  1.90   2.14   2.60
    #>
    #> SMI :
    #> # A tibble: 3 × 5
    #>   stat  tstat   `90`    `95`  `99`
    #>   <fct> <dbl>  <dbl>   <dbl> <dbl>
    #> 1 adf    1.77 -0.437 -0.0900 0.511
    #> 2 sadf   4.28  1.14   1.42   2.04
    #> 3 gsadf  4.49  1.90   2.14   2.60
    #>
    #> CAC :
    #> # A tibble: 3 × 5
    #>   stat  tstat   `90`    `95`  `99`
    #>   <fct> <dbl>  <dbl>   <dbl> <dbl>
    #> 1 adf   0.987 -0.437 -0.0900 0.511
    #> 2 sadf  2.91   1.14   1.42   2.04
    #> 3 gsadf 2.97   1.90   2.14   2.60
    #>
    #> FTSE :
    #> # A tibble: 3 × 5
    #>   stat  tstat   `90`    `95`  `99`
    #>   <fct> <dbl>  <dbl>   <dbl> <dbl>
    #> 1 adf   0.194 -0.437 -0.0900 0.511
    #> 2 sadf  2.56   1.14   1.42   2.04
    #> 3 gsadf 2.67   1.90   2.14   2.60

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)
    #>
    #> ── 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 1% significance level
    #> FTSE:     Rejects H0 at the 1% 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()`. The user still has to specify the critical values, however we can still utilize the package’s critical values by leaving the cv-argument blank.

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

    dstamp_stocks <- datestamp(est_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       30 positive    TRUE
    #>
    #> 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 1997-10-07 1997-11-04       17 positive   FALSE
    #> 2 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   1   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)

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) %>%
      autoplot()