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 x 5
#>   name  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 x 5
#>   name  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 x 5
#>   name  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 x 5
#>   name  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        End Duration
#> 1 1997-02-10 1997-11-04       38
#> 2 1998-01-27       <NA>       30
#> 
#> SMI :
#>        Start        End Duration
#> 1 1993-12-02 1994-02-17       11
#> 2 1997-04-14 1997-09-02       20
#> 3 1997-09-09 1997-11-04        8
#> 4 1997-11-25       <NA>       39
#> 
#> CAC :
#>        Start        End Duration
#> 1 1997-07-08 1997-08-19        6
#> 2 1998-03-10 1998-08-12       22
#> 
#> FTSE :
#>        Start        End Duration
#> 1 1997-07-08 1997-11-04       17
#> 2 1998-02-10 1998-06-24       19

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)
#> Warning: Removed 2 rows containing missing values (geom_rect).

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()
#> Warning: Removed 2 rows containing missing values (geom_segment).