We present the empirical study of Holý and Zouhar (2022) which analyzes the results of the Ice Hockey World Championships. Our main object of interest is the annual ranking of 16 teams participating in the championships. While there exists a comprehensive statistical toolkit for ranking data, as described e.g. by Alvo and Yu (2014), it is worth noting that the time perspective is often overlooked in the ranking literature, as highlighted by Yu et al. (2019)}. This is precisely where the GAS model emerges as a valuable tool in our analysis.
Our analyzed data are supplied in the
ice_hockey_championships
dataset. We restrict ourselves to
years 1998–2019 just as Holý and Zouhar (2022). In 1998, the number of
teams in the tournament increased from 12 to 16. In 2020, the
championship was canceled due to Covid-19 pandemic. We start by creating
two variables – the final ranking of 16 participating teams in each year
y
and the dummy variable indicating which country (or
countries) hosted the championship in each year x
.
We look at some basic statistics. In our sample, nine countries have participated each year.
participate <- colSums(is.finite(y))
names(participate)[participate == t]
#> [1] "CAN" "CHE" "CZE" "FIN" "LVA" "RUS" "SVK" "SWE" "USA"
The following countries hosted the championships, some of them multiple times.
host <- sapply(x, FUN = sum)
host[host > 0L]
#> AUT BLR CAN CHE CZE DEU FIN FRA LVA NOR RUS SVK SWE
#> 1 1 1 2 2 3 3 1 1 1 3 2 3
In the years under analysis, the gold medals were awarded to the following countries.
The gasmodel
package provides a single distribution on
rankings – the Plackett–Luce distribution.
distr(filter_type = "ranking")
#> distr_title param_title distr param type dim orthog default
#> 32 Plackett-Luce Worth pluce worth ranking multi FALSE TRUE
It is a convenient and simple probability distribution on rankings utilizing a worth parameter for each item to be ranked. It originates from Luce’s choice axiom and is also related to the Thurstone’s theory of comparative judgment, see Luce (1977) and Yellott (1977). For more details on this distribution, see Plackett (1975), Stern (1990), and Critchlow et al. (1991).
We consider three different model specifications. We incorporate
x
as an explanatory variable in our model to capture
possible home advantage. For each model specification, we assume a
panel-like structure where each worth parameter has its own intercept,
while the regression and dynamics parameters remain the same for all
worth parameters. In the gasmodel
package, this structure
can be achieved using the coef_fix_value
and
coef_fix_other
arguments. Alternatively, for convenience,
the value panel_structure
can be included in the
coef_fix_special
argument. It is important to note that the
worth parameters in the Plackett–Luce distribution are not identifiable,
and it is common practice to impose a standardizing condition. In our
model, we enforce the condition that the sum of all \(\omega_i\) is 0. This can be accomplished
by including the value zero_sum_intercept
in the
coef_fix_special
argument.
First, we estimate the static model where there are no dynamics involved. In this case, we set both the autoregressive and score orders to zero. Either a single integer can be provided to determine the order for all parameters, or a vector of integers can be supplied to specify the order for individual parameters.
est_static <- gas(y = y, x = x, distr = "pluce", p = 0, q = 0,
coef_fix_special = c("zero_sum_intercept", "panel_structure"))
Second, we estimate the standard mean-reverting GAS model of order one. In order to expedite the numerical optimization process, we incorporate starting values based on the static model.
est_stnry <- gas(y = y, x = x, distr = "pluce",
coef_fix_special = c("zero_sum_intercept", "panel_structure"),
coef_start = as.vector(rbind(est_static$fit$par_unc / 2, 0, 0.5, 0.5)))
Third, we estimate the random walk model. In other words, we set the
autoregressive coefficient to 1. The easiest way to specify this is by
including the value random_walk
in the
coef_fix_special
argument. In our random walk model, we
consider the initial values of the worth parameters to be parameters to
be estimated. While the par_init
argument does not directly
support this, we can set regress = "sep"
and use cumulative
sums of exogenous variables to achieve this initialization for this
particular model. However, it is generally not recommended to estimate
initial parameter values as it introduces additional variables, lacks
reasonable asymptotics, and can lead to overfitting in finite samples.
It is important to approach the random walk model with caution, as it is
not stationary and the standard maximum likelihood asymptotics are not
valid.
est_walk <- gas(y = y, x = lapply(x, cumsum), distr = "pluce", regress = "sep",
coef_fix_special = c("zero_sum_intercept", "panel_structure", "random_walk"),
coef_start = as.vector(rbind(est_static$fit$par_unc, 0, 0.5, 1)))
To avoid redundancy, we will omit the output of the
gas()
function, which contains rows for each coefficient of
each worth parameter. Since most coefficients are the same due to the
assumed panel structure, it is unnecessary to display them all. Instead,
we print only one set of the home advantage and dynamics
coefficients.
cbind(est_static = c("beta1" = unname(coef(est_static)[2]), "alpha1" = 0, "phi1" = 0),
est_stnry = coef(est_stnry)[2:4],
est_walk = coef(est_walk)[2:4])
#> est_static est_stnry est_walk
#> beta1 0.1707329 0.2274380 0.09873333
#> alpha1 0.0000000 0.3919431 0.34300134
#> phi1 0.0000000 0.5062478 1.00000000
In all three models, coefficient \(\alpha_1\) representing the home advantage is positive but not significant.
cbind(est_static = c("beta1" = unname(est_static$fit$coef_pval)[2], "alpha1" = 0, "phi1" = 0),
est_stnry = est_stnry$fit$coef_pval[2:4],
est_walk = est_walk$fit$coef_pval[2:4])
#> est_static est_stnry est_walk
#> beta1 0.514887 3.772811e-01 5.995826e-01
#> alpha1 0.000000 2.141363e-06 2.634614e-09
#> phi1 0.000000 6.463390e-04 0.000000e+00
We compare the models using the Akaike information criterion (AIC).
The gas
class allows for generic function
AIC()
. In terms of AIC, the mean-reverting model
outperformed the remaining two by a wide margin.
Our models enable us to construct the ‘ultimate’ or long-run ranking. The rankings produced by both models are in agreement for all but the first three positions. However, the long-term strength estimates for these three teams are very close to each other, making the final ranking less clear-cut.
tibble(team = colnames(y)) %>%
mutate(stnry_strength = est_stnry$fit$par_unc) %>%
mutate(stnry_rank = rank(-stnry_strength)) %>%
mutate(static_strength = est_static$fit$par_unc) %>%
mutate(static_rank = rank(-static_strength)) %>%
arrange(stnry_rank)
#> # A tibble: 24 × 5
#> team stnry_strength stnry_rank static_strength static_rank
#> <chr> <dbl> <dbl> <dbl> <dbl>
#> 1 FIN 3.76 1 3.69 3
#> 2 CAN 3.74 2 3.72 2
#> 3 SWE 3.71 3 3.86 1
#> 4 CZE 3.51 4 3.42 4
#> 5 RUS 3.31 5 3.19 5
#> 6 USA 1.83 6 2.18 6
#> 7 CHE 1.72 7 1.78 7
#> 8 SVK 1.69 8 1.56 8
#> 9 LVA 0.883 9 0.830 9
#> 10 DEU 0.343 10 0.334 10
#> 11 BLR 0.275 11 0.116 11
#> 12 NOR 0.0544 12 -0.0665 12
#> 13 DNK -0.0732 13 -0.175 13
#> 14 FRA -0.384 14 -0.501 14
#> 15 AUT -0.812 15 -0.878 15
#> 16 ITA -1.02 16 -1.10 16
#> 17 UKR -1.34 17 -1.52 17
#> 18 SVN -1.75 18 -1.64 18
#> 19 KAZ -1.83 19 -1.78 19
#> 20 JPN -1.99 20 -1.94 20
#> 21 HUN -3.28 21 -3.20 21
#> 22 GBR -3.92 22 -3.89 22
#> 23 POL -3.95 23 -3.90 23
#> 24 KOR -3.96 24 -3.91 24
Additionally, we can examine the evolution of the worth parameters
for individual teams over the years. The point estimates of time-varying
parameter values can be directly obtained from the gas()
function. Using the generic plot()
function allows us to
visualize the time-varying parameters of individual models. When
multiple parameters are time-varying, as in our scenario, the function
plots them in sequence. For the purpose of this document, we will only
display figures specific to the Canada team.
However, it is important to note that these estimates are subject to
uncertainty. To capture the uncertainty, we can utilize simulations by
leveraging the gas_filter()
function, which accepts the
output of the gas()
function as an argument. This allows us
to obtain the standard deviations and quantiles for the worth parameter
estimates, providing a more comprehensive understanding of the parameter
dynamics over time.
To visualize time-varying parameters with confidence band, we can use
the plot()
on the gas_filter
object.
Finally, we perform one-year-ahead forecasts. We use the
gas_forecast()
function, which can again take the estimated
model as an argument.
fcst_stnry <- gas_forecast(est_stnry, t_ahead = 1, x_ahead = 0)
tibble(team = colnames(y)) %>%
mutate(fcst_strength = fcst_stnry$forecast$par_tv_ahead_mean[1, ]) %>%
mutate(fcst_gold = exp(fcst_strength) / sum(exp(fcst_strength))) %>%
mutate(fcst_rank = rank(-fcst_strength)) %>%
mutate(real_rank = ice_hockey_championships$rankings[24, ]) %>%
arrange(real_rank)
#> # A tibble: 24 × 5
#> team fcst_strength fcst_gold fcst_rank real_rank
#> <chr> <dbl> <dbl> <dbl> <dbl>
#> 1 CAN 3.97 0.234 2 1
#> 2 FIN 3.97 0.235 1 2
#> 3 USA 2.09 0.0356 6 3
#> 4 DEU 0.742 0.00929 10 4
#> 5 RUS 3.43 0.137 3 5
#> 6 CHE 1.82 0.0272 7 6
#> 7 CZE 3.41 0.134 4 7
#> 8 SVK 1.58 0.0214 8 8
#> 9 SWE 3.40 0.133 5 9
#> 10 KAZ -2.05 0.000569 19 10
#> 11 LVA 0.978 0.0117 9 11
#> 12 DNK 0.229 0.00556 11 12
#> 13 NOR 0.125 0.00501 12 13
#> 14 GBR -3.55 0.000127 22 14
#> 15 BLR -0.704 0.00219 14 15
#> 16 ITA -0.957 0.00170 16 16
#> 17 AUT -0.832 0.00192 15 Inf
#> 18 FRA -0.490 0.00271 13 Inf
#> 19 HUN -3.31 0.000162 21 Inf
#> 20 JPN -2.21 0.000486 20 Inf
#> 21 KOR -3.81 0.0000982 23 Inf
#> 22 POL -3.99 0.0000821 24 Inf
#> 23 SVN -1.95 0.000631 18 Inf
#> 24 UKR -1.69 0.000813 17 Inf
The forecasted values can be displayed using the generic
plot()
function.
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