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Introduction to the mtarm Package

Luis Hernando Vanegas

Sergio Alejandro Calderón

Luz Marina Rondón

1 Multivariate Threshold Autoregressive (TAR) models

The mtarm package provides a computational tool designed for Bayesian estimation, inference, and forecasting in multivariate Threshold Autoregressive (TAR) models. These models provide a versatile approach for modeling nonlinear multivariate time series and include multivariate Self-Exciting Threshold Autoregressive (SETAR) and Vector Autoregressive (VAR) models as particular cases (Vanegas, Calderón V, and Rondón 2025). The package accommodates a broad class of innovation distributions beyond the Gaussian assumption, such as Student-\(t\), slash, symmetric hyperbolic, Laplace, contaminated normal, skew-normal, and skew-\(t\) distributions, thereby enabling robust modeling of heavy tails, asymmetry, and other non-Gaussian characteristics.

1.1 Installation

1.1.1 Install from GitHub

remotes::install_github("lhvanegasp/mtarm")

1.1.2 Install from CRAN

install.packages("mtarm")

1.2 Application: Temperature, precipitation, and two river flows in Iceland

1.2.1 Dataset

The data are available in the object `iceland.rf` and were obtained from (Tong 1990), who provided a detailed description of the geographical and meteorological characteristics of the rivers and analyzed each series individually. Subsequently, (Tsay 1998) conducted a bivariate analysis of the same dataset. The focus is on the bivariate time series \(\{(Y_{1,t},Y_{2,t})^{\top}\}_{t\geq 1}\), where \(Y_{1,t}\) and \(Y_{2,t}\) denote the daily river flow (in cubic meters per second, \({m}^3/{s}\)) of the Jökulsá Eystri and Vatnsdalsá rivers, respectively. The sample covers the period from 1972 to 1974, comprising 1095 observations. The exogenous variables include daily precipitation \(X_t\), measured in millimeters (\({mm}\)), and temperature \(Z_t\), measured in degrees Celsius (\(^\circ\mathrm{C}\)), both recorded at the meteorological station in Hveravellir. Precipitation corresponds to the accumulated rainfall from 9:00 A.M. of the previous day to 9:00 A.M. of the current day.

library(mtarm)
data(iceland.rf)       
str(iceland.rf)     
#> 'data.frame':    1096 obs. of  5 variables:
#>  $ Vatnsdalsa   : num  16.1 19.2 14.5 11 13.6 12.5 10.5 10.1 9.68 9.02 ...
#>  $ Jokulsa      : num  30.2 29 28.4 27.8 27.8 27.8 27.8 27.8 27.8 27.3 ...
#>  $ Precipitation: num  8.1 4.4 7 0 0 0 1.9 1.2 0 0.1 ...
#>  $ Temperature  : num  0.9 1.6 0.1 0.6 2 0.8 1.4 1.3 2.2 0.1 ...
#>  $ Date         : Date, format: "1972-01-01" "1972-01-02" ...
summary(iceland.rf[,-5])
#>    Vatnsdalsa        Jokulsa       Precipitation     Temperature      
#>  Min.   : 3.670   Min.   : 22.00   Min.   : 0.000   Min.   :-22.4000  
#>  1st Qu.: 6.100   1st Qu.: 26.70   1st Qu.: 0.000   1st Qu.: -4.2000  
#>  Median : 7.500   Median : 31.40   Median : 0.300   Median :  0.3000  
#>  Mean   : 8.938   Mean   : 41.15   Mean   : 2.519   Mean   : -0.4407  
#>  3rd Qu.: 9.240   3rd Qu.: 50.90   3rd Qu.: 2.500   3rd Qu.:  3.9000  
#>  Max.   :54.000   Max.   :143.00   Max.   :79.300   Max.   : 13.9000
plot(ts(as.matrix(iceland.rf[,-5])), main="Iceland")

1.2.2 Model specification

Following (Tsay 1998), the series are modeled using a \(\mathrm{TAR}(2; p=(15,15), q=(4,4), d=(2,2))\) specification given by

\[ Y_t= \sum_{j=1}^{2} I\!\left(Z_{t-h}\in(c_{j-1},c_j]\right) \left(\! \phi_0^{^{(j)}} +\sum_{i=1}^{15}\boldsymbol{\phi}_i^{^{(j)}}Y_{t-i} +\sum_{i=1}^{4}\boldsymbol{\beta}_i^{^{(j)}}X_{t-i} +\sum_{i=1}^{2}\delta_i^{^{(j)}}Z_{t-i} +\epsilon_t^{^{(j)}} \!\right) \]

where \(\epsilon_t^{^{(j)}}\) is the error term. The last 55 observations (from November 7 to December 31, 1974), corresponding to \(5\%\) of the sample, are excluded from the estimation stage and reserved for out-of-sample forecast evaluation. The following code requests the estimation for the \(\mathrm{TAR}(2; p=(15,15), q=(4,4), d=(2,2))\) specification under Gaussian, Student-\(t\), and Laplace error distributions.

1.2.3 Parameter estimation

set.seed(09102)
fits <- mtar_grid(~ Jokulsa + Vatnsdalsa | Temperature | Precipitation,
                  data=iceland.rf, subset={Date<="1974-11-06"},                           
                  row.names=Date, nregim.min=2, nregim.max=2, p.min=15,                 
                  p.max=15, q.min=4, q.max=4, d.min=2, d.max=2,                           
                  n.burnin=500, n.sim=400, n.thin=2, ssvs=TRUE,
                  dist=c("Gaussian","Student-t","Laplace"),
                  plan_strategy="multisession")

fits
#> 
#> 
#> Sample size          : 1026 time points (1972-01-16 to 1974-11-06)
#> 
#> Output Series        : Jokulsa    |    Vatnsdalsa
#> 
#> Threshold Series (TS): Temperature
#> 
#> Exogenous Series (ES): Precipitation
#> 
#> Error Distribution   : Gaussian, Laplace, Student-t
#> 
#> Number of regimes    : 2
#> 
#> Deterministics       : Intercept
#> 
#> Autoregressive order : 15
#> 
#> Maximum lag for ES   : 4
#> 
#> Maximum lag for TS   : 2

1.2.4 Model comparison using forecast accuracy measures

1.2.4.1 Adjusted within-sample

The following code requests Deviance Information Criterion (DIC) (Spiegelhalter et al. 2002, 2014) and Watanabe-Akaike Information Criterion (WAIC) (Watanabe 2010) values.

DIC(fits)
#>                          DIC
#> Gaussian.2.15.4.2  10073.665
#> Laplace.2.15.4.2    7088.548
#> Student-t.2.15.4.2  7570.670

WAIC(fits)
#>                         WAIC
#> Gaussian.2.15.4.2  10267.211
#> Laplace.2.15.4.2    8172.876
#> Student-t.2.15.4.2  7624.427

1.2.4.2 Out-of-sample with standard \(h\)-step-ahead forecasting

In addition, the following code provides the median of the log-score (Good 1952), the Energy Score (ES) (Gneiting et al. 2008)—a multivariate extension of the Continuous Ranked Probability Score (CRPS)(Matheson and Winkler 1976; Grimit et al. 2006)—and the Absolute Percentage Error (APE), all computed from the observed and forecasted values for the last 55 observations.

newdata <- subset(iceland.rf, Date>"1974-11-06") 
set.seed(09102)

oos <- out_of_sample(fits, newdata=newdata, n.ahead=nrow(newdata), FUN=median) 
oos[,c(1,2,5,6)]
#>                    Log.Score Energy.Score Jokulsa.APE Vatnsdalsa.APE
#> Gaussian.2.15.4.2   3.756526     2.768467    5.791811       34.22062
#> Laplace.2.15.4.2    3.486000     2.027685    4.032976       19.60633
#> Student-t.2.15.4.2  3.508241     2.396514    3.783218       20.11439

1.2.4.3 Out-of-sample with rolling-origin forecasting and fixed parameters

set.seed(09102)
oos2 <- out_of_sample(fits, newdata=newdata, n.ahead=nrow(newdata), 
                      rolling=5, FUN=median) 

for(i in 1:length(oos2)){                   
   cat("\n",i,"-step-ahead\n")   
   print(oos2[[i]][,c(1,2,5,6)]) 
}                            
#> 
#>  1 -step-ahead
#>                    Log.Score Energy.Score Jokulsa.APE Vatnsdalsa.APE
#> Gaussian.2.15.4.2  2.5752867    1.2867011    1.577961       7.751834
#> Laplace.2.15.4.2   1.5069745    0.8707850    1.021102       4.374278
#> Student-t.2.15.4.2 0.9977087    0.7287759    1.093024       5.228132
#> 
#>  2 -step-ahead
#>                    Log.Score Energy.Score Jokulsa.APE Vatnsdalsa.APE
#> Gaussian.2.15.4.2   3.017372     1.658087    2.436912      12.468417
#> Laplace.2.15.4.2    2.336255     1.267278    1.614512       5.781262
#> Student-t.2.15.4.2  1.914611     1.166369    1.633090       5.845511
#> 
#>  3 -step-ahead
#>                    Log.Score Energy.Score Jokulsa.APE Vatnsdalsa.APE
#> Gaussian.2.15.4.2   3.182215     1.837573    2.754749      15.365773
#> Laplace.2.15.4.2    2.625043     1.486705    1.995235       7.677927
#> Student-t.2.15.4.2  2.236478     1.399870    1.984019       7.513583
#> 
#>  4 -step-ahead
#>                    Log.Score Energy.Score Jokulsa.APE Vatnsdalsa.APE
#> Gaussian.2.15.4.2   3.321731     1.967156    3.001013      17.236090
#> Laplace.2.15.4.2    2.870681     1.603429    2.299671       8.466708
#> Student-t.2.15.4.2  2.565933     1.621374    2.591758       7.240398
#> 
#>  5 -step-ahead
#>                    Log.Score Energy.Score Jokulsa.APE Vatnsdalsa.APE
#> Gaussian.2.15.4.2   3.382073     2.068730    3.284366      19.211759
#> Laplace.2.15.4.2    2.967714     1.692586    2.243715       9.666655
#> Student-t.2.15.4.2  2.777403     1.773609    2.757114       9.341327

1.2.5 Summary of the best model

summary(fits[["Laplace.2.15.4.2"]])
#> 
#> 
#> Sample size          : 1026 time points (1972-01-16 to 1974-11-06)
#> 
#> Output Series (OS)   : Jokulsa    |    Vatnsdalsa
#> 
#> Threshold Series (TS): Temperature with a estimated delay equal to 0
#> 
#> Exogenous Series (ES): Precipitation
#> 
#> Error Distribution   : Laplace
#> 
#> Number of regimes    : 2
#> 
#> Deterministics       : Intercept
#> 
#> Autoregressive orders: 15 in each regime
#> 
#> Maximum lags for ES  : 4 in each regime
#> 
#> Maximum lags for TS  : 2 in each regime
#> 
#> 
#> Thresholds (Mean, HDI.Lower, HDI.Upper)
#>                                                 
#> Regime 1 (-Inf,0.24] (-Inf,0.101] (-Inf,0.35403]
#> Regime 2  (0.24,Inf)  (0.101,Inf)  (0.35403,Inf)
#> 
#> 
#> Regime1:
#>      OS.lag(1) OS.lag(2) OS.lag(3) OS.lag(4) OS.lag(5) OS.lag(6) OS.lag(7)
#> SSVS         1         1      0.11      0.29         0         0         0
#>      OS.lag(8) OS.lag(9) OS.lag(10) OS.lag(11) OS.lag(12) OS.lag(13) OS.lag(14)
#> SSVS         0      0.44       0.15       0.16          0          0          0
#>      OS.lag(15) ES.lag(1) ES.lag(2) ES.lag(3) ES.lag(4) TS.lag(1) TS.lag(2)
#> SSVS       0.24         0         0         0         0         0      0.01
#> 
#> Autoregressive coefficients
#>                        Mean  2(1-PD)  HDI.Lower HDI.Upper             Mean
#> (Intercept)         4.71928     1e-05   3.59591   5.96216    |     1.46940
#> Jokulsa.lag( 1)     0.77745     1e-05   0.65459   0.87914    |    -0.08784
#> Vatnsdalsa.lag( 1)  0.33704     1e-05   0.21744   0.45593    |     1.03659
#> Jokulsa.lag( 2)    -0.01746     6e-01  -0.07278   0.04219    |     0.04701
#> Vatnsdalsa.lag( 2) -0.27120     1e-05  -0.39015  -0.12786    |    -0.20697
#>                     2(1-PD)  HDI.Lower HDI.Upper
#> (Intercept)            1e-05   0.99726   1.95633
#> Jokulsa.lag( 1)        1e-05  -0.13183  -0.04396
#> Vatnsdalsa.lag( 1)     1e-05   0.93730   1.11482
#> Jokulsa.lag( 2)        1e-05   0.01860   0.07639
#> Vatnsdalsa.lag( 2)     1e-05  -0.28657  -0.13284
#> 
#> Scale parameter (Mean, HDI.Lower, HDI.Upper)
#>            Jokulsa Vatnsdalsa      Jokulsa Vatnsdalsa      Jokulsa Vatnsdalsa
#> Jokulsa    0.24937    0.04570    . 0.19658    0.02911    . 0.29087    0.06407
#> Vatnsdalsa 0.04570    0.08887    . 0.02911    0.06767    . 0.06407    0.10777
#> 
#> 
#> Regime2:
#>      OS.lag(1) OS.lag(2) OS.lag(3) OS.lag(4) OS.lag(5) OS.lag(6) OS.lag(7)
#> SSVS         1         1         0         0         0         0         0
#>      OS.lag(8) OS.lag(9) OS.lag(10) OS.lag(11) OS.lag(12) OS.lag(13) OS.lag(14)
#> SSVS         0         0          0          0          0          0          0
#>      OS.lag(15) ES.lag(1) ES.lag(2) ES.lag(3) ES.lag(4) TS.lag(1) TS.lag(2)
#> SSVS          0      0.01         0         0         0         1         1
#> 
#> Autoregressive coefficients
#>                        Mean  2(1-PD)  HDI.Lower HDI.Upper             Mean
#> (Intercept)         1.30758     1e-02   0.34610   2.28749    |     0.45310
#> Jokulsa.lag( 1)     1.08490     1e-05   1.01553   1.15346    |    -0.00644
#> Vatnsdalsa.lag( 1)  0.61082     1e-05   0.28481   0.93183    |     1.29627
#> Jokulsa.lag( 2)    -0.22325     1e-05  -0.28360  -0.17030    |     0.00160
#> Vatnsdalsa.lag( 2) -0.26323     1e-01  -0.50575   0.07505    |    -0.32087
#> Temperature.lag(1)  1.18878     1e-05   0.99705   1.38429    |     0.04785
#> Temperature.lag(2) -0.72583     1e-05  -0.90614  -0.50919    |    -0.05406
#>                     2(1-PD)  HDI.Lower HDI.Upper
#> (Intercept)          0.00001   0.24315   0.72736
#> Jokulsa.lag( 1)      0.34000  -0.01748   0.00474
#> Vatnsdalsa.lag( 1)   0.00001   1.16829   1.40942
#> Jokulsa.lag( 2)      0.74500  -0.00759   0.01096
#> Vatnsdalsa.lag( 2)   0.00001  -0.41281  -0.20274
#> Temperature.lag(1)   0.01000   0.01232   0.08129
#> Temperature.lag(2)   0.01500  -0.09224  -0.02052
#> 
#> Scale parameter (Mean, HDI.Lower, HDI.Upper)
#>            Jokulsa Vatnsdalsa      Jokulsa Vatnsdalsa      Jokulsa Vatnsdalsa
#> Jokulsa    5.81841    0.27859    . 4.92348    0.15094    . 6.82294    0.41989
#> Vatnsdalsa 0.27859    0.28275    . 0.15094    0.24123    . 0.41989    0.33833
#>

1.2.6 Residuals

res <- residuals(fits[["Laplace.2.15.4.2"]])  
par(mfrow=c(1,2)) 
qqnorm(res[["full"]], pch=20, col="blue", main="") 
abline(0, 1, lty=3) 
hist(res[["full"]], freq=FALSE, xlab="Quantile-type residual",
     ylab="Density", main="") 
curve(dnorm(x), col="blue", add=TRUE)

par(mfrow=c(1,2))  
acf(res[["full"]], col="blue", main="")
pacf(res[["full"]], col="blue", main="")

1.2.7 Forecasting

pred <- predict(fits[["Laplace.2.15.4.2"]], newdata=newdata,
                n.ahead=nrow(newdata), row.names=Date, credible=0.8)

head(pred$summary)
#>            Jokulsa.Mean Jokulsa.Lower Jokulsa.Upper Vatnsdalsa.Mean
#> 1974-11-07     22.24887      14.80440      29.71098        7.065749
#> 1974-11-08     23.26888      17.07110      29.95938        7.468197
#> 1974-11-09     24.56545      19.57061      29.75290        7.602840
#> 1974-11-10     25.07271      20.53018      29.32400        7.452584
#> 1974-11-11     25.20357      21.53071      29.01823        7.231594
#> 1974-11-12     25.32326      22.85621      29.72021        7.061325
#>            Vatnsdalsa.Lower Vatnsdalsa.Upper
#> 1974-11-07         5.570865         8.656971
#> 1974-11-08         5.363028         9.101836
#> 1974-11-09         5.299086         9.449402
#> 1974-11-10         4.942358         9.212046
#> 1974-11-11         4.797265         9.265275
#> 1974-11-12         5.144482         9.462583
tail(pred$summary)
#>            Jokulsa.Mean Jokulsa.Lower Jokulsa.Upper Vatnsdalsa.Mean
#> 1974-12-26     25.79234      22.67541      29.20841        6.107449
#> 1974-12-27     25.90488      22.38146      29.01598        6.098012
#> 1974-12-28     25.94826      22.61811      28.87847        6.095481
#> 1974-12-29     25.95476      22.75529      28.69989        6.015718
#> 1974-12-30     25.91199      22.33841      28.47903        5.914708
#> 1974-12-31     25.75638      23.24709      29.06307        5.910189
#>            Vatnsdalsa.Lower Vatnsdalsa.Upper
#> 1974-12-26         3.901664         8.291431
#> 1974-12-27         3.992384         8.462892
#> 1974-12-28         3.676926         8.205371
#> 1974-12-29         3.960235         8.159962
#> 1974-12-30         4.202140         8.192281
#> 1974-12-31         3.702814         7.604186

1.2.8 Summary statistics

fitmcmc <- coda::as.mcmc(fits[["Laplace.2.15.4.2"]])
summary(fitmcmc)
#> 
#> 
#>  Iterations = 501:1299
#> 
#>  Thinning interval = 2
#> 
#>  Sample size per chain = 400
#> 
#> Thresholds:
#>             Mean       Sd Sd(Mean)    2.5%    25%     50%     75%   97.5%
#> Threshold.1 0.24 0.069355  0.02148 0.10709 0.2063 0.24164 0.28712 0.37538
#> 
#> 
#> Regime 1
#> 
#> 
#> Autoregressive coefficients:
#>                                    Mean       Sd  Sd(Mean)      2.5%       25%
#> Jokulsa:(Intercept)            4.719282 0.639500 0.1795684  3.700281  4.241557
#> Vatnsdalsa:(Intercept)         1.469398 0.268241 0.0677283  1.000085  1.273678
#> Jokulsa:Jokulsa.lag( 1)        0.777453 0.063010 0.0141672  0.633180  0.738324
#> Vatnsdalsa:Jokulsa.lag( 1)    -0.087844 0.024333 0.0048510 -0.133022 -0.105813
#> Jokulsa:Vatnsdalsa.lag( 1)     0.337044 0.062506 0.0051981  0.223531  0.294181
#> Vatnsdalsa:Vatnsdalsa.lag( 1)  1.036586 0.044641 0.0048578  0.941195  1.006997
#> Jokulsa:Jokulsa.lag( 2)       -0.017464 0.031735 0.0031497 -0.072485 -0.040201
#> Vatnsdalsa:Jokulsa.lag( 2)     0.047013 0.016309 0.0017976  0.017332  0.036022
#> Jokulsa:Vatnsdalsa.lag( 2)    -0.271201 0.069195 0.0165978 -0.400936 -0.319687
#> Vatnsdalsa:Vatnsdalsa.lag( 2) -0.206967 0.039197 0.0031167 -0.286568 -0.230777
#>                                     50%        75%     97.5%
#> Jokulsa:(Intercept)            4.596442  5.1348531  6.069274
#> Vatnsdalsa:(Intercept)         1.447402  1.6705781  1.968240
#> Jokulsa:Jokulsa.lag( 1)        0.790344  0.8264448  0.871635
#> Vatnsdalsa:Jokulsa.lag( 1)    -0.086233 -0.0694346 -0.044908
#> Jokulsa:Vatnsdalsa.lag( 1)     0.335304  0.3811853  0.469796
#> Vatnsdalsa:Vatnsdalsa.lag( 1)  1.038449  1.0665587  1.119978
#> Jokulsa:Jokulsa.lag( 2)       -0.018531  0.0036256  0.042678
#> Vatnsdalsa:Jokulsa.lag( 2)     0.046598  0.0596761  0.076146
#> Jokulsa:Vatnsdalsa.lag( 2)    -0.273804 -0.2249351 -0.136540
#> Vatnsdalsa:Vatnsdalsa.lag( 2) -0.207859 -0.1808252 -0.132843
#> 
#> 
#> Scale parameter:
#>                           Mean        Sd  Sd(Mean)     2.5%      25%      50%
#> Jokulsa.Jokulsa       0.249370 0.0251663 0.0023048 0.201676 0.233108 0.248983
#> Jokulsa.Vatnsdalsa    0.045699 0.0088572 0.0016123 0.029299 0.039962 0.044792
#> Vatnsdalsa.Vatnsdalsa 0.088868 0.0107010 0.0010622 0.069301 0.081832 0.088942
#>                            75%   97.5%
#> Jokulsa.Jokulsa       0.267048 0.30054
#> Jokulsa.Vatnsdalsa    0.051385 0.06437
#> Vatnsdalsa.Vatnsdalsa 0.095470 0.11105
#> 
#> 
#> Regime 2
#> 
#> 
#> 
#> Autoregressive coefficients:
#>                                     Mean        Sd   Sd(Mean)       2.5%
#> Jokulsa:(Intercept)            1.3075820 0.5018816 0.02693731  0.1764215
#> Vatnsdalsa:(Intercept)         0.4530976 0.1245738 0.00992205  0.2431474
#> Jokulsa:Jokulsa.lag( 1)        1.0849030 0.0383974 0.00345264  1.0180005
#> Vatnsdalsa:Jokulsa.lag( 1)    -0.0064415 0.0061333 0.00040984 -0.0174777
#> Jokulsa:Vatnsdalsa.lag( 1)     0.6108226 0.1660698 0.01063256  0.2729072
#> Vatnsdalsa:Vatnsdalsa.lag( 1)  1.2962716 0.0650681 0.01599231  1.1512708
#> Jokulsa:Jokulsa.lag( 2)       -0.2232455 0.0308198 0.00268918 -0.2835984
#> Vatnsdalsa:Jokulsa.lag( 2)     0.0016049 0.0049368 0.00030936 -0.0075856
#> Jokulsa:Vatnsdalsa.lag( 2)    -0.2632336 0.1457999 0.01109848 -0.5391868
#> Vatnsdalsa:Vatnsdalsa.lag( 2) -0.3208660 0.0562924 0.01253762 -0.4105189
#> Jokulsa:Temperature.lag(1)     1.1887787 0.1065221 0.00910691  0.9970505
#> Vatnsdalsa:Temperature.lag(1)  0.0478535 0.0187300 0.00108803  0.0114049
#> Jokulsa:Temperature.lag(2)    -0.7258338 0.1018480 0.00791009 -0.9209181
#> Vatnsdalsa:Temperature.lag(2) -0.0540590 0.0189246 0.00137457 -0.0906691
#>                                      25%        50%        75%      97.5%
#> Jokulsa:(Intercept)            1.0088174  1.3369085  1.6442004  2.2046857
#> Vatnsdalsa:(Intercept)         0.3679797  0.4475482  0.5186073  0.7273559
#> Jokulsa:Jokulsa.lag( 1)        1.0569074  1.0851120  1.1128026  1.1604548
#> Vatnsdalsa:Jokulsa.lag( 1)    -0.0106835 -0.0063279 -0.0020743  0.0047436
#> Jokulsa:Vatnsdalsa.lag( 1)     0.5065260  0.6123568  0.7163413  0.9294226
#> Vatnsdalsa:Vatnsdalsa.lag( 1)  1.2578740  1.3062128  1.3418977  1.3979920
#> Jokulsa:Jokulsa.lag( 2)       -0.2432853 -0.2220171 -0.2002674 -0.1703035
#> Vatnsdalsa:Jokulsa.lag( 2)    -0.0017068  0.0015996  0.0048872  0.0109577
#> Jokulsa:Vatnsdalsa.lag( 2)    -0.3559644 -0.2729285 -0.1780783  0.0536621
#> Vatnsdalsa:Vatnsdalsa.lag( 2) -0.3609004 -0.3282851 -0.2856788 -0.1994312
#> Jokulsa:Temperature.lag(1)     1.1115798  1.1875205  1.2645726  1.3842905
#> Vatnsdalsa:Temperature.lag(1)  0.0357031  0.0477983  0.0614416  0.0811665
#> Jokulsa:Temperature.lag(2)    -0.8000215 -0.7236439 -0.6593145 -0.5206088
#> Vatnsdalsa:Temperature.lag(2) -0.0660588 -0.0537316 -0.0418908 -0.0143790
#> 
#> 
#> Scale parameter:
#>                          Mean       Sd  Sd(Mean)    2.5%     25%     50%
#> Jokulsa.Jokulsa       5.81841 0.501736 0.0341567 4.88681 5.47334 5.81445
#> Jokulsa.Vatnsdalsa    0.27859 0.071145 0.0038619 0.15094 0.22734 0.27809
#> Vatnsdalsa.Vatnsdalsa 0.28275 0.024618 0.0015403 0.23297 0.26596 0.28190
#>                           75%   97.5%
#> Jokulsa.Jokulsa       6.10752 6.80022
#> Jokulsa.Vatnsdalsa    0.32812 0.41989
#> Vatnsdalsa.Vatnsdalsa 0.29836 0.33451

1.2.9 Convergence diagnostics

1.2.9.1 Geweke statistic

geweke_diagTAR(fits[["Laplace.2.15.4.2"]])
#> 
#> Fraction in 1st window = 0.1
#> 
#> Fraction in 2nd window = 0.5
#> Thresholds:
#> Threshold.1 
#>      1.8944
#> 
#> 
#> Regime 1
#> 
#> 
#> Autoregressive coefficients:
#>                    Jokulsa Vatnsdalsa
#> (Intercept)        -2.9592   -1.82894
#> Jokulsa.lag( 1)     1.8198    0.34278
#> Vatnsdalsa.lag( 1)  3.6611    4.64634
#> Jokulsa.lag( 2)     1.9775   -1.75061
#> Vatnsdalsa.lag( 2) -2.1832    0.23539
#> 
#> 
#> Scale parameter:
#>            Jokulsa Vatnsdalsa
#> Jokulsa     1.6637     1.0644
#> Vatnsdalsa  1.0644     4.0612
#> 
#> 
#> Regime 2
#> 
#> 
#> 
#> Autoregressive coefficients:
#>                      Jokulsa Vatnsdalsa
#> (Intercept)         0.275425   -0.21867
#> Jokulsa.lag( 1)     0.451632   -2.51149
#> Vatnsdalsa.lag( 1)  2.914177    2.30717
#> Jokulsa.lag( 2)    -0.660319    1.57149
#> Vatnsdalsa.lag( 2) -3.671382   -2.13820
#> Temperature.lag(1)  0.041469   -0.21647
#> Temperature.lag(2)  0.316697    0.55146
#> 
#> 
#> Scale parameter:
#>            Jokulsa Vatnsdalsa
#> Jokulsa    0.10636    0.39787
#> Vatnsdalsa 0.39787   -1.51946

1.2.9.2 Effective sample size

effectiveSize_TAR(fits[["Laplace.2.15.4.2"]]) 
#> Thresholds:
#> Threshold.1 
#>      10.425
#> 
#> 
#> Regime 1
#> 
#> 
#> Autoregressive coefficients:
#>                    Jokulsa Vatnsdalsa
#> (Intercept)         12.683     15.686
#> Jokulsa.lag( 1)     19.781     25.160
#> Vatnsdalsa.lag( 1) 144.595     84.447
#> Jokulsa.lag( 2)    101.521     82.312
#> Vatnsdalsa.lag( 2)  17.380    158.163
#> 
#> 
#> Scale parameter:
#>            Jokulsa Vatnsdalsa
#> Jokulsa    119.225     30.178
#> Vatnsdalsa  30.178    101.499
#> 
#> 
#> Regime 2
#> 
#> 
#> 
#> Autoregressive coefficients:
#>                    Jokulsa Vatnsdalsa
#> (Intercept)         347.13    157.634
#> Jokulsa.lag( 1)     123.68    223.959
#> Vatnsdalsa.lag( 1)  243.95     16.554
#> Jokulsa.lag( 2)     131.35    254.664
#> Vatnsdalsa.lag( 2)  172.58     20.159
#> Temperature.lag(1)  136.82    296.343
#> Temperature.lag(2)  165.78    189.547
#> 
#> 
#> Scale parameter:
#>            Jokulsa Vatnsdalsa
#> Jokulsa     215.77     339.39
#> Vatnsdalsa  339.39     255.44
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