Package ‘dccmidas’

February 21, 2024

Type Package
Title DCC Models with GARCH and GARCH-MIDAS Specifications in the
Univariate Step, RiskMetrics, Moving Covariance and Scalar and
Diagonal BEKK Models
Version 0.1.2
Description Estimates a variety of Dynamic Conditional Correlation (DCC) models. More in de-
tail, the 'dccmidas' package allows the estimation of the cor-
rected DCC (cDCC) of Aielli (2013) <doi:10.1080/07350015.2013.771027>, the DCC-
MIDAS of Colacito et al. (2011) <doi:10.1016/j.jeconom.2011.02.013>, the Asymmet-
ric DCC of Cappiello et al. <doi:10.1093/jjfinec/nbl005>, and the Dynamic Equicorrela-
tion (DECO) of Engle and Kelly (2012) <doi:10.1080/07350015.2011.652048>. 'dccmidas' of-
fers the possibility of including standard GARCH <doi:10.1016/0304-4076(86)90063-
1>, GARCH-MIDAS <doi:10.1162/REST_a_00300> and Double Asymmetric GARCH-
MIDAS <doi:10.1016/j.econmod.2018.07.025> models in the univariate estimation. More-
over, also the scalar and diagonal BEKK <doi:10.1017/S0266466600009063> models can be es-
timated. Finally, the package calculates also the var-cov matrix under two non-parametric mod-
els: the Moving Covariance and the RiskMetrics specifications.
License GPL-3
LinkingTo Rcpp, RcppArmadillo
Encoding UTF-8
LazyData true
RoxygenNote 7.2.3
RdMacros Rdpack
Depends R (>= 4.0.0)
Imports maxLik (>= 1.3-8), rumidas (>= 0.1.1), rugarch (>= 1.4-4),
roll (>= 1.1.4), xts (>= 0.12.0), Rdpack (>= 1.0.0), zoo (>=
1.8.8), stats (>= 4.0.2), utils (>= 4.0.2)
Suggests knitr, rmarkdown
NeedsCompilation yes
Author Vincenzo Candila [aut, cre]
Maintainer Vincenzo Candila <vcandila@unisa.it>

1
2 bekk_fit

Repository CRAN
Date/Publication 2024-02-21 14:10:02 UTC

R topics documented:
bekk_fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
cov_eval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
dcc_fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Det . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
ftse100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
indpro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Inv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
moving_cov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
nasdaq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
plot_dccmidas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
riskmetrics_mat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
sp500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Index 16

bekk_fit BEKK fit

Description
Obtains the estimation the scalar and diagonal BEKK model

Usage
bekk_fit(r_t, model = "sBEKK", R = 100, out_of_sample = NULL)

Arguments
r_t List of daily returns. At the moment, at most 5 assets can be considered
model Valid choices are: ’sBEKK’(scalar BEKK) and ’dBEKK’ (diagonal BEKK)
R optional Number of random samples drawn from a Uniform distribution used
to inizialize the log-likelihood. Equal to 100 by default
out_of_sample optional A positive integer indicating the number of periods before the last to
keep for out of sample forecasting

Details
Function bekk_fit implements the estimation of scalar and diagonal BEKK models. For details on
BEKK models, see Engle and Kroner (1995)
cov_eval 3

Value
bekk_fit returns a list containing the following components:
• assets: Names of the assets considered.
• mat_coef: Matrix of estimated coefficients of the model, with the QML standard errors.
• obs: The number of daily observations used for the estimation.
• period: The period of the estimation.
• H_t: Conditional covariance matrix, reported as an array. It refers to the in-sample period.
• est_time: Time of estimation.
• llk: The value of the log-likelihood at the maximum.
• H_t_oos: Conditional covariance matrix, reported as an array, for the out-of-sample period, if
the param ’out_of_sample’ is used.
• Days: Days of the (in-)sample period.

References
Engle RF, Kroner KF (1995). “Multivariate simultaneous generalized ARCH.” Econometric theory,
11(1), 122–150. doi:10.1017/S0266466600009063.

Examples

require(xts)
# close to close daily log-returns
r_t_s<-diff(log(sp500['2010/2019'][,3]))
r_t_s[1]<-0
r_t_n<-diff(log(nasdaq['2010/2019'][,3]))
r_t_n[1]<-0
r_t_f<-diff(log(ftse100['2010/2019'][,3]))
r_t_f[1]<-0
db_m<-merge.xts(r_t_s,r_t_n,r_t_f)
db_m<-db_m[complete.cases(db_m),]
colnames(db_m)<-c("S&P500","NASDAQ","FTSE100")
# list of returns
r_t<-list(db_m[,1],db_m[,2],db_m[,3])
bekk_est<-bekk_fit(r_t,model="sBEKK")
bekk_est$mat_coef

cov_eval Var-cov matrix evaluation

Description
Evaluates the estimated var-cov matrix H_t with respect to a covariance proxy, under different
robust loss functions (Laurent et al. 2013). The losses considered are also used in Amendola et al.
(2020).
4 cov_eval

Usage
cov_eval(H_t, cov_proxy = NULL, r_t = NULL, loss = "FROB")

Arguments
H_t Estimated covariance matrix, formatted as array
cov_proxy optional Covariance matrix, formatted as array
r_t optional List of daily returns used to calculate H_t. If parameter ’cov_proxy’
is not provided, then r_t must be included. In this case, a (noise) proxy will be
automatically used
loss Robust loss function to use. Valid choices are: "FROB" for Frobenius (by de-
fault), "SFROB" for Squared Frobenius, "EUCL" for Euclidean, "QLIKE" for
QLIKE and "RMSE" for Root Mean Squared Errors

Value
The value of the loss for each t

References
Amendola A, Braione M, Candila V, Storti G (2020). “A Model Confidence Set approach to the
combination of multivariate volatility forecasts.” International Journal of Forecasting, 36(3), 873 -
891. doi:10.1016/j.ijforecast.2019.10.001.

Laurent S, Rombouts JV, Violante F (2013). “On loss functions and ranking forecasting perfor-
mances of multivariate volatility models.” Journal of Econometrics, 173(1), 1–10. doi:10.1016/
j.jeconom.2012.08.004.

Examples

require(xts)
# close to close daily log-returns
r_t_s<-diff(log(sp500['2010/2019'][,3]))
r_t_s[1]<-0
r_t_n<-diff(log(nasdaq['2010/2019'][,3]))
r_t_n[1]<-0
r_t_f<-diff(log(ftse100['2010/2019'][,3]))
r_t_f[1]<-0
db_m<-merge.xts(r_t_s,r_t_n,r_t_f)
db_m<-db_m[complete.cases(db_m),]
colnames(db_m)<-c("S&P500","NASDAQ","FTSE100")
# list of returns
r_t<-list(db_m[,1],db_m[,2],db_m[,3])
# estimation
K_c<-144
N_c<-36
cdcc_est<-dcc_fit(r_t,univ_model="sGARCH",distribution="norm",
corr_model="DCCMIDAS",N_c=N_c,K_c=K_c)
cov_eval(cdcc_est$H_t,r_t=r_t)[(K_c+1):dim(cdcc_est$H_t)[3]]
dcc_fit 5

dcc_fit DCC fit (first and second steps)

Description
Obtains the estimation of a variety of DCC models, using as univariate models both GARCH and
GARCH-MIDAS specifications.

Usage
dcc_fit(
r_t,
univ_model = "sGARCH",
distribution = "norm",
MV = NULL,
K = NULL,
corr_model = "cDCC",
lag_fun = "Beta",
N_c = NULL,
K_c = NULL,
out_of_sample = NULL
)

Arguments
r_t List of daily returns on which estimate a DCC model. Each daily return must be
an ’xts’ object. Note that the sample period of the returns should be the same.
Otherwise, a merge is performed
univ_model Specification of the univariate model. Valid choices are: some of the specifica-
tions used in the rugarch (ugarchspec) and rumidas (ugmfit) packages. More
in detail, the models coming from rugarch are: model Valid models (currently
implemented) are ’sGARCH’, ’eGARCH’, ’gjrGARCH’, ’iGARCH’, and ’cs-
GARCH’. The models implemented from rumidas are: ’GM_skew’,’GM_noskew’,
’DAGM_skew’, and ’DAGM_noskew’
distribution optional Distribution chosen for the univariate estimation. Valid choices are:
"norm" (by default) and "std", respectively, for the Normal and Student’s t dis-
tributions
MV optional MIDAS variable to include in the univariate estimation, if the model
specificied is a GARCH-MIDAS (GM, Engle et al. (2013)) or a Double Asym-
metric GM (DAGM, Engle et al. (2013)). In the case of MIDAS-based models,
please provide a list of the MIDAS variables obtained from the mv_into_mat
function. If the same MV variable is used, then provide always a list, with the
same (transformed) variable repeated
6 dcc_fit

K optional The number of lagged realization of MV variable to use, if ’univ_model’

has a MIDAS term
corr_model Correlation model used. Valid choices are: "cDCC" (the corrected DCC of
Aielli (2013)), "aDCC" (the asymmetric DCC model of Cappiello et al. (2006)),
"DECO" (Dynamic equicorrelation of Engle and Kelly (2012)), and "DCCMI-
DAS" (the DCC-MIDAS of Colacito et al. (2011)). By detault, it is "cDCC"
lag_fun optional. Lag function to use. Valid choices are "Beta" (by default) and "Al-
mon", for the Beta and Exponential Almon lag functions, respectively, if ’univ_model’
has a MIDAS term and/or if ’corr_model’ is "DCCMIDAS"
N_c optional Number of (lagged) realizations to use for the standarized residuals
forming the long-run correlation, if ’corr_model’ is "DCCMIDAS"
K_c optional Number of (lagged) realizations to use for the long-run correlation, if
’corr_model’ is "DCCMIDAS"
out_of_sample optional A positive integer indicating the number of periods before the last to
keep for out of sample forecasting

Details
Function dcc_fit implements the two-steps estimation of the DCC models. In the first step,
a variety of univariate models are considered. These models can be selected using for the pa-
rameter ’univ_model’ one of the following choices: ’sGARCH’ (standard GARCH of Bollerslev
(1986)), ’eGARCH’ of Nelson (1991), ’gjrGARCH’ of Glosten et al. (1993), ’iGARCH’ (Integrated
GARCH of Engle and Bollerslev (1986)), ’csGARCH’ (the Component GARCH of Engle and Lee
(1999)), ’GM_noskew’ and ’GM_skew’ (the GARCH-MIDAS model of Engle et al. (2013), respec-
tively, without and with the asymmetric term in the short-run component), and ’DAGM_noskew’
and ’DAGM_skew’ (the Double Asymmetric GARCH-MIDAS model of Amendola et al. (2019),
respectively, without and with the asymmetric term in the short-run component).

Value
dcc_fit returns an object of class ’dccmidas’. The function summary.dccmidas can be used to
print a summary of the results. Moreover, an object of class ’dccmidas’ is a list containing the
following components:

• assets: Names of the assets considered.

• model: Univariate model used in the first step.
• est_univ_model: List of matrixes of estimated coefficients of the univariate model, with the
QML (Bollerslev and Wooldridge 1992) standard errors.
• corr_coef_mat: Matrix of estimated coefficients of the correlation model, with the QML stan-
dard errors.
• mult_model: Correlation model used in the second step.
• obs: The number of daily observations used for the estimation.
• period: The period of the (in-sample) estimation.
• H_t: Conditional covariance matrix, reported as an array.
• R_t: Conditional correlation matrix, reported as an array.
dcc_fit 7

• R_t_bar: Conditional long-run correlation matrix, reported as an array, if the correlation ma-
trix includes a MIDAS specification.
• H_t_oos: Conditional covariance matrix, reported as an array, for the out-of-sample period, if
present.
• R_t_oos: Conditional correlation matrix, reported as an array, for the out-of-sample period, if
present.
• R_t_bar_oos: Conditional long-run correlation matrix, reported as an array, if the correlation
matrix includes a MIDAS specification, for the out-of-sample period, if present.
• est_time: Time of estimation.
• Days: Days of the (in-)sample period.
• llk: The value of the log-likelihood (for the second step) at the maximum.

References
Aielli GP (2013). “Dynamic conditional correlation: on properties and estimation.” Journal of
Business & Economic Statistics, 31(3), 282–299. doi:10.1080/07350015.2013.771027.

Amendola A, Candila V, Gallo GM (2019). “On the asymmetric impact of macro–variables on

volatility.” Economic Modelling, 76, 135–152. doi:10.1016/j.econmod.2018.07.025.

Bollerslev T (1986). “Generalized autoregressive conditional heteroskedasticity.” Journal of Econo-

metrics, 31(3), 307–327. doi:10.1016/03044076(86)900631.

Bollerslev T, Wooldridge JM (1992). “Quasi-maximum likelihood estimation and inference in dy-

namic models with time-varying covariances.” Econometric Reviews, 11, 143–172. doi:10.1080/
07474939208800229.

Cappiello L, Engle RF, Sheppard K (2006). “Asymmetric dynamics in the correlations of global
equity and bond returns.” Journal of Financial Econometrics, 4(4), 537–572. doi:10.1093/jjfinec/
nbl005.

Colacito R, Engle RF, Ghysels E (2011). “A component model for dynamic correlations.” Journal
of Econometrics, 164(1), 45–59. doi:10.1016/j.jeconom.2011.02.013.

Engle R, Kelly B (2012). “Dynamic equicorrelation.” Journal of Business & Economic Statis-
tics, 30(2), 212–228. doi:10.1080/07350015.2011.652048.

Engle RF, Bollerslev T (1986). “Modelling the persistence of conditional variances.” Econometric
Reviews, 5(1), 1–50. doi:10.1080/07474938608800095.

Engle RF, Ghysels E, Sohn B (2013). “Stock market volatility and macroeconomic fundamen-
tals.” Review of Economics and Statistics, 95(3), 776–797. doi:10.1162/REST_a_00300.

Engle RF, Lee GJ (1999). “A Long-run and Short-run Component Model of Stock Return Volatil-
ity.” In Engle RF, White H (eds.), Cointegration, Causality, and Forecasting: A Festschrift in Honor
of Clive W. J. Granger, 475–497. Oxford University Press, Oxford.
8 Det

Glosten LR, Jagannathan R, Runkle DE (1993). “On the relation between the expected value and
the volatility of the nominal excess return on stocks.” The Journal of Finance, 48(5), 1779–1801.
doi:10.1111/j.15406261.1993.tb05128.x.

Nelson DB (1991). “Conditional heteroskedasticity in asset returns: A new approach.” Econo-

metrica, 59(2), 347–370. doi:10.2307/2938260.

Examples

require(xts)
# daily log-returns
# close to close daily log-returns
r_t_s<-diff(log(sp500['2010/2019'][,3]))
r_t_s[1]<-0
r_t_n<-diff(log(nasdaq['2010/2019'][,3]))
r_t_n[1]<-0
r_t_f<-diff(log(ftse100['2010/2019'][,3]))
r_t_f[1]<-0
db_m<-merge.xts(r_t_s,r_t_n,r_t_f)
db_m<-db_m[complete.cases(db_m),]
colnames(db_m)<-c("S&P500","NASDAQ","FTSE100")
# list of returns
r_t<-list(db_m[,1],db_m[,2],db_m[,3])
# MV transformation (same MV for all the stocks)
require(rumidas)
mv_m<-mv_into_mat(r_t[[1]],diff(indpro),K=12,"monthly")
# list of MV
MV<-list(mv_m,mv_m,mv_m)
# estimation
K_c<-144
N_c<-36
dccmidas_est<-dcc_fit(r_t,univ_model="GM_noskew",distribution="norm",
MV=MV,K=12,corr_model="DCCMIDAS",N_c=N_c,K_c=K_c)
dccmidas_est
summary.dccmidas(dccmidas_est)

Det Matrix determinant

Description
Calculates the determinant of a numeric matrix.

Usage
Det(x)
ftse100 9

Arguments
x a numeric matrix

Value
The determinant of x.

Examples
x<-matrix(sample(1:25,25,replace=TRUE),ncol=5)
Det(x)

ftse100 FTSE 100 data

Description
Daily data on FTSE 100 collected from the realized library of the Oxford-Man Institute (Heber et
al. 2009).

Usage
data(ftse100)

Format
An object of class "xts".

Details
ftse100 includes the open price (open_price), the realized variance (rv5), and the close price (close_price).
The realized variance has been calculated using intradaily intervals of five minutes (Andersen and
Bollerslev 1998).

Source
Realized library of the Oxford-Man Institute

References
Andersen TG, Bollerslev T (1998). “Answering the Skeptics: Yes, Standard Volatility Models do
Provide Accurate Forecasts.” International Economic Review, 39, 885–905. doi:10.2307/2527343.

Heber G, Lunde A, Shephard N, Sheppard K (2009). “OMI’s realised library, version 0.1.” Oxford–
Man Institute, University of Oxford.

Examples
head(ftse100)
summary(ftse100)
10 indpro

indpro Monthly U.S. Industrial Production

Description

Monthly data on the U.S. Industrial Production index (IP, index 2012=100, seasonally adjusted)
collected from the Federal Reserve Economic Data (FRED) archive. The IP has been used as
MIDAS term in different contributions (see, for instance, Engle et al. (2013), Conrad and Loch
(2015), and Amendola et al. (2017)).

Usage

data(indpro)

Format

An object of class "xts".

Source

Archive of the Federal Reserve Economic Data (FRED)

References

Amendola A, Candila V, Scognamillo A (2017). “On the influence of US monetary policy on crude
oil price volatility.” Empirical Economics, 52(1), 155–178. doi:10.1007/s0018101610695.

Conrad C, Loch K (2015). “Anticipating Long-Term Stock Market Volatility.” Journal of Applied
Econometrics, 30(7), 1090–1114. doi:10.1002/jae.2404.

Engle RF, Ghysels E, Sohn B (2013). “Stock market volatility and macroeconomic fundamentals.”
Review of Economics and Statistics, 95(3), 776–797. doi:10.1162/REST_a_00300.

Examples

head(indpro)
summary(indpro)
plot(indpro)
Inv 11

Inv Inverse of a matrix

Description
Calculates the inverse of a numeric matrix

Usage
Inv(x)

Arguments
x a numeric matrix

Value
The inverse of x.

Examples
x<-matrix(sample(1:25,25,replace=TRUE),ncol=5)
Inv(x)

moving_cov Moving Covariance model

Description
Obtains the matrix H_t, under the Moving Covariance model.

Usage
moving_cov(r_t, V = 22)

Arguments
r_t List of daily returns
V Length of the rolling window adopted. By default, V is 22

Value
A list with the Ht matrix, for each t.
12 nasdaq

Examples

require(xts)
# close to close daily log-returns
r_t_s<-diff(log(sp500['2010/2019'][,3]))
r_t_s[1]<-0
r_t_n<-diff(log(nasdaq['2010/2019'][,3]))
r_t_n[1]<-0
r_t_f<-diff(log(ftse100['2010/2019'][,3]))
r_t_f[1]<-0
db_m<-merge.xts(r_t_s,r_t_n,r_t_f)
db_m<-db_m[complete.cases(db_m),]
colnames(db_m)<-c("S&P500","NASDAQ","FTSE100")
# list of returns
r_t<-list(db_m[,1],db_m[,2],db_m[,3])
MC<-moving_cov(r_t,V=60)

nasdaq NASDAQ data

Description

Daily data on NASDAQ collected from the realized library of the Oxford-Man Institute (Heber et
al. 2009).

Usage

data(nasdaq)

Format

An object of class "xts".

Details

nasdaq includes the open price (open_price), the realized variance (rv5), and the close price (close_price).
The realized variance has been calculated using intradaily intervals of five minutes (Andersen and
Bollerslev 1998).

Source

Realized library of the Oxford-Man Institute

plot_dccmidas 13

References
Andersen TG, Bollerslev T (1998). “Answering the Skeptics: Yes, Standard Volatility Models do
Provide Accurate Forecasts.” International Economic Review, 39, 885–905. doi:10.2307/2527343.

Heber G, Lunde A, Shephard N, Sheppard K (2009). “OMI’s realised library, version 0.1.” Oxford–
Man Institute, University of Oxford.

Examples
head(nasdaq)
summary(nasdaq)

plot_dccmidas Plot method for ’dccmidas’ class

Description
Plots of the conditional volatilities on the main diagonal and of the conditional correlations on the
extra-diagonal elements.

Usage
plot_dccmidas(
x,
K_c = NULL,
vol_col = "black",
long_run_col = "red",
cex_axis = 0.75,
LWD = 2,
asset_sub = NULL
)

Arguments
x An object of class ’dccmidas’, that is the result of a call to dcc_fit.
K_c optional Number of (lagged) realizations to use for the long-run correlation, , if
’corr_model’ is "DCCMIDAS"
vol_col optional Color of the volatility and correlation plots. "black" by default
long_run_col optional Color of the long-run correlation plots, if present. "red" by default
cex_axis optional Size of the x-axis. Default to 0.75
LWD optional Width of the plotted lines. Default to 2
asset_sub optional Numeric vector of selected assets to consider for the plot. NULL by
default
14 riskmetrics_mat

Value
No return value, called for side effects

Examples

require(xts)
# close to close daily log-returns
r_t_s<-diff(log(sp500['2010/2019'][,3]))
r_t_s[1]<-0
r_t_n<-diff(log(nasdaq['2010/2019'][,3]))
r_t_n[1]<-0
r_t_f<-diff(log(ftse100['2010/2019'][,3]))
r_t_f[1]<-0
db_m<-merge.xts(r_t_s,r_t_n,r_t_f)
db_m<-db_m[complete.cases(db_m),]
colnames(db_m)<-c("S&P500","NASDAQ","FTSE100")
# list of returns
r_t<-list(db_m[,1],db_m[,2],db_m[,3])
# estimation
K_c<-144
N_c<-36
cdcc_est<-dcc_fit(r_t,univ_model="sGARCH",distribution="norm",
corr_model="DCCMIDAS",N_c=N_c,K_c=K_c)
plot_dccmidas(cdcc_est,K_c=144)

riskmetrics_mat RiskMetrics model

Description
Obtains the matrix H_t, under the RiskMetrics model.

Usage
riskmetrics_mat(r_t, lambda = 0.94)

Arguments
r_t List of daily returns
lambda optional Decay parameter. Default to 0.94

Value
A list with the Ht matrix, for each t.
sp500 15

sp500 S&P 500 data

Description
Daily data on S&P 500 collected from the realized library of the Oxford-Man Institute (Heber et al.
2009).

Usage
data(sp500)

Format
An object of class "xts".

Details
sp500 includes the open price (open_price), the realized variance (rv5), and the close price (close_price).
The realized variance has been calculated using intradaily intervals of five minutes (Andersen and
Bollerslev 1998).

Source
Realized library of the Oxford-Man Institute

References
Andersen TG, Bollerslev T (1998). “Answering the Skeptics: Yes, Standard Volatility Models do
Provide Accurate Forecasts.” International Economic Review, 39, 885–905. doi:10.2307/2527343.

Heber G, Lunde A, Shephard N, Sheppard K (2009). “OMI’s realised library, version 0.1.” Oxford–
Man Institute, University of Oxford.

Examples
head(sp500)
summary(sp500)
Index

∗ datasets
ftse100, 9
indpro, 10
nasdaq, 12
sp500, 15

bekk_fit, 2

cov_eval, 3

dcc_fit, 5, 13
Det, 8

ftse100, 9

indpro, 10
Inv, 11

moving_cov, 11
mv_into_mat, 5

nasdaq, 12

plot_dccmidas, 13

riskmetrics_mat, 14

sp500, 15
summary.dccmidas, 6

ugarchspec, 5
ugmfit, 5

16