Using mlrv to anaylze data

Data analysis in the paper of Bai and Wu (2023b).

Loading data

Hong Kong circulatory and respiratory data.

library(mlrv)
library(foreach)
library(magrittr)

data(hk_data)
colnames(hk_data) = c("SO2","NO2","Dust","Ozone","Temperature",
                      "Humidity","num_circu","num_respir","Hospital Admission",
                      "w1","w2","w3","w4","w5","w6")
n = nrow(hk_data)
t = (1:n)/n
hk = list()

hk$x = as.matrix(cbind(rep(1,n), scale(hk_data[,1:3])))
hk$y = hk_data$`Hospital Admission`

Test for long memory

pvmatrix = matrix(nrow=2, ncol=4)
###inistialization
setting = list(B = 5000, gcv = 1, neighbour = 1)
setting$lb = floor(10/7*n^(4/15)) - setting$neighbour 
setting$ub = max(floor(25/7*n^(4/15))+ setting$neighbour,             
                  setting$lb+2*setting$neighbour+1)

Using the plug-in estimator for long-run covariance matrix function.

setting$lrvmethod =0. 

i=1
# print(rule_of_thumb(y= hk$y, x = hk$x))
for(type in c("KPSS","RS","VS","KS")){
  setting$type = type
  print(type)
  result_reg = heter_covariate(list(y= hk$y, x = hk$x), setting, mvselect = -2)
  print(paste("p-value",result_reg))
  pvmatrix[1,i] = result_reg
  i = i + 1
}

## [1] "KPSS"
## [1] "p-value 0.3048"
## [1] "RS"
## [1] "p-value 0.302"
## [1] "VS"
## [1] "p-value 0.1542"
## [1] "KS"
## [1] "p-value 0.4268"

Debias difference-based estimator for long-run covariance matrix function.

setting$lrvmethod =1

i=1
for(type in c("KPSS","RS","VS","KS"))
{
  setting$type = type
  print(type)
  result_reg = heter_covariate(list(y= hk$y, x = hk$x), setting, mvselect = -2)
  print(paste("p-value",result_reg))
  pvmatrix[2,i] = result_reg
  i = i + 1
}

## [1] "KPSS"
## [1] "p-value 0.7348"
## [1] "RS"
## [1] "p-value 0.8366"
## [1] "VS"
## [1] "p-value 0.7162"
## [1] "KS"
## [1] "p-value 0.8914"

Output

rownames(pvmatrix) = c("plug","diff")
colnames(pvmatrix) = c("KPSS","RS","VS","KS")
knitr::kable(pvmatrix,type="latex")

	KPSS	RS	VS	KS
plug	0.3048	0.3020	0.1542	0.4268
diff	0.7348	0.8366	0.7162	0.8914

xtable::xtable(pvmatrix, digits = 3)

## % latex table generated in R 4.6.1 by xtable 1.8-8 package
## % Wed Oct  7 12:13:10 2026
## \begin{table}[ht]
## \centering
## \begin{tabular}{rrrrr}
##   \hline
##  & KPSS & RS & VS & KS \\ 
##   \hline
## plug & 0.305 & 0.302 & 0.154 & 0.427 \\ 
##   diff & 0.735 & 0.837 & 0.716 & 0.891 \\ 
##    \hline
## \end{tabular}
## \end{table}

Sensitivity Check

Using parameter `shift’ to multiply the GCV selected bandwidth by a factor. - Shift = 1.2 with plug-in estimator.

pvmatrix = matrix(nrow=2, ncol=4)
setting$lrvmethod = 0
i=1
for(type in c("KPSS","RS","VS","KS")){
  setting$type = type
  print(type)
  result_reg = heter_covariate(list(y= hk$y, x = hk$x),
                                             setting,
                                        mvselect = -2, shift = 1.2)
  print(paste("p-value",result_reg))
  pvmatrix[1,i] = result_reg
  i = i + 1
}

## [1] "KPSS"
## [1] "p-value 0.4408"
## [1] "RS"
## [1] "p-value 0.3874"
## [1] "VS"
## [1] "p-value 0.1092"
## [1] "KS"
## [1] "p-value 0.5976"

Shift = 1.2 with debias difference-based estimator. Choosing verbose_dist = TRUE can print more intermediate results, the exact values of smoothing parameters, the test statistic, the summary of bootstrap distribution and the p-value.

setting$lrvmethod =1
i=1
for(type in c("KPSS","RS","VS","KS"))
{
  setting$type = type
  print(type)
  result_reg = heter_covariate(list(y= hk$y, x = hk$x),
                                             setting,
                                        mvselect = -2, verbose_dist = TRUE, shift = 1.2)
  print(paste("p-value",result_reg))
  pvmatrix[2,i] = result_reg
  i = i + 1
}

## [1] "KPSS"
## [1] "gcv 0.193398841583897"
## [1] "m 8 tau_n 0.382134206312301"
## [1] "test statistic: 141.654657280932"
## [1] "Bootstrap distribution"
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   12.38   71.14  138.15  226.45  282.30 3201.93 
## [1] "p-value 0.4912"
## [1] "RS"
## [1] "gcv 0.193398841583897"
## [1] "m 18 tau_n 0.382134206312301"
## [1] "test statistic: 1067.76713443354"
## [1] "Bootstrap distribution"
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   568.3  1099.6  1314.7  1381.3  1595.3  3510.7 
## [1] "p-value 0.7874"
## [1] "VS"
## [1] "gcv 0.193398841583897"
## [1] "m 16 tau_n 0.382134206312301"
## [1] "test statistic: 103.342038019402"
## [1] "Bootstrap distribution"
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   17.06   68.76  111.65  154.08  190.58 1740.79 
## [1] "p-value 0.5364"
## [1] "KS"
## [1] "gcv 0.193398841583897"
## [1] "m 17 tau_n 0.382134206312301"
## [1] "test statistic: 671.676091515896"
## [1] "Bootstrap distribution"
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   339.8   720.0   922.1  1003.2  1203.2  2727.0 
## [1] "p-value 0.807"

Results of shift = 1.2

rownames(pvmatrix) = c("plug","diff")
colnames(pvmatrix) = c("KPSS","RS","VS","KS")
knitr::kable(pvmatrix,type="latex")

	KPSS	RS	VS	KS
plug	0.4408	0.3874	0.1092	0.5976
diff	0.4912	0.7874	0.5364	0.8070

xtable::xtable(pvmatrix, digits = 3)

## % latex table generated in R 4.6.1 by xtable 1.8-8 package
## % Wed Oct  7 12:13:47 2026
## \begin{table}[ht]
## \centering
## \begin{tabular}{rrrrr}
##   \hline
##  & KPSS & RS & VS & KS \\ 
##   \hline
## plug & 0.441 & 0.387 & 0.109 & 0.598 \\ 
##   diff & 0.491 & 0.787 & 0.536 & 0.807 \\ 
##    \hline
## \end{tabular}
## \end{table}

Shift = 0.8 with plug-in estimator.

pvmatrix = matrix(nrow=2, ncol=4)
setting$lrvmethod =0

i=1
for(type in c("KPSS","RS","VS","KS")){
  setting$type = type
  print(type)
  result_reg = heter_covariate(list(y= hk$y, x = hk$x),
                                             setting,
                                        mvselect = -2,  shift = 0.8)
  print(paste("p-value",result_reg))
  pvmatrix[1,i] = result_reg
  i = i + 1
}

## [1] "KPSS"
## [1] "p-value 0.2192"
## [1] "RS"
## [1] "p-value 0.1606"
## [1] "VS"
## [1] "p-value 0.1036"
## [1] "KS"
## [1] "p-value 0.2752"

Shift = 0.8 with the debias difference-based estimator.

setting$lrvmethod =1

i=1
for(type in c("KPSS","RS","VS","KS"))
{
  setting$type = type
  print(type)
  result_reg = heter_covariate(list(y= hk$y, x = hk$x),
                                             setting,
                                        mvselect = -2, verbose_dist = TRUE, shift = 0.8)
  print(paste("p-value",result_reg))
  pvmatrix[2,i] = result_reg
  i = i + 1
}

## [1] "KPSS"
## [1] "gcv 0.128932561055931"
## [1] "m 9 tau_n 0.382134206312301"
## [1] "test statistic: 166.543448031108"
## [1] "Bootstrap distribution"
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   13.09  103.24  205.86  339.32  427.67 3681.34 
## [1] "p-value 0.5738"
## [1] "RS"
## [1] "gcv 0.128932561055931"
## [1] "m 18 tau_n 0.382134206312301"
## [1] "test statistic: 998.08124125936"
## [1] "Bootstrap distribution"
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##     735    1257    1520    1592    1851    4315 
## [1] "p-value 0.9456"
## [1] "VS"
## [1] "gcv 0.128932561055931"
## [1] "m 18 tau_n 0.382134206312301"
## [1] "test statistic: 78.0587445148257"
## [1] "Bootstrap distribution"
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   17.96   98.25  161.02  220.96  272.86 1921.61 
## [1] "p-value 0.8502"
## [1] "KS"
## [1] "gcv 0.128932561055931"
## [1] "m 9 tau_n 0.332134206312301"
## [1] "test statistic: 709.345279801765"
## [1] "Bootstrap distribution"
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   290.6   695.6   908.1   996.3  1225.2  3088.6 
## [1] "p-value 0.732"

Results of shift = 0.8

rownames(pvmatrix) = c("plug","diff")
colnames(pvmatrix) = c("KPSS","RS","VS","KS")
knitr::kable(pvmatrix,type="latex")

	KPSS	RS	VS	KS
plug	0.2192	0.1606	0.1036	0.2752
diff	0.5738	0.9456	0.8502	0.7320

xtable::xtable(pvmatrix, digits = 3)

## % latex table generated in R 4.6.1 by xtable 1.8-8 package
## % Wed Oct  7 12:14:18 2026
## \begin{table}[ht]
## \centering
## \begin{tabular}{rrrrr}
##   \hline
##  & KPSS & RS & VS & KS \\ 
##   \hline
## plug & 0.219 & 0.161 & 0.104 & 0.275 \\ 
##   diff & 0.574 & 0.946 & 0.850 & 0.732 \\ 
##    \hline
## \end{tabular}
## \end{table}

Test for structure stability

Test if the coefficient function of “SO2”,“NO2”,“Dust” of the second year is constant.

hk$x = as.matrix(cbind(rep(1,n), (hk_data[,1:3])))
hk$y = hk_data$`Hospital Admission`
setting$type = 0
setting$bw_set = c(0.1, 0.35)
setting$eta = 0.2
setting$lrvmethod = 1
setting$lb  = 10
setting$ub  = 15
hk1 = list()
hk1$x = hk$x[366:730,]
hk1$y = hk$y[366:730]
p1 <- heter_gradient(hk1, setting, mvselect = -2, verbose = T)

## [1] "m 11 tau_n 0.364293094094381"
## [1] 10464.35
##        V1       
##  Min.   : 1454  
##  1st Qu.: 3413  
##  Median : 4422  
##  Mean   : 4793  
##  3rd Qu.: 5789  
##  Max.   :17099

p1

## [1] 0.0098