Distributed-lag linear structural equation models

Description

Inference functionalities for distributed-lag linear structural equation models (DLSEMs). DLSEMs are Markovian structural causal models where each factor of the joint probability distribution is a distributed-lag linear regression with constrained lag shapes (Magrini, 2018; Magrini et. al, 2019; Magrini, 2020). DLSEMs account for temporal delays in the dependence relationships among the variables through a single parameter per covariate, thus allowing to perform dynamic causal inference in a feasible fashion. Endpoint-constrained quadratic ('ecq'), quadratic decreasing ('qd'), linearly decreasing ('ld') and gamma ('gam') lag shapes are available. The main functions of the package are:

• dlsem, to perform parameter estimation;

• causalEff, to compute all the pathwise causal lag shapes and the overall one connecting two or more variables;

• lagPlot, to display a pathwise or an overall causal lag shape.

Details

Author(s)

Alessandro Magrini <alessandro.magrini@unifi.it>

References

A. Magrini (2018). Linear Markovian models for lag exposure assessment. Biometrical Letters, 55(2): 179-195. DOI: 10.2478/bile-2018-0012.

A. Magrini, F. Bartolini, A. Coli, B. Pacini (2019). A structural equation model to assess the impact of agricultural research expenditure on multiple dimensions. Quality and Quantity, 53(4): 2063-2080. DOI: 10.1007/s11135-019-00855-z

A. Magrini (2020). A family of theory-based lag shapes for distributed-lag linear regression. To be appeared on Italian Journal of Applied Statistics.

European agricultural data

Description

Data on research expenditure, research activity, productivity and impacts of Agriculture from 1990 to 2014 in EU 15 countries.

Usage

data(agres)

Format

A data frame with a total of 350 observations on the following 20 variables:

COUNTRY

Code of the country. Belgium and Luxembourg are merged together with code BL.

YEAR

Year.

GDP

Gross domestic product (million euro PPS). Source: Eurostat.

EMPL_AGR

Persons employed in Agriculture (count). Source: Eurostat.

UAA

Utilized agricultural area (hectares). Source: Eurostat.

PATENT_OTHER

Mechanical, chemical and environment-related patent applications (count). Source: OECD.

GBAORD_AGR

Government research expenditure (million euro PPS). Source: OECD.

BERD_AGR

Business enterprise research expenditure (million euro PPS). Source: Eurostat.

RD_EDU_AGR

Agricultural researchers with tertiary education (count). Source: OECD.

PATENT_AGR

Agricultural patent applications (count). Source: OECD.

TFP_AGR

Total factor productivity (index 2005=100). Source: USDA.

GVA_AGR

Gross value added of Agriculture (international dollars). Source: Faostat.

PPI_AGR

Producer price of agricultural output (index 2005=100). Source: Faostat.

ENTR_INCOME_AGR

Net entrepreneurial income of Agriculture (index 2005=100). Source: Eurostat.

GHG_AGR

Greenhouse gas emissions due to Agriculture (gigagrams CO2 equivalent). Source: Faostat.

NBAL_AGR

Nitrogen gross nutrient balance (kg of nutrient per hectare). Source: Eurostat.

INCOME_RURAL

Mean familiar income in rural areas (international dollars).

UNEMPL_RURAL

Unemployment rate in rural areas (index 2005=100).

Conversion into the graphNEL class

Description

An object of class dlsem is converted into an object of class graphNEL.

Usage

as.graphNEL(x, conf = 0.95, use.ns = FALSE)

Arguments

Value

An object of class graphNEL.

See Also

dlsem.

Examples

data(industry)
indus.code <- list(
  Consum~ecq(Job,0,5),
  Pollution~ecq(Job,1,8)+ecq(Consum,1,7)
  )
indus.mod <- dlsem(indus.code,group="Region",exogenous=c("Population","GDP"),data=industry,
  log=TRUE)
as.graphNEL(indus.mod)

Automated plot of lag shapes

Description

All the single-edge pathwise causal lag shapes are saved as pdf files.

Usage

auto.lagPlot(x, cumul = FALSE, conf = 0.95, plotDir = NULL)

Arguments

See Also

dlsem; lagPlot.

Examples

data(industry)
indus.code <- list(
  Consum~ecq(Job,0,5),
  Pollution~ecq(Job,1,8)+ecq(Consum,1,7)
  )
indus.mod <- dlsem(indus.code,group="Region",exogenous=c("Population","GDP"),data=industry,
  log=TRUE)
## NOT RUN:
# auto.lagPlot(indus.mod,plotDir=getwd())

Automated model code

Description

Given a set of variable names, a model code including all the possible edges is built.

Usage

autoCode(var.names, lag.type = "ecq")

Arguments

Value

A list of formulas to be passed as argument model.code in dlsem().

See Also

dlsem.

Examples

autoCode(c("Job","Consum","Pollution"),lag.type="ecq")

Assessment of dynamic causal effects

Description

All the pathwise causal lag shapes and the overall one connecting two or more variables are computed.

Usage

causalEff(x, from = NULL, to = NULL, lag = NULL, cumul = FALSE, conf = 0.95,
  use.ns = FALSE)

Arguments

Details

A pathwise causal lag shape is the set of causal effects associated to a path at different time lags. An overall causal lag shape is the set of overall causal effects of a variable on another one at different time lags.

Note that, due to the properties of the multiple linear regression model, causal effects are net of the influence of the group factor and exogenous variables.

Value

A list containing several matrices including point estimates, standard errors and asymptotic confidence intervals (at level conf) for all the pathwise causal lag shapes and the overall one connecting the starting variables to the ending variable.

Note

Value NULL is returned if one of the following occurs: (i) no significant path at confidence level conf exists connecting the starting variables to the ending variable; (ii) the requested path does not exist or is not significant at confidence level conf. Note that the edges between the starting variables and their respective parents are deleted as a consequence of intervention. See Magrini (2018) for technical details on causal inference in distributed-lag linear structural equation models.

References

A. Magrini (2018). Linear Markovian models for lag exposure assessment. Biometrical Letters, 55(2): 179-195. DOI: 10.2478/bile-2018-0012.

See Also

dlsem; lagPlot.

Examples

data(industry)
indus.code <- list(
  Consum~ecq(Job,0,5),
  Pollution~ecq(Job,1,8)+ecq(Consum,1,7)
  )
indus.mod <- dlsem(indus.code,group="Region",exogenous=c("Population","GDP"),data=industry,
  log=TRUE)
causalEff(indus.mod,from="Job",to="Pollution",lag=c(0,5,10,15),cumul=TRUE)

Comparison among different distributed-lag linear structural equation models

Description

Several competing distributed-lag linear structural equation models are compared based on information criteria.

Usage

compareModels(x)

Arguments

Value

A data.frame with one record for each model in x on the following quantities: log-likelihood, number of parameters, Akaike Information Criterion (AIC), Bayesian Information criterion (BIC).

Note

In order to keep the sample size constant, only the non-missing residuals across all the models are considered (see Magrini, 2020, for details).

References

H. Akaike (1974). A New Look at the Statistical Identification Model. IEEE Transactions on Automatic Control, 19, 716-723. DOI: 10.1109/TAC.1974.1100705

A. Magrini (2020). A family of theory-based lag shapes for distributed-lag linear regression. To be appeared on Italian Journal of Applied Statistics.

G. Schwarz (1978). Estimating the Dimension of a Model. Annals of Statistics, 6, 461-464. DOI: 10.1214/aos/1176344136

See Also

dlsem.

Examples

data(industry)

# model with endpoint-contrained quadratic lag shapes
indus.code <- list(
  Consum~ecq(Job,0,5),
  Pollution~ecq(Job,1,8)+ecq(Consum,1,7)
  )
indus.mod <- dlsem(indus.code,group="Region",exogenous=c("Population","GDP"),data=industry,
  log=TRUE)
  
# model with gamma lag shapes
indus.code_2 <- list(
  Consum~gam(Job,0.85,0.2),
  Pollution~gam(Job,0.95,0.05)+gam(Consum,0.9,0.15)
  )
indus.mod_2 <- dlsem(indus.code_2,group="Region",exogenous=c("Population","GDP"),data=industry,
  log=TRUE)
  
compareModels(list(indus.mod,indus.mod_2))

Parameter estimation

Description

Parameter estimation is performed for a distributed-lag linear structural equation model. A single group factor may be taken into account.

Usage

dlsem(model.code, group = NULL, time = NULL, exogenous = NULL, data, hac = TRUE,
  gamma.by = 0.05, global.control = NULL, local.control = NULL, log = FALSE,
  diff.options = list(test=NULL, maxdiff=2, ndiff=NULL),
  imput.options = list(tol=0.0001, maxiter=500, maxlag=2, no.imput=NULL), quiet = FALSE)
  

Arguments

Details

Each regression model in a distributed-lag linear structural equation model has the form:

y_t = \beta_0+\sum_{i=1}^p \sum_{l=0}^{L_i} \beta_{i,l} ~ x_{i,t-l}+\epsilon_t

where y_t is the value of the response variable at time t, x_{i,t-l} is the value of the i-th covariate at l time lags before t, and \epsilon_t is the random error at time t uncorrelated with the covariates and with \epsilon_{t-1}. The set (\beta_{i,0},\beta_{i,1},\ldots,\beta_{i,L_i}) is the lag shape of the i-th covariate. Currently available lag shapes are the endpoint-constrained quadratic lag shape:

\beta_{i,l} = \theta_i \left[-\frac{4}{(b_i-a_i+2)^2} l^2+\frac{4(a_i+b_i)}{(b_i-a_i+2)^2} l-\frac{4(a_i-1)(b_i+1)}{(b_i-a_i+2)^2} \right] \hspace{1cm} a_i \leq l \leq b_i

(otherwise, \beta_{i,l}=0); the quadratic decreasing lag shape:

\beta_{i,l} = \theta_i \frac{l^2-2(b_i+1)l+(b_i+1)^2}{(b_i-a_i+1)^2} \hspace{1cm} a_i \leq l \leq b_i

(otherwise, \beta_{i,l}=0); the linearly decreasing lag shape:

\beta_{i,l} = \theta_i \frac{b_i+1-l}{b_i+1-a_i} \hspace{1cm} a_i \leq l \leq b_i

(otherwise, \beta_{i,l}=0); the gamma lag shape:

\beta_{i,l} = \theta_i (l+1)^\frac{a_i}{1-a_i}b_i^l \left[\left(\frac{a_i}{(a_i-1)\log(b_i)}\right)^\frac{a_i}{1-a_i}b_i^{\frac{a_i}{(a_i-1)\log(b_i)}-1}\right]^{-1}

0<a_i<1 \hspace{1cm} 0<b_i<1

See Magrini (2020) for details on these constrained lag shapes.

Formulas cannot contain neither qualitative variables or interaction terms (no ':' or '*' symbols), nor functions excepting the following operators for the specification of lag shapes:

• ecq: quadratic (2nd order polynomial) lag shape with endpoint constraints;

• qd: quadratic (2nd order polynomial) decreasing lag shape;

• ld: linearly decreasing lag shape;

• gam: gamma lag shape.

Each operator must have the following three arguments (provided within brackets and separated by commas):

1. the name of the covariate to which the lag is applied;

2. parameter a_i;

3. parameter b_i;

4. the group factor (optional). If not provided and argument group is not NULL, this is found automatically.

The formula of regression models with no endogenous covariates may be omitted from argument model.code. The group factor and exogenous variables must not appear in any formula.

Argument local.control must be a named list containing one or more among the following components:

• adapt: a named vector of logical values, where each component must have the name of one endogenous variable and indicate whether adaptation of lag shapes should be performed for the regression model of that variable.

• min.gestation: a named list. Each component of the list must have the name of one endogenous variable and be a named vector. Each component of the named vector must have the name of one covariate in the regression model of the endogenous variable above and include the minimum gestation lag for its lag shape.

• max.gestation: the same as min.gestation, with the exception that the named vector must include the maximum gestation lag.

• max.lead: the same as min.gestation, with the exception that the named vector must include the maximum lead lag.

• min.width: the same as min.gestation, with the exception that the named vector must include the minimum lag width.

• sign: the same as min.gestation, with the exception that the named vector must include the lag sign (either '+' for positive or '-' for negative). Local control options have no default values, and global ones are applied in their absence. If some local control options conflict with global ones, only the former are applied.

Value

An object of class dlsem, with the following components:

S3 methods available for class dlsem are:

References

A. P. Dempster, N. M. Laird, and D. B. Rubin (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39(1): 1-38.

C. W. J. Granger, and P. Newbold (1974). Spurious regressions in econometrics. Journal of Econometrics, 2(2): 111-120.

A. Magrini (2019). A family of theory-based lag shapes for distributed-lag linear regression. To be appeared on Italian Journal of Applied Statistics.

W. K. Newey, and K. D. West (1978). A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica, 55(3), 703-708.

See Also

unirootTest; causalEff; compareModels.

Examples

data(industry)

# Estimation without adaptation of lag shapes
indus.code <- list(
  Consum~ecq(Job,0,5),
  Pollution~ecq(Job,1,8)+ecq(Consum,1,7)
  )
indus.mod <- dlsem(indus.code,group="Region",time="Year",exogenous=c("Population","GDP"),
  data=industry,log=TRUE)

# Adaptation of lag shapes (estimation takes some seconds more)
indus.global <- list(adapt=TRUE,max.gestation=5,max.lead=15,min.width=3,sign="+")
## NOT RUN:
# indus.mod <- dlsem(indus.code,group="Region",time="Year",exogenous=c("Population","GDP"),
#   data=industry,global.control=indus.global,log=TRUE)

# Summary of estimation
summary(indus.mod)$endogenous

# DAG with edges coloured according to the sign
plot(indus.mod)

# DAG disregarding statistical significance
plot(indus.mod,style=0)  


### Comparison among alternative models

# Model 2: quadratic decreasing lag shapes
indus.code_2 <- list(
  Job ~ 1,
  Consum~qd(Job),
  Pollution~qd(Job)+qd(Consum)
  )
## NOT RUN:
# indus.mod_2 <- dlsem(indus.code_2,group="Region",time="Year",exogenous=c("Population","GDP"),
#   data=industry,global.control=indus.global,log=TRUE)

# Model 3: linearly decreasing lag shapes
indus.code_3 <- list(
  Job ~ 1,
  Consum~ld(Job),
  Pollution~ld(Job)+ld(Consum)
  )
## NOT RUN:
# indus.mod_3 <- dlsem(indus.code_3,group="Region",time="Year",exogenous=c("Population","GDP"),
#   data=industry,global.control=indus.global,log=TRUE)

# Model 4: gamma lag shapes
indus.code_4 <- list(
  Job ~ 1,
  Consum~gam(Job),
  Pollution~gam(Job)+gam(Consum)
  )
## NOT RUN:
# indus.mod_4 <- dlsem(indus.code_4,group="Region",time="Year",exogenous=c("Population","GDP"),
#   data=industry,global.control=indus.global,log=TRUE)

# comparison of the three models
## NOT RUN:
# compareModels(list(indus.mod,indus.mod_2,indus.mod_3,indus.mod_4))


### A more complex model

data(agres)

# Qualitative exogenous variable
agres$POLICY <- factor(1*(agres$YEAR>=2005))
levels(agres$POLICY) <- c("no","yes")

# Causal levels
context.var <- c("GDP","EMPL_AGR","UAA","PATENT_OTHER","POLICY")
investment.var <- c("GBAORD_AGR","BERD_AGR")
research.var <- c("RD_EDU_AGR","PATENT_AGR")
impact.var <-  c("GVA_AGR","PPI_AGR")
agres.var <- c(context.var,investment.var,research.var,impact.var)

# Constraints on lag shapes
agres.global <- list(adapt=TRUE,max.gestation=5,max.lead=15,sign="+")
agres.local <- list(
  sign=list(
    PPI_AGR=c(GBAORD_AGR="-",BERD_AGR="-",RD_EDU_AGR="-",PATENT_AGR="-")
    )
  )

# Endpoint-constrained quadratic lag shapes (estimation takes a couple of minutes)
auxcode <- c(paste(investment.var,"~1",sep=""),
  paste(research.var,"~",paste("ecq(",investment.var,",,)",
    collapse="+",sep=""),sep=""),
  paste(impact.var,"~",paste("ecq(",c(investment.var,research.var),",,)",
    collapse="+",sep=""),sep=""))
agres.code <- list()
for(i in 1:length(auxcode)) {
  agres.code[[i]] <- formula(auxcode[i])
  }
## NOT RUN:
# agres.mod <- dlsem(agres.code,group="COUNTRY",time="YEAR",exogenous=context.var,
#   data=agres,global.control=agres.global,local.control=agres.local,log=TRUE)
# summary(agres.mod)$endogenous

## Gamma lag shapes (estimation takes some minutes)
auxcode_2 <- c(paste(investment.var,"~1",sep=""),
  paste(research.var,"~",paste("gam(",investment.var,",,)",
    collapse="+",sep=""),sep=""),
  paste(impact.var,"~",paste("gam(",c(investment.var,research.var),",,)",
    collapse="+",sep=""),sep=""))
agres.code_2 <- list()
for(i in 1:length(auxcode_2)) {
  agres.code_2[[i]] <- formula(auxcode_2[i])
  }
## NOT RUN:
# agres.mod_2 <- dlsem(agres.code_2,group="COUNTRY",time="YEAR",exogenous=context.var,
#  data=agres,global.control=agres.global,local.control=agres.local,log=TRUE)
# summary(agres.mod_2)$endogenous
# compareModels(list(agres.mod,agres.mod_2))

Sampling from a distributed-lag linear structural equation model

Description

A future sample from a distributed-lag linear structural equation model is drawn.

Usage

drawSample(x, n)

Arguments

Value

An object of class data.frame.

Note

Sampling is conditioned to the most recent observed value of all the variables.

For variables subdued to logarithmic transformation and/or differencing, the sampled values are in logarithmic scale and/or after differencing, as well.

If a group factor was specified for the model, a sample of size n is drawn for each group.

See Also

dlsem.

Examples

data(industry)
indus.code <- list(
  Consum~ecq(Job,0,5),
  Pollution~ecq(Job,1,8)+ecq(Consum,1,7)
  )
indus.mod <- dlsem(indus.code,group="Region",time="Year",exogenous=c("Population","GDP"),
  data=industry,log=TRUE)
drawSample(indus.mod,10)

Lag shape constructors

Description

Lag shape constructors to be used in model formulas.

Usage

ecq(x, a, b, x.group = NULL, nlag = NULL)
qd(x, a, b, x.group = NULL, nlag = NULL)
ld(x, a, b, x.group = NULL, nlag = NULL)
gam(x, a, b, x.group = NULL, nlag = NULL)

Arguments

References

A. Magrini (2020). A family of theory-based lag shapes for distributed-lag linear regression. To be appeared on Italian Journal of Applied Statistics.

Examples

data(industry)
# example in linear regression
m1 <- lm(Consum ~ -1+Region+ecq(Job,0,5,x.group=Region), data=industry)
m2 <- lm(Consum ~ -1+Region+gam(Job,0.85,0.2,x.group=Region), data=industry)

Industrial development

Description

Simulated data on industrial development from 1983 to 2014 in 10 imaginary regions.

Usage

data(industry)

Format

A data frame with a total of 320 observations on the following 7 variables:

Region

ID of the region.

Year

Year.

Population

Population (number of inhabitants).

GDP

Gross domestic product (1000 international dollars).

Job

Employment in industry (Index, 1980=100).

Consum

Private consumption (Index, 1980=100).

Pollution

Greenhouse gas emissions (tonnes of CO2 equivalent).

Conditional independence check

Description

Conditional independence between two variables is checked using the d-separation criterion (Pearl, 2000, page 16 and following).

Usage

isIndep(x, var1 = NULL, var2 = NULL, given = NULL, conf = 0.95, use.ns = FALSE)

Arguments

Value

Logical

Note

Conditional independence is checked statically, that is the whole history of conditioning variables is supposed to be known.

The result is unchanged if arguments var1 and var2 are switched.

Dependence is a necessary but not sufficient condition for causation: see the discussion in Pearl (2000).

References

J. Pearl (2000). Causality: Models, Reasoning, and Inference. Cambridge University Press. Cambridge, UK. ISBN: 978-0-521-89560-6

See Also

dlsem.

Examples

data(industry)
indus.code <- list(
  Consum~ecq(Job,0,5),
  Pollution~ecq(Job,1,8)+ecq(Consum,1,7)
  )
indus.mod <- dlsem(indus.code,group="Region",exogenous=c("Population","GDP"),data=industry,
  log=TRUE)
isIndep(indus.mod,"Job","Pollution",given=c("Consum"))

Plot of lag shapes

Description

A pathwise or an overall causal lag shape is displayed.

Usage

lagPlot(x, from = NULL, to = NULL, path = NULL, maxlag = NULL, cumul = FALSE,
  conf = 0.95, use.ns = FALSE, ylim = NULL, title = NULL)

Arguments

Note

Value NULL is returned if one of the following occurs: (i) no significant path at confidence level conf exists connecting the starting variables to the ending variable; (ii) the requested path does not exist or is not significant at confidence level conf.

See Also

dlsem; lagShapes; causalEff.

Examples

data(industry)
indus.code <- list(
  Consum~ecq(Job,0,5),
  Pollution~ecq(Job,1,8)+ecq(Consum,1,7)
  )
indus.mod <- dlsem(indus.code,group="Region",exogenous=c("Population","GDP"),data=industry,
  log=TRUE)

# the lag shape of the causal effect associated to specific paths
lagPlot(indus.mod,path="Job*Pollution")
lagPlot(indus.mod,path="Job*Consum*Pollution")

# the lag shape of an overall causal effect
lagPlot(indus.mod,from="Job",to="Pollution")

Estimated lag shapes

Description

Estimated lag shapes and their standard errors are provided.

Usage

lagShapes(x, cumul = FALSE)

Arguments

Value

A list of lists, one for each endogenous variable, each containing several matrices including the estimated lag shapes and their standard errors.

See Also

dlsem; causalEff; lagPlot.

Examples

data(industry)
indus.code <- list(
  Consum~ecq(Job,0,5),
  Pollution~ecq(Job,1,8)+ecq(Consum,1,7)
  )
indus.mod <- dlsem(indus.code,group="Region",exogenous=c("Population","GDP"),data=industry,
  log=TRUE)
lagShapes(indus.mod)

Heteroskedasticty and autocorrelation consistent covariance matrix

Description

The heteroskedasticty and autocorrelation consistent (HAC) covariance matrix of least square estimates (Newey & West, 1978) is applied to an object of class lm. A single group factor may be taken into account.

Usage

lmHAC(x, group = NULL)

Arguments

Value

An object of class hac and lm. The HAC covariance matrix is stored into the component vcov of the object, which is taken into account by the summary and the vcov methods. The HAC covariance matrix has the attribute max.lag, indicating the maximum lag of autocorrelation, which is automatically computed based on fit to data.

Note

If group is not NULL, the HAC covariance matrix is computed within each group. Residuals are assumed to be temporally ordered within each group.

References

W. K. Newey, and K. D. West (1978). A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica, 55(3), 703-708.

Examples

data(industry)

m0 <- lm(Consum ~ -1+Region+ecq(Job,0,5,x.group=Region), data=industry)
summary(m0)
confint(m0)

m0_hac <- lmHAC(m0,group="Region")
summary(m0_hac)
confint(m0_hac)

Plot for diagnostics of the residuals in a distributed-lag structural equation model.

Description

Several different types of plot for diagnostics of the residuals in a distributed-lag structural equation model are displayed.

Usage

residualPlot(x, type = "fr")

Arguments

Note

If type is equal to 'ts' or 'ac' and a group factor was specified for x, the results are displayed as minimum, 1st quartile, median, 3rd quartile and maximum by group.

See Also

dlsem.

Examples

data(industry)
indus.code <- list(
  Consum~ecq(Job,0,5),
  Pollution~ecq(Job,1,8)+ecq(Consum,1,7)
  )
indus.mod <- dlsem(indus.code,group="Region",exogenous=c("Population","GDP"),data=industry,
  log=TRUE)
residualPlot(indus.mod,type="fr")
residualPlot(indus.mod,type="qq")
residualPlot(indus.mod,type="ts")
residualPlot(indus.mod,type="ac")

Unit root test

Description

Unit root test is performed on a set of quantitative variables. A single group factor may be taken into account.

Usage

unirootTest(x = NULL, group = NULL, time = NULL, data, test = NULL, log = FALSE)

Arguments

Value

An object of class unirootTest, consisting of a list with one component for each variable tested. Each list contains the following components:

Note

The null hypothesis of the ADF test is the presence of a unit root. The lag order to calculate the statistic of the ADF test is automatically selected according to the precedure by Ng \& Perron (2001).

The null hypothesis of the KPSS test is stationarity. The statistic of the KPSS test is calculated at the lag order 4*(n/100)^0.25.

If the group factor is specified, p-values of each group are combined using the method proposed by Demetrescu (2006).

References

M. Demetrescu, U. Hassler, and A. Tarcolea (2006). Combining Significance of Correlated Statistics with Application to Panel Data. Oxford Bulletin of Economics and Statistics, 68(5), 647-663. DOI: 10.1111/j.1468-0084.2006.00181.x

D. A. Dickey, and W. A. Fuller (1981). Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root. Econometrica, 49: 1057-1072. DOI: 10.2307/1912517

D. Kwiatkowski, P. C. B. Phillips, P. Schmidt and Y. Shin (1992). Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics, 54(1-3): 159-178.

S. Ng, and W. P. Perron (2001). Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power. Econometrica, 60: 1519-1554. DOI: 10.1111/1468-0262.00256.

Examples

data(industry)
indus.urt <- unirootTest(c("Job","Consum","Population","GDP"),
  group="Region",time="Year",data=industry,log=TRUE)
indus.urt      ## p-values
indus.urt$Job  ## details for variable 'Job'