KSS: KSS-Routine

In phtt: Panel Data Analysis with Heterogeneous Time Trends

Description

Estimation of Panel Data Models with Heterogeneous Time Trends

Usage

KSS(formula,
    additive.effects = c("none", "individual", "time", "twoways"),
    consult.dim.crit = FALSE,
    d.max            = NULL,
    sig2.hat         = NULL,
    factor.dim       = NULL,
    level            = 0.01,
    spar             = NULL,
    CV               = FALSE,
    convergence      = 1e-6,
    restrict.mode    = c("restrict.factors","restrict.loadings"), ...)

Arguments

formula	An object of class 'formula'.
additive.effects	Type of Data Transformations: "none": for no transformation "individual": for within transformation "time": for between transformation "twoways": for twoways transformation
consult.dim.crit	logical. If consult.dim.crit is FALSE (default) and factor.dim is NULL: Only the dimensionality criterion of Kneip, Sickles & Song 2012 is used. If consult.dim.crit is TRUE and factor.dim is NULL: All implemented dimensionality criteria as implemented in the function OptDim() are computed and the user has to select one proposed dimension via a GUI.
d.max	A maximal dimension needed for some dimensionality-criteria that are implemented in the function OptDim(). The default (d.max=NULL) yields to an internal selection of d.max.
sig2.hat	Standard deviation of the error-term. The default (sig2.hat=NULL) yields to an internal estimation of sig2.hat.
factor.dim	Dimension of Factor-Structure. The default (factor.dim=NULL) yields to an internal estimation of factor.dim.
level	Significance-level for Dimensionality-Criterion of Kneip, Sickles & Song 2012.
spar	Smoothing parameter for spline smoothing of the residuals. If (spar=NULL) (default) and CV=FALSE spar is determined via generalized cross validation (GCV).
CV	logical. Selects the procedure for the determination of the smoothing parameter spar. If CV=FALSE (default) and spar=NULL: The smoothing parameter spar is determined by GCV. If CV=TRUE and spar=NULL: The smoothing parameter spar is determined by Leave-one-out cross validation (CV).
convergence	Convergence criterion for the CV-optimization of the smoothing parameter spar. Default is convergence=1e-6.
restrict.mode	Type of Restriction on the Factor-Structure: "restrict.factors": Factors are restricted to have an euclidean norm of 1. "restrict.loadings": Factor-Loadings are restricted to have an euclidean norm of 1.
...	Additional arguments to be passed to the low level functions.

Details

'KSS' is a function to estimate panel data models with unobserved heterogeneous time trends v_i(t). The considered model in Kneip, Sickles & Song (2012) is given by Y_{it}=θ_{t}+∑_{j=1}^Pβ_{j} X_{itj}+v_i(t)+ε_{it}, i=1,...,n; t=1,...,T. Where the individual time trends, v_i(t), are assumed to come from a finite dimensional factor model v_i(t)=∑_{l=1}^dλ_{il}f_l(t), λ_{il}\in R, f_l\in L^2[0,T]. The unobserved functions v_i(t) can be interpreted as smooth functions of a continuous argument t, as well as stochastic processes for discrete argument t.

• formula Usual 'formula'-object. If you wish to estimate a model without an intercept use '-1' in the formula-specification. Each Variable has to be given as a TxN-matrix. Missing values are not allowed.

• additive.effects

  • "none": The data is not transformed, except for an eventually subtraction of the overall mean; if the model is estimated with an intercept. The assumed model can be written as Y_{it}=μ+∑_{j=1}^Pβ_{j} X_{itj}+v_i(t)+ε_{it}, i=1,...,n; t=1,...,T. The parameter 'mu' is set to zero if '-1' is used in formula.

  • "individual": This is the "within"-model, which assumes that there are time-constant individual effects, tau_i, besides the individual time trends v_i(t). The model can be written as Y_{it}=μ+∑_{j=1}^Pβ_{j} X_{itj}+v_i(t)+α_{i}+ε_{it}, i=1,...,n; t=1,...,T. The parameter 'mu' is set to zero if '-1' is used in formula.

  • "time": This is the "between"-model, which assumes that there is a common (for all individuals) time trend, beta_0(t). The model can be written as Y_{it}=μ+θ_{t}+∑_{j=1}^Pβ_{j} X_{itj}+v_i(t)+ε_{it}, i=1,...,n; t=1,...,T. The parameter 'mu' is set to zero if '-1' is used in formula.

  • "twoways": This is the "twoways"-model ("within" & "between"), which assumes that there are time-constant individual effects, tau_i, and a common time trend, beta_0(t). The model can be written as Y_{it}=μ+θ_{t}+∑_{j=1}^Pβ_{j} X_{itj}+α_i+v_i(t)+ε_{it}, i=1,...,n; t=1,...,T. The parameter 'mu' is set to zero if '-1' is used in formula.

Value

'KSS' returns an object of 'class' '"KSS"'.

An object of class '"KSS"' is a list containing at least the following components:

• dat.matrix: Whole data set stored within a (N*T)x(p+1)-Matrix, where P is the number of independent variables without the intercept.

• dat.dim: Vector of length 3: c(T,N,p)

• slope.para: Beta-parameters

• beta.V: Covariance matrix of the beta-parameters.

• names: Names of the dependent and independent variables.

• is.intercept: Used an intercept in the formula?: TRUE or FALSE

• additive.effects: Additive effect type. One of: "none","individual","time", "twoways".

• Intercept: Intercept-parameter

• Add.Ind.Eff: Estimated values of additive individual effects.

• Add.Tim.Eff: Estimated values of additive time effects.

• unob.factors: Txd-matrix of estimated unobserved common factors, where 'd' is the number of used factors.

• ind.loadings: Nxd-matrix of loadings parameters.

• unob.fact.stru: TxN-matrix of the estimated factor structure. Each column represents an estimated individual unobserved time trend.

• used.dim: Used dimensionality of the factor structure.

• optimal.dim: List of proposed dimensionalities.

• fitted.values: Fitted values.

• orig.Y: Original values of the dependent variable.

• residuals: Residuals

• sig2.hat: Estimated variance of the error term.

• degrees.of.freedom: Degrees of freedom of the residuals.

• call

Author(s)

Dominik Liebl

References

• Kneip, A., Sickles, R. C., Song, W., 2012 “A New Panel Data Treatment for Heterogeneity in Time Trends”, Econometric Theory

See Also

Eup

Examples

## See the example in 'help(Cigar)' in order to take a look at the
## data set Cigar

##########
## DATA ##
##########

data(Cigar)
## Panel-Dimensions:
N <- 46
T <- 30
## Dependent variable:
  ## Cigarette-Sales per Capita
  l.Consumption    <- log(matrix(Cigar$sales, T,N))
## Independent variables:
  ## Consumer Price Index
  cpi        <- matrix(Cigar$cpi, T,N)
  ## Real Price per Pack of Cigarettes 
  l.Price  <- log(matrix(Cigar$price, T,N)/cpi)
  ## Real Disposable Income per Capita  
  l.Income    <- log(matrix(Cigar$ndi,   T,N)/cpi)

## Estimation:
KSS.fit      <- KSS(l.Consumption~l.Price+l.Income, CV=TRUE)
(KSS.fit.sum <- summary(KSS.fit))
plot(KSS.fit.sum)

Example output

Progress: CV-Optimization is running.
.........
 CV-Optimization converged.
Call:
KSS.default(formula = l.Consumption ~ l.Price + l.Income, CV = TRUE)

Residuals:
   Min     1Q Median     3Q    Max 
 -0.11  -0.01   0.00   0.01   0.13 


 Slope-Coefficients:
            Estimate  StdErr z.value    Pr(>z)    
(Intercept)   4.0300  0.1720   23.40 < 2.2e-16 ***
l.Price      -0.2590  0.0216  -12.00 < 2.2e-16 ***
l.Income      0.1610  0.0372    4.32  1.54e-05 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Additive Effects Type:  none  

Used Dimension of the Unobserved Factors: 6  

Residual standard error: 0.00074 on 921 degrees of freedom 
R-squared: 0.99 

phtt documentation built on May 2, 2019, 5:54 p.m.