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KSS-Routine

KSS.Rd

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 \($Y_{it}=\theta_{t}+\sum_{j=1}^P\beta_{j} X_{itj}+v_i(t)+\epsilon_{it}\quad i=1,\dots,n; t=1,\dots,T.$\) Where the individual time trends, v_i(t), are assumed to come from a finite dimensional factor model \($v_i(t)=\sum_{l=1}^d\lambda_{il}f_l(t),\quad\lambda_{il}\in R,\quad f_l\in L^2[0,T].$\) 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 \($Y_{it}=\mu+\sum_{j=1}^P\beta_{j} X_{itj}+v_i(t)+\epsilon_{it}\quad i=1,\dots,n; t=1,\dots,T.$\) 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 \($Y_{it}=\mu+\sum_{j=1}^P\beta_{j} X_{itj}+v_i(t)+\alpha_{i}+\epsilon_{it}\quad i=1,\dots,n; t=1,\dots,T.$\) 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 \($Y_{it}=\mu+\theta_{t}+\sum_{j=1}^P\beta_{j} X_{itj}+v_i(t)+\epsilon_{it}\quad i=1,\dots,n; t=1,\dots,T.$\) 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 \($Y_{it}=\mu+\theta_{t}+\sum_{j=1}^P\beta_{j} X_{itj}+\alpha_i+v_i(t)+\epsilon_{it}\quad i=1,\dots,n; t=1,\dots,T.$\) 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

References

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

Author

Dominik Liebl

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)
#> Progress: CV-Optimization is running.
#> .........
#>  CV-Optimization converged.
(KSS.fit.sum <- summary(KSS.fit))
#> 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 
plot(KSS.fit.sum)