McSpatial (version 1.1.1)

cubespline: Smooth cubic spline estimation

Description

Estimates a smooth cubic spline for a model of the form y = f(z) + XB +u. The function divides the range of z into knots+1 equal intervals. The regression is $y=cons+\lambda_1 (z-z_0) + \lambda_2 (z-z_0)^2 + \lambda_3 (z-z_0)^3 + \sum_{k=1}^{K} \gamma_k(z-z_k)^3 D_k + X \beta + u$ where $z_0 = min(z), z_1...z_K$ are the knots, and $D_k=1$ if $z \geq z_k$. Estimation can be carried out for a fixed value of K or for a range of K. In the latter case, the function indicates the value of K that produces the lowest value of one of the following criteria: the AIC, the Schwarz information criterion, or the gcv.

Usage

cubespline(form,knots=1,mink=1,maxk=1,crit="gcv",data=NULL)

Arguments

form

Model formula. The spline is used with the first explanatory variable.

knots

If knots is specified, fits a cubic spline with K = knots. Default is knots=1, mink = 1, and maxk = 1, which implies a cubic spline with a single knot.

mink

The lower bound to search for the value of K that minimizes crit. mink can take any value greater than zero. The default is mink = 1

maxk

The upper bound to search for the value of K that minimizes crit. maxk must be great than or equal to mink. The default is maxk = 1.

crit

The selection criterion. Must be in quotes. The default is the generalized cross-validation criterion, or "gcv". Options include the Akaike information criterion, "aic", and the Schwarz criterion, "sc". Let nreg be the number of explanatory variable

data

A data frame containing the data. Default: use data in the current working directory

Value

• yhat
  The predicted values of the dependent variable at the original data points
• rss
  The residual sum of squares
• sig2
  The estimated error variance
• aic
  The value for AIC
• sc
  The value for sc
• gcv
  The value for gcv
• coef
  The estimated coefficient vector, B
• splinehat
  The predicted values for z alone, normalized to have the same mean as the dependent variable. If no X variables are included in the regression, splinehat = yhat.
• knots
  The vector of knots

References

McMillen, Daniel P., "Testing for Monocentricity," in Richard J. Arnott and Daniel P. McMillen, eds., A Companion to Urban Economics, Blackwell, Malden MA (2006), 128-140. McMillen, Daniel P., "Issues in Spatial Data Analysis," Journal of Regional Science 50 (2010), 119-141. Suits, Daniel B., Andrew Mason, and Louis Chan, "Spline Functions Fitted by Standard Regression Methods," Review of Economics and Statistics 60 (1978), 132-139.

See Also

cparlwr fourier lwr lwrgrid semip

Examples

data(cookdata)
fardata <- cookdata[!is.na(cookdata$LNFAR),]
par(ask=TRUE)

# single variable
o <- order(fardata$DCBD)
fit1 <- cubespline(LNFAR~DCBD, mink=1, maxk=10,data=fardata)
c(fit1$rss, fit1$sig2, fit1$aic, fit1$sc, fit1$gcv, fit1$knots)
plot(fardata$DCBD[o], fardata$LNFAR[o], xlab="Distance from CBD", ylab="Log FAR")
lines(fardata$DCBD[o], fit1$splinehat[o], col="red")

# multiple explanatory variables
fit2 <- cubespline(fardata$LNFAR~fardata$DCBD+fardata$AGE,  mink=1, maxk=10)
c(fit2$rss, fit2$sig2, fit2$aic, fit2$sc, fit2$gcv, fit2$knots)
plot(fardata$DCBD[o], fardata$LNFAR[o], xlab="Distance from CBD", ylab="Log FAR")
lines(fardata$DCBD[o], fit2$splinehat[o], col="red")

# pre-specified number of knots
fit3 <- cubespline(LNFAR~DCBD+AGE,  knots=4, data=fardata)