tvLM: Time-Varying Coefficients Linear Models

Description

tvLM is used to fit a time-varying coefficients linear model

Usage

tvLM(
  formula,
  z = NULL,
  ez = NULL,
  data,
  bw = NULL,
  cv.block = 0,
  est = c("lc", "ll"),
  tkernel = c("Triweight", "Epa", "Gaussian"),
  singular.ok = TRUE
)

Value

An object of class tvlm

The object of class tvlm have the following components:

coefficients: A matrix of dimensions
fitted: The fitted values.
residuals: Estimation residuals.
x: A matrix with the regressors data.
y: A vector with the dependent variable data.
z: A vector with the smoothing variable.
ez: A vector with the smoothing estimation variable.
bw: Bandwidth of mean estimation.
est: Nonparametric estimation methodology.
tkernel: Kernel used in estimation.
level: Confidence interval range.
runs: Number of bootstrap replications.
tboot: Type of bootstrap.
BOOT: List with all bootstrap replications of coefficients, if done.

Arguments

formula: An object of class formula.
z: A vector with the smoothing variable.
ez: (optional) A scalar or vector with the smoothing values. If values are not included then the vector z is used instead.
data: An optional data frame or matrix.
bw: An opcional scalar. It represents the bandwidth in the estimation of trend coefficients. If NULL, it is selected by cross validation.
cv.block: A positive scalar with the size of the block in leave one block out cross-validation. By default 'cv.block=0' meaning leave one out cross-validation.
est: The nonparametric estimation method, one of "lc" (default) for linear constant or "ll" for local linear.
tkernel: A character, either "Triweight" (default), "Epa" or "Gaussian" kernel function.
singular.ok: Logical. If FALSE, a singular model is an error.

Details

Models for tvLM are specified symbolically using the same formula format than function lm. A typical model has the form response ~ terms where response is the (numeric) response vector and terms is a series of terms which specifies a linear predictor for response. A terms specification of the form first + second indicates all the terms in first together with all the terms in second with duplicates removed. A specification of the form first:second indicates the set of terms obtained by taking the interactions of all terms in first with all terms in second. The specification first*second indicates the cross of first and second. This is the same as first + second + first:second.

A formula has an implied intercept term. To remove this use either y ~ x - 1 or y ~ 0 + x.

References

Bollerslev, T., Patton, A. J. and Quaedvlieg, R. (2016) Exploiting the errors: A simple approach for improved volatility forecasting. Journal of Econometrics, 192, 1-18.

Casas, I., Mao, X. and Veiga, H. (2018) Reexamining financial and economic predictability with new estimators of realized variance and variance risk premium. Url= http://pure.au.dk/portal/files/123066669/rp18_10.pdf

Examples

Run this code

## Simulate a linear process with time-varying coefficient
## as functions of scaled time.
set.seed(42)
tau <- seq(1:200)/200
beta <- data.frame(beta1 = sin(2*pi*tau), beta2= 2*tau)
X1 <- rnorm(200)
X2 <- rchisq(200, df = 4)
error <- rt(200, df = 10)
y <- apply(cbind(X1, X2)*beta, 1, sum) + error
data <- data.frame(y = y, X1 = X1, X2 = X2)
## Estimate coefficients with lm and tvLM for comparison

coef.lm <- stats::lm(y ~ 0 + X1 + X2, data = data)$coef
tvlm.fit <- tvLM(y ~ 0 + X1 + X2, data = data, bw = 0.29)

## Estimate coefficients of different realized variance models
data("RV")
RV2 <- head(RV, 2000)
##Bollerslev t al. (2016) HARQ model
HARQ <- with(RV2, lm(RV ~ RV_lag + I(RV_lag * RQ_lag_sqrt) + RV_week + RV_month))

#Casas et al. (2018) TVHARQ model
TVHARQ <- with(RV2, tvLM (RV ~ RV_lag + RV_week + RV_month, z = RQ_lag_sqrt, 
                         bw = 0.0061))
boxplot(data.frame(TVHARQ = TVHARQ$coefficients[,2] * RV2$RV_lag,
                   HARQ = (HARQ$coef[2] + HARQ$coef[3] * RV2$RQ_lag_sqrt)*RV2$RV_lag),
                   main = expression (RV[t-1]), outline = FALSE)

Run the code above in your browser using DataLab