tvLM {tvReg}

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
)

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.

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.

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

See Also

bw, tvAR, confint, plot, print and summary

Examples

## 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)