These functions compute the different terms required for tcor() to compute the confidence
interval around the time-varying correlation coefficient. These terms are defined in Choi & Shin (2021).
Arguments
- smoothed_obj
an object created with
calc_rho.- H
an object created with
calc_H.- e
an object created with
calc_e.- l
a scalar indicating a number of time points.
- L
a scalar indicating a bandwidth parameter.
- AR.method
character string specifying the method to fit the autoregressive model used to compute \(\hat{\gamma}_1\) in \(L_{And}\) (see
stats::arfor details).- h
a scalar indicating the bandwidth used by the smoothing function.
Value
calc_H()returns a 5 x 5 x \(t\) array of elements of class numeric, which corresponds to \(\hat{H_t}\) in Choi & Shin (2021).calc_e()returns a \(t\) x 5 matrix of elements of class numeric storing the residuals, which corresponds to \(\hat{e}_t\) in Choi & Shin (2021).calc_Gamma()returns a 5 x 5 matrix of elements of class numeric, which corresponds to \(\hat{\Gamma}_l\) in Choi & Shin (2021).calc_GammaINF()returns a 5 x 5 matrix of elements of class numeric, which corresponds to \(\hat{\Gamma}^\infty\) in Choi & Shin (2021).calc_L_And()returns a scalar of class numeric, which corresponds to \(L_{And}\) in Choi & Shin (2021).calc_D()returns a \(t\) x 5 matrix of elements of class numeric storing the residuals, which corresponds to \(D_t\) in Choi & Shin (2021).calc_SE()returns a vector of length \(t\) of elements of class numeric, which corresponds to \(se(\hat{\rho}_t(h))\) in Choi & Shin (2021).
Functions
calc_H(): computes the \(\hat{H_t}\) array.\(\hat{H_t}\) is a component needed to compute confidence intervals; \(H_t\) is defined in eq. 6 from Choi & Shin (2021).
calc_e(): computes \(\hat{e}_t\).\(\hat{e}_t\) is defined in eq. 9 from Choi & Shin (2021).
calc_Gamma(): computes \(\hat{\Gamma}_l\).\(\hat{\Gamma}_l\) is defined in eq. 9 from Choi & Shin (2021).
calc_GammaINF(): computes \(\hat{\Gamma}^\infty\).\(\hat{\Gamma}^\infty\) is the long run variance estimator, defined in eq. 9 from Choi & Shin (2021).
calc_L_And(): computes \(L_{And}\).\(L_{And}\) is defined in Choi & Shin (2021, p 342). It also corresponds to \(S_T^*\), eq 5.3 in Andrews (1991).
calc_D(): computes \(D_t\).\(D_t\) is defined in Choi & Shin (2021, p 338).
calc_SE(): computes \(se(\hat{\rho}_t(h))\).The standard deviation of the time-varying correlation (\(se(\hat{\rho}_t(h))\)) is defined in eq. 8 from Choi & Shin (2021). It depends on \(D_{Lt}\), \(D_{Mt}\) & \(D_{Ut}\), themselves defined in Choi & Shin (2021, p 337 & 339). The \(D_{Xt}\) terms are all computed within the function since they all rely on the same components.
References
Choi, JE., Shin, D.W. Nonparametric estimation of time varying correlation coefficient. J. Korean Stat. Soc. 50, 333–353 (2021). doi:10.1007/s42952-020-00073-6
Andrews, D. W. K. Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica: Journal of the Econometric Society, 817-858 (1991).
Examples
rho_obj <- with(na.omit(stockprice),
calc_rho(x = SP500, y = FTSE100, t = DateID, h = 20, kernel = "box"))
head(rho_obj)
#> x y x2 y2 xy t
#> 1 0.004361280 -0.010096362 1.902076e-05 1.019365e-04 -4.403306e-05 2000-04-03
#> 2 -0.007980069 -0.003332639 6.368150e-05 1.110648e-05 2.659469e-05 2000-04-04
#> 3 -0.004479051 -0.007449493 2.006189e-05 5.549494e-05 3.336666e-05 2000-04-05
#> 4 0.008375645 0.011874107 7.015143e-05 1.409944e-04 9.945331e-05 2000-04-06
#> 5 0.009697455 0.015107515 9.404064e-05 2.282370e-04 1.465045e-04 2000-04-07
#> 6 -0.007420214 -0.004377977 5.505958e-05 1.916668e-05 3.248553e-05 2000-04-10
#> x_smoothed y_smoothed x2_smoothed y2_smoothed xy_smoothed
#> 1 -0.002369705 -0.003425687 7.493522e-05 0.0001296218 4.822802e-05
#> 2 -0.008177062 -0.005608043 4.327807e-04 0.0001804120 1.960199e-04
#> 3 -0.008177062 -0.005608043 4.327807e-04 0.0001804120 1.960199e-04
#> 4 -0.008177062 -0.005608043 4.327807e-04 0.0001804120 1.960199e-04
#> 5 -0.004666980 -0.006882863 4.776387e-04 0.0001990454 1.238864e-04
#> 6 -0.002107705 -0.003205704 4.367050e-04 0.0001841147 1.015342e-04
#> sd_x_smoothed sd_y_smoothed rho_smoothed
#> 1 0.008325846 0.01085755 0.4437047
#> 2 0.019128940 0.01220499 0.6431811
#> 3 0.019128940 0.01220499 0.6431811
#> 4 0.019128940 0.01220499 0.6431811
#> 5 0.021350832 0.01231550 0.3489847
#> 6 0.020790924 0.01318477 0.3457474
## Computing \eqn{\hat{H_t}}
H <- calc_H(smoothed_obj = rho_obj)
H[, , 1:2] # H array for the first two time points
#> , , 1
#>
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] -3.945960e-05 -3.300682e-05 0.008325846 0.004817547 -3.993791e-05
#> [2,] 0.000000e+00 -6.666556e-05 0.000000000 0.009730246 -2.305781e-05
#> [3,] 6.931972e-05 2.320876e-05 0.000000000 0.000000000 4.011015e-05
#> [4,] 0.000000e+00 -9.467769e-05 0.000000000 0.000000000 0.000000e+00
#> [5,] 0.000000e+00 9.375183e-05 0.000000000 0.000000000 8.101253e-05
#>
#> , , 2
#>
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] -0.0003128371 -8.804651e-05 0.01912894 0.007850022 -1.714660e-04
#> [2,] 0.0000000000 -1.048203e-04 0.00000000 0.009345535 -7.641902e-05
#> [3,] 0.0003659163 6.162284e-05 0.00000000 0.000000000 1.501626e-04
#> [4,] 0.0000000000 -8.733903e-05 0.00000000 0.000000000 0.000000e+00
#> [5,] 0.0000000000 1.467253e-04 0.00000000 0.000000000 1.787702e-04
#>
## Computing \eqn{\hat{e}_t}
e <- calc_e(smoothed_obj = rho_obj, H = H)
head(e) # e matrix for the first six time points
#> x2_resid y2_resid x_resid y_resid xy_resid
#> [1,] 0.80844454 -1.0858300496 -0.3464174 -0.1790269 -0.8778333739
#> [2,] 0.01029819 0.2348247345 -0.9998939 0.9448573 0.0024182690
#> [3,] 0.19332027 -0.3594249430 -0.9626273 0.8708137 -0.0694841264
#> [4,] 0.86532278 1.1437918620 -0.2512165 -0.3082598 0.9897491561
#> [5,] 0.67278103 1.6548373386 -0.5473657 -1.7384866 1.1133431705
#> [6,] -0.25552057 -0.0006027114 -0.9347092 0.9999996 0.0001540051
## Computing \eqn{\hat{\Gamma}_l}
calc_Gamma(e = e, l = 3)
#> x2_resid y2_resid x_resid y_resid xy_resid
#> x2_resid -0.08426588 0.00120430 -0.016571795 -0.0486144700 -0.028603802
#> y2_resid 0.02366280 -0.09087723 -0.036021416 -0.0104226203 0.003046219
#> x_resid 0.03505289 0.02239173 -0.130886043 0.0113781767 0.008721415
#> y_resid -0.00935183 -0.01810924 -0.002669087 -0.1306788327 -0.026723552
#> xy_resid -0.01353423 -0.01153453 -0.009324456 -0.0003183333 -0.059677907
## Computing \eqn{\hat{\Gamma}^\infty}
calc_GammaINF(e = e, L = 2)
#> x2_resid y2_resid x_resid y_resid xy_resid
#> x2_resid 0.89131796 0.03994933 -0.045481299 0.01885230 0.022464392
#> y2_resid 0.03994933 0.87487207 -0.016957014 0.04726965 -0.013540548
#> x_resid -0.04548130 -0.01695701 1.574267210 0.07364551 0.003709538
#> y_resid 0.01885230 0.04726965 0.073645507 1.61624702 0.036153472
#> xy_resid 0.02246439 -0.01354055 0.003709538 0.03615347 0.819391952
## Computing \eqn{L_{And}}
calc_L_And(e = e)
#> [1] 6.097187
sapply(c("yule-walker", "burg", "ols", "mle", "yw"),
function(m) calc_L_And(e = e, AR.method = m)) ## comparing AR.methods
#> yule-walker burg ols mle yw
#> 6.097187 6.097427 6.097239 6.096643 6.097187
## Computing \eqn{D_t}
D <- calc_D(smoothed_obj = rho_obj)
head(D) # D matrix for the first six time points
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] -3200.4218 -1881.9156 22.727368 13.320330 11062.153
#> [2,] -878.8636 -2158.8784 9.647499 10.810085 4283.231
#> [3,] -878.8636 -2158.8784 9.647499 10.810085 4283.231
#> [4,] -878.8636 -2158.8784 9.647499 10.810085 4283.231
#> [5,] -382.7778 -1150.4619 22.603106 1.911857 3803.059
#> [6,] -399.9275 -994.4519 10.008519 1.313051 3647.990
## Computing \eqn{se(\hat{\rho}_t(h))}
# nb: takes a few seconds to run
run <- FALSE ## change to TRUE to run the example
if (in_pkgdown() || run) {
SE <- calc_SE(smoothed_obj = rho_obj, h = 50)
head(SE) # SE vector for the first six time points
}
#> [1] 0.09718916 0.07043209 0.06966624 0.06889940 0.10133594 0.10045678