## More details about the models can be found in the article
## "The statistical content and empirical testing of the MIDAS restrictions"
## by Virmantas Kvedaras and Vaidotas Zemlys
library(midasr)
data("USunempr")
data("USrealgdp")
y <- diff(log(USrealgdp))
x <- window(diff(USunempr), start = 1949)
trend <- 1:length(y)
allk <- lapply(c(12, 15, 18, 24) - 1, function(k) {
update(midas_r(y ~ trend + fmls(x, k, 12, nealmon), start = list(x = rep(0, 3))), Ofunction = "nls")
})
#### Compute the derivative test
dtest <- lapply(allk, deriv_tests)
### The first derivative tests, gradient is zero
sapply(dtest, with, first)
### The second derivative tests, hessian is positive definite
sapply(dtest, with, second)
### The minimal eigenvalue of hessian is borderline zero, yet positive.
sapply(dtest, with, min(eigenval))
### Apply hAh test
lapply(allk, hAh_test)
### Apply robust hAh test
lapply(allk, hAhr_test)
### View summaries
lapply(allk, summary)
## Plot the coefficients
dev.new()
par(mfrow = c(2, 2))
plot_info <- lapply(allk, function(x) {
k <- length(coef(x, midas = TRUE, term = "x"))
ttl <- sprintf("d = %.0f: p-val.(hAh_HAC) < %.2f", k, max(hAhr_test(x)$p.value, 0.01))
plot_midas_coef(x, title = ttl)
})
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