nonparametric_tests.R

#' Returns the result of given event study nonparametric tests.
#'
#' Performs main nonparametric tests for each date in the event window and
#' returns a data frame of their statistics and significance.
#'
#' \code{nonparametric_tests} performs given tests among \code{\link{sign_test}},
#' \code{\link{generalized_sign_test}}, \code{\link{corrado_sign_test}},
#' \code{\link{rank_test}}, \code{\link{modified_rank_test}},
#' \code{\link{wilcoxon_test}}, and merge result to a single data frame. If
#' \code{all = TRUE} (the default value), the function ignores the value of
#' \code{tests}.
#'
#' @param list_of_returns a list of objects of S3 class \code{returns}, each
#' element of which is treated as a security.
#' @param event_start an object of \code{Date} class giving the first date of
#' the event period.
#' @param event_end an object of \code{Date} class giving the last date of the
#' event period.
#' @param all a logical vector of length one indicating whether all tests should
#' be performed. The default value is \code{TRUE}.
#' @param tests a list of tests' functions among \code{\link{sign_test}},
#' \code{\link{generalized_sign_test}}, \code{\link{corrado_sign_test}},
#' \code{\link{rank_test}}, \code{\link{modified_rank_test}}, and
#' \code{\link{wilcoxon_test}}.
#' @return A data frame of the following columns:
#' \itemize{
#'     \item \code{date}: a calendar date
#'     \item \code{weekday}: a day of the week
#'     \item \code{percentage}: a share of non-missing observations for a given
#'           day
#'     \item Various tests' statistics and significance
#' }
#'
#' @references \itemize{
#' \item Corrado C.J., Zivney T.L. \emph{The Specification and Power of
#' the Sign Test in Event Study Hypothesis Tests Using Daily Stock Returns}.
#' Journal of Financial and Quantitative Analysis, 27(3):465-478, 1992.
#' \item McConnell J.J., Muscarella C.J. \emph{Capital expenditure plans and
#' firm value} Journal of Financial Economics, 14:399-422, 1985.
#' \item Boehmer E., Musumeci J., Poulsen A.B. \emph{Event-study methodology
#' under conditions of event-induced variance}. Journal of Financial Economics,
#' 30(2):253-272, 1991.
#' \item Cowan A.R. \emph{Nonparametric Event Study Tests}. Review of
#' Quantitative Finance and Accounting, 2:343-358, 1992.
#' \item Corrado C.J. \emph{A Nonparametric Test for Abnormal Security-Price
#' Performance in Event Studies}. Journal of Financial Economics 23:385-395,
#' 1989.
#' \item Campbell C.J., Wasley C.E. \emph{Measuring Security Price Performance
#' Using Daily NASDAQ Returns}. Journal of Financial Economics 33:73-92, 1993.
#' \item Savickas R. \emph{Event-Induced Volatility and Tests for Abnormal
#' Performance}. The Journal of Financial Research, 26(2):156-178, 2003.
#' \item Kolari J.W., Pynnonen S. \emph{Event Study Testing with Cross-sectional
#' Correlation of Abnormal Returns}. The Review of Financial Studies,
#' 23(11):3996-4025, 2010.
#' \item Wilcoxon F. \emph{Individual Comparisons by Ranking Methods}.
#' Biometrics Bulletin 1(6):80-83, 1945.
#' \item Lehmann E.L, \emph{Nonparametrics: Statistical Methods Based on Ranks}.
#' San Francisco: Holden-Day, 1975.
#' \item Hollander M., Wolfe D.A. \emph{Nonparametric Statistical Methods}.
#' New York: John Wiley & Sons, 1973.
#' }
#'
#' @seealso \code{\link{sign_test}}, \code{\link{generalized_sign_test}},
#' \code{\link{corrado_sign_test}}, \code{\link{rank_test}},
#' \code{\link{modified_rank_test}}, and \code{\link{wilcoxon_test}}.
#'
#' @examples
#' \dontrun{
#' library("magrittr")
#' rates_indx <- get_prices_from_tickers("^GSPC",
#'                                       start = as.Date("2019-04-01"),
#'                                       end = as.Date("2020-04-01"),
#'                                       quote = "Close",
#'                                       retclass = "zoo") %>%
#'     get_rates_from_prices(quote = "Close",
#'                           multi_day = TRUE,
#'                           compounding = "continuous")
#' tickers <- c("AMZN", "ZM", "UBER", "NFLX", "SHOP", "FB", "UPWK")
#' nparam <- get_prices_from_tickers(tickers,
#'                                   start = as.Date("2019-04-01"),
#'                                   end = as.Date("2020-04-01"),
#'                                   quote = "Close",
#'                                   retclass = "zoo") %>%
#'     get_rates_from_prices(quote = "Close",
#'                           multi_day = TRUE,
#'                           compounding = "continuous") %>%
#'     apply_market_model(regressor = rates_indx,
#'                        same_regressor_for_all = TRUE,
#'                        market_model = "sim",
#'                        estimation_method = "ols",
#'                        estimation_start = as.Date("2019-04-01"),
#'                        estimation_end = as.Date("2020-03-13")) %>%
#'     nonparametric_tests(event_start = as.Date("2020-03-16"),
#'                         event_end = as.Date("2020-03-20"))
#' }
#' ## The result of the code above is equivalent to:
#' data(securities_returns)
#' nparam <- nonparametric_tests(list_of_returns = securities_returns,
#'                               event_start = as.Date("2020-03-16"),
#'                               event_end = as.Date("2020-03-20"))
#'
#' @export
nonparametric_tests <- function(list_of_returns, event_start, event_end,
                                all = TRUE, tests) {
    if(missing(tests)) {
        if(all) {
            tests <- list(sign_test, generalized_sign_test, corrado_sign_test,
                          rank_test, modified_rank_test, wilcoxon_test)
        } else {
            stop("Specify at least one test.")
        }
    } else {
        message("Argument all will be ignored.")
        for(i in seq_along(tests)) {
            tests[[i]] <- match.fun(tests[[i]])
        }
    }
    result <- NULL
    for(test in tests) {
        if(is.null(result)) {
            result <- test(list_of_returns, event_start, event_end)
        } else {
            result <- merge(x = result, y = test(list_of_returns, event_start,
                                                 event_end)[, c(1, 4, 5)],
                            by = "date", all = TRUE)
        }
    }
    return(result)
}

#' An event study simple binomial sign test.
#'
#' A binomial sign test which determines whether the frequency of positive
#' abnormal returns in the event period is significantly different from
#' one-half.
#'
#' This test is application of the simple binomial test to the event study,
#' which indicates whether the cross-sectional frequency of positive abnormal
#' returns is significantly different from 0.5. This test is stable
#' to outliers, in other words allows for checking if the result is driven by
#' few companies with extremely large abnormal performance. For this test the
#' estimation period and the event period must not overlap, otherwise an error
#' will be thrown. The test statistic is assumed to have a normal distribution
#' in approximation under a null hypothesis, if the number of securities is
#' large. Typically the test is used together with parametric tests.
#' The test is well-specified for the case, when cross-sectional abnormal
#' returns are not symmetric. Also this procedure is less sensitive to extreme
#' returns than the rank test. The significance levels of \eqn{\alpha} are 0.1,
#' 0.05, and 0.01 (marked respectively by *, **, and ***).
#'
#' @param list_of_returns a list of objects of S3 class \code{returns}, each
#' element of which is treated as a security.
#' @param event_start an object of \code{Date} class giving the first date of
#' the event period.
#' @param event_end an object of \code{Date} class giving the last date of the
#' event period.
#' @return A data frame of the following columns:
#' \itemize{
#'     \item \code{date}: a calendar date
#'     \item \code{weekday}: a day of the week
#'     \item \code{percentage}: a share of non-missing observations for a given
#'           day
#'     \item \code{sign_stat}: a sign test statistic
#'     \item \code{sign_signif}: a significance of the statistic
#' }
#'
#' @references Boehmer E., Musumeci J., Poulsen A.B. \emph{Event-study
#' methodology under conditions of event-induced variance}. Journal of Financial
#' Economics, 30(2):253-272, 1991.
#'
#' @seealso \code{\link{nonparametric_tests}}, \code{\link{generalized_sign_test}},
#' \code{\link{corrado_sign_test}}, \code{\link{rank_test}},
#' \code{\link{modified_rank_test}}, and \code{\link{wilcoxon_test}}.
#'
#' @examples
#' \dontrun{
#' library("magrittr")
#' rates_indx <- get_prices_from_tickers("^GSPC",
#'                                       start = as.Date("2019-04-01"),
#'                                       end = as.Date("2020-04-01"),
#'                                       quote = "Close",
#'                                       retclass = "zoo") %>%
#'     get_rates_from_prices(quote = "Close",
#'                           multi_day = TRUE,
#'                           compounding = "continuous")
#' tickers <- c("AMZN", "ZM", "UBER", "NFLX", "SHOP", "FB", "UPWK")
#' get_prices_from_tickers(tickers,
#'                         start = as.Date("2019-04-01"),
#'                         end = as.Date("2020-04-01"),
#'                         quote = "Close",
#'                         retclass = "zoo") %>%
#'     get_rates_from_prices(quote = "Close",
#'                           multi_day = TRUE,
#'                           compounding = "continuous") %>%
#'     apply_market_model(regressor = rates_indx,
#'                        same_regressor_for_all = TRUE,
#'                        market_model = "sim",
#'                        estimation_method = "ols",
#'                        estimation_start = as.Date("2019-04-01"),
#'                        estimation_end = as.Date("2020-03-13")) %>%
#'     sign_test(event_start = as.Date("2020-03-16"),
#'               event_end = as.Date("2020-03-20"))
#' }
#' ## The result of the code above is equivalent to:
#' data(securities_returns)
#' sign_test(list_of_returns = securities_returns,
#'           event_start =  as.Date("2020-03-16"),
#'           event_end = as.Date("2020-03-20"))
#'
#' @export
sign_test <- function(list_of_returns, event_start, event_end) {
    # check event_start and event_end for class and value validity
    if(!inherits(event_start, "Date")) {
        stop("event_start must be an object of class Date.")
    }
    if(!inherits(event_end, "Date")) {
        stop("event_end must be an object of class Date.")
    }
    if(event_start > event_end) {
        stop("event_start must be earlier than event_end.")
    }


    # zoo objects of abnormal returns
    event_binary <- NULL
    for(i in seq_along(list_of_returns)) {
        # check whether each element of list_of_returns is returns
        if(!inherits(list_of_returns[[i]], "returns")) {
            stop("Each element of list_of_rates must have class returns.")
        }

        if(list_of_returns[[i]]$estimation_end >= event_start) {
            stop(paste0("For ", as.character(i), "-th company estimation",
                        " period overlaps with event period."))
        }

        company_event_abnormal <- zoo::as.zoo(list_of_returns[[i]]$abnormal[
            zoo::index(list_of_returns[[i]]$abnormal) >= event_start &
                zoo::index(list_of_returns[[i]]$abnormal) <= event_end])

        company_event_binary <- zoo::zoo(as.numeric(company_event_abnormal > 0),
                                         zoo::index(company_event_abnormal))

        if(is.null(event_binary)) {
            event_binary <- company_event_binary
        } else {
            event_binary <- merge(event_binary, company_event_binary,
                                  all = TRUE)
        }
    }
    event_number_of_companies <- rowSums(!is.na(event_binary))
    event_binary_sums <- rowMeans(event_binary, na.rm = TRUE) * ncol(event_binary)
    event_binary_sums[is.nan(event_binary_sums)] <- NA
    result <- data.frame(date = zoo::index(event_binary),
                         weekday = weekdays(zoo::index(event_binary)),
                         percentage = event_number_of_companies /
                             ncol(event_binary) * 100)
    event_binary <- as.matrix(event_binary)
    event_number_of_companies[event_number_of_companies == 0] <- NA
    statistics <- (event_binary_sums - event_number_of_companies * 0.5) /
        sqrt(event_number_of_companies * 0.25)
    statistics[is.nan(statistics)] <- NA
    significance <- rep("", length(statistics))
    significance[abs(statistics) >= const_q1] <- "*"
    significance[abs(statistics) >= const_q2] <- "**"
    significance[abs(statistics) >= const_q3] <- "***"
    result <- cbind(result, data.frame(sign_stat = statistics,
                                       sign_signif = significance))
    rownames(result) <- NULL
    return(result)
}

#' An event study binomial sign test.
#'
#' A binomial sign test which determines whether the frequency of positive
#' abnormal returns in the event period is significantly different from the
#' frequency in the estimation period.
#'
#' This test is application of the binomial test to the event study,
#' which indicates whether the cross-sectional frequency of positive abnormal
#' returns is significantly different from the expected. This test is stable
#' to outliers, in other words allows for checking if the result is driven by
#' few companies with extremely large abnormal performance. For this test the
#' estimation period and the event period must not overlap, otherwise an error
#' will be thrown. This test uses an estimate from the estimation period instead
#' of using naive value of expected frequency 0.5. The test statistic is assumed
#' to have a normal distribution. Typically the test is used together with
#' parametric tests. The test is well-specified for the case, when
#' cross-sectional abnormal returns are not symmetric. Also this procedure is
#' less sensitive to extreme returns than the rank test. The significance levels
#' of \eqn{\alpha} are 0.1, 0.05, and 0.01 (marked respectively by *, **, and
#' ***).
#'
#' @param list_of_returns a list of objects of S3 class \code{returns}, each
#' element of which is treated as a security.
#' @param event_start an object of \code{Date} class giving the first date of
#' the event period.
#' @param event_end an object of \code{Date} class giving the last date of the
#' event period.
#' @return A data frame of the following columns:
#' \itemize{
#'     \item \code{date}: a calendar date
#'     \item \code{weekday}: a day of the week
#'     \item \code{percentage}: a share of non-missing observations for a given
#'           day
#'     \item \code{gsign_stat}: a generalized sign test statistic
#'     \item \code{gsign_signif}: a significance of the statistic
#' }
#'
#' @references \itemize{
#' \item McConnell J.J., Muscarella C.J. \emph{Capital expenditure plans and
#' firm value} Journal of Financial Economics, 14:399-422, 1985.
#' \item Cowan A.R. \emph{Nonparametric Event Study Tests}. Review of
#' Quantitative Finance and Accounting, 2:343-358, 1992.
#' }
#'
#' @seealso \code{\link{nonparametric_tests}}, \code{\link{sign_test}},
#' \code{\link{corrado_sign_test}}, \code{\link{rank_test}},
#' \code{\link{modified_rank_test}}, and \code{\link{wilcoxon_test}}.
#'
#' @examples
#' \dontrun{
#' library("magrittr")
#' rates_indx <- get_prices_from_tickers("^GSPC",
#'                                       start = as.Date("2019-04-01"),
#'                                       end = as.Date("2020-04-01"),
#'                                       quote = "Close",
#'                                       retclass = "zoo") %>%
#'     get_rates_from_prices(quote = "Close",
#'                           multi_day = TRUE,
#'                           compounding = "continuous")
#' tickers <- c("AMZN", "ZM", "UBER", "NFLX", "SHOP", "FB", "UPWK")
#' get_prices_from_tickers(tickers,
#'                         start = as.Date("2019-04-01"),
#'                         end = as.Date("2020-04-01"),
#'                         quote = "Close",
#'                         retclass = "zoo") %>%
#'     get_rates_from_prices(quote = "Close",
#'                           multi_day = TRUE,
#'                           compounding = "continuous") %>%
#'     apply_market_model(regressor = rates_indx,
#'                        same_regressor_for_all = TRUE,
#'                        market_model = "sim",
#'                        estimation_method = "ols",
#'                        estimation_start = as.Date("2019-04-01"),
#'                        estimation_end = as.Date("2020-03-13")) %>%
#'     generalized_sign_test(event_start = as.Date("2020-03-16"),
#'                           event_end = as.Date("2020-03-20"))
#' }
#' ## The result of the code above is equivalent to:
#' data(securities_returns)
#' generalized_sign_test(list_of_returns = securities_returns,
#'                       event_start =  as.Date("2020-03-16"),
#'                       event_end = as.Date("2020-03-20"))
#'
#' @export
generalized_sign_test <- function(list_of_returns, event_start, event_end) {
    # check event_start and event_end for class and value validity
    if(!inherits(event_start, "Date")) {
        stop("event_start must be an object of class Date.")
    }
    if(!inherits(event_end, "Date")) {
        stop("event_end must be an object of class Date.")
    }
    if(event_start > event_end) {
        stop("event_start must be earlier than event_end.")
    }

    # zoo objects of abnormal returns
    estimation_binary <- NULL
    event_binary <- NULL
    for(i in seq_along(list_of_returns)) {

        # check whether each element of list_of_returns is returns
        if(!inherits(list_of_returns[[i]], "returns")) {
            stop("Each element of list_of_rates must have class returns.")
        }

        if(list_of_returns[[i]]$estimation_end >= event_start) {
            stop(paste0("For ", as.character(i), "-th company estimation",
                        " period overlaps with event period."))
        }

        company_estimation_abnormal <- zoo::as.zoo(list_of_returns[[i]]$abnormal[
            zoo::index(list_of_returns[[i]]$abnormal) >=
                list_of_returns[[i]]$estimation_start &
                zoo::index(list_of_returns[[i]]$abnormal) <=
                list_of_returns[[i]]$estimation_end])
        company_event_abnormal <- zoo::as.zoo(list_of_returns[[i]]$abnormal[
            zoo::index(list_of_returns[[i]]$abnormal) >= event_start &
                zoo::index(list_of_returns[[i]]$abnormal) <= event_end])

        company_estimation_binary <- zoo::zoo(
            as.numeric(company_estimation_abnormal > 0),
            zoo::index(company_estimation_abnormal))

        company_event_binary <- zoo::zoo(as.numeric(company_event_abnormal > 0),
                                         zoo::index(company_event_abnormal))

        if(is.null(estimation_binary)) {
            estimation_binary <- company_estimation_binary
        } else {
            estimation_binary <- merge(estimation_binary,
                                       company_estimation_binary, all = TRUE)
        }
        if(is.null(event_binary)) {
            event_binary <- company_event_binary
        } else {
            event_binary <- merge(event_binary, company_event_binary,
                                    all = TRUE)
        }
    }
    p_hat <- mean(as.matrix(estimation_binary), na.rm = TRUE)
    event_number_of_companies <- rowSums(!is.na(event_binary))
    event_binary_sums <- rowMeans(event_binary, na.rm = TRUE) * ncol(event_binary)
    event_binary_sums[is.nan(event_binary_sums)] <- NA
    result <- data.frame(date = zoo::index(event_binary),
                         weekday = weekdays(zoo::index(event_binary)),
                         percentage = event_number_of_companies /
                             ncol(event_binary) * 100)

    # estimation_binary <- as.matrix(estimation_binary)
    event_binary <- as.matrix(event_binary)
    event_number_of_companies[event_number_of_companies == 0] <- NA
    statistics <- (event_binary_sums - event_number_of_companies * p_hat) /
        sqrt(event_number_of_companies * p_hat * (1 - p_hat))
    statistics[is.nan(statistics)] <- NA
    significance <- rep("", length(statistics))
    significance[abs(statistics) >= const_q1] <- "*"
    significance[abs(statistics) >= const_q2] <- "**"
    significance[abs(statistics) >= const_q3] <- "***"
    result <- cbind(result, data.frame(gsign_stat = statistics,
                                       gsign_signif = significance))
    rownames(result) <- NULL
    return(result)
}

#' Corrado's sign test (1992).
#'
#' An event study nonparametric test described in Corrado and Zivney 1992.
#'
#' Performs a nonparametric test for the event study, which is described in
#' Corrado and Zivney 1992. This test is similar to procedure, described in
#' Brown and Warner 1985 (t-ratio), but instead of using abnormal
#' returns, the test uses \eqn{G_{i,t} = sign(A_{i,t} - median(A_i))}.
#' \code{sign} and \code{median} are ones, which have the same definition as R
#' functions. For this test the estimation period and the event period must not
#' overlap, otherwise an error will be thrown. The sign test procedure avoids
#' the misspecification of tests, which assumes symmetry around zero of abnormal
#' returns (the median equals to zero). For a single day the performance of this
#' test is proven to be better than classical Brown and Warner's test (without
#' event-induced variance). This test is dominated by rank test. The
#' significance levels of \eqn{\alpha} are 0.1, 0.05, and 0.01 (marked
#' respectively by *, **, and ***).
#'
#' @param list_of_returns a list of objects of S3 class \code{returns}, each
#' element of which is treated as a security.
#' @param event_start an object of \code{Date} class giving the first date of
#' the event period.
#' @param event_end an object of \code{Date} class giving the last date of the
#' event period.
#' @return A data frame of the following columns:
#' \itemize{
#'     \item \code{date}: a calendar date
#'     \item \code{weekday}: a day of the week
#'     \item \code{percentage}: a share of non-missing observations for a given
#'           day
#'     \item \code{csign_stat}: a Corrado's sign test statistic
#'     \item \code{csign_signif}: a significance of the statistic
#' }
#'
#' @references Corrado C.J., Zivney T.L. \emph{The Specification and Power of
#' the Sign Test in Event Study Hypothesis Tests Using Daily Stock Returns}.
#' Journal of Financial and Quantitative Analysis, 27(3):465-478, 1992.
#'
#' @seealso \code{\link{nonparametric_tests}}, \code{\link{sign_test}},
#' \code{\link{generalized_sign_test}}, \code{\link{rank_test}},
#' \code{\link{modified_rank_test}}, and \code{\link{wilcoxon_test}}.
#'
#' @examples
#' \dontrun{
#' library("magrittr")
#' rates_indx <- get_prices_from_tickers("^GSPC",
#'                                       start = as.Date("2019-04-01"),
#'                                       end = as.Date("2020-04-01"),
#'                                       quote = "Close",
#'                                       retclass = "zoo") %>%
#'     get_rates_from_prices(quote = "Close",
#'                           multi_day = TRUE,
#'                           compounding = "continuous")
#' tickers <- c("AMZN", "ZM", "UBER", "NFLX", "SHOP", "FB", "UPWK")
#' get_prices_from_tickers(tickers,
#'                         start = as.Date("2019-04-01"),
#'                         end = as.Date("2020-04-01"),
#'                         quote = "Close",
#'                         retclass = "zoo") %>%
#'     get_rates_from_prices(quote = "Close",
#'                           multi_day = TRUE,
#'                           compounding = "continuous") %>%
#'     apply_market_model(regressor = rates_indx,
#'                        same_regressor_for_all = TRUE,
#'                        market_model = "sim",
#'                        estimation_method = "ols",
#'                        estimation_start = as.Date("2019-04-01"),
#'                        estimation_end = as.Date("2020-03-13")) %>%
#'     corrado_sign_test(event_start = as.Date("2020-03-16"),
#'                       event_end = as.Date("2020-03-20"))
#' }
#' ## The result of the code above is equivalent to:
#' data(securities_returns)
#' corrado_sign_test(list_of_returns = securities_returns,
#'                   event_start =  as.Date("2020-03-16"),
#'                   event_end = as.Date("2020-03-20"))
#'
#' @export
corrado_sign_test <- function(list_of_returns, event_start, event_end) {
    # check event_start and event_end for class and value validity
    if(!inherits(event_start, "Date")) {
        stop("event_start must be an object of class Date.")
    }
    if(!inherits(event_end, "Date")) {
        stop("event_end must be an object of class Date.")
    }
    if(event_start > event_end) {
        stop("event_start must be earlier than event_end.")
    }

    # zoo objects of signs
    event_sign <- NULL
    full_sign <- NULL
    delta_full <- numeric(length(list_of_returns))
    for(i in seq_along(list_of_returns)) {

        # check whether each element of list_of_returns is returns
        if(!inherits(list_of_returns[[i]], "returns")) {
            stop("Each element of list_of_rates must have class returns.")
        }

        if(list_of_returns[[i]]$estimation_end >= event_start) {
            stop(paste0("For ", as.character(i), "-th company estimation",
                        " period overlaps with event period."))
        }
        company_full_abnormal <- zoo::as.zoo(c(
            list_of_returns[[i]]$abnormal[
                zoo::index(list_of_returns[[i]]$abnormal) >=
                    list_of_returns[[i]]$estimation_start &
                    zoo::index(list_of_returns[[i]]$abnormal) <=
                    list_of_returns[[i]]$estimation_end],
            list_of_returns[[i]]$abnormal[
                zoo::index(list_of_returns[[i]]$abnormal) >= event_start &
                    zoo::index(list_of_returns[[i]]$abnormal) <= event_end]))

        company_event_abnormal <- zoo::as.zoo(list_of_returns[[i]]$abnormal[
            zoo::index(list_of_returns[[i]]$abnormal) >= event_start &
                zoo::index(list_of_returns[[i]]$abnormal) <= event_end])

        company_median <- stats::median(zoo::coredata(company_full_abnormal), na.rm = TRUE)
        company_full_sign <- sign(company_full_abnormal - company_median)
        company_event_sign <- sign(company_event_abnormal - company_median)

        if(is.null(full_sign)) {
            full_sign <- company_full_sign
        } else {
            full_sign <- merge(full_sign, company_full_sign, all = TRUE)
        }

        if(is.null(event_sign)) {
            event_sign <- company_event_sign
        } else {
            event_sign <- merge(event_sign, company_event_sign, all = TRUE)
        }

        delta_full[i] <-
            length(company_full_abnormal[!is.na(company_full_abnormal)])

    }
    event_number_of_companies <- rowSums(!is.na(event_sign))
    result <- data.frame(date = zoo::index(event_sign),
                         weekday = weekdays(zoo::index(event_sign)),
                         percentage = event_number_of_companies /
                             ncol(event_sign) * 100)

    full_sign <- as.matrix(full_sign)
    event_sign <- as.matrix(event_sign)
    number_of_companies <- rowSums(!is.na(full_sign))
    number_of_companies[number_of_companies == 0] <- NA
    full_sign_sums <- rowMeans(full_sign, na.rm = TRUE) * ncol(full_sign)
    full_sign_sums[is.nan(full_sign_sums)] <- NA
    sd_full <- sqrt(1 / mean(delta_full, na.rm = TRUE) *
                        sum((1 / sqrt(number_of_companies) * full_sign_sums)^2, na.rm = TRUE))

    event_sign_sums <- rowMeans(event_sign, na.rm = TRUE) * ncol(event_sign)
    event_sign_sums[is.nan(event_sign_sums)] <- NA
    event_number_of_companies[event_number_of_companies == 0] <- NA
    statistics <- 1 / sqrt(event_number_of_companies) *
        event_sign_sums / sd_full
    statistics[is.nan(statistics)] <- NA
    significance <- rep("", length(statistics))
    significance[abs(statistics) >= const_q1] <- "*"
    significance[abs(statistics) >= const_q2] <- "**"
    significance[abs(statistics) >= const_q3] <- "***"
    result <- cbind(result, data.frame(csign_stat = statistics,
                                       csign_signif = significance))
    rownames(result) <- NULL
    return(result)
}

#' An event study rank test.
#'
#' An original rank test applied to an event study, which is based on
#' Wilcoxon (1945) rank test.
#'
#' This procedure uses ranks of abnormal returns to examine significance of each
#' day in the event window. In order to get ranks of corresponding abnormal
#' returns, the procedure uses regular R function \code{\link{rank}} with
#' parameter \code{ties.method = "average"} and \code{na.last = "keep"}. For
#' this test the estimation period and the event period must not overlap,
#' otherwise an error will be thrown. The test statistic is assumed to have a
#' normal distribution (as an approximation). The test is well-specified for the
#' case, when cross-sectional abnormal returns are not symmetric. The test is
#' stable to variance increase during event window. This test is more sensitive
#' to extreme values than sign test. For data with missing data see the
#' \code{\link{modified_rank_test}}. The significance levels of \eqn{\alpha} are
#' 0.1, 0.05, and 0.01 (marked respectively by *, **, and ***).
#'
#' @param list_of_returns a list of objects of S3 class \code{returns}, each
#' element of which is treated as a security.
#' @param event_start an object of \code{Date} class giving the first date of
#' the event period.
#' @param event_end an object of \code{Date} class giving the last date of the
#' event period.
#' @return A data frame of the following columns:
#' \itemize{
#'     \item \code{date}: a calendar date
#'     \item \code{weekday}: a day of the week
#'     \item \code{percentage}: a share of non-missing observations for a given
#'           day
#'     \item \code{rank_stat}: a rank test statistic
#'     \item \code{rank_signif}: a significance of the statistic
#' }
#'
#' @references \itemize{
#' \item Corrado C.J. \emph{A Nonparametric Test for Abnormal Security-Price
#' Performance in Event Studies}. Journal of Financial Economics 23:385-395,
#' 1989.
#' \item Cowan A.R. \emph{Nonparametric Event Study Tests}. Review of
#' Quantitative Finance and Accounting, 2:343-358, 1992.
#' \item Campbell C.J., Wasley C.E. \emph{Measuring Security Price Performance
#' Using Daily NASDAQ Returns}. Journal of Financial Economics 33:73-92, 1993.
#' \item Savickas R. \emph{Event-Induced Volatility and Tests for Abnormal
#' Performance}. The Journal of Financial Research, 26(2):156-178, 2003.
#' }
#'
#' @seealso \code{\link{nonparametric_tests}},\code{\link{sign_test}},
#' \code{\link{generalized_sign_test}}, \code{\link{corrado_sign_test}},
#' \code{\link{modified_rank_test}}, and \code{\link{wilcoxon_test}}.
#'
#' @examples
#' \dontrun{
#' library("magrittr")
#' rates_indx <- get_prices_from_tickers("^GSPC",
#'                                       start = as.Date("2019-04-01"),
#'                                       end = as.Date("2020-04-01"),
#'                                       quote = "Close",
#'                                       retclass = "zoo") %>%
#'     get_rates_from_prices(quote = "Close",
#'                           multi_day = TRUE,
#'                           compounding = "continuous")
#' tickers <- c("AMZN", "ZM", "UBER", "NFLX", "SHOP", "FB", "UPWK")
#' get_prices_from_tickers(tickers,
#'                         start = as.Date("2019-04-01"),
#'                         end = as.Date("2020-04-01"),
#'                         quote = "Close",
#'                         retclass = "zoo") %>%
#'     get_rates_from_prices(quote = "Close",
#'                           multi_day = TRUE,
#'                           compounding = "continuous") %>%
#'     apply_market_model(regressor = rates_indx,
#'                        same_regressor_for_all = TRUE,
#'                        market_model = "sim",
#'                        estimation_method = "ols",
#'                        estimation_start = as.Date("2019-04-01"),
#'                        estimation_end = as.Date("2020-03-13")) %>%
#'     rank_test(event_start = as.Date("2020-03-16"),
#'               event_end = as.Date("2020-03-20"))
#' }
#' ## The result of the code above is equivalent to:
#' data(securities_returns)
#' rank_test(list_of_returns = securities_returns,
#'           event_start =  as.Date("2020-03-16"),
#'           event_end = as.Date("2020-03-20"))
#'
#' @export
rank_test <- function(list_of_returns, event_start, event_end) {

    # check event_start and event_end for class and value validity
    if(!inherits(event_start, "Date")) {
        stop("event_start must be an object of class Date.")
    }
    if(!inherits(event_end, "Date")) {
        stop("event_end must be an object of class Date.")
    }
    if(event_start > event_end) {
        stop("event_start must be earlier than event_end.")
    }

    # zoo objects of abnormal returns
    full_rank <- NULL
    event_rank <- NULL
    delta_full <- numeric(length(list_of_returns))
    avg_rank <- numeric(length(list_of_returns))
    for(i in seq_along(list_of_returns)) {

        # check whether each element of list_of_returns is returns
        if(!inherits(list_of_returns[[i]], "returns")) {
            stop("Each element of list_of_rates must have class returns.")
        }

        if(list_of_returns[[i]]$estimation_end >= event_start) {
            stop(paste0("For ", as.character(i), "-th company estimation",
                        " period overlaps with event period."))
        }

        company_full_abnormal <- zoo::as.zoo(c(
            list_of_returns[[i]]$abnormal[
                zoo::index(list_of_returns[[i]]$abnormal) >=
                    list_of_returns[[i]]$estimation_start &
                    zoo::index(list_of_returns[[i]]$abnormal) <=
                    list_of_returns[[i]]$estimation_end],
            list_of_returns[[i]]$abnormal[
                zoo::index(list_of_returns[[i]]$abnormal) >= event_start &
                zoo::index(list_of_returns[[i]]$abnormal) <= event_end]))

        company_full_rank <- zoo::zoo(rank(x = zoo::coredata(company_full_abnormal),
                                           na.last = "keep",
                                           ties.method = "average"),
                                      zoo::index(company_full_abnormal))
        company_event_rank <- zoo::as.zoo(company_full_rank[
            zoo::index(company_full_rank) >= event_start &
            zoo::index(company_full_rank) <= event_end])

        if(is.null(full_rank)) {
            full_rank <- company_full_rank
        } else {
            full_rank <- merge(full_rank, company_full_rank, all = TRUE)
        }

        if(is.null(event_rank)) {
            event_rank <- company_event_rank
        } else {
            event_rank <- merge(event_rank, company_event_rank, all = TRUE)
        }

        delta_full[i] <-
            length(company_full_abnormal[!is.na(company_full_abnormal)])
        avg_rank[i] <- mean(company_full_rank, na.rm = TRUE)
    }

    event_number_of_companies <- rowSums(!is.na(event_rank))
    result <- data.frame(date = zoo::index(event_rank),
                         weekday = weekdays(zoo::index(event_rank)),
                         percentage = event_number_of_companies /
                             ncol(event_rank) * 100)

    full_rank <- as.matrix(full_rank)
    event_rank <- as.matrix(event_rank)

    number_of_companies <- rowSums(!is.na(full_rank))
    number_of_companies[number_of_companies == 0] <- NA


    event_number_of_companies[event_number_of_companies == 0] <- NA
    avg_rank_full <- matrix(rep(avg_rank, nrow(full_rank)),
                              nrow = nrow(full_rank), ncol = ncol(full_rank),
                              byrow = TRUE)
    avg_rank_event <- matrix(rep(avg_rank, nrow(event_rank)),
                             nrow = nrow(event_rank), ncol = ncol(event_rank),
                             byrow = TRUE)

    full_differences <- rowMeans(full_rank - avg_rank_full, na.rm = TRUE) * ncol(full_rank)
    full_differences[is.nan(full_differences)] <- NA

    event_differences <- rowMeans(event_rank - avg_rank_event, na.rm = TRUE) * ncol(event_rank)
    event_differences[is.nan(event_differences)] <- NA

    sd_full <- sqrt(1 / mean(delta_full) * sum((1 / number_of_companies * full_differences)^2, na.rm = TRUE))

    statistics <- 1 / event_number_of_companies * event_differences / sd_full
    statistics[is.nan(statistics)] <- NA
    significance <- rep("", length(statistics))
    significance[abs(statistics) >= const_q1] <- "*"
    significance[abs(statistics) >= const_q2] <- "**"
    significance[abs(statistics) >= const_q3] <- "***"
    result <- cbind(result, data.frame(rank_stat = statistics,
                                       rank_signif = significance))
    rownames(result) <- NULL
    return(result)
}

#' An event study modified rank test.
#'
#' The test is the modification of the original rank test, proposed by Corrado
#' 1989. This test is adapted to missing values in abnormal returns.
#'
#' In addition to the original rank test, the procedure divides corresponding
#' ranks by the number of nonmissing returns plus one for each security. This
#' leads to order statistics with uniform distribution. In limit overall
#' statistics under a null hypothesis is approximately normally distributed. For
#' this test the estimation period and the event period must not overlap,
#' otherwise an error will be thrown. The test is well-specified for the case,
#' when cross-sectional abnormal returns are not symmetric. The test is stable
#' to variance increase during the event window. This test is more sensitive to
#' extreme values than the sign test. The significance levels of \eqn{\alpha}
#' are 0.1, 0.05, and 0.01 (marked respectively by *, **, and ***).
#'
#' @param list_of_returns a list of objects of S3 class \code{returns}, each
#' element of which is treated as a security.
#' @param event_start an object of \code{Date} class giving the first date of
#' the event period.
#' @param event_end an object of \code{Date} class giving the last date of the
#' event period.
#' @return A data frame of the following columns:
#' \itemize{
#'     \item \code{date}: a calendar date
#'     \item \code{weekday}: a day of the week
#'     \item \code{percentage}: a share of non-missing observations for a given
#'           day
#'     \item \code{mrank_stat}: a modified rank test statistic
#'     \item \code{mrank_signif}: a significance of the statistic
#' }
#'
#' @references \itemize{
#' \item Corrado C.J., Zivney T.L. \emph{The Specification and Power of
#' the Sign Test in Event Study Hypothesis Tests Using Daily Stock Returns}.
#' Journal of Financial and Quantitative Analysis, 27(3):465-478, 1992.
#' \item Kolari J.W., Pynnonen S. \emph{Event Study Testing with Cross-sectional
#' Correlation of Abnormal Returns}. The Review of Financial Studies,
#' 23(11):3996-4025, 2010.
#' }
#'
#' @seealso \code{\link{nonparametric_tests}},\code{\link{sign_test}},
#' \code{\link{generalized_sign_test}}, \code{\link{corrado_sign_test}},
#' \code{\link{rank_test}}, and \code{\link{wilcoxon_test}}.
#'
#' @examples
#' \dontrun{
#' library("magrittr")
#' rates_indx <- get_prices_from_tickers("^GSPC",
#'                                       start = as.Date("2019-04-01"),
#'                                       end = as.Date("2020-04-01"),
#'                                       quote = "Close",
#'                                       retclass = "zoo") %>%
#'     get_rates_from_prices(quote = "Close",
#'                           multi_day = TRUE,
#'                           compounding = "continuous")
#' tickers <- c("AMZN", "ZM", "UBER", "NFLX", "SHOP", "FB", "UPWK")
#' get_prices_from_tickers(tickers,
#'                         start = as.Date("2019-04-01"),
#'                         end = as.Date("2020-04-01"),
#'                         quote = "Close",
#'                         retclass = "zoo") %>%
#'     get_rates_from_prices(quote = "Close",
#'                           multi_day = TRUE,
#'                           compounding = "continuous") %>%
#'     apply_market_model(regressor = rates_indx,
#'                        same_regressor_for_all = TRUE,
#'                        market_model = "sim",
#'                        estimation_method = "ols",
#'                        estimation_start = as.Date("2019-04-01"),
#'                        estimation_end = as.Date("2020-03-13")) %>%
#'     modified_rank_test(event_start = as.Date("2020-03-16"),
#'                        event_end = as.Date("2020-03-20"))
#' }
#' ## The result of the code above is equivalent to:
#' data(securities_returns)
#' modified_rank_test(list_of_returns = securities_returns,
#'                    event_start =  as.Date("2020-03-16"),
#'                    event_end = as.Date("2020-03-20"))
#'
#' @export
modified_rank_test <- function(list_of_returns, event_start, event_end) {
    # Corrado Zivney 1992
    # Kolari Pynnonen 2010

    # check event_start and event_end for class and value validity
    if(!inherits(event_start, "Date")) {
        stop("event_start must be an object of class Date.")
    }
    if(!inherits(event_end, "Date")) {
        stop("event_end must be an object of class Date.")
    }
    if(event_start > event_end) {
        stop("event_start must be earlier than event_end.")
    }


    # zoo objects of abnormal returns
    full_rank_modif <- NULL
    event_rank_modif <- NULL
    delta_full <- numeric(length(list_of_returns))
    for(i in seq_along(list_of_returns)) {

        # check whether each element of list_of_returns is returns
        if(!inherits(list_of_returns[[i]], "returns")) {
            stop("Each element of list_of_rates must have class returns.")
        }

        if(list_of_returns[[i]]$estimation_end >= event_start) {
            stop(paste0("For ", as.character(i), "-th company estimation",
                        " period overlaps with event period."))
        }

        company_full_abnormal <- zoo::as.zoo(c(
            list_of_returns[[i]]$abnormal[
                zoo::index(list_of_returns[[i]]$abnormal) >=
                    list_of_returns[[i]]$estimation_start &
                    zoo::index(list_of_returns[[i]]$abnormal) <=
                    list_of_returns[[i]]$estimation_end],
            list_of_returns[[i]]$abnormal[
                zoo::index(list_of_returns[[i]]$abnormal) >= event_start &
                    zoo::index(list_of_returns[[i]]$abnormal) <= event_end]))

        company_full_rank_modif <- zoo::zoo(rank(x =
                                                zoo::coredata(company_full_abnormal),
                                                na.last = "keep",
                                                ties.method = "average") /
                                    (1 + sum(!is.na(company_full_abnormal))),
                                            zoo::index(company_full_abnormal))
        company_event_rank_modif <- zoo::as.zoo(company_full_rank_modif[
            zoo::index(company_full_rank_modif) >= event_start &
                zoo::index(company_full_rank_modif) <= event_end])

        if(is.null(full_rank_modif)) {
            full_rank_modif <- company_full_rank_modif
        } else {
            full_rank_modif <- merge(full_rank_modif, company_full_rank_modif,
                                     all = TRUE)
        }

        if(is.null(event_rank_modif)) {
            event_rank_modif <- company_event_rank_modif
        } else {
            event_rank_modif <- merge(event_rank_modif,
                                      company_event_rank_modif, all = TRUE)
        }

        delta_full[i] <-
            length(company_full_abnormal[!is.na(company_full_abnormal)])
    }

    event_number_of_companies <- rowSums(!is.na(event_rank_modif))
    result <- data.frame(date = zoo::index(event_rank_modif),
                         weekday = weekdays(zoo::index(event_rank_modif)),
                         percentage = event_number_of_companies /
                             ncol(event_rank_modif) * 100)

    full_rank_modif <- as.matrix(full_rank_modif)
    event_rank_modif <- as.matrix(event_rank_modif)

    number_of_companies <- rowSums(!is.na(full_rank_modif))
    number_of_companies[number_of_companies == 0] <- NA
    event_number_of_companies[event_number_of_companies == 0] <- NA



    full_differences <- rowMeans(full_rank_modif - 0.5, na.rm = TRUE) * ncol(full_rank_modif)
    full_differences[is.nan(full_differences)] <- NA

    event_differences <- rowMeans(event_rank_modif - 0.5, na.rm = TRUE) * ncol(event_rank_modif)
    event_differences[is.nan(event_differences)] <- NA

    sd_full <- sqrt(1 / mean(delta_full) * sum((1 / sqrt(number_of_companies) * full_differences)^2, na.rm = TRUE))

    statistics <- 1 / sqrt(event_number_of_companies) * event_differences / sd_full
    statistics[is.nan(statistics)] <- NA
    significance <- rep("", length(statistics))
    significance[abs(statistics) >= const_q1] <- "*"
    significance[abs(statistics) >= const_q2] <- "**"
    significance[abs(statistics) >= const_q3] <- "***"
    result <- cbind(result, data.frame(mrank_stat = statistics,
                                       mrank_signif = significance))
    rownames(result) <- NULL
    return(result)
}


#' An event study Wilcoxon signed rank test.
#'
#' Performs Wilcoxon test on the event period for abnormal returns
#' (abnormal returns are considered as differences).
#'
#' The estimation periods can overlap with event windows, because the procedure
#' takes into account only abnormal returns from the event window. The test has
#' the same algorithm as built-in \code{R} \code{\link{wilcox.test}}. The
#' critical values are exact values, which are obtained from
#' \code{\link{qsignrank}}. The algorithm is the following: for each day in
#' event window the cross-sectional abnormal returns treated as sample of
#' differences. Firstly the absolute value of these differences are computed,
#' and corresponding ranks of non-zero values are calculated. The test statistic
#' is the sum of ranks, corresponding to positive abnormal returns. The
#' significance levels of \eqn{\alpha} are 0.1, 0.05, and 0.01 (marked
#' respectively by *, **, and ***).
#'
#' @param list_of_returns a list of objects of S3 class \code{returns}, each
#' element of which is treated as a security.
#' @param event_start an object of \code{Date} class giving the first date of
#' the event period.
#' @param event_end an object of \code{Date} class giving the last date of the
#' event period.
#' @return A data frame of the following columns:
#' \itemize{
#'     \item \code{date}: a calendar date
#'     \item \code{weekday}: a day of the week
#'     \item \code{percentage}: a share of non-missing observations for a given
#'           day
#'     \item \code{wlcx_stat}: a Wilcoxon signed rank test statistic
#'     \item \code{wlcx_signif}: a significance of the statistic
#' }
#'
#' @references \itemize{
#' \item Wilcoxon F. \emph{Individual Comparisons by Ranking Methods}.
#' Biometrics Bulletin 1(6):80-83, 1945.
#' \item Kolari J.W., Pynnonen S. \emph{Event Study Testing with Cross-sectional
#' Correlation of Abnormal Returns}. The Review of Financial Studies,
#' 23(11):3996-4025, 2010.
#' \item Lehmann E.L, \emph{Nonparametrics: Statistical Methods Based on Ranks}.
#' San Francisco: Holden-Day, 1975.
#' \item Hollander M., Wolfe D.A. \emph{Nonparametric Statistical Methods}.
#' New York: John Wiley & Sons, 1973.
#' }
#'
#' @seealso \code{\link{nonparametric_tests}}, \code{\link{sign_test}},
#' \code{\link{generalized_sign_test}}, \code{\link{corrado_sign_test}},
#' \code{\link{rank_test}}, and \code{\link{modified_rank_test}}.
#'
#' @examples
#' \dontrun{
#' library("magrittr")
#' rates_indx <- get_prices_from_tickers("^GSPC",
#'                                       start = as.Date("2019-04-01"),
#'                                       end = as.Date("2020-04-01"),
#'                                       quote = "Close",
#'                                       retclass = "zoo") %>%
#'     get_rates_from_prices(quote = "Close",
#'                           multi_day = TRUE,
#'                           compounding = "continuous")
#' tickers <- c("AMZN", "ZM", "UBER", "NFLX", "SHOP", "FB", "UPWK")
#' get_prices_from_tickers(tickers,
#'                         start = as.Date("2019-04-01"),
#'                         end = as.Date("2020-04-01"),
#'                         quote = "Close",
#'                         retclass = "zoo") %>%
#'     get_rates_from_prices(quote = "Close",
#'                           multi_day = TRUE,
#'                           compounding = "continuous") %>%
#'     apply_market_model(regressor = rates_indx,
#'                        same_regressor_for_all = TRUE,
#'                        market_model = "sim",
#'                        estimation_method = "ols",
#'                        estimation_start = as.Date("2019-04-01"),
#'                        estimation_end = as.Date("2020-03-13")) %>%
#'     wilcoxon_test(event_start = as.Date("2020-03-16"),
#'                   event_end = as.Date("2020-03-20"))
#' }
#' ## The result of the code above is equivalent to:
#' data(securities_returns)
#' wilcoxon_test(list_of_returns = securities_returns,
#'               event_start =  as.Date("2020-03-16"),
#'               event_end = as.Date("2020-03-20"))
#'
#' @export
wilcoxon_test <- function(list_of_returns, event_start, event_end) {
    # event and estimation period could overlap
    # should be applied only for large N
    # check event_start and event_end for class and value validity
    if(!inherits(event_start, "Date")) {
        stop("event_start must be an object of class Date.")
    }
    if(!inherits(event_end, "Date")) {
        stop("event_end must be an object of class Date.")
    }
    if(event_start > event_end) {
        stop("event_start must be earlier than event_end.")
    }


    # zoo objects of abnormal returns
    event_abnormal <- NULL

    for(i in seq_along(list_of_returns)) {

        # check whether each element of list_of_returns is returns
        if(!inherits(list_of_returns[[i]], "returns")) {
            stop("Each element of list_of_rates must have class returns.")
        }

        if(list_of_returns[[i]]$estimation_end >= event_start) {
            message(paste0("For ", as.character(i), "-th company estimation",
                           " period overlaps with event period."))
        }

        company_event_abnormal <- zoo::as.zoo(list_of_returns[[i]]$abnormal[
            zoo::index(list_of_returns[[i]]$abnormal) >= event_start &
                zoo::index(list_of_returns[[i]]$abnormal) <= event_end])

        if(is.null(event_abnormal)) {
            event_abnormal <- company_event_abnormal
        } else {
            event_abnormal <- merge(event_abnormal, company_event_abnormal,
                                    all = TRUE)
        }
    }
    result <- data.frame(date = zoo::index(event_abnormal),
                         weekday = weekdays(zoo::index(event_abnormal)),
                         percentage = rowSums(!is.na(as.matrix(event_abnormal)),
                                              na.rm = TRUE) /
                             ncol(event_abnormal) * 100)

    event_abnormal <- as.matrix(event_abnormal)
    event_abs <- abs(event_abnormal)
    event_abs[event_abs == 0] <- NA
    event_rank <- t(apply(event_abs, 1, rank, na.last = "keep",
                          ties.method = "average"))
    event_rank[event_abnormal < 0] <- 0
    N <- rowSums(!is.na(event_abs))
    N[N == 0] <- NA
    statistics <- rowMeans(event_rank, na.rm = TRUE) * ncol(event_rank)
    statistics[is.nan(statistics)] <- NA
    significance <- rep("", length(statistics))
    significance[statistics >= stats::qsignrank(1 - 0.1, n = N) |
            statistics <= N * (N + 1) / 2 - stats::qsignrank(1 - 0.1 / 2, n = N)] <- "*"
    significance[statistics >= stats::qsignrank(1 - 0.05, n = N) |
            statistics <= N * (N + 1) / 2 - stats::qsignrank(1 - 0.05 / 2, n = N)] <- "**"
    significance[statistics >= stats::qsignrank(1 - 0.01, n = N) |
            statistics <= N * (N + 1) / 2 - stats::qsignrank(1 - 0.01 / 2, n = N)] <- "***"

    result <- cbind(result, data.frame(wlcx_stat = statistics,
                                       wlcx_signif = significance))
    rownames(result) <- NULL
    return(result)
}