Otneim and Tjøstheim (2017) describes a new method for estimating multivariate density functions using the concept of local Gaussian correlations. Otneim and Tjøstheim (2018) expands the idea to the estimation of conditional density functions. This package, written for the R programming language, provides a simple interface for implementing these methods in practical problems.
Let us illustrate the use of this package by looking at the built-in data set of daily closing prices of 4 major European stock indices in the period 1991-1998. We load the data and transform them to daily returns:
data(EuStockMarkets)
x <- apply(EuStockMarkets, 2, function(x) diff(log(x)))
# Remove the days where at least one index did not move at all
x <- x[!apply(x, 1, function(x) any(x == 0)),]When using this package, the first task is always to create an lg-object using the lg_main()-function. This object contains all the estimation parameters that will be used in the estimation step, including the bandwidths. There are three main parameters that one can tune:
See the documentation of the lg_main()-function for further details. We can now construct the lg-object using the default configuration by running
library(lg)
lg_object <- lg_main(x)We can then specify a set of grid points and estimate the probability density function of x using the dlg()-function. We choose a set of grid ponts that go diagonally through R^4, estimate, and plot the result as follows:
grid <- matrix(rep(seq(-.03, .03, length.out = 100), 4), ncol = 4, byrow = FALSE)
density_estimate <- dlg(lg_object = lg_object, grid = grid)
# plot(grid[,1], density_estimate$f_est, type = "l",
# xlab = "Diagonal grid point", ylab = "Estimated density")If we want to calculate conditional density functions, we must take care to notice the order of the columns in our data set. This is because the estimation routine, implemented in the clg()-function, will always assume that the independent variables come first. Looking at the top of our data set:
head(x)
#> DAX SMI CAC FTSE
#> [1,] -0.009326550 0.006178360 -0.012658756 0.006770286
#> [2,] -0.004422175 -0.005880448 -0.018740638 -0.004889587
#> [3,] 0.009003794 0.003271184 -0.005779182 0.009027020
#> [4,] -0.001778217 0.001483372 0.008743353 0.005771847
#> [5,] -0.004676712 -0.008933417 -0.005120160 -0.007230164
#> [6,] 0.012427042 0.006737244 0.011714353 0.008517217we see that DAX comes first. Say that we want to estimate the conditional density of DAX, given that SMI = CAC = FTSE = 0. We do that by running
grid <- matrix(seq(-.03, .03, length.out = 100), ncol = 1) # The grid must be a matrix
condition <- c(0, 0, 0) # Value of dependent variables
cond_dens_est <- clg(lg_object = lg_object,
grid = grid,
condition = condition)
# plot(grid, cond_dens_est$f_est, type = "l",
# xlab = "DAX", ylab = "Estimated conditional density")If we want to estimate the conditional density of CAC and FTSE given DAX and SMI, for example, we must first shuffle the data so that CAC and FTSE come first, and supply the conditional value for DAX and SMI through the vector condition, now having two elements.
The following statistical tests are available:
Let us quickly demonstrate their implementation. For the first test, we generate some data from a bivariate (t)-distribution. They are uncorrelated, but not independent. In order to test for independence, we create an lg-object, and apply the function ind_test(). One may of course change the estimation method, the bandwidths and so in the call to lg_main(), and there are furher options available in ind_test(). For this illustration we use only 20 bootstrap replications, but this must of course be significantly higher in practical applications.
set.seed(1)
x <- mvtnorm::rmvt(n = 100, df = 2)
lg_object <- lg_main(x, est_method = "5par")
test_result <- ind_test(lg_object, n_rep = 20)
test_result$p_value
#> [1] 0The test for serial dependence in a time series X(t) can be performed
in exactly the same way by collecting X(t) and X(t-k)
as columns in the data set, for some lag k. For a test for
serial cross dependence between X(t) and Y(t) one must
collect X(t) and Y(t-k) as columns in the data set,
but also choose either bootstrap_type = "block" or
bootstrap_type = "stationary" in order to correctly
resample under the null hypothesis. We refer to the original article
Lacal and Tjøstheim (2017b) for details on this.
In order to perform the test for financial contagion, one must collect the non-crisis and crisis data in separate data sets, and create separate lg-objects, and then apply the cont_test()-function.
Berentsen, Geir Drage, Tore Selland Kleppe, and Dag Tjøstheim. “Introducing localgauss, an R package for estimating and visualizing local Gaussian correlation.” Journal of Statistical Software 56.1 (2014): 1-18.
Berentsen, Geir Drage, and Dag Tjøstheim. “Recognizing and visualizing departures from independence in bivariate data using local Gaussian correlation.” Statistics and Computing 24.5 (2014): 785-801.
Lacal, Virginia, and Dag Tjøstheim. “Local Gaussian autocorrelation and tests for serial independence.” Journal of Time Series Analysis 38.1 (2017a): 51-71.
Lacal, Virginia, and Dag Tjøstheim. “Estimating and testing nonlinear local dependence between two time series.” Journal of Business & Economic Statistics just-accepted (2017b).
Otneim, Håkon, and Dag Tjøstheim. “The locally gaussian density estimator for multivariate data.” Statistics and Computing 27.6 (2017): 1595-1616.
Otneim, Håkon, and Dag Tjøstheim. “Conditional density estimation using the local Gaussian correlation.” Statistics and Computing 28.2 (2018): 303-321.
Støve, Bård, Dag Tjøstheim, and Karl Ove Hufthammer. “Using local Gaussian correlation in a nonlinear re-examination of financial contagion.” Journal of Empirical Finance 25 (2014): 62-82.
Tjøstheim, Dag, & Hufthammer, Karl Ove (2013). Local Gaussian correlation: a new measure of dependence. Journal of Econometrics, 172(1), 33-48.