Function to generate an AR(1) variance-covariance matrix with parameter rho s.t. \lvert \rho\rvert < 1.

Description

Function to generate an AR(1) variance-covariance matrix with parameter rho s.t. \lvert \rho\rvert < 1.

Usage

ARcov(rho, n)

Arguments

Value

A variance-covariance matrix of size n x n based on the AR(1) process.

Function to generate an ARIMA(1,1,0) variance-covariance matrix for the Litterman method with parameter \rho such that \lvert \rho \rvert < 1.

Description

Function to generate an ARIMA(1,1,0) variance-covariance matrix for the Litterman method with parameter \rho such that \lvert \rho \rvert < 1.

Usage

ARcov_lit(rho, n)

Arguments

Value

A variance-covariance matrix of size (n x n) for the ARIMA(1,1,0) process, used in the Litterman method.

GHG Emissions and Financial Data for IBM

Description

This dataset contains time series data on greenhouse gas (GHG) emissions and financial variables for IBM covering the period from Q3 2005 to Q3 2021. It is designed for use in demonstrating temporal disaggregation and adaptive LASSO methods for estimating high-frequency GHG emissions from low-frequency data.

Usage

Data

Format

A data frame with 68 rows (representing quarters) and 113 variables:

time

Numeric vector representing the time index, spanning from Q3 2005 to Q3 2021

GHG

Numeric vector of annual greenhouse gas emissions for IBM, recorded annually and repeated quarterly

financial_variables

A matrix or data frame of 112 financial variables, extracted from quarterly balance sheets, income statements, and cash flow statements for each company

Source

Original data collected from financial statements and GHG reports of IBM.

High and Low-Frequency Data Generating Processes

Description

This function generates a high-frequency response vector y, following the relationship y = X\beta + \epsilon, where X is a matrix of indicator series and \beta is a potentially sparse coefficient vector. The low-frequency vector Y is generated by aggregating y according to a specified aggregation method.

Usage

TempDisaggDGP(
  n_l,
  n,
  aggRatio = 4,
  p = 1,
  beta = 1,
  sparsity = 1,
  method = "Chow-Lin",
  aggMat = "sum",
  rho = 0,
  mean_X = 0,
  sd_X = 1,
  sd_e = 1,
  simul = FALSE,
  sparse_option = "random",
  setSeed = 42
)

Arguments

Details

The aggregation ratio (aggRatio) determines the ratio between the low and high-frequency series (e.g., aggRatio = 4 for annual-to-quarterly). If the number of observations n exceeds aggRatio \times n_l, the aggregation matrix will include zero columns for the extrapolated values.

The function supports several data generating processes (DGP) for the residuals, including 'Denton', 'Denton-Cholette', 'Chow-Lin', 'Fernandez', and 'Litterman'. These methods differ in how they generate the high-frequency data and residuals, with optional autocorrelation specified by rho.

Value

A list containing the following components:

• y_Gen: Generated high-frequency response series (an n \times 1 matrix).

• Y_Gen: Generated low-frequency response series (an n_l \times 1 matrix).

• X_Gen: Generated high-frequency indicator series (an n \times p matrix).

• Beta_Gen: Generated coefficient vector (a p \times 1 matrix).

• e_Gen: Generated high-frequency residual series (an n \times 1 matrix).

Examples

data <- TempDisaggDGP(n_l=25,n=100,p=10,rho=0.5)
X <- data$X_Gen
Y <- data$Y_Gen

Function to perform Chow-Lin temporal disaggregation from Chow and Lin (1971) and its special case counterpart, Litterman Litterman (1983).

Description

Used in disaggregate to find estimates given the optimal rho parameter.

Usage

chowlin(Y, X, rho, aggMat = "sum", aggRatio = 4, litterman = FALSE)

Arguments

Value

A list containing the following elements:

• y: Estimated high-frequency response series (an n \times 1 matrix).

• betaHat: Estimated coefficient vector (a p \times 1 matrix).

• u_l: Estimated aggregate residual series (an n_l \times 1 matrix).

References

Chow GC, Lin A (1971). “Best Linear Unbiased Interpolation, Distribution, and Extrapolation of Time Series by Related Series.” The review of Economics and Statistics, 53(4), 372–375.

Litterman RB (1983). “A random walk, Markov model for the distribution of time series.” Journal of Business & Economic Statistics, 1(2), 169–173.

Likelihood function for Chow-Lin or Litterman temporal disaggregation.

Description

This function computes the likelihood function used in temporal disaggregation to find the optimal \rho parameter. It is used in conjunction with disaggregate to estimate the autocorrelation coefficient \rho.

Usage

chowlin_likelihood(Y, X, vcov)

Arguments

Value

    The log-likelihood value for the given parameters.

Temporal Disaggregation Methods

Description

This function contains the traditional standard-dimensional temporal disaggregation methods proposed by Denton (1971), Dagum and Cholette (2006), Chow and Lin (1971), Fernández (1981) and Litterman (1983), and the high-dimensional methods of Mosley et al. (2022).

Usage

disaggregate(
  Y,
  X = matrix(data = rep(1, times = (nrow(Y) * aggRatio)), nrow = (nrow(Y) * aggRatio)),
  aggMat = "sum",
  aggRatio = 4,
  method = "Chow-Lin",
  Denton = "additive-first-diff"
)

Arguments

Details

Takes in a n_l \times 1 low-frequency series to be disaggregated Y and a n \times p high-frequency matrix of p indicator series X. If n > n_l \times aggRatio where aggRatio is the aggregation ratio (e.g. aggRatio = 4 if annual-to-quarterly disagg, or aggRatio = 3 if quarterly-to-monthly disagg) then extrapolation is done to extrapolate up to n.

Value

y_Est: Estimated high-frequency response series (output is an n \times 1 matrix).

beta_Est: Estimated coefficient vector (output is a p \times 1 matrix).

rho_Est: Estimated residual AR(1) autocorrelation parameter.

ul_Est: Estimated aggregate residual series (output is an n_l \times 1 matrix).

References

Chow GC, Lin A (1971). “Best Linear Unbiased Interpolation, Distribution, and Extrapolation of Time Series by Related Series.” The review of Economics and Statistics, 53(4), 372–375.

Dagum EB, Cholette PA (2006). Benchmarking, Temporal Distribution, and Reconciliation Methods for Time Series. Springer.

Denton FT (1971). “Adjustment of monthly or quarterly series to annual totals: an approach based on quadratic minimization.” Journal of the american statistical association, 66(333), 99–102.

Fernández RB (1981). “A methodological note on the estimation of time series.” The Review of Economics and Statistics, 63(3), 471–476.

Litterman RB (1983). “A random walk, Markov model for the distribution of time series.” Journal of Business & Economic Statistics, 1(2), 169–173.

Mosley L, Eckley IA, Gibberd A (2022). “Sparse Temporal Disaggregation.” Journal of the Royal Statistical Society Series A: Statistics in Society, 185(4), 2203-2233. ISSN 0964-1998, doi:10.1111/rssa.12952, https://academic.oup.com/jrsssa/article-pdf/185/4/2203/49420183/jrsssa_185_4_2203.pdf.

Examples

data <- TempDisaggDGP(n_l=25,n=100,p=10,rho=0.5)
X <- data$X_Gen
Y <- data$Y_Gen
fit_chowlin <- disaggregate(Y=Y,X=X,method='Chow-Lin')
y_hat = fit_chowlin$y_Est

High-dimensional BIC score

Description

This function calculates a BIC score that performs better than the ordinary BIC in high-dimensional scenarios. It uses the variance estimator given in Yu and Bien (2019).

Usage

hdBIC(X, Y, covariance, beta)

Arguments

Value

      The BIC score for model comparison.

References

Yu G, Bien J (2019). “Estimating the error variance in a high-dimensional linear model.” Biometrika, 106(3), 533–546.

Index of support for LARS algorithm in high-dimensional settings

Description

This function returns the index where the support of beta coefficients exceeds n_l/2, preventing the BIC from becoming erratic in high-dimensional scenarios.

Usage

k.index(coef_matrix, n_l)

Arguments

Value

       The index where the support of beta exceeds \eqn{n_l/2}, or the number of rows of the matrix if no such index is found.

Refit LASSO estimate into GLS

Description

This function reduces the bias in LASSO estimates by re-fitting the active set of coefficients back into GLS (Generalized Least Squares).

Usage

refit(X, Y, beta)

Arguments

Value

 A debiased estimate of the beta coefficients (a \eqn{p \times 1} vector).

Simulation Diagnostics

Description

This function provides diagnostics for evaluating the accuracy of simulated data. Specifically, it computes the Mean Squared Error (MSE) between the true and estimated response vectors, and optionally, the sign recovery percentage of the coefficient vector.

Usage

simulDiagnosis(data_Hat, data_True, sgn = FALSE)

Arguments

Details

The function takes in the generated high-frequency data (data_True) and the estimated high-frequency data (data_Hat), and returns the Mean Squared Error (MSE) between the true and estimated values of the response vector. If the sgn parameter is set to TRUE, the function additionally computes the percentage of correctly recovered signs of the coefficient vector.

Value

If sgn is FALSE, the function returns the Mean Squared Error (MSE) between the true and estimated response vectors. If sgn is TRUE, the function returns a list containing both the MSE and the sign recovery percentage.

Examples

true_data <- list(y_Gen = c(1, 2, 3), Beta_Gen = c(1, -1, 0))
est_data <- list(y_Est = c(1.1, 1.9, 2.8), beta_Est = c(1, 1, 0))
mse <- simulDiagnosis(est_data, true_data)
results <- simulDiagnosis(est_data, true_data, sgn = TRUE)

Sparse Temporal Disaggregation

Description

This function performs sparse temporal disaggregation as described in Mosley et al. (2022). It estimates the high-frequency response series using LARS (Least Angle Regression) and applies either a LASSO or adaptive LASSO penalty.

Usage

sptd(Y, X, rho, aggMat = "sum", aggRatio = 4, adaptive = FALSE)

Arguments

Value

A list containing:

• y: Estimated high-frequency response series (n \times 1 matrix).

• betaHat: Estimated coefficient vector (p \times 1 matrix).

• u_l: Estimated aggregate residual series (n_l \times 1 matrix).

References

Mosley L, Eckley IA, Gibberd A (2022). “Sparse Temporal Disaggregation.” Journal of the Royal Statistical Society Series A: Statistics in Society, 185(4), 2203-2233. ISSN 0964-1998, doi:10.1111/rssa.12952, https://academic.oup.com/jrsssa/article-pdf/185/4/2203/49420183/jrsssa_185_4_2203.pdf.

BIC Score for Sparse Temporal Disaggregation

Description

This function calculates the BIC score for sparse temporal disaggregation, as described in Mosley et al. (2022). It uses the LARS algorithm to find the optimal beta coefficients and refits the models to compute BIC scores.

Usage

sptd_BIC(Y, X, vcov)

Arguments

Value

    The minimum BIC score from the refitted models.