Helper function to check parameter vectors for positive signed integer values

Description

Helper function to check parameter vectors for positive signed integer values

Usage

checkBoolean(parameter, name)

Helper function to check parameter matrices

Description

Helper function to check parameter matrices

Usage

checkParameterMatrix(parameter, name, no_of_rows, no_of_cols)

Helper function to check positive double function parameter

Description

Helper function to check positive double function parameter

Usage

checkPositiveDouble(parameter, name)

Helper function to check unsigned integer function parameter

Description

Helper function to check unsigned integer function parameter

Usage

checkPositiveSignedInteger(parameter, name, bit_size = 64)

Helper function to check parameter vectors for positive signed integer values

Description

Helper function to check parameter vectors for positive signed integer values

Usage

checkPositiveSignedParameterVector(parameter, name, size)

Cross-validate SDFM Hyper-Parameters

Description

This function uses time series cross-validation (Hyndman and Athanasopoulos 2018) in combination with random hyper-parameter search (Bergstra and Bengio 2012) to validate the hyper-parameters of a sparse dynamic factor model as described in Franjic D, Schweikert K (2024). “Nowcasting Macroeconomic Variables with a Sparse Mixed Frequency Dynamic Factor Model.” Available at SSRN 4733872.

Usage

crossVal(
  data,
  variable_of_interest,
  fcast_horizon,
  delay,
  frequency,
  no_of_factors,
  seed,
  min_ridge_penalty,
  max_ridge_penalty,
  cv_repetitions,
  cv_size,
  lasso_penalty_type,
  min_max_penalty,
  max_factor_lag_order = 10,
  lag_estim_criterion = "BIC",
  decorr_errors = TRUE,
  max_iterations = 1000,
  weights = NULL,
  comp_null = 1e-15,
  spca_conv_crit = 1e-04,
  parallel = FALSE,
  no_of_cores = 1,
  max_ar_lag_order = 5,
  max_predictor_lag_order = 5,
  jitter = 1e-08,
  svd_method = "precise",
  verbose = TRUE
)

Arguments

Details

fcast_horizon should be set to the target prediction horizon, as hyper-parameters can differ substantially between different horizons. For nowcasting, use fcast_horizon = 0. For backcasting, fcast_horizon can be set to a negative number indicating the step-back backcasting horizon.

Internally, candidates of the hyper-parameters are drawn randomly. However, a regular dense DFM will always be considered by default. The ridge penalty is drawn as \exp(u), where u is uniformly distributed between min_ridge_penalty and max_ridge_penalty. If lasso_penalty_type = "selected", the lasso penalty is drawn as a random vector \bm{v}, where each entry is uniformly distributed. If lasso_penalty_type = "steps", the lasso penalty is drawn as a random value v that is uniformly distributed. If lasso_penalty_type = "penalty", the lasso penalty is drawn as a random vector \exp(\bm{v}), where each entry of \bm{v} is uniformly distributed. In all three cases, the upper and lower bounds of the uniform distributions governing the lasso penalties are given by the first and second entry of min_max_penalty, respectively.

For medium to large data sets in combination with a medium to large cv_size, it can be beneficial to set parallel = TRUE. This will enable parallelisation via the doParallel, doSNOW, foreach, and parallel packages in R. In this case, no_of_cores should be set to the number of physical cores of the user's machine. It is not advisable to use the number of logical cores, as this can considerably deteriorate performance.

This function serves as a direct wrapper to nowcast. For more information on the additional function parameters, see the corresponding help page.

Value

An object of class SDFMcrossVal with main components:

CV

A list with components `CV Results` (matrix of all cross-validation errors and corresponding hyper-parameter values) and `Min. CV` (row of ⁠CV Results⁠ with the minimum cross-validation error).

BIC

A list with components ⁠BIC Results⁠ (matrix of all BIC values and corresponding hyper-parameter values) and ⁠Min. BIC⁠ (row of ⁠BIC Results⁠ with the minimum BIC).

Author(s)

Domenic Franjic

References

Bergstra J, Bengio Y (2012). “Random search for hyper-parameter optimization.” Journal of Machine Learning Research, 13(2).

Hyndman RJ, Athanasopoulos G (2018). Forecasting: principles and practice, 3 edition. OTexts Melbourne.

Franjic D, Schweikert K (2024). “Nowcasting Macroeconomic Variables with a Sparse Mixed Frequency Dynamic Factor Model.” Available at SSRN 4733872.

See Also

sparsePCA: Routine for fitting estimating a sparse factor loading matrix.

kalmanFilterSmoother: Routine for filtering and smoothing latent factors.

twoStepSDFM: Two-step estimation routine for a sparse dynamic factor model.

twoStepDenseDFM: Two-step estimation routine for a dense dynamic factor model.

Examples

data(mixed_freq_factor_model)
no_of_vars <- dim(mixed_freq_factor_model$data)[2]
no_of_factors <- dim(mixed_freq_factor_model$factors)[2]
cv_results <- crossVal(data = mixed_freq_factor_model$data, variable_of_interest = 1, 
                       fcast_horizon = 0, delay = mixed_freq_factor_model$delay, 
                       frequency = mixed_freq_factor_model$frequency,
                       no_of_factors = no_of_factors, seed = 25032026,
                       min_ridge_penalty = 1e-5, max_ridge_penalty = 10, 
                       cv_repetitions = 1, cv_size = 50, lasso_penalty_type = "selected",
                       min_max_penalty = c(5, 45), verbose = FALSE)
print(cv_results)
cv_plots <- plot(cv_results)        
cv_plots$`CV Results`
cv_plots$`BIC Results` 

Example factor model dataset

Description

This is a simulated factor model dataset for demonstration and testing with TwoStepSDFM.

Usage

factor_model

Format

A SimulData containing the following elements:

data

If starting_date is provided, a zoo object, else, a (no_of_vars \times no_of_obs) numeric matrix holding the simulated data.

factors

If starting_date is provided, a zoo object, else a (no_of_factors \times no_of_obs) numeric matrix holding the simulated latent factors.

trans_var_coeff

Numeric (no_of_factors \times (no_of_factors * factor_lag_order)) factor VAR coefficient matrix.

loading_matrix

Numeric factor loading matrix.

meas_error

If starting_date is provided, a zoo object, else a (no_of_vars \times no_of_obs) numeric matrix holding the fundamental measurement errors.

meas_error_var_cov

Numeric measurement error variance-covariance matrix.

trans_error_var_cov

Numeric transition error variance-covariance matrix.

frequency

Integer vector of variable frequencies.

delay

Integer vector of variable delays, measured as the number of months since the latest available observation.

Source

Generated via simFM(). For details see factor_model$call.

Internal forecasting wrapper function

Description

This function is for internal use only and may change in future releases without notice. Users should use nowcast() instead for a stable and supported interface. Helper function to check parameter vectors for positive signed integer values

Usage

forecastWrapper(
  target_variables,
  quarterly_predictors,
  factors,
  target_variable_delay,
  quarterly_delay,
  lag_estim_criterion,
  max_fcast_horizon,
  max_ar_lag_order,
  max_predictor_lag_order,
  jitter
)

Univariate Representation of the Multivariate Kalman Filter and Smoother

Description

Filter and smooth the latent states/factors of a linear Gaussian state-space model, with measurement equation

\bm{x}_t = \bm{\Lambda} \bm{f}_{t} + \bm{\xi}_t,\quad \bm{\xi}_t \sim \mathcal{N}(\bm{0}, \bm{\Sigma}_{\xi}),

and transition equation

\bm{f}_t = \sum_{p=0}^P\bm{\Phi}_p \bm{f}_{t-p} + \bm{\epsilon}_t,\quad \bm{\epsilon}_t \sim \mathcal{N}(\bm{0}, \bm{\Sigma}_{f}).

for t = 1, ..., T. For filtering and smoothing, the univariate representation of the multivariate Kalman Filter and Smoother is implemented according to Koopman SJ, Durbin J (2000). “Fast filtering and smoothing for multivariate state space models.” Journal of Time Series Analysis, 21(3), 281–296..

Usage

kalmanFilterSmoother(
  data,
  delay,
  no_of_factors,
  loading_matrix,
  meas_error_var_cov,
  trans_error_var_cov,
  trans_var_coeff,
  factor_lag_order,
  fcast_horizon = 0,
  decorr_errors = TRUE,
  comp_null = 1e-15,
  parallel = FALSE,
  jitter = 1e-08
)

Arguments

Details

To implement the univariate representation of the Kalman Filter and Smoother, the measurement error term has to be cross-sectionally uncorrelated. If meas_error_var_cov is not diagonal, one should set decorr_errors = TRUE so that the data can be decorrelated internally prior to filtering and smoothing.

When decorrelating, the function first adds jitter to the diagonal elements of meas_error_var_cov and then tries to compute the Cholesky factor via Eigen's standard LLT decomposition (Guennebaud et al. 2010). If the initial decorrelation fails, it silently switches to Eigen's more robust, but slower, LDLT decomposition with pivoting (Guennebaud et al. 2010). If this also fails, it is likely that meas_error_var_cov is not well-behaved. The analysis should be repeated with a larger jitter or a more robust variance-covariance matrix (estimator). The success of the internal Cholesky decomposition is reported by llt_success_code.

Value

An object of class KFSFit with components:

data

Original data matrix.

smoothed_factors

Object containing the smoothed factor estimates. The object inherits its class from data: If data is provided as zoo, smoothed_factors will be a zoo object. If data is provided as matrix, smoothed_factors will be a (no_of_factors \times no_of_obs) matrix.

smoothed_state_variance

(no_of_factors \times (no_of_factors * no_of_obs)) matrix, where each (no_of_factors \times no_of_factors) block represents the smoother uncertainty at time point t

factor_var_lag_order

Integer order of the VAR process in the state equation.

llt_success_code

Integer indicating the status of the Cholesky factorization: 0 = LLT succeeded, -1 = LLT failed but LDLT succeeded, -2 = both failed and errors are treated as uncorrelated.

Author(s)

Domenic Franjic

References

Koopman SJ, Durbin J (2000). “Fast filtering and smoothing for multivariate state space models.” Journal of Time Series Analysis, 21(3), 281–296.

Guennebaud G, Jacob B, others (2010). “Eigen.” https://libeigen.gitlab.io.

Examples

data(factor_model)
no_of_factors <- dim(factor_model$factors)[2]
factor_lag_order <- dim(factor_model$trans_var_coeff)[2] / no_of_factors
filter_fit <- kalmanFilterSmoother(data = factor_model$data, delay = factor_model$delay, 
                                   no_of_factors = no_of_factors, 
                                   loading_matrix = factor_model$loading_matrix, 
                                   meas_error_var_cov = factor_model$meas_error_var_cov, 
                                   trans_error_var_cov = factor_model$trans_error_var_cov,
                                   trans_var_coeff = factor_model$trans_var_coeff, 
                                   factor_lag_order = factor_lag_order, 
                                   fcast_horizon = 5, decorr_errors = TRUE,  
                                   comp_null = 1e-15, parallel = FALSE,  jitter = 1e-8)
print(filter_fit)
filter_plots <- plot(filter_fit)
filter_plots$`Factor Time Series Plots`

Helper function to check positive double function parameter

Description

Helper function to check positive double function parameter

Usage

makeRaggedEdges(data, delay)

Mixed-frequency factor model dataset

Description

This dataset contains simulated mixed-frequency factor model data for examples in TwoStepSDFM.

Usage

mixed_freq_factor_model

Format

A SimulData containing the following elements:

data

If starting_date is provided, a zoo object, else, a (no_of_vars \times no_of_obs) numeric matrix holding the simulated data.

factors

If starting_date is provided, a zoo object, else a (no_of_factors \times no_of_obs) numeric matrix holding the simulated latent factors.

trans_var_coeff

Numeric (no_of_factors \times (no_of_factors * factor_lag_order)) factor VAR coefficient matrix.

loading_matrix

Numeric factor loading matrix.

meas_error

If starting_date is provided, a zoo object, else a (no_of_vars \times no_of_obs) numeric matrix holding the fundamental measurement errors.

meas_error_var_cov

Numeric measurement error variance-covariance matrix.

trans_error_var_cov

Numeric transition error variance-covariance matrix.

frequency

Integer vector of variable frequencies.

delay

Integer vector of variable delays, measured as the number of months since the latest available observation.

Source

Generated via Generated via simFM(). For details see mixed_freq_factor_model$call.

Estimate the number of Factors

Description

Estimate the number of factors of a linear Gaussian latent factor model using via an eigenvalue slope test according to Onatski A (2009). “Testing hypotheses about the number of factors in large factor models.” Econometrica, 77(5), 1447–1479..

Usage

noOfFactors(
  data,
  min_no_factors = 1,
  max_no_factors = 7,
  confidence_threshold = 0.05
)

Arguments

Details

The procedure splits the data matrix along the time dimension into two equally sized (no_of_vars \times cut_off) sub-matrices \bm{X}_{1/2} and \bm{X}_{2/2}. It then proceeds to build \tilde{\bm{X}} := \bm{X}_{1/2} + i\bm{X}_{2/2}, where i=\sqrt{-1}. We then compute eigenvalues of the Gram matrix \tilde{\bm{X}} \tilde{\bm{X}}^{\dagger}, where \tilde{\bm{X}}^{\dagger} represents the adjoint. Finally, a test based on the computed eigenvalues is performed. This test is an iterative testing procedure, starting by testing the null that the true number of factors is min_no_factors. If the test is rejected by comparison of the p-value against confidence_threshold, we test whether the true number of factors is min_no_factors + 1 until we can no longer reject at confidence_threshold or max_no_factors is reached.

As the distribution of the eigenvalues under the null is nonstandard (Onatski 2009), simulated critical values are used. They are retrieved from Onatski A (2009). “Alexey Onatskiy – An old link to some of my papers.” Formerly available at https://www.econ.cam.ac.uk/people/faculty/ao319/papers; Last accessed.. As the range of the simulated critical values is limited, the minimum and maximum number of potential factors is limited such that max_no_factors should be no more than min_no_factors + 17. However, it is recommended to operate well below this maximum as the test size decreases with max_no_factors - min_no_factors.

Value

An object of class NoOfFactorsFit with components:

no_of_factors

Integer estimated number of factors.

p_value

Numeric p-value of the final test.

confidence_threshold

Numeric significance level used.

statistic

Numeric test statistic value of the last test.

eigen_values

Numeric vector of eigenvectors of the complex data Gram matrix.

Author(s)

Domenic Franjic

References

Onatski A (2009). “Testing hypotheses about the number of factors in large factor models.” Econometrica, 77(5), 1447–1479.

Onatski A (2009). “Alexey Onatskiy – An old link to some of my papers.” Formerly available at https://www.econ.cam.ac.uk/people/faculty/ao319/papers; Last accessed.

Examples

data(factor_model)
no_of_factors_estim <- noOfFactors(data = factor_model$data, min_no_factors = 1, 
                                   max_no_factors = 5, confidence_threshold = 0.05)
print(no_of_factors_estim)
factor_estim_plots <- plot(no_of_factors_estim)
factor_estim_plots$`Eigen Value Plot`

Predict Mixed-Frequency Data via Dynamic Factor Models

Description

Backcast, nowcast, and forecast quarterly target variables via a sparse/dense DFM using additional monthly data with ragged edges. Forecasts are produced using all quarterly targets and a quarterly representation of latent monthly factors (Mariano and Murasawa 2003). Final predictions are computed via equally weighted forecast averaging of ARDL models (Marcellino and Schumacher 2010) for each of the targets and quarterfied factors.

Usage

nowcast(
  data,
  variables_of_interest,
  max_fcast_horizon,
  delay,
  selected,
  frequency,
  no_of_factors,
  sparse = TRUE,
  max_factor_lag_order = 10,
  lag_estim_criterion = "BIC",
  decorr_errors = TRUE,
  ridge_penalty = 1e-06,
  lasso_penalty = NULL,
  max_iterations = 1000,
  max_no_steps = NULL,
  weights = NULL,
  comp_null = 1e-15,
  spca_conv_crit = 1e-04,
  parallel = FALSE,
  max_ar_lag_order = 5,
  max_predictor_lag_order = 5,
  jitter = 1e-08,
  svd_method = "precise"
)

Arguments

Details

This function serves as a prediction wrapper for the twoStepDenseDFM and twoStepSDFM functions. data should be a mixed-frequency data set. Currently, only monthly and quarterly data are supported. With respect to the quarterly data, the function expects the realization of the quarterly observations to occur in the last month of the quarter. Indicate quarterly and monthly variables via frequency by setting the corresponding element of frequency to 4 for quarterly and to 12 for monthly data.

This function is only able to compute predictions for quarterly variables. To impute the ragged edges of the monthly observations, and potentially compute additional predictions for the monthly variables, call predict on the SDFMFit object returned by twoStepDenseDFM / twoStepSDFM (see predict.SDFMFit).

max_fcast_horizon sets the maximum number of forecasts predicted starting from the final observation of the data set. For each target, the number of backcasts and whether or not a nowcast should be computed is determined internally. This is done in such a way that every missing quarterly observation of the targets is predicted.

max_ar_lag_order governs the maximum number of lags of the current target used to predict said target in each ARDL model. max_predictor_lag_order governs the maximum number of lags of each additional quarterly predictor, including other potential targets and the aggregated factors, used to predict any given target in each ARDL model. The actual number of lags is internally estimated using the BIC. Setting max_ar_lag_order = 0 disables the use of target lags in its own prediction function.

sparse toggles between a sparse DFM and a dense DFM. If sparse = FALSE, all SPCA stopping criteria and other parameters passed to the sparse estimation routine are ignored (for details on these parameters see twoStepDenseDFM). Parameters governing the Kalman Filter and Smoother are passed directly to twoStepDenseDFM / twoStepSDFM. For details see the corresponding help pages.

Value

The nowcast function returns named list containing the following objects:

Forecasts

Numeric matrix of the target variables and their respective backcasts, nowcasts, and/or forecasts.

SDFM Fit

An SDFMFit object holding the estimates of the model parameters and the latent factors (see twoStepSDFM or twoStepDenseDFM).

Author(s)

Domenic Franjic

References

Mariano RS, Murasawa Y (2003). “A new coincident index of business cycles based on monthly and quarterly series.” Journal of Applied Econometrics, 18(4), 427-443. doi:10.1002/jae.695.

Marcellino M, Schumacher C (2010). “Factor MIDAS for nowcasting and forecasting with ragged-edge data: A model comparison for German GDP.” Oxford Bulletin of Economics and Statistics, 72(4), 518–550.

Franjic D, Schweikert K (2024). “Nowcasting Macroeconomic Variables with a Sparse Mixed Frequency Dynamic Factor Model.” Available at SSRN 4733872.

See Also

sparsePCA: Routine for fitting estimating a sparse factor loading matrix.

kalmanFilterSmoother: Routine for filtering and smoothing latent factors.

twoStepSDFM: Two-step estimation routine for a sparse dynamic factor model.

twoStepDenseDFM: Two-step estimation routine for a dense dynamic factor model.

Examples

data(mixed_freq_factor_model)
no_of_vars <- dim(mixed_freq_factor_model$data)[2]
no_of_factors <- dim(mixed_freq_factor_model$factors)[2]
sparse_nowcast <- nowcast(data = mixed_freq_factor_model$data, variables_of_interest = c(1, 2),
                          max_fcast_horizon = 4, delay = mixed_freq_factor_model$delay,
                          selected = rep(floor(0.5 * no_of_vars), no_of_factors), 
                          frequency = mixed_freq_factor_model$frequency, 
                          no_of_factors = no_of_factors, sparse = TRUE)
print(sparse_nowcast)
dense_nowcast <- nowcast(data = mixed_freq_factor_model$data, variables_of_interest = c(1, 2),
                         max_fcast_horizon = 4, delay = mixed_freq_factor_model$delay,
                         selected = NULL, frequency = mixed_freq_factor_model$frequency, 
                         no_of_factors = no_of_factors, sparse = FALSE)
sparse_plots <- plot(sparse_nowcast)
sparse_plots$`Single Pred. Fcast Density Plots Series 1`

Helper function to wrap the nowcasting routine

Description

Helper function to wrap the nowcasting routine

Usage

nowcastSpecificationHelper(
  cv_repetitions,
  no_of_factors,
  no_of_variables,
  no_of_observations,
  no_of_mtly_variables,
  lasso_penalty_type,
  data,
  variable_of_interest,
  fcast_horizon,
  delay,
  candidates,
  frequency,
  max_factor_lag_order,
  decorr_errors,
  lag_estim_criterion,
  max_iterations,
  comp_null,
  spca_conv_crit,
  max_ar_lag_order,
  max_predictor_lag_order,
  jitter,
  svd_method,
  weights
)

Generic plotting function for KFSFit S3 objects

Description

Create diagnostic plots for a KFSFit object.

Usage

## S3 method for class 'KFSFit'
plot(x, axis_text_size = 20, legend_title_text_size = 20, ...)

Arguments

Value

A named list of patchwork/ggplot objects:

⁠Factor Time Series Plots⁠

patchwork/ggplot object graphing the estimated factors over time with 95% confidence bands based on the smoother uncertainty of the Kalman Filter and Smoother.

Author(s)

Domenic Franjic

Generic plotting function for NoOfFactorsFit S3 objects

Description

Create diagnostic plots for an NoOfFactorsFit object,

Usage

## S3 method for class 'NoOfFactorsFit'
plot(x, axis_text_size = 20, legend_title_text_size = 20, ...)

Arguments

Value

A named list of plot objects:

⁠Eigen Value Plot⁠

ggplot object showing a bar plot of the eigenvalues of the complex data Gram matrix.

Author(s)

Domenic Franjic

Generic plotting function for SDFMFit S3 objects

Description

Generic plotting function for SDFMFit S3 objects

Usage

## S3 method for class 'SDFMFit'
plot(x, axis_text_size = 20, legend_title_text_size = 20, ...)

Arguments

Value

A named list of plot objects:

⁠Factor Time Series Plots⁠

patchwork/ggplot object graphing the estimated factors over time with 95% confidence bands based on the smoother uncertainty of the Kalman Filter and Smoother.

⁠Loading Matrix Heatmap⁠

ggplot object showing a heatmap of the estimated factor loadings. Zeros are highlighted in black.

⁠Meas. Error Var.-Cov. Matrix Heatmap⁠

ggplot object showing a heatmap of the measurement error variance-covariance matrix.

⁠Eigenvalue Plot⁠

ggplot object showing a bar plot of the eigenvalues of the measurement error variance–covariance matrix.

Author(s)

Domenic Franjic

Generic plotting function for SDFMcrossVal S3 objects

Description

Generic plotting function for SDFMcrossVal S3 objects

Usage

## S3 method for class 'SDFMcrossVal'
plot(x, axis_text_size = 20, legend_title_text_size = 20, ...)

Arguments

Value

A named list of ggplot objects:

⁠CV Results⁠

ggplot object of the cross-validation error against the log Ridge penalty. The overall sparsity level of the loading matrix induced by the lasso penalty is indicated by point shapes and colours.

⁠BIC Results⁠

ggplot object of the BIC against the log Ridge penalty. The overall sparsity level of the loading matrix induced by the lasso penalty is indicated by point shapes and colours.

Author(s)

Domenic Franjic

Generic plotting function for SDFMnowcast S3 objects

Description

Generic plotting function for SDFMnowcast S3 objects

Usage

## S3 method for class 'SDFMnowcast'
plot(x, axis_text_size = 20, ...)

Arguments

Value

A named list storing of ggplot objects:

⁠Single Pred. Fcast Density Plots x⁠

patchwork / ggplot objects graphing the distribution of forecasts generated by the predictors for each prediction (backcasts, nowcasts, forecasts) for each target, respectively. Altogether, there will be as many such objects as there are targets, with x replaced by the column name of the target.

Author(s)

Domenic Franjic

Generic plotting function for SPCAFit S3 objects

Description

Create diagnostic plots for an SPCAFit object.

Usage

## S3 method for class 'SPCAFit'
plot(x, axis_text_size = 20, legend_title_text_size = 20, ...)

Arguments

Value

A named list of plot objects:

⁠Factor Time Series Plots⁠

patchwork/ggplot object showing the estimated factors over time.

⁠Loading Matrix Heatmap⁠

ggplot object showing a heatmap of the estimated factor loadings. Zeros are highlighted in black.

⁠Meas. Error Var.-Cov. Matrix Heatmap⁠

ggplot object showing a heatmap of the measurement error variance–covariance matrix.

⁠Eigenvalue Plot⁠

ggplot object showing a bar plot of the eigenvalues of the measurement error variance–covariance matrix.

Author(s)

Domenic Franjic

Generic plotting function for SimulData S3 objects

Description

Create diagnostic plots for an SimulData object.

Usage

## S3 method for class 'SimulData'
plot(x, axis_text_size = 20, legend_title_text_size = 20, ...)

Arguments

Value

A named list of plot objects:

⁠Factor Time Series Plots⁠

patchwork/ggplot object showing the simulated factors over time.

⁠Loading Matrix Heatmap⁠

ggplot object showing a heatmap of the simulated factor loadings. Zeros are highlighted in black.

⁠Meas. Error Var.-Cov. Matrix Heatmap⁠

ggplot object showing a heatmap of the measurement error variance-covariance matrix.

⁠Meas. Error Var.-Cov. Eigenvalue Plot⁠

ggplotobject showing a bar plot of the eigenvalues of the measurement error variance-covariance matrix.

⁠Data Var.-Cov. Matrix Heatmap⁠

ggplot object showing a heatmap of the data variance-covariance matrix.

⁠Data Var.-Cov. Eigenvalue Plot⁠

ggplot object showing a bar plot of the eigenvalues of the data variance-covariance matrix.

Helper for plotting factor time series

Description

Helper for plotting factor time series

Usage

plotFactorEstimates(
  factors,
  smoothed_state_variance,
  no_of_factors,
  axis_text_size
)

Helper for plotting loading matrix heat maps

Description

Helper for plotting loading matrix heat maps

Usage

plotLoadingHeatMap(
  loading_matrix_estim,
  series_names,
  no_of_factors,
  axis_text_size,
  legend_title_text_size
)

Helper for plotting measurement error var.cov. eigenvalues

Description

Helper for plotting measurement error var.cov. eigenvalues

Usage

plotMeasVarCovEigenvalues(
  eigen_values,
  no_of_factors,
  axis_text_size,
  legend_title_text_size
)

Helper for plotting measurement error var.cov. heatmap

Description

Helper for plotting measurement error var.cov. heatmap

Usage

plotMeasVarCovHeatmap(
  measurement_error_var_cov_df,
  series_names,
  axis_text_size,
  legend_title_text_size
)

Generic plotting function for SDFMFit S3 objects

Description

Predict all missing observations due to ragged edges in the data set plus horizon steps ahead.

Usage

## S3 method for class 'SDFMFit'
predict(object, horizon = 0, ...)

Arguments

Value

A named list of plot objects:

data

Object containing the original data. The object inherits its class from object$data: If data is provided as zoo, data will be a zoo object. If data is provided as matrix, data will be a (no_of_factors\timesno_of_obs) matrix.

data_missing_pred

Object containing only the predictions of all missing observations plus the forecasts. Inherits its class from object$data as above.

data_imputed

Object containing the observed data, predictions of all missing observations plus the forecasts. Inherits its class from object$data as above.

Author(s)

Domenic Franjic

Examples

data(factor_model)
no_of_vars <- dim(factor_model$data)[2]
no_of_factors <- dim(factor_model$factors)[2]
sdfm_fit <- twoStepSDFM(data = factor_model$data, delay = factor_model$delay,
                        selected = rep(floor(0.5 * no_of_vars), no_of_factors),
                        no_of_factors = no_of_factors, fcast_horizon = 5)
dfm_fit <- twoStepDenseDFM(data = factor_model$data, delay = factor_model$delay, 
                           no_of_factors = no_of_factors, fcast_horizon = 5)
predict(sdfm_fit, horizon = 5)
predict(dfm_fit, horizon = 5)

Generic printing function for KFSFit S3 objects

Description

Print a compact summary of a KFSFit object.

Usage

## S3 method for class 'KFSFit'
print(x, ...)

Arguments

Value

No return value, called for side effects.

Author(s)

Domenic Franjic

Generic printing function for NoOfFactorsFit S3 objects

Description

Print a compact summary of an NoOfFactorsFit object.

Usage

## S3 method for class 'NoOfFactorsFit'
print(x, ...)

Arguments

Value

No return value; Prints a summary to the console.

Author(s)

Domenic Franjic

Generic printing function for SDFMFit S3 objects

Description

Print a compact summary of an SDFMFit object.

Usage

## S3 method for class 'SDFMFit'
print(x, ...)

Arguments

Value

No return value; Prints a summary to the console.

Author(s)

Domenic Franjic

Generic print function for SDFMcrossVal S3 objects

Description

Generic print function for SDFMcrossVal S3 objects

Usage

## S3 method for class 'SDFMcrossVal'
print(x, ...)

Arguments

Value

No return value; Prints a summary to the console.

Author(s)

Domenic Franjic

Generic print function for SDFMnowcast S3 objects

Description

Generic print function for SDFMnowcast S3 objects

Usage

## S3 method for class 'SDFMnowcast'
print(x, ...)

Arguments

Value

No return value; Prints a summary to the console.

Author(s)

Domenic Franjic

Generic printing function for SPCAFit S3 objects

Description

Print a compact summary of an SPCAFit object.

Usage

## S3 method for class 'SPCAFit'
print(x, ...)

Arguments

Value

No return value; Prints a summary to the console.

Author(s)

Domenic Franjic

Generic printing function for SimulData S3 objects

Description

Print a compact summary of an SimulData object.

Usage

## S3 method for class 'SimulData'
print(x, ...)

Arguments

Value

No return value; Prints a summary to the console.

Author(s)

Domenic Franjic

Simulate Dynamic Factor Models.

Description

Simulate data from a linear Gaussian state-space model (latent factor model), with measurement equation

\bm{x}_t = \bm{\Lambda} \bm{f}_{t} + \bm{\xi}_t,\quad \bm{\xi}_t \sim \mathcal{N}(\bm{\mu}, \bm{\Sigma}_{\xi}),

and transition equation

\bm{f}_t = \sum_{p=1}^P\bm{\Phi}_p \bm{f}_{t-p} + \bm{\epsilon}_t,\quad \bm{\epsilon}_t \sim \mathcal{N}(\bm{0}, \bm{\Sigma}_{f}).

for t = 1, ..., T, as is used in, among others, Franjic D, Schweikert K (2024). “Nowcasting Macroeconomic Variables with a Sparse Mixed Frequency Dynamic Factor Model.” Available at SSRN 4733872..

Usage

simFM(
  no_of_obs,
  no_of_vars,
  no_of_factors,
  loading_matrix,
  meas_error_mean,
  meas_error_var_cov,
  trans_error_var_cov,
  trans_var_coeff,
  factor_lag_order,
  delay = NULL,
  quarterfy = FALSE,
  quarterly_variable_ratio = 0,
  corr = FALSE,
  beta_param = Inf,
  seed = 20022024,
  burn_in = 1000,
  rescale = TRUE,
  starting_date = NULL,
  check_stationarity = FALSE,
  stationarity_check_threshold = 1e-05,
  parallel = FALSE
)

Arguments

Details

The delay vector indicates the number of observations at the end of the sample that will be set to NA for each variable. Here, delay refers to the number of months for monthly data and the number of quarters for quarterly data. For example, consider delay <- c(1, 1) and assume the variable with index 1 will be quarterfied. In that case, the variable with index 1 will be delayed by 1 quarter, i.e., it will be missing 3 observations at the end of the panel. The variable with index 2 will be delayed by 1 month, i.e., it will be missing 1 observation at the end of the panel. This convention differs from the delay object of the SimulData class this function returns. There, delay represents the number of months since the most recent publication. For monthly variables, these values coincide, but for quarterly variables they are inherently different.

If quarterfy = TRUE, floor(quarterly_variable_ratio * no_of_vars) variables will be aggregated to a quarterly representation using the geometric mean according to Mariano RS, Murasawa Y (2003). “A new coincident index of business cycles based on monthly and quarterly series.” Journal of Applied Econometrics, 18(4), 427-443. doi:10.1002/jae.695..

If corr = TRUE, the matrix meas_error_var_cov is internally replaced by a random variance-covariance matrix: \tilde{\bm{\Sigma}}:=\bm{S}\bm{R}\bm{S}, where \bm{S} is a diagonal matrix with entries equal to sqrt(diag(meas_error_var_cov)) and \bm{R} is a random correlation matrix. \bm{R} is drawn according to Lewandowski D, Kurowicka D, Joe H (2009). “Generating random correlation matrices based on vines and extended onion method.” Journal of Multivariate Analysis, 100(9), 1989–2001. (see also https://stats.stackexchange.com/questions/2746/how-to-efficiently-generate-random-positive-semidefinite-correlation-matrices). The parameter beta_param governs the degree of cross-correlation of the off-diagonal elements. For more information see the literature cited above.

The random draws of the fundamental error terms are drawn within the ⁠C++⁠ backend. Therefore, seed must be provided and set.seed() will not guarantee reproduceability.

Value

Returns a SimulData containing the following elements:

data

If starting_date is provided, a zoo object, else, a (no_of_vars \times no_of_obs) numeric matrix holding the simulated data.

factors

If starting_date is provided, a zoo object, else a (no_of_factors \times no_of_obs) numeric matrix holding the simulated latent factors.

trans_var_coeff

Numeric (no_of_factors \times (no_of_factors * factor_lag_order)) factor VAR coefficient matrix.

loading_matrix

Numeric factor loading matrix.

meas_error

If starting_date is provided, a zoo object, else a (no_of_vars \times no_of_obs) numeric matrix holding the fundamental measurement errors.

meas_error_var_cov

Numeric measurement error variance-covariance matrix.

trans_error_var_cov

Numeric transition error variance-covariance matrix.

frequency

Integer vector of variable frequencies.

delay

Integer vector of variable delays, measured as the number of months since the latest available observation.

Author(s)

Domenic Franjic

References

Mariano RS, Murasawa Y (2003). “A new coincident index of business cycles based on monthly and quarterly series.” Journal of Applied Econometrics, 18(4), 427-443. doi:10.1002/jae.695.

Lewandowski D, Kurowicka D, Joe H (2009). “Generating random correlation matrices based on vines and extended onion method.” Journal of Multivariate Analysis, 100(9), 1989–2001.

Franjic D, Schweikert K (2024). “Nowcasting Macroeconomic Variables with a Sparse Mixed Frequency Dynamic Factor Model.” Available at SSRN 4733872.

Examples

seed <- 02102025
set.seed(seed)
no_of_obs <- 100
no_of_vars <- 50
no_of_factors <- 3
trans_error_var_cov <- diag(1, no_of_factors)
loading_matrix <- matrix(round(rnorm(no_of_vars * no_of_factors)), no_of_vars, no_of_factors)
meas_error_mean <- rep(0, no_of_vars)
meas_error_var_cov <- diag(1, no_of_vars)
trans_var_coeff <- cbind(diag(0.5, no_of_factors), -diag(0.25, no_of_factors))
factor_lag_order <- 2
delay <- c(floor(rexp(no_of_vars, 1)))
quarterfy <- FALSE
quarterly_variable_ratio  <- 0
corr <- TRUE
beta_param <- 2
burn_in <- 999
starting_date <- "1970-01-01"
rescale <- TRUE
check_stationarity <- TRUE
stationarity_check_threshold <- 1e-10
factor_model <- simFM(no_of_obs = no_of_obs, no_of_vars = no_of_vars,
                      no_of_factors = no_of_factors, loading_matrix = loading_matrix,
                      meas_error_mean = meas_error_mean, 
                      meas_error_var_cov = meas_error_var_cov,
                      trans_error_var_cov = trans_error_var_cov, 
                      trans_var_coeff = trans_var_coeff,
                      factor_lag_order = factor_lag_order, delay = delay, 
                      quarterfy = quarterfy, 
                      quarterly_variable_ratio  = quarterly_variable_ratio, corr = corr,
                      beta_param = beta_param, seed = seed, burn_in = burn_in, 
                      starting_date = starting_date, rescale = rescale, 
                      check_stationarity = check_stationarity,
                      stationarity_check_threshold = stationarity_check_threshold)
print(factor_model)
spca_plots <- plot(factor_model)
spca_plots$`Factor Time Series Plots`
spca_plots$`Loading Matrix Heatmap`
spca_plots$`Meas. Error Var.-Cov. Matrix Heatmap`
spca_plots$`Meas. Error Var.-Cov. Eigenvalue Plot`
spca_plots$`Data Var.-Cov. Matrix Heatmap`
spca_plots$`Data Var.-Cov. Eigenvalue Plot`

Sparse Principal Components Analysis

Description

Estimate sparse sparse principal components via SPCA according to Zou H, Hastie T, Tibshirani R (2006). “Sparse principal component analysis.” Journal of Computational and Graphical Statistics, 15(2), 265–286..

Usage

sparsePCA(
  data,
  delay,
  selected,
  no_of_factors,
  ridge_penalty = 1e-06,
  lasso_penalty = NULL,
  max_iterations = 1000,
  weights = NULL,
  max_no_steps = NULL,
  comp_null = 1e-15,
  spca_conv_crit = 1e-04,
  parallel = FALSE,
  svd_method = "precise",
  normalise = TRUE,
  comp_var_expl = TRUE
)

Arguments

Details

The function takes three stopping criteria: selected, lasso_penalty, and max_no_steps. With selected the SPCA algorithm stops if each column of the estimated loading matrix has the corresponding number of non-zero loadings. This allows the user to directly control the degree of sparsity of each factor loading. With lasso_penalty, the SPCA algorithm stops as soon as the side-constraints of the inherent elastic-net problem are no longer satisfied. With max_no_steps, the SPCA algorithm only takes that many LARS steps for each factor loading's individual elastic-net problem before stopping. If all criteria are provided, the first one satisfied will stop the algorithm. For details see also (Zou et al. 2006) and (Zou and Hastie 2020).

Loosely, each SPCA algorithm iteration solves an elastic-net type problem for each column of the loading matrix. One can extend this problem to the adaptive elastic-net (Zou and Zhang 2009). The variable weights lets the user provide weights for each observation. These weights must be strictly greater than zero and are normalised internally to represent relative weights. For more information on the computational implementation of the weight extension in the context of SPCA see Zou Q, Zhang P (2024). “On General Weighted Adaptive Sparse Principal Component Analysis.” In Proceedings of the 2024 4th International Conference on Computational Modeling, Simulation and Data Analysis, 335–340..

In each SPCA algorithm iteration, the function executes an SVD. To this end, Eigen provides two alternatives (Guennebaud et al. 2010): Option precise makes use of JacobiSVD. This method is numerically more stable, but computationally costly, especially for medium to large matrices. Option fast makes use of BDCSVD. This divide-and-conquer approach can lead to significant performance gains with respect to large matrices. BDCSVD, however, can be numerically unstable when Eigen is compiled with aggressive speed optimisations. In the context of the R, this should be of no concern. By default, R and most packages are compiled with "mild" -O2 optimisation and without any additional aggressive optimisation flags. Nonetheless, one should checker whether both variants provide reasonably close results before switching to fast. For more information see Guennebaud G, Jacob B, others (2010). “Eigen.” https://libeigen.gitlab.io..

Value

An object of class SPCAFit with components:

data

Original data matrix.

loading_matrix_estim

Numeric matrix of estimated factor loadings.

factor_estim

Object containing the SPCA factor estimates. The object inherits its class from data: If data is provided as zoo, factor_estim will be a zoo object. If data is provided as matrix, factor_estim will be a (no_of_factors \times no_of_obs) matrix.

total_var_expl

Numeric total variance explained.

pct_var_expl

Numeric vector relative variance explained by each factor.

Author(s)

Domenic Franjic

References

Zou H, Hastie T, Tibshirani R (2006). “Sparse principal component analysis.” Journal of Computational and Graphical Statistics, 15(2), 265–286.

Zou H, Zhang HH (2009). “On the adaptive elastic-net with a diverging number of parameters.” Annals of statistics, 37(4), 1733.

Guennebaud G, Jacob B, others (2010). “Eigen.” https://libeigen.gitlab.io.

Zou H, Hastie T (2020). elasticnet: Elastic-Net for Sparse Estimation and Sparse PCA. R package version 1.3, https://CRAN.R-project.org/package=elasticnet.

Zou Q, Zhang P (2024). “On General Weighted Adaptive Sparse Principal Component Analysis.” In Proceedings of the 2024 4th International Conference on Computational Modeling, Simulation and Data Analysis, 335–340.

Examples

data(factor_model)
set.seed(17032026)
no_of_factors <- 3
no_of_vars <- dim(factor_model$data)[2]
selected <- rep(floor(0.5 * no_of_vars), no_of_factors)
lasso_penalty <- exp(runif(no_of_factors, -10, 1))
max_no_steps <- 1000
spca_fit <- sparsePCA(data = factor_model$data, delay = factor_model$delay, 
                      selected = selected, no_of_factors = no_of_factors, 
                      ridge_penalty = 1e-2, lasso_penalty = lasso_penalty,
                      max_iterations = 1000, weights = NULL, 
                      max_no_steps = max_no_steps, comp_null = 1e-15,
                      spca_conv_crit = 1e-04, parallel = FALSE, 
                      svd_method = "precise", normalise = FALSE,
                      comp_var_expl = TRUE)
print(spca_fit)
spca_plots <- plot(spca_fit)
spca_plots$`Factor Time Series Plots`
spca_plots$`Loading Matrix Heatmap`
spca_plots$`Meas. Error Var.-Cov. Matrix Heatmap`
spca_plots$`Eigenvalue Plot`
spca_plots$`Variance Explained Chart`

Two Step Dense Dynamic Factor Model Estimator.

Description

Estimate a dense dynamic factor model with measurement equation

\bm{x}_t = \bm{\Lambda} \bm{f}_{t} + \bm{\xi}_t,\quad \bm{\xi}_t \sim \mathcal{N}(\bm{0}, \bm{\Sigma}_{\xi}),

and transition equation

\bm{f}_t = \sum_{p=0}^P\bm{\Phi}_p \bm{f}_{t-p} + \bm{\epsilon}_t,\quad \bm{\epsilon}_t \sim \mathcal{N}(\bm{0}, \bm{\Sigma}_{f}).

using principal components analysis and the Kalman Filter and Smoother according to Giannone D, Reichlin L, Small D (2008). “Nowcasting: The real-time informational content of macroeconomic data.” Journal of Monetary Economics, 55(4), 665-676. ISSN 0304-3932, doi:10.1016/j.jmoneco.2008.05.010. and Doz C, Giannone D, Reichlin L (2011). “A two-step estimator for large approximate dynamic factor models based on Kalman filtering.” Journal of Econometrics, 164(1), 188-205. ISSN 0304-4076, doi:10.1016/j.jeconom.2011.02.012..

Usage

twoStepDenseDFM(
  data,
  delay,
  no_of_factors,
  max_factor_lag_order = 10,
  lag_estim_criterion = "BIC",
  decorr_errors = TRUE,
  comp_null = 1e-15,
  parallel = FALSE,
  fcast_horizon = 0,
  jitter = 1e-08
)

Arguments

Details

The function performs a two-step estimation procedure for dense dynamic factor models as described in Giannone D, Reichlin L, Small D (2008). “Nowcasting: The real-time informational content of macroeconomic data.” Journal of Monetary Economics, 55(4), 665-676. ISSN 0304-3932, doi:10.1016/j.jmoneco.2008.05.010. and Doz C, Giannone D, Reichlin L (2011). “A two-step estimator for large approximate dynamic factor models based on Kalman filtering.” Journal of Econometrics, 164(1), 188-205. ISSN 0304-4076, doi:10.1016/j.jeconom.2011.02.012.. In the first step, the factor loading matrix is estimated using PCA. In the second step the latent factors are estimated using the univariate representation of the Kalman Filter and Smoother (Koopman and Durbin 2000).

With respect to the univariate representation of the Kalman filter and smoother, decorr_errors indicates whether the data should be decorrelated internally prior to filtering and smoothing. jitter is added to the diagonal elements of the measurement variance–covariance matrix. For more details, see kalmanFilterSmoother.

Value

An object of class SDFMFit with main components:

data

Original data object.

loading_matrix_estim

Numeric matrix of estimated factor loadings.

smoothed_factors

Object containing the SPCA factor estimates. The object inherits its class from data: If data is provided as zoo, factor_estim will be a zoo object. If data is provided as matrix, factor_estim will be a (no_of_factors\timesno_of_obs matrix.

smoothed_state_variance

(no_of_factors\times( no_of_factors * no_of_obs)) matrix, where each (no_of_factors \timesno_of_factors) block represents the smoother uncertainty at time pointt.

factor_var_lag_order

Integer order of the VAR process in the state equation.

error_var_cov_cholesky_factor

Numeric lower-triangular Cholesky factor of the estimated measurement error variance–covariance matrix.

llt_success_code

Integer indicating the status of the Cholesky factorization: 0 = LLT succeeded, -1 = LLT failed but LDLT succeeded, -2 = both failed and errors are treated as uncorrelated.

Author(s)

Domenic Franjic

References

Koopman SJ, Durbin J (2000). “Fast filtering and smoothing for multivariate state space models.” Journal of Time Series Analysis, 21(3), 281–296.

Giannone D, Reichlin L, Small D (2008). “Nowcasting: The real-time informational content of macroeconomic data.” Journal of Monetary Economics, 55(4), 665-676. ISSN 0304-3932, doi:10.1016/j.jmoneco.2008.05.010.

Guennebaud G, Jacob B, others (2010). “Eigen.” https://libeigen.gitlab.io.

Doz C, Giannone D, Reichlin L (2011). “A two-step estimator for large approximate dynamic factor models based on Kalman filtering.” Journal of Econometrics, 164(1), 188-205. ISSN 0304-4076, doi:10.1016/j.jeconom.2011.02.012.

See Also

sparsePCA: Routine for fitting estimating a sparse factor loading matrix.

kalmanFilterSmoother: Routine for filtering and smoothing latent factors.

twoStepSDFM: Two-step estimation routine for a sparse dynamic factor model.

Examples

data(factor_model)
no_of_vars <- dim(factor_model$data)[2]
no_of_factors <- dim(factor_model$factors)[2]
dfm_fit <- twoStepDenseDFM(data = factor_model$data, delay = factor_model$delay, 
                           no_of_factors = no_of_factors)
print(dfm_fit)
dfm_plots <- plot(dfm_fit)
dfm_plots$`Factor Time Series Plots`
dfm_plots$`Loading Matrix Heatmap`
dfm_plots$`Meas. Error Var.-Cov. Matrix Heatmap`
dfm_plots$`Meas. Error Var.-Cov. Eigenvalue Plot`

Two Step Sparse Dynamic Factor Model Estimator.

Description

Estimate a sparse dynamic factor model with measurement equation

\bm{x}_t = \bm{\Lambda} \bm{f}_{t} + \bm{\xi}_t,\quad \bm{\xi}_t \sim \mathcal{N}(\bm{0}, \bm{\Sigma}_{\xi}),

and transition equation

\bm{f}_t = \sum_{p=0}^P\bm{\Phi}_p \bm{f}_{t-p} + \bm{\epsilon}_t,\quad \bm{\epsilon}_t \sim \mathcal{N}(\bm{0}, \bm{\Sigma}_{f}).

using sparse principal components analysis and the Kalman Filter and Smoother according to Franjic D, Schweikert K (2024). “Nowcasting Macroeconomic Variables with a Sparse Mixed Frequency Dynamic Factor Model.” Available at SSRN 4733872..

Usage

twoStepSDFM(
  data,
  delay,
  selected,
  no_of_factors,
  max_factor_lag_order = 10,
  lag_estim_criterion = "BIC",
  decorr_errors = TRUE,
  ridge_penalty = 1e-06,
  lasso_penalty = NULL,
  max_iterations = 1000,
  max_no_steps = NULL,
  weights = NULL,
  comp_null = 1e-15,
  spca_conv_crit = 1e-04,
  parallel = FALSE,
  fcast_horizon = 0,
  jitter = 1e-08,
  svd_method = "precise"
)

Arguments

Details

The function performs a two-step estimation procedure for sparse dynamic factor models as described in Franjic D, Schweikert K (2024). “Nowcasting Macroeconomic Variables with a Sparse Mixed Frequency Dynamic Factor Model.” Available at SSRN 4733872.. In the first step, the factor loading matrix is estimated using SPCA (Zou et al. 2006). This will shrink some of the loadings towards or exactly to zero. In the second step the latent factors are estimated using the univariate representation of the Kalman Filter and Smoother (Koopman and Durbin 2000).

The function takes three stopping criteria for the SPCA algorithm: selected, lasso_penalty, and max_no_steps. The argument weights allows specifying weights for the \ell_1 constraint. svd_method controls the decomposition method for internal SVDs. For a detailed description of these arguments and the SPCA step, see sparsePCA.

With respect to the univariate representation of the Kalman filter and smoother, decorr_errors indicates whether the data should be decorrelated internally prior to filtering and smoothing. jitter is added to the diagonal elements of the measurement variance–covariance matrix. For more details, see kalmanFilterSmoother.

For more information on the two-step estimation procedure see Franjic D, Schweikert K (2024). “Nowcasting Macroeconomic Variables with a Sparse Mixed Frequency Dynamic Factor Model.” Available at SSRN 4733872..

Value

An object of class SDFMFit with main components: #'

data

Original data object.

loading_matrix_estim

Numeric matrix of estimated factor loadings.

smoothed_factors

Object containing the SPCA factor estimates. The object inherits its class from data: If data is provided as zoo, factor_estim will be a zoo object. If data is provided as matrix, factor_estim will be a (no_of_factors\timesno_of_obs matrix.

smoothed_state_variance

(no_of_factors\times( no_of_factors * no_of_obs)) matrix, where each (no_of_factors \timesno_of_factors) block represents the smoother uncertainty at time pointt.

factor_var_lag_order

Integer order of the VAR process in the state equation.

error_var_cov_cholesky_factor

Numeric lower-triangular Cholesky factor of the estimated measurement error variance–covariance matrix.

llt_success_code

Integer indicating the status of the Cholesky factorization: 0 = LLT succeeded, -1 = LLT failed but LDLT succeeded, -2 = both failed and errors are treated as uncorrelated.

Author(s)

Domenic Franjic

References

Koopman SJ, Durbin J (2000). “Fast filtering and smoothing for multivariate state space models.” Journal of Time Series Analysis, 21(3), 281–296.

Zou H, Hastie T, Tibshirani R (2006). “Sparse principal component analysis.” Journal of Computational and Graphical Statistics, 15(2), 265–286.

Guennebaud G, Jacob B, others (2010). “Eigen.” https://libeigen.gitlab.io.

Franjic D, Schweikert K (2024). “Nowcasting Macroeconomic Variables with a Sparse Mixed Frequency Dynamic Factor Model.” Available at SSRN 4733872.

See Also

sparsePCA: Routine for fitting estimating a sparse factor loading matrix.

kalmanFilterSmoother: Routine for filtering and smoothing latent factors.

twoStepDenseDFM: Two-step estimation routine for a dense dynamic factor model.

Examples

data(factor_model)
no_of_vars <- dim(factor_model$data)[2]
no_of_factors <- dim(factor_model$factors)[2]
sdfm_fit <- twoStepSDFM(data = factor_model$data, delay = factor_model$delay,
                        selected = rep(floor(0.5 * no_of_vars), no_of_factors),
                        no_of_factors = no_of_factors)
print(sdfm_fit)
sdfm_plots <- plot(sdfm_fit)
sdfm_plots$`Factor Time Series Plots`
sdfm_plots$`Loading Matrix Heatmap`
sdfm_plots$`Meas. Error Var.-Cov. Matrix Heatmap`
sdfm_plots$`Meas. Error Var.-Cov. Eigenvalue Plot`