FunctionalVAR

This repository implements the Functional VAR (FVAR) approach from Huang (2023). models the joint dynamics of macro aggregates and functional variables within the SVAR framework. It supports three identification schemes: (1) short-run restrictions, (2) internal instrumental variables (IV), and (3) external IV. Inference is performed via a moving-block bootstrap variant.

For theoretical details, refer to the latest draft. For implementation questions, contact jiaming.huang@barcelonagse.eu.

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Minimal Working Example

See MWE.m for minimal implementation of the simulation exercise in the paper. For your application with macro data (Txn) and functional data (TxnGridFcn), you can run the following:

DATASET.data = data;                    % T x n matrix of macro variables
DATASET.gridFcn = gridOfYourFcn;        % 1 x ngridFcn vector of function grid points
DATASET.fcn = fcnData;                  % T x ngridFcn matrix of functional data
DATASET.varsName = colNamesOfYourData;  % Cell array of variable names
modelSpec.varsSel = varsToIncludeInVAR; % e.g., {'x', 'Func', 'y'}
FVAR = doFVAR(DATASET, modelSpec);      % Run estimation

The output FVAR contains:

• Estimated VAR coefficients
• Model specification
• Impulse response functions (IRs)
• Bootstrap confidence intervals

Then you can play with the model to data setup to your needs (see below for supported features).

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Specification

1. DATASET Struct

Field	Description
data	T x n matrix of macro variables
varsName	1 x n cell array of variable names
varsDiff	Cell array of differenced variables (cumulative IRs computed). Default: {}
fcn	T x ngridFcn matrix of functional data evaluations
gridFcn	1 x ngridFcn vector of function domain grid points
dates	T x 1 vector of dates (datetime or numeric)

2. modelSpec Struct

VAR Specification

Parameter	Description
varsSel	Cell array of variables in VAR order (Cholesky ordering if using identification='CHOL') Example: {'x', 'Func', 'y'}
p	VAR lag order. Default: 1
iDET	Deterministic components: 1=constant, 2=linear trend, 3=quadratic trend. Default: 1
dateStart dateEnd	Estimation sample time window

FPCA Specification

Parameter	Description
fdParobj	fdPar object for functional smoothing (Default: Fourier basis): matlab create_fourier_basis([0,1], 21);
nFPCMax	Max FPCs to consider. Selected via cumulative variance. Default: 20

SVAR Specification

Parameter	Description
identification	"CHOL" (Cholesky), "InternalIV", or "SVARIV". Default: "CHOL"
varsShock	Shock variable(s). Default: First entry in varsSel
varsUnitNorm	Variables for unit shock normalization (Internal IV only)
Instrument	Instrument variable (Required for "SVARIV")
nZlags	IV lag order (External IV only). Default: 0
nNWlags	Newey-West covariance lags. Default: floor(1.3*sqrt(T))
irhor	Maximum IRs horizon (total horizons = irhor + 1). Default: 36

Bootstrap

Parameter	Description
nBoot	Bootstrap replications. Default: 500
blkSize	Moving block length. Default: 16
cLevel	Confidence level (%). Default: 95
verbose	Display bootstrap progress. Default: false

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FPCA Implementation

FDA Package Requirement

I use the fdaM MATLAB package for Functional Principal Component Analysis (FPCA):
Download fdaM.

1. Functional Data Smoothing

See Chapter 3 of Ramsay et al. (2009) for details on the basis functions. For example, the Fourier basis can be specified as follows:

% Create Fourier basis
rangeval = [0, 1];       % Function domain
nbasis = 21;             % Number of basis functions
basisobj = create_fourier_basis(rangeval, nbasis);

% Convert raw data to functional objects
fdParobj = fdPar(basisobj);
fdobj = smooth_basis(argvals, f, fdParobj);  % f: nGrid x T matrix

2. FPCA

% Perform FPCA
pcaOut = pca_fd(fdobj, modelSpec.nFPCMax, fdParobj);

% Extract components
mu = eval_fd(gridFcn, pcaOut.meanfd)';     % Mean function
bas = eval_fd(gridFcn, pcaOut.harmfd);    % Eigenfunctions (nGrid x nFPC)
fpcs = pcaOut.harmscr;                  % FPC scores (T x nFPC)

% Retain FPCs explaining ≥95% variance
nFPC = find(cumsum(pcaOut.varprop) >= 0.95, 1);
fhat = mu + fpcs(:,1:nFPC) * bas(:,1:nFPC)';  % Reconstructed functions

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SVAR Implementation

• Short-run restrictions: Recursive identification via Cholesky (Kilian & Lütkepohl, 2017).
• Internal IV: Instrument ordered first in a recursive VAR (Plagborg‐Møller & Wolf, 2021).
• External IV: Reduced-form VAR, plus a separate IV regression (Stock & Watson, 2018).

Bootstrap:

• Recursive SVAR: Brüggemann et al. (2016).
• SVAR-IV: Jentsch & Lunsford (2022).

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References

1. Brüggemann, R., Jentsch, C., & Trenkler, C. (2016). Inference in VARs with conditional heteroskedasticity of unknown form. Journal of Econometrics, 191(1), 69–85.
2. Huang, J. (2023). Functional VAR.
3. Jentsch, C., & Lunsford, K. G. (2022). Asymptotically valid bootstrap inference for proxy SVARs. Journal of Business & Economic Statistics, 40(4), 1876–1891.
4. Kilian, L., & Lütkepohl, H. (2017). Structural vector autoregressive analysis. Cambridge University Press.
5. Mertens, K., & Ravn, M. O. (2013). The dynamic effects of personal and corporate income tax changes in the United States. American Economic Review, 103(4), 1212–1247.
6. Plagborg-Møller, M., & Wolf, C. K. (2021). Local projections and VARs estimate the same impulse responses. Econometrica, 89(2), 955–980.
7. Ramsay, J. O., & Silverman, B. W. (2005). Functional data analysis (2nd ed.). Springer.
8. Ramsay, J. O., Hooker, G., & Graves, S. (2009). Functional data analysis with R and MATLAB. Springer.
9. Stock, J. H., & Watson, M. W. (2018). Identification and estimation of dynamic causal effects in macroeconomics using external instruments. The Economic Journal, 128(610), 917–948.

Citation

If this code contributes to your work, please cite:

@article{huang2023functional,
  title = {Functional VAR},
  author = {Huang, Jiaming},
  year = {2023},
}