GLP

This repository contains the code and data for “Group Local Projection”. You can find the latest draft and supplemental materials on my webpage, and download the raw output files here.

Feel free to contact me (jiaming.huang@barcelonagse.eu) if you have any doubts on implementing the codes.

Replication

For the simulation study, the results can be replicated using SIM_KnownG0.m (Section 6.2), SIM_UnknownG0.m (Section 6.3) and SIM_NoTrueGroup.m (Section 6.4). The folder supp contains additional exercises in the supplemental material (Section S1), and the multinomial logit model in Section S2.3.

For the empirical application, EMP_Main.m gives the baseline results (individual LP-IV, panel LP-IV and the GLP). This is compared with ad hoc grouping criterion (EMP_Adhoc.m) and FAVAR (EMP_FAVAR.m). They both take EMP_data.mat as input.

The FAVAR application involves two additional folders:

1. fred: used to estimate the factor model, which is borrowed from the FRED-MD package by McCracken and Ng (2016). I modify the factors_em.m file to include the Onatski (2010) criterion.

2. svar: used to estimate the SVAR-IV model after factor estimation. Here I use the proxy-SVAR upackage provided by Mertens and Montiel-Olea (2018). Alternative inference methods are included (wild bootstrap, delta method and Jentsch Lunsford MBB)

Auxiliary functions in the routine folder are:

• DGP.m, DGP_NoGroup.m and DGP_FDAH.m: generate simulated data

• GLP.m: general function for the GLP estimation

  • HAC4d.m: compute HAC robust variance estimator for 4-d matrices; see equation (17)
  • GLP_SIM_Infeasible.m: infeasible group-by-group panel LP-IV in which the latent group is known --this is essentially xtivreg2 for each group; related is GLP_SIM_Infeasible1.m where we know the true group assignment but still use our GMM objective function (see SM S1.6)
  • GLP_SIM_KnownG0.m: GLP with known number of groups G0 (Section 6.2)
  • GLP_SIM_UnknownG0.m: GLP that runs for Ghat=1,...Gmax, and select the number of groups by IC; see equation (28)-(29) (I remove the inference as is not reported in simulations)
  • GLP_SIM_KnownG0_Inference.m: same as GLP_SIM_KnownG0.m except that we return results for both large T and small T inference

• eval_GroupLPIV.m: evaluate the performance of the GLP

• ind_LP.m: individual LP-IV, it can be used also for non-IV case

  • HAC.m: HAC standard errors in the individual LP-IV model
  • ind_LP_noc.m: same as ind_LP.m but without constant term (for FDAH)

• lag.m: compute lags of variables

• panel_LP.m: panel LP-IV, also works for non-IV case

• params.m: parameter values in simulation designs

• preEmpData.m: prepare data to be used as input for GLP.m

• resid.m: Compute whether the iteration converges

For your own application

You need to:

1. Prepare your data (balanced panel) in long format

2. Specify your model (see below for an example):

FE = 1;                % unit fixed effects - 1; random effects - 0
par.y_idx = 1;         % dependent variable
par.x_idx = 2;         % policy variable (scalar)
par.c_idx = [3 4 5];   % controls
par.z_idx = [6];       % instrument (preEmpData is written for exogenous control; but we can always specify par.zx_idx and par.zc_idx that instrument z and c separately)
par.nylag = 4;         % specify the number of lags, for y x w and z
par.nxlag = 4;
par.nclag = 4;
par.nzlag = 4;
par.horizon = 24;      % horizons
par.nwtrunc = 25;      % truncation order for Newey-West standard erors (for individual LP-IV)
par.start = '1975-01-01';
par.end   = '2007-12-01';
reg = preEmpData(data, par); % now you have your reg (struct variable) 

3. Supply the data struct to GLP.m. Before that, it is recommended to run individual LP-IV (to get initial guesses), you can otherwise specify your own guesses (and weight matrix).

indOut = ind_LP(reg);  % individual LP-IV

Gmax   = 8;             % maximal number of groups
nInit  = 100;           % number of initializations
bInit  = indOut.b;      % this is the output from individual LP-IV, from which we can draw initial guesses
weight = repmat(mean(indOut.v_hac,3),1,1,par.N,1);  % this is the weight matrix
inference = 1;          % large T inference
[Gr_EST, GIRF, GSE, OBJ, IC] = GLP(reg, Gmax, nInit, bInit, weight, FE, inference);

4. After the GLP estimation, we first look at the number of groups selected by IC

figure;
plot(IC,'b-s','LineWidth',2,'MarkerSize',5,...
    'MarkerEdgeColor','blue',...
    'MarkerFaceColor','blue');
xlabel('Number of Groups');

5. Then we can examine the IRs (e.g. relabel them according to positiveness)

% store relabeled group assignment
Group_relabel = nan(par.N,Gmax);
for Ghat = 1:Gmax
    girf   = squeeze(GIRF{1,Ghat});
    gse    = squeeze(GSE{1,Ghat});
    Ub_GLP = girf + 1.96*gse;
    Lb_GLP = girf - 1.96*gse;

    % order ir by average positiveness
    [~,ord] = sort(mean(girf,2),'descend');
    figure;
    for g = 1:Ghat
        subplot(ceil(Ghat/2),2,g);
        k = ord(g);
        hold on;
        % bands
        fill([1:H, fliplr(1:H)],...
            [Ub_GLP(k,:) fliplr(Lb_GLP(k,:))],...
            BandColors,'EdgeColor','none');
        % IR
        plot(1:H, girf(k,:),'LineWidth',1.2,'color',LineColors);

        yline(0,'k','LineWidth',.7);
        xlabel(strcat({'Group'},{' '},num2str(g)));
        xlim([1 H]); axis tight
        set(gca,'XTick',[1 6 12 18 24],'XTickLabel',cellstr(num2str([1 6 12 18 24]')),...
            'FontSize',8,'Layer','top')
        hold off
    end
    % store ordered group classification
    gr_tmp = zeros(par.N,1);
    for g = 1:Ghat
        gr_tmp = gr_tmp +(Gr_EST(:,Ghat) == ord(g) )*g;
    end
    Group_relabel(:,Ghat) = gr_tmp; % store it for later output
end

The GLP.m function

The GLP.m is a ready-to-use function for implementing the GLP. It is written for balanced panel with exact identification (L=K).

It takes the following input:

• reg: data (reg.LHS, reg.x, reg.c, reg.zx, reg.zc, reg.param)
  • reg.LHS NT by H dependent variables
  • reg.x, NT by K policy variables whose coefs are to be grouped
  • reg.zx, NT by Lx IV for reg.x (optional, use reg.x if not specified)
  • reg.c, NT by P controls whose coefs vary across i (can be empty)
  • reg.zc NT by Lc IV for reg.c (optional, use reg.c if not specified)
  • reg.param.N, reg.param.T
• Gmax: maximal number of groups to be classified
• nInit: number of initializations
• bInit: potential initial values, it is recommended to use IR estimates from individual LP-IV (ind_LP.m) as initial guess, but you can specify your own guess
• weight: either string ('2SLS', 'IV') or user-supplied weights; it is recommended to use the inverse of the covariance matrix of moment conditions (averaged over h), i.e. the horizon-specific weights to account for noises in each horizons
• FE: 1 - fixed effects (include a constant term in controls)
• inference: 1 - large T; 2 - fixed T;

It gives the following output:

• Gr_EST: Group composition, N by Gmax matrix
• GIRF: Group IRF, 1 by Gmax cell, with K by 1 by G by H coefs
• GSE: Group standard errors, 1 by Gmax cell, with K by 1 by G by H SE
• OBJ: minimized objective function for each Ghat, 1 by Gmax vector
• IC: IC: information criterion, 1 by Gmax vector

Note: Notice that you can freely specify the set of variables of interest reg.x whose IRs are grouped and the set of nuisance variables reg.c whose IRs are unit-specific. Moreover, both x and c can be potentially endogenous, as long as we provide the corresponding instruments reg.zx and reg.zc.

References

Huang, J. (2021). Group Local Projections.

McCracken, M. W., & Ng, S. (2016). FRED-MD: A monthly database for macroeconomic research. Journal of Business & Economic Statistics, 34(4), 574-589.

Mertens, K., & Montiel Olea, J. L. (2018). Marginal tax rates and income: New time series evidence. The Quarterly Journal of Economics, 133(4), 1803-1884.

Onatski, A. (2010). Determining the number of factors from empirical distribution of eigenvalues. The Review of Economics and Statistics, 92(4), 1004-1016.