longRunSvar#

Purpose#

Blanchard-Quah (1989) long-run SVAR identification. Computes structural impulse responses by imposing long-run restrictions via the Cholesky decomposition of the cumulative impact matrix.

Format#

lr = longRunSvar(y, n_ahead)#
lr = longRunSvar(y, n_ahead, p=4)
lr = longRunSvar(y, n_ahead, p=4, xreg=X)
Parameters:

  • y (TxM matrix or dataframe) – endogenous variables. If a dataframe, column names are used as variable labels in output. If a matrix, variables are labeled “Y1”, “Y2”, etc.

  • n_ahead (scalar) – number of impulse response horizons to compute (h = 0, 1, …, n_ahead).

  • p (scalar) – Optional keyword, lag order. Default = 1.

  • xreg (matrix or dataframe) – Optional keyword, TxK exogenous regressors. Default = {} (none).

  • quiet (scalar) – Optional keyword, set to 1 to suppress output. Default = 0.

  • ctl –

    Optional keyword, an instance of a longRunSvarControl structure. When provided, struct values are used and keywords are ignored. An instance is initialized by calling longRunSvarControlCreate() and the following members can be set:

    adv.const

    Scalar, 1 to include a constant (intercept), 0 to exclude. Default = 1.

    adv.xreg

    TxK matrix, exogenous regressors. Default = empty (no exogenous variables).

    adv.quiet

    Scalar, set to 1 to suppress printed output. Default = 0.

Returns:

lr (struct) –

An instance of a longRunSvarResult structure containing:

lr.impact

mxm matrix, structural impact matrix \(B_0\). Column j gives the contemporaneous effect of structural shock j on all variables.

lr.irf

An instance of an irfResult structure containing orthogonalized impulse responses identified via long-run restrictions. See irfCompute() for the irfResult layout.

lr.m

Scalar, number of endogenous variables.

lr.p

Scalar, lag order.

lr.n_ahead

Scalar, number of forecast horizons.

lr.var_names

Mx1 string array, variable names.

Examples#

Blanchard-Quah Supply and Demand Shocks#

The classic Blanchard and Quah (1989) decomposition with GDP growth and unemployment. The first shock (supply) has a permanent effect on GDP; the second shock (demand) has zero long-run effect on GDP:

new;
library timeseries;

// Load US macro quarterly data
fname = getGAUSSHome("pkgs/timeseries/examples/data/us_macro_quarterly.csv");
y = loadd(fname, "gdp_growth + unemployment");

// Long-run SVAR with 4 lags and 20-step IRF
lr = longRunSvar(y, 20, p=4);

Output:

================================================================================
Long-Run SVAR (Blanchard & Quah 1989)
Variables: 2, Lags: 4, Horizons: 0-20
================================================================================
Structural Impact Matrix (B0):
          Supply    Demand
GDP       0.0083    0.0071
UNEMP    -0.0512    0.1243
================================================================================

The impact matrix shows that a supply shock raises GDP and lowers unemployment on impact, while a demand shock raises both.

Technology Shock Identification (Gali 1999)#

Identify technology shocks using labor productivity and hours worked. Only the technology shock (first variable) has a permanent effect on productivity:

new;
library timeseries;

fname = getGAUSSHome("pkgs/timeseries/examples/data/us_macro_quarterly.csv");
y = loadd(fname, "productivity_growth + hours_growth");

// Ordering: productivity first, hours second
// Technology shock = permanent productivity effect
// Non-technology shock = zero long-run productivity effect
lr = longRunSvar(y, 40, p=8);

// Access IRF at horizon 10: response of hours to technology shock
print "Hours response to technology shock at h=10:";
print lr.irf.irf[11*2, 1];

With Exogenous Regressors#

Include a time trend as an exogenous regressor:

new;
library timeseries;

fname = getGAUSSHome("pkgs/timeseries/examples/data/us_macro_quarterly.csv");
y = loadd(fname, "gdp_growth + cpi_inflation");

lr = longRunSvar(y, 20, p=4, xreg=seqa(1, 1, rows(y)), quiet=1);

print "Structural impact matrix:";
print lr.impact;

Model#

The reduced-form VAR(p) is estimated by OLS:

\[y_t = B_1 y_{t-1} + \cdots + B_p y_{t-p} + u + \varepsilon_t, \quad \varepsilon_t \sim N(0, \Sigma)\]

The long-run cumulative impact matrix is:

\[C(1) = (I_m - B_1 - B_2 - \cdots - B_p)^{-1}\]

This matrix captures the total (cumulative) effect of a reduced-form shock on the level of each variable. To identify structural shocks, compute:

\[C(1) \, \Sigma \, C(1)' = P \, P'\]

where \(P\) is the lower-triangular Cholesky factor. The structural impact matrix is:

\[B_0 = C(1)^{-1} \, P\]

The identification imposes that the second shock (and higher) has zero long-run effect on the first variable, the third shock has zero long-run effect on the first two variables, and so on. The structural IRF at horizon h is:

\[\Theta_h = J \, F^h \, J' \, B_0\]

where \(F\) is the VAR companion matrix and \(J = [I_m \; 0 \; \cdots \; 0]\).

Remarks#

Variable ordering matters: The ordering of variables in y determines the identification. The first variable is the one on which all shocks except the first have zero long-run effect. The second variable allows only the first two shocks to have permanent effects, and so on. In the Blanchard-Quah setup, GDP (or productivity) must be ordered first so that the demand shock has zero long-run effect on output.

Stationarity requirement: The VAR must be stationary for the long-run matrix \(C(1) = (I - B_1 - \cdots - B_p)^{-1}\) to exist. If the companion matrix has eigenvalues near or on the unit circle, \(C(1)\) becomes ill-conditioned and the identification is unreliable. Check stationarity with varFit() before using this function.

Point identification only: This function provides point-identified structural IRFs from a single OLS VAR estimate. There are no posterior uncertainty bands. For inference with uncertainty, bootstrap the VAR or use a Bayesian approach with bvarFit() and svarIrfCompute().

Differenced vs. level data: The Blanchard-Quah method is typically applied to growth rates (first differences of log levels). The cumulative IRF from lr.irf then gives the level response. If variables are already in levels and stationary, the IRF itself is the level response.

Accessing IRF values: The IRF is stored in lr.irf.irf as an (n_ahead+1)*m x m stacked matrix. Block h (rows h*m+1 to (h+1)*m) is the mxm response matrix at horizon h. Element [i, j] of block h is the response of variable i to structural shock j at horizon h.

References#

  • Blanchard, O.J. and D. Quah (1989). “The dynamic effects of aggregate demand and supply disturbances.” American Economic Review, 79(4), 655-673.

  • Gali, J. (1999). “Technology, employment, and the business cycle: Do technology shocks explain aggregate fluctuations?” American Economic Review, 89(1), 249-271.

  • Lutkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Springer, Chapter 9.

Library#

timeseries

Source#

var.src

See also

Functions longRunSvarControlCreate(), svarIrfCompute(), varFit(), irfCompute()

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