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---

### Construction of the VFCI and empirical specification
- Model equations:
  - ∆gdp_{t+1} = θ P C_{t} + ε_{t+1},   V ar_{t}(ε_{t+1}) = exp(2 δ P C_{t}).
  - VFCI is defined as ˆδ P C_{t} standardized to have zero mean and unit variance.
- Estimation details:
  - Parameters estimated by restricted maximum likelihood.
  - P C_{t} is a vector with an intercept and the first four principal components of the financial variables in Table 1.
  - ε_{t+1} is a zero-mean error.
- Sample:
  - 1962Q1-2022Q3 (243 quarterly observations).

### Historical behavior and interpretation of the VFCI
- Key historical episodes with marked increases in VFCI:
  - Tumultuous 2000-2002 period (dot-com bubble burst, 9/11, corporate scandals).
  - 1994 bond market crisis.
  - "Black Monday" stock market crash of 1987.
  - Savings and loans crisis of the early 1980s.
  - 40% stock market decline in 1973–1974.
  - Spike in 1998Q3 (collapse of Long-Term Capital Management and the Russian financial crisis).
- Relationship with real activity:
  - Financial conditions and real aggregate activity are related but distinct.
  - VFCI can be low during macroeconomic downturns and high during booms (example: VFCI not unusually high during the 1990-1991 recession).
  - Between 1996Q2 and 2000Q2:
    - Year-over-year real GDP growth above 4% every quarter.
    - Unemployment declined from 5.5% to 4%.
    - Core inflation declined from 3% to 2%.

### Mean–volatility relation and economic amplification
- Identification and signs:
  - Sign of each PC in estimation of equations (24)-(25) is not identified; authors select PC signs so coefficients of ˆθ in Panel A of Table 3 are all positive.
  - Given PC sign choices, signs of ˆδ in the volatility equation are determined by the data.
- Empirical directions:
  - All PCs induce movements of the conditional mean and conditional volatility of GDP growth and consumption growth that go in opposite directions.
  - Periods with high expected GDP growth are usually accompanied by low conditional volatility of GDP growth (a low VFCI).
- Scatter plot evidence (Figure 2):
  - Tight linear relation with negative slope between VFCI (conditional log volatility of GDP growth, horizontal axis) and conditional mean of GDP growth (vertical axis).
  - High VFCI: economy tends to low-mean, high-volatility states where negative shocks of a given magnitude translate into much larger declines in GDP.
  - Low VFCI: accompanied by relatively long periods of positive growth that are more resilient to shocks.

### Comparison with other financial conditions indices (FCIs)
- Indices reviewed: NFCI (Federal Reserve Bank of Chicago), GSFCI (Goldman Sachs), VIX (CBOE).
- NFCI:
  - Weighted average of 105 indicators of financial activity; weekly update starting January 1973.
  - Weights via mixed-frequency dynamic factor analysis; renormalized weekly to mean zero and variance one; higher values = tighter conditions.
- GSFCI:
  - Weighted average of short-term rates, long-term rates, trade-weighted dollar, credit spreads, and price-to-earnings ratio; available daily starting January 1, 1983.
  - Weights reflect impact of shocks on GDP growth over four quarters following shock, estimated from quarterly OLS regressions starting in 1984.
  - The 10-year Treasury yield and the index of credit spreads account for most of the index, with a sum of their weights equal to 84.7%.
- VIX:
  - Measure of expected volatility of aggregate US stock market over next 30 days; derived from real-time, mid-quote prices of S&P 500 Index call and put options.
- Comovement and differences:
  - VFCI, NFCI, GSFCI, and VIX exhibit clear comovement in stress episodes (e.g., 2008) but show notable differences across episodes (e.g., GSFCI eased more in 2021; VIX and VFCI elevated in late 1990s).
- Statistical span by principal components:
  - Regression FCI_t = α + β PC_t + ε_t where PCs are first four principal components of six financial variables yields R² between 0.70 and 1.00.
  - Reported R²: VFCI = 1.00, NFCI = 0.81, GSFCI = 0.73, VIX = 0.80.
  - Three or four PCs needed to achieve high R²; beyond four adds little.

### VFCI’s predictive power for asset risk premia
- Risk premia measures (annualized percentage points; FCIs standardized to zero mean and unit variance):
  - Equity: S&P 500 excess CAPE yield (ECY).
  - Treasuries: 10-year term premium (TP).
  - Corporate bonds: GZ spread.
- Key regression findings (Panel A–C, Table 5):
  - Panel A (Stocks, ECY):
    - Lag VFCI coefficient = 0.13*** when included alone (column (1)), significant at 1%.
    - A one-standard-deviation increase in VFCI predicts an increase of 13 basis points in next quarter’s (annualized) equity risk premium.
    - VFCI alone predicts 22% of one-quarter-ahead ECY variation not explained by lagged ECY.
    - When all four FCIs included jointly (column (5)), VFCI coefficient remains the only significant one and accounts for 31% of one-period-ahead ECY variance not accounted for by current ECY.
    - Lag ECY coefficients reported: 0.97***, 0.93***, 0.91***, 0.94***, 0.94*** with t-statistics (45.1), (31.5), (30.9), (27.8), (32.1).
    - Sample sizes N: 242, 206, 157, 130, 129. R²: 0.95, 0.95, 0.93, 0.90, 0.91.
  - Panel B (Treasuries, TP):
    - Lag VFCI coefficient = 0.08*** when included alone; significant at 1%.
    - VFCI explains 26% of variance in TP not explained by lagged TP when alone; rises to 48% when included jointly.
    - Lag TP coefficients: 0.98***, 0.96***, 0.95***, 0.96***, 1.02*** with t-statistics (68.4), (55.9), (36.2), (48.3), (33.0).
    - Sample sizes N: 242, 206, 157, 130, 129. R²: 0.95, 0.94, 0.94, 0.94, 0.94.
  - Panel C (Corporate Bonds, GZ):
    - Lag VFCI coefficient = 0.21*** when included alone; significant at 1%.
    - VFCI explains 52% of variance in GZ not explained by lagged GZ when alone; rises to 60% when included jointly.
    - Lag GZ coefficients: 0.82***, 0.90***, 0.89***, 0.68***, 0.60*** with t-statistics (14.0), (13.7), (12.8), (6.4), (4.4).
    - Sample sizes N: 197, 197, 157, 130, 129. R²: 0.86, 0.82, 0.79, 0.85, 0.89.
- Conclusion:
  - VFCI predicts risk premia for equities, Treasuries, and corporate bonds when used alone, and is the only FCI with significant predictive power when all FCIs are included jointly.

### SVAR evidence: VFCI shocks and macro-financial dynamics
- SVAR specification:
  - y_t = [logGDP_t, logP_t, fedfunds_t, VFCI_t]′, quarterly from 1962Q1 to 2022Q3 (243 observations).
  - logGDP: log real GDP (GDPC1); logP: log PCEPILFE; fedfunds: quarterly average of FEDFUNDS; VFCI as constructed.
- Summary statistics (Table 6):
  - logGDP: Mean 9.23, SD 0.51, Min 8.23, Max 10.00.
  - logP: Mean 3.93, SD 0.62, Min 2.79, Max 4.75.
  - fedfunds: Mean 4.87, SD 3.72, Min 0.06, Max 17.78.
  - VFCI: Mean 0.00, SD 1.00, Min -1.51, Max 4.45.
- Identification through heteroskedasticity:
  - Sample partitioned into volatility subperiods: 1962Q1-1969Q4; 1970Q1-1979Q3; 1979Q4-1982Q4; 1983Q1-1989Q4; 1990Q1-2007Q4; 2008Q1-2010Q4; 2011Q1-2019Q4; 2020Q1-2022Q3.
  - Relative variance posterior medians (examples):
    - 1962Q1-1969Q4 Lowflation: logGDP 1.13, logP 0.17, fedfunds 0.38, VFCI 0.62.
    - 2008Q1-2010Q4 Financial crisis: logGDP 0.80, logP 0.75, fedfunds 0.46, VFCI 1.65.
    - 2020Q1-2022Q3 Covid-19 pandemic: logGDP 1.53, logP 3.01, fedfunds 0.58, VFCI 1.10.
  - Structural shocks assumed Student-t distributed; median degrees of freedom reported as 2.5 in impulse response draws.
  - Marginal data densities (MDD) favor baseline with t-distributed structural shock variances across subperiods:
    - Baseline (structural shock variances, t) MDD = 1,513.
    - Nothing varying, Gaussian MDD = 1,393.
    - Structural shock variances, Gaussian MDD = 1,324.
    - VAR coefficients Aj varying, Gaussian MDD = 958.
- Impulse response findings:
  - Positive one-standard-deviation VFCI shocks (higher risk premia / tighter financial conditions) cause:
    - GDP decreases by around 0.4 percent for over 6 quarters and approaches a 0.5 percent decline over 20 quarters.
    - Federal funds rate lowers by around 20 basis points (annualized) at the peak around 7 quarters after the shock.
    - Price level declines slightly; effect on inflation is weak (does not exit 90 percent error band).
    - Effects on GDP and federal funds rate remain statistically significant (90 percent error bands exclude zero for more than 20 quarters).
  - Positive one-standard-deviation federal funds shocks (contractionary monetary policy) cause:
    - Transitory yet persistent decline in output.
    - VFCI increases strongly: median response around 5 percent on impact, peaking near 7.5 percent over 3 quarters, dissipating after 8 quarters.
  - Interpretation:
    - VFCI shocks are an important causal source of macroeconomic fluctuations.
    - Monetary policy shocks induce substantial changes in the market price of risk (VFCI) for multiple quarters.
    - One-standard-deviation VFCI shock ~ 25 basis points in federal funds rate magnitude.

### Identification alternatives and robustness
- Alternative identification strategies produce similar IRFs:
  - SVAR-IV using external instruments for GDP and fed funds; treating VFCI as exogenous as instrument.
  - Local Projections with IVs (LP-IV) using same instruments.
  - Recursive VAR (Cholesky) ordering federal funds and VFCI last; robust to ordering of those two.
  - Sign restrictions imposing economic sign constraints on impact responses.
- Comparisons:
  - Five identification strategies (Vol-SVAR baseline, SVAR-IV, LP-IV, Cholesky, Sign Restrictions) produce broadly similar IRFs for interactions between VFCI, monetary policy, and GDP.
- Robustness checks summarized:
  - VFCI materially unchanged when replacing GDP with consumption; using all financial variables vs. principal components; using three or five PCs; including SDF news; adding four lags of log GDP.
  - Impulse responses robust when using alternative VFCI constructions; using stationary variables; replacing VFCI with exp(VFCI); expanding MCMC draws from 10,000 to 1,000,000; varying Minnesota prior tightness (1, 2, or 5) and decay (0.3, 0.5, 0.7).
  - Adding a second financial variable (GZ spread, excess CAPE yield, 3-month Libor-Treasury spread, or NFCI) does not change VFCI shock dynamics.
  - Estimating VFCI within the VAR (using implied volatility of GDP growth from the VAR) yields essentially the same VFCI and impulse responses.

### Identification sensitivity: total vs residual volatility
- VFCI defined as the projection of the log market price of risk onto the asset span: ˆδPC t.
- Alternative volatility measures:
  - Residual log volatility: ˆυ t+1 (residual from equation (25)).
  - Total log volatility: ˆδPC t + ˆξ t+1.
- Re-estimating heteroskedastic BVAR:
  - Using residual log volatility instead of VFCI: IRFs differ substantially — slightly positive response in real GDP and no response in federal funds rate or price level.
  - Using total log volatility instead of VFCI: IRFs show little to no impact on macro dynamics.
- Implication:
  - Projecting total log volatility onto returns (asset span) to construct VFCI extracts the economically meaningful signal that drives macro dynamics.

### Policy-relevant implications and recommendations
- Theoretical grounding:
  - VFCI is a theoretically grounded measure of the economy‑wide price of risk derived from asset pricing theory under a representative consumer framework.
- Key empirical properties:
  - Correlated with other FCIs but with distinct features and superior predictive power for equity, Treasury, and corporate bond risk premia.
  - Constructed from widely available financial data, computationally tractable, and with a relatively long time series.
  - Could be computed globally to track financial conditions in real time across countries.
- Causal links and policy takeaways:
  - Shocks that raise the VFCI (higher market price of risk / tighter financial conditions) materially depress output and lead to easing in monetary policy (lower federal funds rate) over multi-year horizons.
  - Contractionary monetary policy raises the VFCI significantly for multiple quarters, implying substantial two-way interactions between policy and the market price of risk.
  - The VFCI can serve as a useful macro-financial indicator for assessing likely evolution of risk premia and transmission of financial shocks to the real economy and policy rates.
- Recommendations and further research:
  - Compute the VFCI for additional countries, conduct asset pricing tests, and embed the VFCI into structural macro‑financial models.

### Supplemental theoretical and technical appendices (selected exact elements)
- Probability space and consumption notation: (Ω,F,fF_tg_{t∈Z},P); consumption C = {C_s}_{s∈Z} with C_s > 0.
- Assumptions A1–A5 (preserved exactly from source).
- Linearized SDF representation (A.32, A.34) and coefficients definitions (α, κ_j, ξ_k, ζ_k) preserved exactly.
- Internal habit formation and Epstein‑Zin linearizations with exact kernel and coefficient patterns preserved.
- Bond prices, yields, term premium definitions (A.37–A.39) preserved exactly.
- SDF news approximation and empirical estimators (A.40–A.49) preserved exactly, with empirical statement: the consistent empirical estimate of the fourth‑order upper bound is "below310
−5
basis points for all maturities."
- Robustness notes on VFCI estimation:
  - Baseline uses four PCs; robustness to three or five PCs; alternative constructions (heteroskedastic regression, inclusion of lags, alternative yields) produce series similar to baseline.
  - Equity-returns-only VFCI differs noticeably in the 1990s and COVID-19 period.
- Table B.1 R-squared (regressions of FCIs on principal components) preserved exactly:
  - Number of PCs 1: VFCI 0.23 0.04 0.05 0.63
  - Number of PCs 2: VFCI 0.68 0.73 0.11 0.71
  - Number of PCs 3: VFCI 0.83 0.81 0.61 0.71
  - Number of PCs 4: VFCI 1.00 0.81 0.73 0.80
  - Number of PCs 5: VFCI 1.00 0.83 0.74 0.82
  - Number of PCs 6: VFCI 1.00 0.83 0.75 0.85
- Residual distribution diagnostics (Table B.2 exact reporting preserved):
  - Log Real GDP: Mardia Kurtosis Statistic 11609  p-value 0.0021; Shapiro-Wilk Statistic 0.680
  - Log Core PCE: Mardia Kurtosis Statistic 1102500.940; Shapiro-Wilk Statistic 0.500
  - VFCI: Mardia Kurtosis Statistic 42033200.760; Shapiro-Wilk Statistic 0.760
  - Fed Funds: Mardia Kurtosis Statistic 82018400.680; Shapiro-Wilk Statistic 0.680
  - Small p-values indicate strong rejection of normality; motivates t-distributed shocks in baseline.

*Source: wpiea2023199-print-pdf - 1990 / 2020 / 2018 (selected content preserved exactly from source).*

### 2020. The VFCI also shows marked increases around other periods of financial distress: the

### wpiea2023199-print-pdf - 2020. The VFCI also shows marked increases around other periods of financial distress: the

### Construction of the VFCI and empirical specification
- Model equations (as presented):
  - ∆gdp_{t+1} = θ P C_{t} + ε_{t+1},   V ar_{t}(ε_{t+1}) = exp(2 δ P C_{t}),
  - VFCI is defined as ˆδ P C_{t} standardized to have zero mean and unit variance.
- Estimation details:
  - We estimate parameters by restricted maximum likelihood.
  - P C_{t} is a vector with an intercept and the first four principal components of the financial variables in Table 1.
  - ε_{t+1} is a zero-mean error.
- Sample:
  - 1962Q1-2022Q3 (243 quarterly observations).

### Historical behavior and interpretation of the VFCI
- Key historical episodes when VFCI shows marked increases:
  - Tumultuous 2000-2002 period (burst of the dot-com bubble, the 9/11 terrorist attacks, corporate scandals of the early 2000’s).
  - 1994 bond market crisis.
  - “Black Monday” stock market crash of 1987.
  - Savings and loans crisis of the early 1980s.
  - 40% stock market decline in 1973–1974.
  - Spike in 1998Q3 during the collapse of Long-Term Capital Management and the Russian financial crisis.
- Relationship with real economic activity:
  - Financial conditions and real aggregate activity are related but distinct.
  - The VFCI can be low during macroeconomic downturns and high during booms (example: the VFCI does not show unusually high levels during the 1990-1991 recession).
  - Between 1996Q2 and 2000Q2 the VFCI increased steadily while real economic performance was strong:
    - Year-over-year real GDP growth above 4% every quarter.
    - Unemployment declined from 5.5% to 4%.
    - Core inflation declined from 3% to 2%.

### Mean-volatility relation and economic amplification
- Signs and identification:
  - The sign of each of the PCs used in the estimation of equations (24)-(25) is not identified, so the signs of the coefficients ˆθ are arbitrary.
  - The authors select the signs of the PCs so that the coefficients of ˆθ in Panel A of Table 3 are all positive.
  - Once the signs of the PCs (and therefore of the elements of ˆθ) are chosen, the signs of the coefficients ˆδ in the volatility equation are determined by the data and are not arbitrary.
- Empirical finding on directions:
  - Comparing signs of ˆδ in Panel A and ˆθ in Panel B shows that all of the PCs induce movements of the conditional mean and the conditional volatility of GDP growth and consumption growth that go in opposite directions.
  - Interpretation: periods with high expected GDP growth are usually accompanied by a low conditional volatility of GDP growth—a low VFCI.
- Scatter plot evidence (Figure 2):
  - There is a tight linear relation with a negative slope between the VFCI (conditional log volatility of GDP growth on the horizontal axis) and the conditional mean of GDP growth (vertical axis).
  - In periods of tight financial conditions (high VFCI), the economy tends to be in low-mean, high-volatility states where negative shocks of a given magnitude translate into much larger declines in GDP.
  - Conversely, a low VFCI tends to be accompanied by relatively long periods of positive growth that are more resilient to shocks.

### Comparison with other financial conditions indices (FCIs)
- Indices reviewed: NFCI (Federal Reserve Bank of Chicago), GSFCI (Goldman Sachs financial conditions index), VIX (Chicago Board Options Exchange).
- NFCI:
  - Weighted average of 105 indicators of financial activity.
  - Provides a “weekly update on US financial conditions in money markets, debt and equity markets, and the traditional and shadow banking systems” starting in January 1973 (Brave and Butters (2011)).
  - Weights obtained through a mixed-frequency dynamic factor analysis.
  - Renormalized each week so that its sample mean and variance are zero and one; higher values indicate tighter financial conditions.
- GSFCI:
  - Weighted average of short-term interest rates, long-term interest rates, a trade-weighted dollar exchange rate, an index of credit spreads, and the ratio of aggregate stock market prices to the 10-year average of earnings per share.
  - Available daily starting January 1, 1983.
  - Weights reflect the impact that a shock to the variable has on GDP growth over the four quarters following the shock, estimated from quarterly OLS regressions starting in 1984.
  - Designed to be a measure of conditional GDP growth, with higher values indicating tighter financial conditions and lower expected GDP growth over the following year.
  - The 10-year Treasury yield and the index of credit spreads account for most of the index, with a sum of their weights equal to 84.7%.
- VIX:
  - Measure of expected volatility of the aggregate US stock market over the next 30 days.
  - Available daily (and at higher frequencies) starting in January 1,  (text truncated in source).

*Source: wpiea2023199-print-pdf - 2020.*

### 1990. It is derived from real-time, mid-quote prices of S&P 500 Index call and put options.

### wpiea2023199-print-pdf - 1990. It is derived from real-time, mid-quote prices of S&P 500 Index call and put options.

### VFCI and comparison with other financial conditions indices
- The VIX is described as a “fear gauge” and an indicator of the price of risk of the aggregate stock market.
- Four FCIs are compared: VFCI, NFCI, GSFCI, and VIX; they exhibit clear comovement especially in stress episodes (e.g., around the 2008 global financial crisis) but also notable differences:
  - GSFCI eased more than the other FCIs in 2021.
  - VIX and VFCI were more elevated around the tech bubble in the late 1990s.
  - NFCI declined faster after the 1982 recession.
  - VFCI exhibits higher volatility during more recent recessions than alternative FCIs, but lower volatility in times of financial stress before the 2000s (e.g., early 1980s recession).
- Principal components (PCs) of financial variables explain a large fraction of variation in FCIs:
  - Regression specification: FCI_t = α + β PC_t + ε_t, where PC_t are the first four principal components of six financial variables.
  - R² values for regressions of FCIs on the first four PCs lie between 0.70 and 1.00 (table reports R²: VFCI = 1.00, NFCI = 0.81, GSFCI = 0.73, VIX = 0.80).
  - All but two coefficients are statistically significant at the 1% level.
  - Three or four PCs are needed to achieve the high R²s; increasing PCs beyond four adds little additional information.

### VFCI’s predictive power for asset risk premia
- VFCI predicts risk premia for equities, Treasuries, and corporate bonds, and generally outperforms NFCI, GSFCI, and VIX when considered jointly.
- Risk premia measures used (all in annualized percentage points; FCIs standardized to zero mean and unit variance):
  - Equity: S&P 500 excess CAPE yield (ECY) of Shiller (2000).
  - Treasuries: 10-year term premium (TP) of Adrian, Crump, and Moench (2013).
  - Corporate bonds: GZ spread of Gilchrist, Yankov, and Zakrajšek (2009).
- Key regression findings (Panel A–C, Table 5):
  - Panel A (Stocks, ECY):
    - Lag VFCI coefficient = 0.13*** when included alone (column (1)), significant at the 1% level.
    - A one-standard-deviation increase in VFCI predicts an increase of 13 basis points in next quarter’s (annualized) equity risk premium.
    - VFCI alone predicts 22% of one-quarter-ahead ECY variation not explained by lagged ECY.
    - When all four FCIs included jointly (column (5)), VFCI coefficient remains the only significant one and accounts for 31% of one-period-ahead ECY variance not accounted for by current ECY.
    - Lag ECY coefficients reported: 0.97***, 0.93***, 0.91***, 0.94***, 0.94*** with t-statistics (45.1), (31.5), (30.9), (27.8), (32.1).
    - Sample sizes N reported across columns: 242, 206, 157, 130, 129. R² reported: 0.95, 0.95, 0.93, 0.90, 0.91.
  - Panel B (Treasuries, TP):
    - Lag VFCI coefficient = 0.08*** when included alone; significant at the 1% level.
    - VFCI explains 26% of the variance in TP not explained by lagged TP when alone; rises to 48% when included jointly with other FCIs.
    - Lag TP coefficients reported: 0.98***, 0.96***, 0.95***, 0.96***, 1.02*** with t-statistics (68.4), (55.9), (36.2), (48.3), (33.0).
    - Sample sizes N: 242, 206, 157, 130, 129. R²: 0.95, 0.94, 0.94, 0.94, 0.94.
  - Panel C (Corporate Bonds, GZ):
    - Lag VFCI coefficient = 0.21*** when included alone; significant at the 1% level.
    - VFCI explains 52% of the variance in GZ not explained by lagged GZ when alone; rises to 60% when included jointly with other FCIs.
    - Lag GZ coefficients reported: 0.82***, 0.90***, 0.89***, 0.68***, 0.60*** with t-statistics (14.0), (13.7), (12.8), (6.4), (4.4).
    - Sample sizes N: 197, 197, 157, 130, 129. R²: 0.86, 0.82, 0.79, 0.85, 0.89.
- Conclusion: VFCI predicts risk premia for equities, Treasuries, and corporate bonds when used alone, and is the only FCI with significant predictive power when all FCIs are included jointly.

### VFCI shocks and macro-financial dynamics (SVAR evidence)
- SVAR specification:
  - y_t = [logGDP_t, logP_t, fedfunds_t, VFCI_t]′, quarterly from 1962Q1 to 2022Q3 (243 observations).
  - logGDP: log real GDP (FRED code GDPC1); logP: log of PCEPILFE; fedfunds: quarterly average of FEDFUNDS; VFCI as constructed in Section III.
  - Summary statistics (Table 6):
    - logGDP: Mean 9.23, SD 0.51, Min 8.23, Max 10.00.
    - logP: Mean 3.93, SD 0.62, Min 2.79, Max 4.75.
    - fedfunds: Mean 4.87, SD 3.72, Min 0.06, Max 17.78.
    - VFCI: Mean 0.00, SD 1.00, Min -1.51, Max 4.45.
- Identification through heteroskedasticity:
  - Sample partitioned into volatility subperiods (1962Q1-1969Q4; 1970Q1-1979Q3; 1979Q4-1982Q4; 1983Q1-1989Q4; 1990Q1-2007Q4; 2008Q1-2010Q4; 2011Q1-2019Q4; 2020Q1-2022Q3).
  - Relative variance posterior medians (Table 7) reported relative to average normalized to 1; examples:
    - 1962Q1-1969Q4 Lowflation: logGDP 1.13, logP 0.17, fedfunds 0.38, VFCI 0.62.
    - 2008Q1-2010Q4 Financial crisis: logGDP 0.80, logP 0.75, fedfunds 0.46, VFCI 1.65.
    - 2020Q1-2022Q3 Covid-19 pandemic: logGDP 1.53, logP 3.01, fedfunds 0.58, VFCI 1.10.
  - Structural shocks assumed Student-t distributed (fat tails); median degrees of freedom reported as 2.5 in impulse response draws.
  - Marginal data densities (MDD) for VAR specifications (Table 8) favor baseline with t-distributed structural shock variances across subperiods:
    - Baseline (structural shock variances, t) MDD = 1,513.
    - Nothing varying, Gaussian MDD = 1,393.
    - Structural shock variances, Gaussian MDD = 1,324.
    - VAR coefficients Aj varying, Gaussian MDD = 958.
  - t-distributed shocks improve fit and identification relative to Gaussian shocks.
- Impulse response findings (Figure 4 and text):
  - Positive one-standard-deviation VFCI shocks (higher risk premia / tighter financial conditions) cause:
    - Economically large reductions in real GDP: GDP decreases by around 0.4 percent for over 6 quarters and approaches a 0.5 percent decline over 20 quarters.
    - Persistent reductions in the federal funds rate: federal funds rate lowers by around 20 basis points (annualized) at the peak around 7 quarters after the shock.
    - Price level declines slightly; effect on inflation is weak (does not exit 90 percent error band).
    - Effects on GDP and federal funds rate remain statistically significant (90 percent error bands exclude zero for more than 20 quarters).
  - Positive one-standard-deviation federal funds shocks (contractionary monetary policy) cause:
    - Transitory yet persistent decline in output.
    - VFCI increases strongly: median response around 5 percent on impact, peaking near 7.5 percent over 3 quarters, dissipating after 8 quarters.
  - Interpretation:
    - VFCI shocks are an important causal source of macroeconomic fluctuations.
    - Monetary policy shocks induce significant changes in the price of risk (VFCI) for multiple quarters.
    - The VFCI is comparable in magnitude to a 25 basis point change in the federal funds rate (one-standard-deviation VFCI shock ~ 25 bps).

### Identification alternatives and robustness
- Alternative identification strategies considered produce similar impulse responses:
  - Instrumental Variables (SVAR-IV) using:
    - GDP instrument: growth news shock from Cieslak and Pang (1983Q1–2022Q3; converted to quarterly).
    - Fed funds instrument: Romer and Romer (1969Q1–1994Q4) and Nakamura and Steinsson (1995Q1–2022Q3) series spliced and converted.
    - VFCI instrument: treating VFCI as exogenous (all reduced form innovations to VFCI are structural VFCI shocks).
  - Local Projections with IVs (LP-IV) using same instruments.
  - Recursive VAR (Cholesky) ordering federal funds and VFCI last; results robust to ordering of those two.
  - Sign restrictions imposing economic sign constraints on impact responses.
- Comparisons (Figures 5 and 6):
  - Five identification strategies (Vol-SVAR baseline, SVAR-IV, LP-IV, Cholesky, Sign Restrictions) produce broadly similar IRFs for interactions between VFCI, monetary policy, and GDP.
- Robustness checks (Appendix references summarized in text):
  - VFCI remains materially unchanged when: replacing GDP with consumption; using all financial variables vs. principal components; using three or five PCs; including SDF news; adding four lags of log GDP.
  - Impulse responses remain robust when: using alternative VFCI constructions; using stationary variables; replacing VFCI with exp(VFCI); expanding MCMC draws from 10,000 to 1,000,000; varying Minnesota prior tightness (1, 2, or 5) and decay (0.3, 0.5, 0.7).
  - Adding a second financial variable (GZ spread, excess CAPE yield, 3-month Libor-Treasury spread, or NFCI) does not change VFCI shock dynamics.
  - Estimating VFCI within the VAR (using implied volatility of GDP growth from the VAR) yields essentially the same VFCI and impulse responses.

### Analytical conclusions and policy-relevant implications
- The VFCI is a theoretically grounded measure of the economy-wide price of risk derived from asset pricing theory under a representative consumer framework.
- Key empirical properties of the VFCI:
  - Correlated with other FCIs but with distinct features and superior predictive power for equity, Treasury, and corporate bond risk premia.
  - Constructed from widely available financial data, computationally tractable, and with a relatively long time series.
  - Could be computed globally to track financial conditions in real time across countries.
- Causal links and policy takeaways:
  - Shocks that raise the VFCI (higher market price of risk / tighter financial conditions) materially depress output and lead to easing in monetary policy (lower federal funds rate) over multi-year horizons.
  - Monetary policy tightening raises the VFCI significantly for multiple quarters, implying substantial two-way interactions between policy and market price of risk.
  - The VFCI can serve as a useful macro-financial indicator for assessing the likely evolution of risk premia and the transmission of financial shocks to the real economy and policy rates.

*wpiea2023199-print-pdf - 1990. It is derived from real-time, mid-quote prices of S&P 500 Index call and put options.*

### conclusions remain the same: a tightening of financial conditions based on the VFCI leads

### wpiea2023199-print-pdf - conclusions remain the same: a tightening of financial conditions based on the VFCI leads

### Key findings
- "a tightening of financial conditions based on the VFCI leads to an immediate easing of monetary policy and a persistent contraction in output."
- "Con- versely, contractionary monetary policy shocks lead to a tightening of financial conditions."
- "In contrast, output shocks do not move the VFCI much, and inflation seems to be mostly unrelated to it."
- "These results are encouraging, as they suggest a step forward in estimating financial conditions based on economic theory, with broad applicability and uses in policy- making."

### Policy implications
- Tightening in financial conditions (as measured by the VFCI) is associated with an immediate monetary policy response in the form of easing.
- Such financial-tightening episodes are associated with a persistent contraction in output, implying that policymakers should account for financial conditions when setting policy.
- Contractionary monetary policy can itself tighten financial conditions, indicating feedback between monetary policy and financial conditions.

### Recommendations and further research
- "Further research could compute the VFCI for additional countries, conduct asset pricing tests, and embed the VFCI into structural macro-financial models."

*Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023199-print-pdf.pdf*

### References

### wpiea2023199-print-pdf - References

### Supplemental Appendix — Theoretical framework and notation
- Probability space and consumption paths: (Ω,F,fF_tg_{t∈Z},P); consumption C = {C_s}_{s∈Z} with C_s > 0.
- Constant path: ̄C > 0 and log‑deviation g_s := log C_s − log ̄C; one‑period log‑growth ∆c_s := g_s − g_{s−1}.
- One‑sided exponentially weighted L^2 norm: kxk := [E(Σ_{t∈Z} β^{|t|} |x_t|^2)]^{1/2} for fixed β ∈ (0,1).
- Assumptions (A1–A5) (preserved exactly from source):
  - A1. Differentiability of U: continuation value U_t is twice continuously Fréchet differentiable in an open neighborhood of the constant path ̄C.
  - A2. Conditional Riesz representations for DU_t: DU_t[̄C] admits a conditional Riesz representation with F_{max{t,s}}‑measurable kernel G_{t,s}(C) such that DU_t[̄C]·h = Σ_{s∈Z} E_t[G_{t,s}(̄C) h_s] for all h ∈ L^2.
  - A3. Conditional Riesz representations for DM_{t+1}: DM_{t+1}[̄C] admits conditional Riesz representations with F_{max{t+1,s}}‑measurable kernels K_{t+1,s}(̄C) and Z_{t+1,s}(̄C) such that DM_{t+1}[̄C]·h = Σ_{s∈Z} E_{t+1}[K_{t+1,s}(̄C) h_s] − Σ_{s∈Z} E_t[Z_{t+1,s}(̄C) h_s] for all h ∈ L^2.
  - A4. K_{t+1,s}(̄C) and Z_{t+1,s}(̄C) are deterministic, absolutely summable, and time‑homogeneous: K_{t+1,s}(̄C)=K_{s−(t+1)}, Z_{t+1,s}(̄C)=Z_{s−(t+1)}.
  - A5. Level of C determined by growth: g_{t−m} → 0 as m → 1 in L^2 (implying g_t = Σ_{j=0}^∞ ∆c_{t−j}).

### Linearized stochastic discount factor (SDF)
- First‑order expansion around ̄C yields (re‑indexed):
  - M_{t+1}(C) = Λ_0 + ̄C Σ_{p∈Z} K_p E_{t+1}[g_{t+1+p}] − ̄C Σ_{p∈Z} Z_p E_t[g_{t+1+p}] + o(kgk). (A.32)
- Representation in terms of one‑period growth rates and news:
  - Define news N_{t+1,m} := E_{t+1}[∆c_{t+1+m}] − E_t[∆c_{t+1+m}].
  - Linearized SDF (A.34):
    - M_{t+1}(C) = α + Σ_{j=1}^∞ κ_j ∆c_{t+1−j} + Σ_{k=0}^∞ ξ_k E_t[∆c_{t+1+k}] + Σ_{k=0}^∞ ζ_k N_{t+1,k} + o(k∆ck),
    - with coefficients α = Λ_0, κ_j = ̄C Σ_{p=−j}^∞ (K_p − Z_p) for j ≥ 1, ξ_k = ̄C Σ_{p=k}^∞ (K_p − Z_p) for k ≥ 0, ζ_k = ̄C Σ_{p=k}^∞ K_p for k ≥ 0.

### Internal habit formation — linearization and SDF representation
- Utility with internal habits: U_t(C) = E_t Σ_{s=t}^∞ β^{s−t} u(C_s,H_s), H_s = Σ_{j=1}^∞ α_j C_{s−j}, with 0 < β < 1 and Σ_{j=1}^∞ j α_j^j < 1.
- First‑order condition and SDF definition: M_t(C) := ∂U_t(C)/∂C_t and FOC M_{t+1}(C) = β M_{t+1}(C) / M_t(C). (A.35)
- Linearization yields cumulative impact coefficients Ξ_j and representation:
  - SDF_{t+1} (equation A.36) expressed in terms of φ_j, Ξ_j, E_t[∆c_{·}], and news terms; specific definitions preserved in source:
    - ̄M = u_C + Σ_{k=1}^∞ β^k α_k u_H; φ_0 = ̄C (u_{CC} + u_{HH} Σ_{k=1}^∞ β^k α_k^2); etc.
- One‑period internal habits, CRRA surplus consumption special case:
  - For u(C,H) = (C − H)^{1−γ}/(1−γ), α_1 = α, α_j = 0 for j > 1:
    - SDF_{t+1} = β + [βγ/((1−α)(1−αβ))] [ α ∆c_t + ((1−α)αβ − 1) ∆c_{t+1} + αβ (E_{t+1} ∆c_{t+2} − E_t ∆c_{t+1}) ].

### Epstein‑Zin utility — linearization and kernels
- Epstein‑Zin recursion: U_t = [(1−β) C_t^ρ + β S_t^ρ]^{1/ρ}, S_t = [−E_t U_{t+1}^{1−γ}]^{1/(1−γ)}, with β ∈ (0,1), γ > 0 (risk aversion), ψ > 0 (EIS), ρ := 1 − 1/ψ.
- First‑order condition: M_{t+1} = β (C_{t+1}/C_t)^{−1/ψ} (U_{t+1}/S_t)^{1/ψ − γ}.
- Log‑linearized FOC (A.7): D log M_{t+1} = −(1/ψ) ∆c_{t+1} − θ Σ_{j=1}^∞ β^j (E_{t+1} − E_t)[∆c_{t+1+j}], where θ := γ − 1/ψ.
- Resulting kernel patterns (K_p, Z_p) and explicit simplifications:
  - κ_j = 0 for all j ≥ 1, ξ_0 = −β/ψ, ξ_k = 0 for k ≥ 1, ζ_0 = −β/ψ, ζ_k = −β^{k+1} (γ − 1/ψ) for k ≥ 1.
  - Alternative compact form: M_{t+1}(C) = β − (β/ψ) ∆c_{t+1} − β (γ − 1/ψ) Σ_{k=1}^∞ β^k (E_{t+1} − E_t)[∆c_{t+1+k}] + o(k∆ck).

### Rational expectations equation and existence/uniqueness
- FOC rewritten (A.5 form): SDF_{t+1} = ̄g + Σ_{s=1}^∞ g_{−s} ∆c_{t+1−s} + g_0 ∆c_{t+1} + Σ_{s=1}^∞ g_s E_t[∆c_{t+1+s}] + Σ_{s=1}^∞ k_s E_{t+1}[∆c_{t+1+s}], where ̄g, g_s, k_s are functions of α, κ_j, ζ_k.
- Solution existence and uniqueness conditions:
  - Exists if coefficient system (g_s, k_s) admits a canonical Wiener–Hopf factorization.
  - Unique if B(z) has no zeros on the unit circle.
  - When it exists, solution is measurable w.r.t. filtration σ{SDF_s, ∆c_s : s ≤ t}.

### Bond prices, yields, term premium
- One‑period no‑arbitrage bond price: P_t^{(1)} = E_t SDF_{t+1}. (A.37)
- Log‑price p_t^{(n)} := log P_t^{(n)}, with P_t^{(0)} := 1.
- Continuously compounded yield: y_t^{(n)} := −(1/n) p_t^{(n)}. (A.38)
- Term premium: TP_t^{(n)} := y_t^{(n)} − (1/n) Σ_{j=0}^{n−1} E_t y_{t+j}^{(1)}. (A.39)

### SDF news — approximation and empirical estimation
- SDF news as bond‑price news: ∆_{t+1} SDF_{t+1+i} = ∆_{t+1} P_{t+i}^{(1)} = ∆_{t+1} e^{p_{t+i}^{(1)}}. (A.40)
- Second‑order approximation (A.40):
  - ∆_{t+1} SDF_{t+1+i} = e^{E_t p_{t+i}^{(1)}} [ ∆_{t+1} p_{t+i}^{(1)} + (1/2) (∆_{t+1} p_{t+i}^{(1)})^2 ] − B_i,
  - with B_i := (1/2) E[ e^{E_t p_{t+i}^{(1)} − ∆_{t+1} p_{t+i}^{(1)}} (∆_{t+1} p_{t+i}^{(1)})^2 ]. (A.41)
- Bond‑price news identity via yields and term premia (A.43–A.45): exact finite‑horizon algebra expressing ∆_{t+1} p_{t+i}^{(1)} in terms of y^{(m)} and TP^{(m)} for m = i−1, i, i+1.
- Empirical estimators (A.46–A.49):
  - ˆn_{i,t} := i y_t^{(i)} − (i+1) y_t^{(i+1)} + (i+1) TP_t^{(i+1)} − i TP_t^{(i)}.
  - ˆn_{i−1,t+1} := (i−1) y_{t+1}^{(i−1)} − i y_{t+1}^{(i)} + i TP_{t+1}^{(i)} − (i−1) TP_{t+1}^{(i−1)}.
  - ˆN_{i,t+1} := ˆn_{i−1,t+1} − ˆn_{i,t}.
  - ˆB_i := (1/2)(1/(T−i)) Σ_{t=1}^{T−i} e^{ˆn_{i,t}} ˆN_{i,t+1}^2.
- Empirical approximate SDF news:
  - ∆_{t+1} SDF_{t+1+i} ≈ e^{ˆn_{i,t}} [ ˆN_{i,t+1} + (1/2) ˆN_{i,t+1}^2 ] − ˆB_i.

### Statistical properties of the SDF news approximation
- Unbiasedness of approximation error (A.54):
  - Define error_{i,t+1} := ∆_{t+1} e^{p_{t+i}^{(1)}} − e^{E_t p_{t+i}^{(1)}} [ ∆_{t+1} p_{t+i}^{(1)} + (1/2)(∆_{t+1} p_{t+i}^{(1)})^2 ] + B_i.
  - E[error_{i,t+1}] = 0 (the approximation is unbiased by construction).
- Variance bound (fourth‑order upper bound) (A.56 and following):
  - Var(error_{i,t+1}) is bounded above by a term of fourth order in deviations of p_{t+i}^{(1)} − E_{t+1} p_{t+i}^{(1)}; algebra yields an explicit inequality introducing a 1/4 factor and sup_t e^{2 E_t p_{t+i}^{(1)}} multiplicative term.
- Empirical upper bound construction (A.57–A.58):
  - Var(error_{i,t+1}) ≤ (1/4) bc_i b_k_i,
  - b_k_i := (1/(T−i)) Σ_{t=1}^{T−i} ( − y_t^{(1+i)} − ˆn_{i−1,t+1} )^4,
  - bc_i := max_{1≤t≤T} e^{2 ˆn_{i,t}}.
- Empirical statement in source: the consistent empirical estimate of the fourth‑order upper bound is "below310
−5
basis points for all maturities." (text preserved exactly).

### Robustness and VFCI estimation notes
- VFCI (volatility financial conditions index) estimation choices and robustness checks:
  - Principal component span: analysis indicates more than one principal component is needed to span indexes NFCI, GSFCI, and VIX; after four principal components additional PCs add little — baseline uses four PCs but robustness to three or five is shown.
  - Alternate GDP measure: VFCI computed using real PCE instead of real GDP yields very similar coefficients and virtually indistinguishable series.
  - Transformations: VFCI is a log‑volatility measure; exp(VFCI) (rescaled to mean zero and standard deviation one) preserves shape but amplifies peaks.
  - Using three or five PCs: differences are minor; using five PCs reduces the 1970s peak but series remains similar.
  - Variations including realized volatility, SDF news horizons (1, 5, 9 years), four lags of real GDP growth, using all financial variables, and using consumption growth are shown graphically in Figure B.1 (figure referenced as in source).

*Italic line: Source: wpiea2023199-print-pdf - References (supplemental appendix content preserved exactly from source).*

### 1. The light blue lines show variations of the VFCI estimated for robustness. All series are scaled to

### wpiea2023199-print-pdf - 1. The light blue lines show variations of the VFCI estimated for robustness. All series are scaled to be mean zero with a standard deviation of one.

### Construction and alternative VFCI definitions
- Baseline VFCI uses six financial variables (Table 1) and PCA-based estimation; all series are scaled to be mean zero with a standard deviation of one.
- Alternative constructions considered:
  - Heteroskedastic regression of GDP growth on the six financial variables (yields a VFCI virtually indistinguishable from PCA-based VFCI, with a smaller peak during the 1970s).
  - Inclusion of four lags of the dependent variable (real GDP growth) in both the mean and volatility equations (series similar to baseline; notable overshooting and rebound during COVID-19 when lags included in the mean).
  - Inclusion of one-, five-, and ten-year U.S. Treasury yields in the mean (result nearly identical to baseline).
  - Exclusion of realized volatility (use first four principal components of the remaining five traded financial variables) (result similar to baseline).
  - VFCI constructed using only equity returns (returns-only VFCI is noticeably different from baseline, particularly during the 1990s and the COVID-19 period).

### Empirical relationship between FCIs and principal components
- Table B.1: R-squared of regressions of financial conditions indices on principal components of financial variables (preserve exact table values):
  - Number of PCs 1: VFCI 0.23 0.04 0.05 0.63
  - Number of PCs 2: VFCI 0.68 0.73 0.11 0.71
  - Number of PCs 3: VFCI 0.83 0.81 0.61 0.71
  - Number of PCs 4: VFCI 1.00 0.81 0.73 0.80
  - Number of PCs 5: VFCI 1.00 0.83 0.74 0.82
  - Number of PCs 6: VFCI 1.00 0.83 0.75 0.85
- Notes: Regressions of FCI_t = α + β PC_t + ε_t, where PC_t is vector of first n principal components of financial variables. Columns use different FCI measures—NFCI, GSFCI, and VIX. All series are quarterly and use the longest sample available for each FCI.

### BVAR heteroskedastic-regimes robustness (IRF evidence)
- Setup: four-variable BVAR with log real GDP, log core PCE, federal funds rate, and VFCI; identified using heteroskedastic regimes.
- Key robustness findings (IRF comparisons shown in Figures B.3–B.6, B.10, B.12):
  - Alternate VFCI specifications (multiple purple lines) identify a similar VFCI shock to baseline (green), except:
    - Two specifications show milder federal funds rate responses: (1) VFCI using five principal components, and (2) VFCI using all six financial variables. These two have a lower VFCI measure during the 1970s compared to baseline.
  - VFCI estimated with only equity returns produces almost no federal funds rate response; responses of VFCI, real GDP, and prices are similar to baseline for other variables.
  - Truncating sample to end in 2007Q4 (pre-Global Financial Crisis and pre-Covid) yields very similar average causal effects; response of log real GDP is attenuated.
  - Stationary-variable BVAR (real GDP growth, core PCE inflation, federal funds rate, baseline VFCI) yields cumulative IRFs similar to baseline levels-based model.
  - Bayesian prior robustness: baseline Minnesota prior tightness = 3 and decay = 0.5; tested tightness = 1, 2, 5 and decay = 0.3, 0.7; also increased draws to 1,000,000 from 10,000. Dynamic causal effects remain almost identical across these variations.
  - Estimating heteroskedastic BVAR with Gaussian residuals (instead of t-distributed) yields somewhat more pronounced responses of monetary policy and GDP to VFCI shocks, but conclusions remain the same: tightening of VFCI leads to persistent contraction of output and immediate easing of monetary policy; contractionary monetary policy shocks cause tighter financial conditions.
  - VAR-consistent VFCI (constructed using VAR reduced-form residuals of log GDP in heteroskedasticity step) is almost the same as baseline VFCI; IRFs show a much larger VFCI impulse but similar impacts on the other three BVAR variables.

### Adding additional financial variables
- Five-variable BVARs: adding in turn Excess CAPE Yield (ECY), Gilchrist and Zakrajšek (GZ) corporate bond risk premium, TED spread, and NFCI:
  - Inclusion of a fifth financial variable does not change implications of baseline four-variable case; magnitudes of responses of monetary policy, output, and prices to VFCI shocks remain similar.
- Three-financial-variable BVAR (VFCI, GZ spread, TED rate):
  - VFCI shock recreates baseline results: prolonged negative reaction in real GDP and federal funds rate.
  - Core PCE shows a larger negative response here (contrast to insignificant baseline result).
  - VFCI shock creates a positive response in the GZ spread with a delay of a few quarters; VFCI shock has no impact on TED rate.
  - GZ shock (when VFCI and TED rate included) has little impact on VFCI, real GDP, or federal funds rate.
  - TED rate shock has some long-run impact on real GDP and federal funds rate; no impact on VFCI or GZ spread.

### Residual distribution diagnostics and identification choices
- Residual normality diagnostics:
  - Q-Q plots of VAR residuals show strong nonlinear patterns; residuals deviate from 45-degree normal line.
  - Table B.2: Univariate normality tests of the VAR residuals (preserve exact reported values and p-values):
    - Log Real GDP: Mardia Kurtosis Statistic 11609  p-value 0.0021; Shapiro-Wilk Statistic 0.680
    - Log Core PCE: Mardia Kurtosis Statistic 1102500.940; Shapiro-Wilk Statistic 0.500
    - VFCI: Mardia Kurtosis Statistic 42033200.760; Shapiro-Wilk Statistic 0.760
    - Fed Funds: Mardia Kurtosis Statistic 82018400.680; Shapiro-Wilk Statistic 0.680
  - Notes: Small p-values indicate non-normality. Mardia tests kurtosis and skewness; Shapiro-Wilk tests normality. All tests reject normality with high significance.
- Implication: t-distributed shocks in baseline heteroskedastic BVAR are more efficient given non-normal residuals, but Gaussian-shock estimation yields similar substantive conclusions.

### Alternative identification: external instruments and SVAR-IV / LP-IV
- External instruments used:
  - VFCI instrument: treat VFCI as exogenous; for SVAR-IV use Cholesky with VFCI ordered first; for LP-IV use VFCI time series as instrument.
  - Monetary policy instrument: Romer and Romer monetary policy shock (1969Q1–1994Q4) interpolated with Nakamura and Steinsson shock (1995Q1–2022Q3).
  - GDP growth instrument: external instrument from Cieslak and Pang 2021 (1983Q1–2022Q3), estimated via sign-restricted VAR identifying growth news shocks; used as external instrument for GDP growth in stationary model.
- SVAR-IV framework summary:
  - Reduced-form B(L) y_t = η_t with η_t = H ε_t and structural shocks ε_t distributed ε_t ~ (0, σ_ε).
  - Impact matrix H = Θ_0 (the contemporaneous impulse response), with usual relationships Θ(L) = C(L) H and C(L) = [B(L)]^{-1}.
  - External instrument approaches are used as alternative identification strategies to estimate dynamic causal effects in SVAR-IV and LP-IV frameworks.

*Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023199-print-pdf.pdf*

### 2018. The SVAR-IV identification problem is to identifyΘ

### wpiea2023199-print-pdf - 2018. The SVAR-IV identification problem is to identifyΘ

### SVAR-IV identification: instrument conditions and estimation steps
- Instrument conditions:
  - Eε1,t z′t = α 6= 0 (B.61) — instrument relevance (contemporaneous correlation with ε1,t).
  - Eε2:n,t z′t = 0 (B.62) — instrument exogeneity (uncorrelated with other structural shocks).
- Estimation idea (summary as in Stock and Watson 2018 and text):
  - Fit reduced form system in equation B.59 to estimate vector of innovations ηt.
  - Regress all reduced-form innovations apart from the first, η2:n,t, on η1,t using zt as an instrument; residuals form κ2:n,t.
  - Regress η1,t sequentially on η2:n,t using κ2:n,t as instruments to identify ε2:n,t.
  - Using identified structural shocks εt, estimate dynamic causal effects (footnote 21 notes relation to external-instrument recursive VAR approaches such as Romer and Romer 2004 and discussion in Ramey 2016).

### Empirical SVAR-IV results (comparisons and main signs)
- VFCI shock (identified via external instrument / penalty function approach):
  - A tightening of financial conditions by a positive VFCI shock triggers an immediate easing of monetary policy and a contraction in output.
  - Dynamic responses of output and monetary policy are somewhat less persistent than in the heteroskedastic BVAR, but negative responses on impact are highly significant and similar in magnitude.
- Monetary policy and growth shocks:
  - A surprise increase in real GDP leads to an immediate easing of financial conditions; VFCI tightens in following quarters as it reverts to the mean.
  - Loosening of financial conditions in response to growth shock may reflect better identification of these shocks using an external instrument.

### Identification through LP-IV (Local Projections with instruments)
- LP setup and instrument conditions:
  - Moving average representation Yt = Θ(L) εt; impulse response of Yi at horizon h estimated from regression:
    - yi,t+h = Θh,i1 y1,t + uh i,t+h (B.63)
  - OLS invalid because y1,t correlated with uh i,t+h; LP-IV requires instrument conditions B.61 and B.62 and an additional lead-lag exogeneity condition:
    - Eεt+j,t z′t = 0, j 6= 0 (B.64) — instrument uncorrelated with historical and future shocks.
  - A separate LP-IV regression is estimated for each horizon h; serial correlation in errors is modeled; control variables can be included (see Stock and Watson 2018).
- LP-IV empirical findings:
  - Identified monetary policy shock has insignificant effects.
  - GDP growth shock leads to a tightening of the Fed Funds rate and a loosening of financial conditions upon impact.
  - VFCI shock leads to a significant easing of monetary policy and a significant contraction in output; dynamic effects are somewhat less persistent than in the heteroskedastic BVAR.

### Identification through a Recursive VAR
- Baseline ordering: output, prices, monetary policy, financial conditions (VFCI ordered last).
- Robustness check: alternative ordering with Federal Funds rate last.
- Motivation: financial conditions and monetary policy may be endogenous (citing empirical evidence), so alternative forcings are assessed.

### Identification through Sign Restrictions (penalty function approach)
- Sign restriction strategy:
  - Three sets of sign restrictions identify three shocks; each shock identified from its own model (non-exclusionary across shocks).
  - Monetary policy shock: restrict response of prices and output to be negative on impact.
  - Real GDP shock: restrict response of federal funds rate to be positive on impact.
  - VFCI shock: restrict response of real GDP and the federal funds rate to be negative on impact.
  - Unlisted variables are unrestricted; sign restrictions impose only the immediate sign on impact, not magnitudes.

### Total and Residual Volatility: robustness checks for VFCI
- VFCI definition: ˆδPC t — measure of conditional log volatility of real GDP growth (estimated according to equation (25)).
- Alternative volatility measures considered:
  - Residual log volatility: ˆυ t+1 (residual from equation (25)).
  - Total log volatility: ˆδPC t + ˆξ t+1 (total from equation (25)).
- Re-estimating heteroskedastic BVAR with alternative measures yields:
  - Using residual log volatility instead of VFCI:
    - IRFs differ substantially: slightly positive response in real GDP and no response in federal funds rate or price level.
    - Implies VFCI—the projection of the log market price of risk onto the asset span—is the only part of total log volatility with economically meaningful effects on macro dynamics.
  - Using total log volatility instead of VFCI:
    - IRFs show little to no impact on macroeconomic dynamics.
    - Implies projecting total log volatility onto returns (asset span) to construct VFCI extracts a signal not immediately available in total log volatility.
- Figure B.13 description (summary of plotting conventions):
  - Green line: impulse responses to VFCI structural shock in volatility-identified BVAR with 68 percent (dark green) and 90 percent (light green) posterior error bands.
  - Red line: impulse responses to Total Log Volatility Shock in volatility-identified BVAR estimated with total log volatility replacing VFCI.
  - Blue line: impulse responses to Residual Log Volatility Shock in volatility-identified BVAR estimated with residual log volatility replacing VFCI.

*The Market Price of Risk and Macro-Financial Dynamics — Working Paper No. WP/2023/199*

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_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023199-print-pdf.pdf_