## The U.S. Personal Saving Rate (WP/18/128, Section 1–3)

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---

### Introduction and summary
- Paper title and authors: The U.S. Personal Saving Rate. Authors: S. Ouliaris and C. Rochon. Authorized for distribution by Nigel Chalk. June 2018.
- Aim:
  - Develop a time series model for aggregate consumption to predict the U.S. personal saving rate.
  - Test for a structural break in consumption behavior because of the 2008 financial crisis.
- Key sample split:
  - Estimation (pre-crisis): 1961Q1-2007Q4.
  - Evaluation / prediction period: 2008Q1-2017Q4; focused out-of-sample forecast for 2017Q1-2017Q4 and out-of-sample predictions for 2018Q1-2019Q4.
- Stylized facts:
  - From 1975 to 2007 there was a clear downward trend in the personal saving rate.
  - In 2008 there was a significant rise in the saving rate that continued until the end of 2012.
  - The saving rate resumed its decline around 2013 and appears to be returning to its pre-2008 level.
  - The rise in the saving rate during 2008-2012 coincided with marked declines in real net worth and weak/volatile growth in real disposable income.

### Data, model specification, and empirical approach
- Data:
  - National Income and Product Accounts (NIPA, quarterly) data spanning 1961Q1 to 2017Q4. Data release referenced as March 29, 2018.
- Key variables in the long-run real consumption equation:
  - real consumption (rc)
  - real disposable income (rdy)
  - real household net worth (rnw)
  - dependency ratio (dep_ratio)
  - real federal funds rate (r_fed_rate)
- Long-run real consumption equation (preserved specification):
  - Equation (1.1)
    0    1234
    ln(  )ln(  )   ln(   )   _   _ln(  _   )
    tttttt
    rcrdyrnwr   fed   ratedep  ratioβ  ββββϵ=  ++++ +
- Rationale for inclusion:
  - Permanent Income Hypothesis and wealth effects motivate rdy and rnw (positive elasticities).
  - Inter-temporal substitution motivates r_fed_rate (proxied by the real federal funds rate).
  - Life-cycle hypothesis motivates dep_ratio (effect expected to be negative).
- Short-run dynamics:
  - Handled via a multi-equation error-correction model (VECM).
- Personal saving rate identity (preserved as in source):
  - Equation (1.2) (variables gov_transfers, i_payments and y_deflator treated as exogenous and, for empirical analysis, held at zero).

### Time series properties and cointegration
- Unit root evidence (ADF tests) for 1961Q1-2007Q4 suggests nonstationarity for all series in the long-run regression:
  - Real Consumption ln(rc): ADF (Constant) = -1.28, p-Value = 0.64; ADF (Constant, Trend) = -2.52, p-Value = 0.32.
  - Real Disposable Income ln(rdy): ADF (Constant) = -1.85, p-Value = 0.36; ADF (Constant, Trend) = -2.64, p-Value = 0.26.
  - Real Net Worth ln(rnw): ADF (Constant) = 0.38, p-Value = 0.98; ADF (Constant, Trend) = -2.23, p-Value = 0.47.
  - Dependency Ratio ln(dep_ratio): ADF (Constant) = -1.28, p-Value = 0.64; ADF (Constant, Trend) = -2.74, p-Value = 0.22.
  - Real Fed Funds Rate r_fed_rate: ADF (Constant) = -2.39, p-Value = 0.15; ADF (Constant, Trend) = -2.37, p-Value = 0.39.
- Cointegration:
  - Johansen tests indicate two long-run cointegrating relationships.
  - Gregory-Hansen (1996) Zt statistic = -6.49 rejects the no-cointegration null at 5 percent significance; “C/S” statistic also indicates cointegration.

### Main empirical findings and interpretation (structural break assessment)
- Baseline test:
  - Baseline VECM estimated on 1961Q1-2007Q4 used to predict 2008Q1-2017Q4 and to test for structural breaks.
- Main conclusion:
  - Statistical evidence suggests there was no structural break in household saving behavior after 2007Q4.
  - The rise in the saving rate during 2008-2012 was caused by negative shocks to income, employment and wealth (sizable but transitory shocks), not by changes in underlying elasticities.
- Crisis and recovery dynamics:
  - Decline in consumption during the crisis associated with a pronounced decline in real net worth and weak/volatile real disposable income growth; the estimated model validates these observations.
  - The saving rate resumed its decline in 2013 as real disposable income, employment and net worth recovered.
- Current account implications:
  - If real growth in determinants remains strong, the estimated model predicts continued negative pressures on the current account deficit and further external imbalances attributable to the U.S. household sector.
  - Decomposition observations (2012-2017): 2016 gross saving declined mostly due to decline in personal and public saving; 2017 increase in current account deficit attributed to decline in personal saving and increase in gross investment (public and private), partially offset by increase in private saving (excluding personal).

### Johansen test results and VECM analysis (long-run identification and estimates)
- Prior: two long-run relationships:
  - (a) long-run real consumption expenditure equation.
  - (b) evolution of real net worth equation.
- Identifying long-run restrictions:
  - Dependency ratio affects consumption indirectly via its impact on real net worth; real consumption does not affect real net worth directly.
- Estimated long-run equations (Table 3, preserved parameter presentation):
  - (1.3) (as presented in source)
    - Long-run consumption equation and long-run net worth equation estimated; estimated parameters statistically significant at the 5 percent level and of expected sign given the identifying restrictions.
- Variable ranking (effect of a 1 percent shock on consumption, largest to smallest):
  - real disposable income
  - real net worth
  - dependency ratio
  - real-interest rate
- Notes:
  - Imposing a zero restriction on r_fed_rate instead of dep_ratio does not change qualitative results; dependency ratio becomes insignificant in the long-run consumption relation.
  - Net worth is the second most important variable conditional on disposable income; pre-crisis borrowing against rising net worth contributed to a falling saving rate before the crisis.

### Short-run VECM specification, estimation, and interpretation
- Short-run specification:
  - VECM (VAR in first differences plus lagged ECM terms).
  - Stationary exogenous short-run variables: consumer_sentiment, unemp (change in unemployment rate), VIX (change in VIX). Stationarity confirmed by ADF tests.
  - Number of lags chosen by Schwarz information criterion; reported estimation uses one lag.
- Key estimation results:
  - Δlog(rc) equation: coefficient on its own error-correction term (CointEq1) is negative and significant; implied adjustment takes around 5 quarters for a shock to consumption to dissipate.
  - Δlog(rnw) equation: coefficient on CointEq2 is small — weak response of rnw to shocks; change in VIX has a significant negative effect on rnw.
  - Insignificance of CointEq2 for consumption and disposable income indicates near weak exogeneity of rnw.
  - Short-run restrictions joint test: Chi-square with 6 degrees of freedom = 4.40, p-value ≈ 0.62 (insignificant); restrictions acceptable.
- Historical decomposition (2008-2013):
  - Key drivers of rnw and rdy were their own structural shocks; these shocks were principal drivers of real consumption and hence the saving rate.

### Out-of-sample predictions, dynamic simulation, and fan charts
- Simulation setup:
  - VECM solved to generate predicted path of real consumption for 2008Q1-2017Q4 using dynamic simulation (prediction errors accumulate).
  - Dynamic simulation baseline: exogenous variables set to their actual values from 2008Q1 onwards.
  - Stochastic fan chart: exogenous variables held at their 2007Q4 value for 2008Q1-2017Q4; thousands of dynamic simulations with bootstrapped shocks produce a 90 percent confidence interval for the predicted saving rate.
- Key dynamic simulation findings:
  - Dynamic simulation overpredicts actual real disposable income and real net wealth after 2007Q4 and therefore overpredicts real consumption.
  - Overprediction arises from unit-root characteristics, limited serial correlation in innovations, and pre-2008 upward trends in rdy and rnw; dynamic simulation continues pre-crisis trends and misses significant negative impact of the financial crisis.
  - Dynamic predicted saving rate is systematically below the actual saving rate for much of 2008-2016.
  - Despite overprediction, the dynamic 90 percent fan chart encompasses the actual saving rate during 2008Q1-2017Q4, indicating the model remains useful for prediction.
  - Exception: saving rate for 2012Q4 lies outside the fan due to once-off lump sum social security benefit payments and accelerated dividend and bonus payments ahead of tax changes scheduled for January 1, 2013.
- Inference:
  - Dynamic results support inference that there has not been a structural break in consumption behavior arising from the financial crisis.

### Structural break tests and results (multiple-point tests)
- Tests applied to recursive residuals (one-step-ahead predictive errors from expanding-window VECMs for 2008Q1-2017Q4, 40 periods):
  - Quandt-Andrews (1993) Average Likelihood Ratio F-Statistic = 0.39; p-Value = 0.74.
  - Bai - Perron (2003) Sequential 0 versus 1 break F statistic = 4.72 > 0.05 (reported as > 0.05).
  - Null hypothesis: no structural break within 15 percent trimmed data. Twenty-nine (29) break points compared.
- Conclusion:
  - Both statistics strongly support the null hypothesis of no structural break during 2008Q1-2017Q4.
  - Result differs from studies using Markov-switching VARs (e.g., Bianchi, Lettau, and Ludvigson (2018)); differences stem from methodology.

### Static simulation comparison and implications
- Static simulation:
  - Resets endogenous explanatory variables to their lagged, actual values before predicting next period’s saving rate; forecast errors do not accumulate.
- Static simulation findings:
  - Static simulation is more accurate than dynamic simulation; predicted saving rate is near actual saving rate throughout 2008Q1-2017Q4.
  - Static 90 percent fan chart is significantly narrower and mimics realized volatility.
- Interpretation:
  - Superiority of static approach implies dynamic simulation error is largely due to errors in predicting rdy, rnw, and dep_ratio — not due to changes in underlying consumption parameters.
  - Conclusion: post-crisis saving-rate behavior is not due to a permanent change in consumption behavior.
  - Estimating on 1961Q1-2001Q4 and projecting forward yields fan charts that encompass actual saving rate for 2002Q1-2017Q4.

### Ranking of drivers for predicted saving rate during the crisis (2008Q1-2009Q2)
- Method: for predicted saving rate in 2009Q2, set each explanatory variable to its 2007Q4 value for 2008Q1-2009Q2 while keeping others at actuals; rank by absolute change in predicted saving rate.
- Ranked drivers (largest to smallest effect on predicted saving rate during 2008Q1-2009Q2):
  - real disposable income
  - unemployment rate
  - real net worth
- Interpretation:
  - Large negative impact of higher unemployment reflects weaker employment prospects and higher income uncertainty during the crisis.

### Effect of each variable on predicted saving rate (2008Q1-2009Q2) — key table values
- Predicted saving rate in 2009Q2:
  - Static simulation predicted value = 6.33 percent; actual saving rate = 7.1 percent.
- Table 6 entries (preserved values):
  - Real Disposable Income — Rank 1 — Actual Change (%), 2007Q4-2009Q2: 1.65 — Predicted Saving Rate, 2009Q2: 5.50 — Absolute Change (%), Predicted Saving Rate, 2007Q4-2009Q2: 13.1
  - Unemployment — Rank 2 — Actual Change (%), 2007Q4-2009Q2: 4.5 — Predicted Saving Rate, 2009Q2: 6.07 — Absolute Change (%), Predicted Saving Rate, 2007Q4-2009Q2: 4.04
  - Real Net Worth — Rank 3 — Actual Change (%), 2007Q4-2009Q2: -16.8 — Predicted Saving Rate, 2009Q2: 6.24 — Absolute Change (%), Predicted Saving Rate, 2007Q4-2009Q2: 1.50
  - Consumer Sentiment — Rank 4 — Actual Change (%), 2007Q4-2009Q2: -6.2 — Predicted Saving Rate, 2009Q2: 6.25 — Absolute Change (%), Predicted Saving Rate, 2007Q4-2009Q2: 1.30
  - Dependency Ratio — Rank 5 — Actual Change (%), 2007Q4-2009Q2: 2.11 — Predicted Saving Rate, 2009Q2: 6.28 — Absolute Change (%), Predicted Saving Rate, 2007Q4-2009Q2: 0.75
  - VIX — Rank 6 — Actual Change (%), 2007Q4-2009Q2: -30.5 — Predicted Saving Rate, 2009Q2: 6.35 — Absolute Change (%), Predicted Saving Rate, 2007Q4-2009Q2: 0.30
  - Real Federal Funds Rate — Rank 7 — Actual Change (%), 2007Q4-2009Q2: -0.16 — Predicted Saving Rate, 2009Q2: 6.34 — Absolute Change (%), Predicted Saving Rate, 2007Q4-2009Q2: 0.21
- Interpretation:
  - Higher saving rate after 2007Q4 mainly due to large negative shocks to rnw, rdy, and unemployment rather than changes in long-run elasticities.

### Forecasting performance for 2017 and dynamic projections beyond 2017Q4
- Out-of-sample setup:
  - Model re-estimated using data to 2016Q4; predict 2017Q1-2017Q4 using static simulation; leave 4 observations for true out-of-sample assessment.
- Exogenous-variable forecast assumptions for 2017Q1-2017Q4:
  - Assumption 1 (Blue): X(t) = X(2016Q4) for t = 2017Q1-2017Q4.
  - Assumption 2 (Orange): exogenous variables take their actual values for 2017Q1-2017Q4.
- Forecast results for 2017:
  - In both assumptions the model produces reasonable forecasts except for 2017Q4 when actual saving rate was 2.6 percent (its lowest since 2005Q3).
  - Median (average) saving rate for the December quarter during 2010-2017: median = 5.60 percent, average = 5.42 percent.
- Dynamic projections beyond 2017Q4:
  - Model re-estimated using all available data (1961Q1-2017Q4).
  - From 2017Q4 onwards: unemployment set to IMF WEO forecast; consumer sentiment forecast using unemployment and WEO; VIX assumed constant.
  - Sensitivity to projection start:
    - Start in 2018Q1 (actual saving rate below long-term value because 2017Q4 = 2.6 percent) yields an increase toward the long-run path — projections suggest saving rate during 2018-2019 will increase toward 3 percent relative to 2017Q4 value of 2.6 percent.
    - Start in 2017Q1 (saving rate closer to long-run path) yields a decline toward long-term path.
  - BEA data (latest available as of 30 April 2018) indicates saving rate = 3.1 percent for 2018Q1.

### Conclusions, policy-relevant implications, and relevance
- Core conclusions:
  - No evidence of a permanent structural break in household saving behavior after 2007Q4.
  - The 2008-2012 rise in the saving rate is attributed to negative, largely transitory shocks to income, employment and wealth.
  - The estimated model accurately predicts the saving rate for 2012Q1-2017Q4.
- Policy-relevant implications:
  - The resumption of a declining saving rate after 2013 is consistent with recovery in rdy, employment and rnw.
  - If determinants grow strongly going forward, the model predicts continued negative pressures on the U.S. current account deficit and greater external imbalances driven by the household sector.
  - Findings cast doubt on the secular stagnation hypothesis insofar as the underlying propensity to save has not increased in the estimated model.
  - Wealth/credit mechanism: pre-2008 borrowing against housing gains reversed during the crisis; recovery in rdy and rnw may lead households to revert to prior consumption norms and borrowing, reducing the personal saving rate.
- Relevance:
  - Results inform assessments of U.S. consumption and GDP outlook, financial market stability, and persistence of the gap between saving and investment.

### Robustness and sensitivity analyses (appendix summary)
- 1. Adding a trend in the long-run model:
  - Adding a trend does not change number of long-run relationships but is harder to motivate economically.
  - Net worth significance on consumption disappears with a trend; net worth may act as an instrument in trend-augmented VECM.
  - Trend introduces negative bias in dynamic predictions (positive trend coefficient lowers saving rate across forecast).
  - Dynamic and static fan charts with trend still encompass actual saving-rate values during 2008Q1-2017Q4.
  - Multiple-point break tests: Quandt-Andrews = 0.39 (p-Value = 0.74); Bai-Perron Sequential 0 versus 1 break F = 4.72 (p-Value > 0.05). Twenty-nine (29) break points compared.
- 2. Choice of estimation period:
  - Re-estimation on 1961Q1-2001Q4 and projection forward yields fan charts that encompass actual saving rate during 2002Q1-2017Q4.
  - Multiple-point break tests (shorter sample 2002Q1-2017Q4): Quandt-Andrews = 0.82 (p-Value = 0.42); Bai-Perron Sequential 0 versus 1 break F = 1.61 (p-Value > 0.05). Forty-five (45) break points compared.
- 3. Identification choices:
  - Preferred identification: dependency ratio affects consumption only indirectly via real net worth.
  - Alternative identification: zero restriction on r_fed_rate instead of dep_ratio yields similar qualitative results; dynamic prediction 2008Q1-2017Q4 captures overall trend.
  - Multiple-point break tests under alternative identification: Quandt-Andrews = 1.24 (p-Value = 0.25); Bai-Perron Sequential 0 versus 1 break F = 4.21 (p-Value > 0.05). Twenty-nine (29) break points compared.

*Source: IMF Working Paper WP/18/128, "The U.S. Personal Saving Rate," S. Ouliaris and C. Rochon (Sections 1–3).*

### Section 1

### The U.S. Personal Saving Rate

### Introduction and summary
- Paper title: The U.S. Personal Saving Rate. Authors: S. Ouliaris and C. Rochon. Authorized for distribution by Nigel Chalk. June 2018.
- Aim: develop a time series model for aggregate consumption to predict the U.S. personal saving rate and test for a structural break in consumption behavior because of the 2008 financial crisis.
- Key sample split:
  - Estimation (pre-crisis): 1961Q1-2007Q4.
  - Evaluation / prediction period: 2008Q1-2017Q4; focused out-of-sample forecast for 2017Q1-2017Q4 and out-of-sample predictions for 2018Q1-2019Q4 (figures/tables referenced in the paper).
- Stylized facts highlighted:
  - From 1975 to 2007 there was a clear downward trend in the personal saving rate.
  - In 2008 there was a significant rise in the saving rate that continued until the end of 2012, suggesting a possible permanent change in household behavior.
  - The saving rate resumed its decline around 2013 and appears to be returning to its pre-2008 level.
  - The rise in the saving rate during 2008-2012 coincided with marked declines in real net worth and weak/volatile growth in real disposable income.

### Data, model specification, and empirical approach
- Data: National Income and Product Accounts (NIPA, quarterly) data spanning 1961Q1 to 2017Q4. Data release referenced as March 29, 2018.
- Key variables used in the long-run real consumption equation:
  - real consumption (rc)
  - real disposable income (rdy)
  - real household net worth (rnw)
  - dependency ratio (dep_ratio)
  - real federal funds rate (r_fed_rate)
- Long-run real consumption equation (equation numbering and specification preserved):
  - Equation (1.1)
    0    1234
    ln(  )ln(  )   ln(   )   _   _ln(  _   )
    tttttt
    rcrdyrnwr   fed   ratedep  ratioβ  ββββϵ=  ++++ +
- Rationale for variable inclusion:
  - Permanent Income Hypothesis and wealth effects motivate real disposable income and real net wealth; both should have positive elasticities.
  - Inter-temporal substitution arguments motivate inclusion of the real interest rate (proxied by the real federal funds rate).
  - Life-cycle hypothesis supports adding the dependency ratio; effect expected to be negative.
  - Relationship between real consumption and the real federal funds rate is theoretically indeterminate.
- Short-run dynamics handled via a multi-equation error-correction model (VECM) to predict consumption and the personal saving rate.
- Personal saving rate identity used (preserved as in source):
  - Equation (1.2)
    _100 * ((    ((    __) /  _))) /sa vin gra terd yrcg o v    tra n sfersip a ym en tsyd efla to rrd y=−    +++
  - gov_transfers, i_payments and y_deflator are treated as exogenous variables. Empirical analysis holds under assumption these variables are zero (noting they represent a small proportion of the saving rate).

### Time series properties and cointegration
- Unit root evidence (Augmented Dickey-Fuller tests) for 1961Q1-2007Q4 suggests all time series in the long-run regression have a unit root (nonstationary).
- Table 1: Unit Root Analysis, Augmented Dickey-Fuller Tests, 1961Q1-2007Q4 (values preserved)
  - Real Consumption ln(rc): ADF (Constant) = -1.28, p-Value = 0.64; ADF (Constant, Trend) = -2.52, p-Value = 0.32.
  - Real Disposable Income ln(rdy): ADF (Constant) = -1.85, p-Value = 0.36; ADF (Constant, Trend) = -2.64, p-Value = 0.26.
  - Real Net Worth ln(rnw): ADF (Constant) = 0.38, p-Value = 0.98; ADF (Constant, Trend) = -2.23, p-Value = 0.47.
  - Dependency Ratio ln(dep_ratio): ADF (Constant) = -1.28, p-Value = 0.64; ADF (Constant, Trend) = -2.74, p-Value = 0.22.
  - Real Fed Funds Rate r_fed_rate: ADF (Constant) = -2.39, p-Value = 0.15; ADF (Constant, Trend) = -2.37, p-Value = 0.39.
- Cointegration:
  - Johansen tests indicate there appear to be two long-run cointegrating relationships amongst these variables (Table 2 referenced).
  - Gregory-Hansen (1996) statistics applied to test no-cointegration allowing for shifts; Zt statistic = -6.49 rejects null at 5 percent significance. Evidence of cointegration also found with their “C/S” statistic.

### Main empirical findings and interpretation
- Structural break assessment:
  - Estimation of the baseline VECM using data 1961Q1-2007Q4 and prediction for 2008Q1-2017Q4 provides the basis for formal testing whether the post-2007 rise in the saving rate was structural or driven by transitory shocks.
  - Statistical evidence suggests there was no structural break in household saving behavior after 2007Q4. The rise in the saving rate during 2008-2012 was caused by negative shocks to income, employment and wealth (sizable but transitory shocks), not by changes in underlying elasticities.
- Dynamics during crisis and recovery:
  - The marked decline in consumption during the crisis is associated with a very pronounced decline in real net worth and weak/volatile real disposable income growth; the estimated model validates these observations.
  - The saving rate resumed its decline in 2013 as real disposable income, employment and net worth recovered.
- Current account and external balances:
  - Assuming real growth in determinants remains strong, the estimated model predicts continued negative pressures on the current account deficit and further external imbalances attributable to the U.S. household sector.
  - Figure 2 decomposition observations (2012-2017): changes in gross saving and gross investment are split into private and public components; 2016 gross saving declined mostly due to decline in personal and public saving; 2017 increase in current account deficit attributed to decline in personal saving and increase in gross investment (public and private), partially offset by increase in private saving (excluding personal).

### Forecasting performance and robustness exercises (framework overview)
- Forecasting approach:
  - Baseline model estimated on pre-crisis data (1961Q1-2007Q4); used to generate dynamic and static simulations for 2008Q1-2017Q4 and out-of-sample forecasts for 2017Q1-2017Q4 and 2018Q1-2019Q4.
  - Multiple-point structural break tests are conducted for 2008Q1-2017Q4 (Table 5) and robustness checks include allowing for trend in long-run model, shorter sample (2002Q1-2017Q4), and alternative identification schemes (Appendix figures and tables).
- Key model elements:
  - Baseline VECM, cointegrating rank of two, and VECM parameter estimates reported (Table 3, Table 4 referenced).
  - Historical decompositions and dynamic simulations investigate contributions of real net worth, real disposable income and other determinants to consumption and the saving rate (Figures referenced).

### Concluding remarks and policy implications
- Core conclusion: No evidence of a permanent structural break in household saving behavior after 2007Q4; the 2008-2012 rise in the saving rate is attributed to negative, largely transitory shocks to income, employment and wealth.
- Implications:
  - The resumption of a declining saving rate after 2013 is consistent with recovery in real disposable income, employment and net worth.
  - If real growth in determinants remains strong, the model predicts continued negative pressures on the U.S. current account deficit and greater external imbalances driven by the household sector.
- Relevance:
  - Findings inform assessments of U.S. consumption and GDP outlook, financial market stability, and persistence of the gap between saving and investment.

*Source: IMF Working Paper WP/18/128, "The U.S. Personal Saving Rate," S. Ouliaris and C. Rochon (Section 1).*

### Section 2

### Section 2 — Johansen Test Results and VECM Analysis

### Long-run relationships (identification and estimates)
- Prior: two long-run relationships:
  - (a) an equation explaining long-run real consumption expenditure
  - (b) an equation explaining the evolution of real net worth
- Imposed identifying long-run restrictions: dependency ratio affects consumption indirectly via its impact on real net worth; real consumption expenditure does not affect real net worth directly.
- Estimated long-run equations (Table 3):
  - 1
    ln(  )   1.04  0.95ln(   )  0.08ln(   )  0.003 _   _
    ln(   )  13.22 1.81ln(   )  4.07ln(   _)  0.007 _   _
    ttttt
    ttttt
    rcrdyrnwr    fed   rate
    rnwrdydep   ratior    fed   rate
    
    
        
    
    (1.3)
  - Given the restrictions, the estimated parameters are statistically significant at the 5 percent level and of the expected sign.
- Variables ranked in terms of influence on consumption (given a 1 percent shock), from largest to smallest:
  - real disposable income
  - real net worth
  - dependency ratio
  - real-interest rate
- Notes:
  - Imposing a zero restriction on real Fed Fund rate instead of the dependency ratio does not change qualitative results; the estimated parameter on the dependency ratio is insignificant in the long-run consumption relation.
  - Net worth is the second most important variable conditional on disposable income; households borrowed on increasing net wealth to raise consumption, contributing to a falling saving rate before the crisis.

### Short-run model: VECM specification and exogenous variables
- Method: Vector error-correction model (VECM) to account for stochastic non-stationarity and cointegration; VECM is a VAR in first differences plus lagged error-correction (ECM) terms.
- Additional stationary exogenous variables included for short-run movements in consumption:
  - household confidence index (consumer_sentiment)
  - change in the unemployment rate (unemp)
  - change in the VIX (proxy for financial volatility)
- Stationarity of these exogenous variables confirmed using the Augmented Dickey Fuller test.
- Short-run consumption equation estimated (equation (1.4); omitted dynamics from other endogenous variables):
  - 01 1  12  2  1
    0
    00
    ln(  )...()
    ln(_)ln()
    p
    ttttit   i
    i
    kh
    jt   jlt   lt
    jl
    rcunemp
    consumer   sentimentVIX
             
    
    
    
    
    
        
    +
    
    
    (1.4)
- Similar VECM equations estimated for rdy, rnw, dep_ratio, r_fed_rate.
- Number of lags chosen using Schwarz information criterion; in reported estimation the number of lags is one.

### Estimation results and interpretation
- Δlog(rc) equation:
  - Coefficient on its own error-correction term (CointEq1) is negative and significant; estimated value implies it takes around 5 quarters for a shock to consumption to dissipate (i.e., for the model to return to its long-run equilibrium).
- Δlog(rnw) equation:
  - Coefficient on the error-correction term associated with real net worth (CointEq2) is small, suggesting weak responses of real net worth to economic shocks.
  - The change in VIX has a significant negative effect on real net wealth (and hence consumption), but not directly on other endogenous variables.
- Insignificance of CointEq2 for real consumption and disposable income points to near weak exogeneity of real net worth — real net worth does not respond greatly to long-run shocks in consumption (consistent with distinction between a flow variable and a stock variable).
- Short-run restrictions:
  - Six short-run restrictions imposed (the first two lines associated with CointEq1 and CointEq2 are the lagged error terms). Joint test for validity of these restrictions: Chi-square with 6 degrees of freedom = 4.40 and p-value around 0.62 (insignificant; restrictions acceptable).
- Historical decomposition (baseline VECM) for 2008-2013:
  - Key drivers of real net worth and real disposable income were their own respective structural shocks — consistent with view that the financial crisis was driven by a sequence of negative exogenous shocks.
  - Those shocks were principal drivers of real consumption expenditure and hence the saving rate.

### Out-of-sample predictions, dynamic simulation, and fan charts
- VECM solved simultaneously to generate predicted path of real consumption for 2008Q1-2017Q4; dynamic simulation used (allowing prediction errors to build on themselves).
- Dynamic simulation assumptions:
  - all exogenous variables assume their actual values from 2008Q1 onwards (for baseline dynamic simulation).
  - For stochastic fan chart, exogenous variables assumed equal to their 2007Q4 value during 2008Q1-2017Q4; thousands of dynamic simulations with bootstrapped shocks produce a (90 percent) confidence interval for the predicted saving rate.
- Key dynamic simulation findings:
  - Dynamic simulation overpredicts actual real disposable income and real net wealth significantly after 2007Q4, and therefore overpredicts real consumption.
  - Overprediction arises from unit-root characteristics of explanatory variables, limited serial correlation in their innovations, and pre-2008 upward trends in real disposable income and real net worth; dynamic simulation continues pre-crisis trends and misses significant negative impact of the financial crisis.
  - Dynamic predicted saving rate (orange line, Figure 5) is systematically below the actual saving rate for much of 2008-2016.
  - Despite overprediction, the dynamic fan chart (90 percent band) encompasses the actual outcomes of the saving rate during 2008Q1-2017Q4, providing statistical evidence that the underlying model is useful for predicting the actual saving rate.
  - Exception: the saving rate for 2012Q4 is outside the fan due to once-off lump sum social security benefit payments and accelerated dividend and bonus payments ahead of tax rate increases scheduled for January 1, 2013.
- Inference from dynamic results: one can infer that there has not been a structural break in consumption behavior arising from the financial crisis.

### Structural break tests and results
- Multi-point structural break tests applied to recursive residuals from the VECM:
  - Quandt-Andrews (1993) and Bai and Perron (2003) tests (allow unknown, multiple break points).
  - Recursive residuals are the one-step-ahead predictive errors from the VECM when the estimation period is extended one period at a time from 2008Q1 to 2017Q4 (40 periods), derived from 40 distinct VECM models.
- Reported multiple-point break test results (Table 5):
  - Quandt-Andrews (1993) Average Likelihood Ratio F-Statistic: 0.39; p-Value 0.74
  - Bai - Perron (2003) Sequential 0 versus 1 break F statistic: 4.72 > 0.05
  - Note: Null hypothesis is no structural break within 15 percent trimmed data. Twenty-nine (29) break points were compared.
- Conclusion: Both statistics strongly support the null hypothesis of no structural break during 2008Q1-2017Q4.
- Contrast with other findings: result differs from Bianchi, Lettau, and Ludvigson (2018), who find regime shifts using a Markov-switching VAR; differences stem from methodology (VECM approach here allows for changes in unconditional moments).

### Static simulation comparison and implications for structural break
- Static simulation resets endogenous explanatory variables to their lagged, actual values before predicting next period’s saving rate — forecast errors of endogenous variables do not accumulate.
- Static simulation findings:
  - Static simulation is more accurate than dynamic simulation; predicted saving rate is near actual saving rate throughout 2008Q1-2017Q4.
  - Static fan chart (90 percent band) mimics realized volatility in saving rate and is significantly narrower than dynamic counterpart.
- Interpretation:
  - Superiority of static approach implies dynamic simulation error in consumption equation is largely due to errors in predicting real disposable income, net worth, and dependency ratio — not due to changes in underlying parameters of consumption function.
  - Conclusion: post-crisis behavior of saving rate is not due to a structural/permanent change in consumption behavior.
  - Estimating the model with data from 1960 to 2001 (to avoid 2002-2007 low saving-rate period) does not change parameter estimates significantly; predictions for 2002Q1-2017Q4 (in appendix) have fan charts that encompass actual saving rate.

### Ranking of drivers for predicted saving rate during the crisis (2008Q1-2009Q2)
- Method: for predicted saving rate in 2009Q2, set each explanatory variable separately to its 2007Q4 value for 2008Q1-2009Q2 while keeping other variables at actuals; ranking based on absolute change in predicted saving rate (sensitive to sample period).
- Ranked main drivers (highest to lowest effect on predicted saving rate during 2008Q1-2009Q2):
  - real disposable income
  - unemployment rate
  - real net worth
- Interpretation: large negative impact of higher unemployment reflects weaker employment prospects and higher income uncertainty during crisis period.

*Source: wp18128 - Section 2*

### Section 3

### wp18128 - Section 3

### Effect of Each Variable on the Predicted Saving Rate (2008Q1-2009Q2)
- Predicted value for the saving rate in 2009Q2 is 6.33 percent using a static simulation and unconstrained explanatory variables; the actual saving rate was 7.1 percent.
- Table 6: Effect of Each Variable on the Predicted Saving Rate, 2008Q1-2009Q2 (ranked by absolute change in predicted saving rate when holding each variable at its 2007Q4 value for 2007Q4-2009Q2)
  - Real Disposable Income — Rank 1 — Actual Change (%), 2007Q4-2009Q2: 1.65 — Predicted Saving Rate, 2009Q2: 5.50 — Absolute Change (%), Predicted Saving Rate, 2007Q4-2009Q2: 13.1
  - Unemployment — Rank 2 — Actual Change (%), 2007Q4-2009Q2: 4.5 — Predicted Saving Rate, 2009Q2: 6.07 — Absolute Change (%), Predicted Saving Rate, 2007Q4-2009Q2: 4.04
  - Real Net Worth — Rank 3 — Actual Change (%), 2007Q4-2009Q2: -16.8 — Predicted Saving Rate, 2009Q2: 6.24 — Absolute Change (%), Predicted Saving Rate, 2007Q4-2009Q2: 1.50
  - Consumer Sentiment — Rank 4 — Actual Change (%), 2007Q4-2009Q2: -6.2 — Predicted Saving Rate, 2009Q2: 6.25 — Absolute Change (%), Predicted Saving Rate, 2007Q4-2009Q2: 1.30
  - Dependency Ratio — Rank 5 — Actual Change (%), 2007Q4-2009Q2: 2.11 — Predicted Saving Rate, 2009Q2: 6.28 — Absolute Change (%), Predicted Saving Rate, 2007Q4-2009Q2: 0.75
  - VIX — Rank 6 — Actual Change (%), 2007Q4-2009Q2: -30.5 — Predicted Saving Rate, 2009Q2: 6.35 — Absolute Change (%), Predicted Saving Rate, 2007Q4-2009Q2: 0.30
  - Real Federal Funds Rate — Rank 7 — Actual Change (%), 2007Q4-2009Q2: -0.16 — Predicted Saving Rate, 2009Q2: 6.34 — Absolute Change (%), Predicted Saving Rate, 2007Q4-2009Q2: 0.21
- Interpretation: The higher saving rate after 2007Q4 was mainly due to large negative shocks to key drivers during the financial crisis (notably real net worth, real disposable income, and unemployment), rather than changes in long-run elasticities of consumption.

### Forecasting Performance of the Baseline VECM Model
- Out-of-sample forecasting setup:
  - Model re-estimated using data available to 2016Q4 (sample shortened to 2016Q4 to leave 4 observations for a true out-of-sample forecast assessment).
  - Predict saving rate for 2017Q1-2017Q4 using static simulation.
- Exogenous-variable forecast assumptions for 2017Q1-2017Q4:
  - Assumption 1 (Blue line, Figure 11): Exogenous variables remain constant from 2017Q1 onwards: X(t) = X(2016Q4), for t = 2017Q1-2017Q4.
  - Assumption 2 (Orange line, Figure 11): Exogenous variables take their actual values for t = 2017Q1-2017Q4.
- Forecast results for 2017:
  - In both assumptions the model produces reasonable forecasts for the actual saving rate, except for 2017Q4 when the actual saving rate was exceptionally low at 2.6 percent (its lowest value since 2005Q3).
  - The median (average) saving rate for the December quarter was 5.60 (5.42) percent during 2010-2017.
- Dynamic projections beyond 2017Q4:
  - Model re-estimated using all available data (196Q1-2017Q4).
  - From 2017Q4 onwards: unemployment set to IMF WEO forecast; consumer sentiment forecast using unemployment rate and WEO forecasts; VIX assumed constant.
  - Mean-reverting nature of VECM implies starting date matters:
    - Projection start in 2018Q1 (actual saving rate below long-term value because 2017Q4 was 2.6 percent) yields an increase toward the long-run path — projections suggest saving rate during 2018-2019 will increase toward 3 percent relative to 2017Q4 value of 2.6 percent.
    - Projection start in 2017Q1 (saving rate closer to long-run path) yields a decline toward long-term path.
  - BEA data (latest available as of 30 April 2018) indicates a saving rate of 3.1 percent for 2018Q1.

### Conclusions and Policy-Relevant Implications
- Main conclusions:
  - Statistical evidence suggests no structural change in the U.S. consumption function and saving behavior since the financial crisis.
  - The increase in the saving rate during 2008Q1-2011Q4 can be predicted well using lagged actual values of real income and real net worth from a model estimated only with data to 2007Q4.
  - The estimated model accurately predicts the saving rate for 2012Q1-2017Q4, supporting absence of a structural break.
- Implications:
  - Rise in saving rate during the crisis reflects severity of negative shocks to real net worth and unemployment and weak/volatile real disposable income, not a fundamental change in household consumption/saving propensity.
  - The saving rate resumed decline in 2013 as real disposable income, employment and net worth recovered; if these determinants grow strongly going forward, results imply continued negative pressures on the current account deficit.
  - Results cast doubt on the secular stagnation hypothesis (which posits a structural increase in saving propensity) because the underlying propensity to save has not increased in the estimated model.
  - Wealth/credit market mechanism: pre-2008 borrowing secured by unrealized housing gains reversed during the crisis; as real income and net worth recover, households may revert to prior consumption norms and borrow, reducing the personal saving rate.

### Robustness and Sensitivity Analyses (Appendix)
- 1. Allowing for a Trend in the Long-Run Model
  - Adding a trend term does not change the number of long-run relationships but is harder to motivate economically.
  - Net worth significance on consumption disappears when a trend is included, suggesting net worth may be acting as an instrument in the trend-augmented VECM.
  - Adding a trend introduces negative bias in dynamic predictions for the saving rate compared to the model without a trend (Figure 13, green versus orange lines), due to a positive coefficient on the trend term lowering the saving rate across the forecast horizon.
  - Dynamic and static fan charts with the trend still encompass actual saving-rate values during 2008Q1-2017Q4 (Figure 14).
  - Multiple-point break tests for the trend-augmented VECM (2008Q1-2017Q4) support the null hypothesis of no structural break:
    - Table 7: Multiple-Point Break Tests: Trend, 2008Q1-2017Q4
      - Quandt-Andrews (1993) Average Likelihood Ratio F-Statistic — Value: 0.39 — p-Value: 0.74
      - Bai - Perron (2003) Sequential 0 versus 1 break F statistic — Value: 4.72 — p-Value: > 0.05
    - Note: Null hypothesis is no structural break within 15 percent trimmed data. Twenty-nine (29) break points were compared.
- 2. Choice of Estimation Period
  - Re-estimated model on 1961Q1-2001Q4 and projected forward with exogenous variables taking actuals after 2001Q4.
  - Fan charts for this model encompass actual saving rate during 2002Q1-2017Q4.
  - Multiple-point structural break tests are insignificant, indicating finding of no structural break is not sensitive to estimation period:
    - Table 8: Multiple-Point Break Tests: Shorter Sample, 2002Q1-2017Q4
      - Quandt-Andrews (1993) Average Likelihood Ratio F-Statistic — Value: 0.82 — p-Value: 0.42
      - Bai - Perron (2003) Sequential 0 versus 1 break F statistic — Value: 1.61 — p-Value: > 0.05
    - Note: Null hypothesis is no structural break within 15 percent trimmed data. Forty-five (45) break points were compared.
- 3. Identification Choices
  - Preferred identification imposed that dependency ratio affects real consumption only indirectly via real net worth.
  - Alternative identification: zero restriction on the real federal funds rate instead of dependency ratio.
    - Under alternative identification, dynamic prediction for 2008Q1-2017Q4 captures the overall trend in the actual saving rate (Figure 17).
    - Dynamic and static fan charts continue to include actual saving-rate values for 2008Q1-2017Q4 (Figure 18).
    - Multiple-point structural break tests under alternative identification support null of no structural break:
      - Table 9: Multiple-Point Break Tests: Alternative Identification, 2008Q1-2017Q4
        - Quandt-Andrews (1993) Average Likelihood Ratio F-Statistic — Value: 1.24 — p-Value: 0.25
        - Bai - Perron (2003) Sequential 0 versus 1 break F statistic — Value: 4.21 — p-Value: > 0.05
      - Note: Null hypothesis is no structural break within 15 percent trimmed data. Twenty-nine (29) break points were compared.

*Source: wp18128 - Section 3 (IMF Working Paper PDF).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2018/wp18128.pdf_