## 1. Estimated Response Function of the SARB ## Source details **Canonical URL:** [1. Estimated Response Function of the SARB](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12177.pdf) ## Other formats - [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12177.pdf.md) - [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12177.pdf.json) --- ### I. Objective and model - Objective: “reveal” the SARB’s ex-post inflation target point and assess its variation since IT introduction by estimating a simple Taylor rule in a state-space approach that allows the inflation target to follow a pre-determined “random walk” process. - Empirical specification: - Baseline: Taylor rule augmented for interest rate smoothing (Eq. (1)–(3)); implicit inflation target modeled as a random walk; disturbances mean zero, serially uncorrelated Gaussian. - Signal-to-noise ratio parameter ( ) links policy rate variance and implicit inflation target variance; implies a constant implicit target over time. - Data and sample: - Quarterly data for 2001Q1–2011Q4; baseline estimation sample 2001Q4–2011Q4. - Policy/repo rate and real GDP from the SARB website. - Output gap: log real GDP de-trended with Hodrick-Prescott filter; observed fluctuation to 3.2 percent (2008Q2) and -2 percent (2009Q3); projected to close by end-2014. - Natural real rate treatments: (i) constant at historical average ~3.25 percent; (ii) time-varying alternatives. - Inflation expectations: Reuters Econometer Consensus four-quarters-ahead (CPIX until 2008Q4, then headline CPI). ### II. Baseline estimation results (constant natural real rate r = 3.25 percent) - Estimated coefficients (Table 1, baseline): - Interest rate smoothing parameter: 0.8 (reported as 0.795 (0.071)). - Output gap coefficient: 1.9 (reported as 1.913 (0.627)). - Inflation gap coefficient: 3.9 (reported as 3.940 (1.559)). - Signal-to-noise ratio ( ): 0.00965 (baseline). - Implied ratio (inflation weight/output gap weight): 2.1. - Key empirical findings: - Although the official inflation target range is 3–6 percent, estimated implicit inflation target indicates the SARB in practice aimed for the upper segment of the band (4 1/2 –6 percent) for most of the period. - The implicit inflation target exhibits substantial time variation; a Wald test rejects the null that the target is constant with 99 percent confidence. - Estimated implicit-target trajectory (selected points): - Around 6 percent in 2001–02 (rand crisis period). - Moderation to 4 1/2 percent in 2003. - Broadly stable around 5 percent in 2004–07. - Steady increase from 2007Q4 to around 5 1/2 percent in 2008Q3. - Further increase toward 6 percent during the global financial crisis (late-2008/early-2009) and remaining near the upper limit through 2010–2011. - Drivers of 2008 increase: inflation expectations rose to 7.4 percent in 2008Q2 and output gap to 3.2 percent above potential; policy rate hikes were smaller than implied by the reaction function, yielding an elevated estimated target. ### III. Sensitivity and robustness analyses - Dimensions varied: statistical process for implicit target (random walk vs. stationary), constant natural real rate level, and signal-to-noise ratio ( ). A. Random walk vs. stationary process - Relaxation: include coefficient so the target process can be stationary. - Result: Wald test cannot reject = 1; implicit target follows a random walk. Trajectory close to baseline; average target for 2010–11 marginally lower than baseline. B. Different constant natural real rate - Alternative constant r values: - r = 4 percent (75 basis points above historical average). - r = 2.5 percent (75 basis points below historical average). - Result: Changing the constant natural real rate produces a vertical shift in the implicit target by approximately the same magnitude throughout the period; the upward drift in 2008–09 is unchanged, while the level is determined by assumed r. C. Alternative signal-to-noise ratios - Higher implies greater variability in the estimated implicit target; lower implies greater stability. - High scenario: - Implicit target peaks at 7 percent in 2009Q3 and moderates to around 6.6 percent. - Lower interest rate smoothing (smaller smoothing parameter). - Average inflation target for 2010–11: 6.5 percent. - Low scenario: - Implicit target more stable but still shows upward drift in 2007–09. - Average inflation target for 2010–11: 6 percent. - Selected parameter examples (Table 3): - Smoothing: Baseline 0.795 (0.071); high 0.823 (0.077); low 0.742 (0.061). - Inflation coefficient: Baseline 3.940 (1.559); high 4.028 (1.842); low 3.351 (0.991). - Signal-to-noise: Baseline 0.00965; high 0.00526; low 0.0554. - Implied ratios: Baseline 2.1; high 1.9; low 2.0. ### IV. Time-varying natural real rate specifications - Two approaches: - Full variation: Laubach and Williams (2003)-style state-space linking natural real rate to trend growth of output. - Partial variation: natural real rate constant at 3.25 percent until 2008Q4, then declines to 2 percent thereafter. - Estimated natural real rate (full variation): - Increased to about 4.5 percent in mid-2008 from around 3 percent earlier. - Declined significantly with the global financial crisis to 2 percent. - Average reported as 3.2 percent in Figure 7. - Impact on implicit inflation target: - Inclusion of time-varying natural real rate shifts the implicit target trajectory slightly downward but preserves central finding that the SARB targeted the upper segment of the official range (4 1/2 –6 percent). - Both full and partial variation scenarios show upward drift in 2007–09. - Selected estimated coefficients (Table 4): - Smoothing: baseline 0.795 (0.071); partial variation 0.779 (0.059); full variation 0.799 (0.072). - Output gap coefficient: baseline 1.913 (0.627); partial 1.640 (0.519); full 1.525 (0.637). - Inflation coefficient: baseline 3.940 (1.559); partial 3.429 (1.133); full 3.813 (1.547). - Signal-to-noise: baseline 0.00965; partial 0.0089; full 0.0098. - Implied ratio (inflation weight/output gap weight): baseline 2.1; partial 2.1; full 2.5. ### V. Actual versus predicted policy rate and forecast performance - Dynamic forecasts start at 2005Q1 using baseline (constant r), full variation r, partial variation r specifications. - Fitted rates replicate actual SARB policy rate dynamics well: - Predict monetary tightening in 2006–08 and monetary relaxation from 2009 onward. - Forecast performance (sum of squared errors, SSE): - SSE (partial variation): 9.0. - SSE (full variation): 11.3. - SSE (baseline): 13.7. - Partial time-varying natural rate specification outperforms full variation and baseline in SSE. ### VI. Descriptive statistics (Table 1A, 2001Q1–2011Q4) - Variables: Inflation; Inflation Expectations; Output Gap; Policy Rate - Mean: 5.217; 5.532; -0.004; 8.886 - Median: 5.315; 5.600; -0.389; 8.000 - Maximum: 12.830; 7.400; 3.164; 13.500 - Minimum: -2.060; 4.500; -1.951; 5.500 - Std. Dev.: 3.486; 0.562; 1.389; 2.449 - Skewness: -0.056; 0.613; 0.881; 0.366 - Kurtosis: 2.777; 4.455; 2.831; 1.952 - Observations: 44; 44; 44; 44 ### VII. Key conclusions - Methodology: Taylor rule in state-space with implicit inflation target modeled as a time-varying random walk. - Core conclusions: - Empirical evidence indicates the SARB targeted the upper segment of the official inflation range (4 1/2 –6 percent) since IT introduction, despite the official 3–6 percent range. - The implicit inflation target varied over time, with a pronounced upward drift in 2007–09 and remaining near the upper limit (around 6 percent) through 2010–2011. - Results are sensitive to assumptions about the natural real rate and the signal-to-noise ratio, but the main qualitative finding (targeting the upper segment) is robust across specifications. - Allowing a time-varying natural real rate shifts the implicit target slightly downward and, under full variation, increases the relative weight on inflation stabilization (implied ratio about 2.5). - Robustness: the partial time-varying natural rate specification provided the best fit to actual policy rate dynamics (lowest SSE). *Source: _wp12177 - 1. Estimated Response Function of the SARB* ### 1. Estimated Response Function of the SARB .....................................................................7 ### 1. Estimated Response Function of the SARB ### Major components of the content unit - Section 1: Estimated Response Function of the SARB .....................................................................7 - Section 2: Estimated Response Function with a Relaxation of the Random Walk Assumption .......9 - Section 3: Estimated Response Function with Different Signal-to-Noise Ratios ............................10 - Section 4: Estimated response function with time-varying natural real rate ...................................12 ### Figures referenced in this content unit (by caption and figure number) - Figure 1. Inflation Targets in Selected Emerging Markets ..............................................................3 - Figure 2. South Africa: Output Gap by Hodric-Presscot Filter ........................................................5 - Figure 3. Inflation and Inflation Expectations ..................................................................................6 - Figure 4. The Estimated Implicit Inflation Target ...........................................................................7 - Figure 5. The Estimated Implicit Inflation Target with Different Levels of Natural Real Rate ......9 - Figure 6. The Estimated Implicit Inflation Target with Alternative Signal-to-Noise Ratios .........10 - Figure 7. The Estimated Time-Varying Natural Interest Rate .......................................................11 - Figure 8. The Estimated Implicit Inflation Target with Time-Varying Natural Interest Rate .......11 - Figure 9. Actual and Predicted Policy Rate ...................................................................................12 ### Appendix material - Appendix Table 1.A Labor Descriptive Statistics, 2001-Q1-2011-Q4 ............................................................15 ### Structural summary for ingestion - Primary focus: estimation of the South African Reserve Bank's implicit policy-response function (sectioning indicates robustness checks and alternative assumptions). - Robustness and sensitivity analyses signaled by sections on relaxing the random walk assumption, varying signal-to-noise ratios, and allowing a time-varying natural real rate. - Visual materials: nine figures documenting targets, gaps, expectations, estimated implicit inflation targets under alternative assumptions, estimated time-varying natural interest rate, and actual vs predicted policy rate. - Time coverage indicated in appendix: labor descriptive statistics for 2001-Q1-2011-Q4. *Source: _wp12177 - 1. Estimated Response Function of the SARB* ### Appendix Figures ### _wp12177 - Appendix Figures ### I. Introduction and objective - South Africa adopted inflation targeting (IT) as a formal framework for monetary policy in February 2000. - The official inflation target range specified from 2002 was between 3 and 6 percent; the target measure was re-defined several times. - The South African Reserve Bank (SARB) has not announced a mid-point (point inflation target), providing flexibility but adding uncertainty about the “true” inflation target point. - Paper objective: “reveal” the SARB’s ex-post inflation target point and assess its variation since IT introduction by estimating a simple Taylor rule in a state-space approach that allows the inflation target to follow a pre-determined “random walk” process. ### II. Empirical model and data - Monetary policy reaction function: Taylor rule augmented for interest rate smoothing. - Core equations: baseline Taylor rule (Eq. (1)), smoothing specification (Eq. (2)), combined state-space equation (Eq. (3)). The implicit inflation target is modeled as a random walk; disturbances are mean zero, serially uncorrelated Gaussian. - Signal-to-noise ratio parameter ( ) links policy rate variance and implicit inflation target variance; implies a constant implicit target over time. - Data and sample period: - Quarterly data for 2001Q1–2011Q4. - Policy/repo rate and real GDP from the SARB website. - Output gap: natural logarithm of real GDP de-trended with Hodrick-Prescott filter; computed output gap fluctuated to 3.2 percent (2008Q2) and -2 percent (2009Q3); projected to close by end-2014. - Natural real rate treatments: (i) constant at historical average ~3.25 percent; (ii) time-varying alternatives. - Inflation expectations: Reuters Econometer Consensus for four quarters ahead (CPIX until 2008Q4, then headline CPI). ### III. Estimation results under a constant natural real rate (r = 3.25 percent) - Sample: 2001Q4–2011Q4 (baseline assumes natural real rate constant at 3.25 percent). - Estimated coefficients (Table 1, baseline): - Interest rate smoothing parameter: 0.8 (reported as 0.795 (0.071) in Table 1). - Output gap coefficient: 1.9 (reported as 1.913 (0.627)). - Inflation gap coefficient: 3.9 (reported as 3.940 (1.559)) — greater than one, indicating aggressive response to inflation deviations. - Signal-to-noise ratio ( ): 0.00965 (baseline). - Ratio (inflation weight/output gap weight) implied as 2.1. - Main findings: - Although the official range is 3–6 percent, the SARB in practice aimed for the upper segment of the band (4 1/2 –6 percent) for most of the period. - The implicit inflation target exhibits substantial time variation; a Wald test rejects the null that the target is constant with 99 percent confidence. - Estimated implicit target trajectory highlights: - Around 6 percent in 2001–02 (rand crisis period). - Moderation to 4 1/2 percent in 2003. - Broadly stable at around 5 percent in 2004–07. - Steady increase from 2007Q4 to around 5 1/2 percent in 2008Q3. - Further increase toward 6 percent during the global financial crisis (late-2008/early-2009) and remaining near the upper limit through 2010–2011. - Drivers of 2008 increase: inflation expectations rose to 7.4 percent in 2008Q2 and output gap to 3.2 percent above potential; policy rate hikes were smaller than implied by the reaction function, yielding an elevated estimated target. ### IV. Sensitivity analysis to pre-determined parameters - Elements varied: statistical process of implicit target (random walk vs. stationary), constant natural real rate level, and signal-to-noise ratio ( ). A. Random walk vs. stationary process - Relaxation: include coefficient so target process can be stationary. - Result: Wald test cannot reject = 1; implicit target follows a random walk. Trajectory close to baseline; average target for 2010–11 marginally lower than baseline. B. Different constant natural real rate - Two alternative constant rates: - r = 4 percent (75 basis points above historical average). - r = 2.5 percent (75 basis points below historical average). - Result: Changing the constant natural real rate produces a vertical shift in the implicit target by approximately the same magnitude throughout the period; upward drift in 2008–09 unchanged, level determined by assumed r. C. Alternative signal-to-noise ratios - Higher implies greater variability in the estimated implicit target; lower implies greater stability. - Under high : - Implicit target peaks at 7 percent in 2009Q3 and moderates to around 6.6 percent. - Lower interest rate smoothing (smaller smoothing parameter). - Average inflation target for 2010–11: 6.5 percent. - Under low : - Implicit target more stable but still an upward drift in 2007–09. - Average inflation target for 2010–11: 6 percent. - Table 3 parameter examples (selected): - Baseline smoothing: 0.795 (0.071); high smoothing: 0.823 (0.077); low smoothing: 0.742 (0.061). - Baseline inflation coefficient: 3.940 (1.559); high : 4.028 (1.842); low : 3.351 (0.991). - Baseline 0.00965; high : 0.00526; low : 0.0554. - Implied ratios: Baseline 2.1; high 1.9; low 2.0. ### V. Time-varying natural real rate - Two approaches: - Full variation: natural real rate varies through time using Laubach and Williams (2003)-style state-space system linking natural real rate to trend growth of output. - Partial variation: natural real rate constant at 3.25 percent until 2008Q4, then declines to 2 percent thereafter. - Estimated natural real rate (full variation): - Increased to about 4.5 percent in mid-2008 from around 3 percent earlier. - Declined significantly with the global financial crisis to 2 percent. - Average reported as 3.2 percent in Figure 7. - Impact on implicit inflation target: - Inclusion of time-varying natural real rate shifts the implicit target trajectory slightly downward but preserves the central finding that the SARB targeted the upper segment of the official range (4 1/2 –6 percent). - Both full and partial variation scenarios show upward drift in 2007–09. - Estimated coefficients under time-varying specifications (Table 4, selected): - Smoothing: baseline 0.795 (0.071); partial variation 0.779 (0.059); full variation 0.799 (0.072). - Output gap coefficient: baseline 1.913 (0.627); partial 1.640 (0.519); full 1.525 (0.637). - Inflation coefficient: baseline 3.940 (1.559); partial 3.429 (1.133); full 3.813 (1.547). - Signal-to-noise: baseline 0.00965; partial 0.0089; full 0.0098. - Implied ratio (inflation weight/output gap weight): baseline 2.1; partial 2.1; full 2.5 — suggesting higher relative importance of inflation stabilization in the full-variation specification. ### VI. Actual versus predicted policy rate (dynamic forecast) - Dynamic forecasts start at 2005Q1 using specifications: baseline (constant r), full variation r, partial variation r. - Fitted rates follow the actual SARB policy rate well: - Predict monetary tightening in 2006–08 and monetary relaxation from 2009 onward. - Forecast performance: - Partial time-varying natural rate specification outperforms full variation and baseline in terms of sum of squared errors (SSE): - SSE (partial variation): 9.0. - SSE (full variation): 11.3. - SSE (baseline): 13.7. - Visual comparison (Figure 9) shows predicted series tracking actual policy rate under each specification. ### VII. Conclusions — key findings - Estimation approach: Taylor rule in state-space with implicit inflation target modeled as a time-varying random walk. - Core conclusions: - Despite an official inflation target range of 3–6 percent, empirical estimates indicate the SARB in practice aimed for the upper segment of the range (4 1/2 –6 percent) since IT introduction. - The implicit inflation target varied over time; a pronounced upward drift occurred in 2007–09 and the target remained near the upper limit (around 6 percent) through 2010–2011. - Results are sensitive to assumptions about the natural real rate and the signal-to-noise ratio, but the main qualitative finding (targeting the upper segment) is robust across specifications. - Allowing a time-varying natural real rate shifts the implicit target slightly downward and, under full variation, increases the relative weight on inflation stabilization (ratio to about 2.5). - Robustness: specifications with a partial time-varying natural rate provided the best fit to actual policy rate dynamics (lowest SSE). *Source: _wp12177 - Appendix Figures (extracted content).* ### introduction of the inflation target framework. In addition, the results show that the implicit ### _wp12177 - introduction of the inflation target framework. In addition, the results show that the implicit ### Key empirical finding on the implicit inflation target - The implicit inflation target varied over time. - In recent years it drifted upward toward the upper limit of the inflation target range (6 percent). - This perhaps implies that since the outbreak of the financial crisis in 2008 the SARB has become more tolerant toward higher levels of inflation to better support economic activity in the face of an extremely challenging global environment. ### Descriptive statistics (Table 1A, 2001q1–2011q4) - Variables: Inflation; Inflation Expectations; Output Gap; Policy Rate - Mean: 5.217; 5.532; -0.004; 8.886 - Median: 5.315; 5.600; -0.389; 8.000 - Maximum: 12.830; 7.400; 3.164; 13.500 - Minimum: -2.060; 4.500; -1.951; 5.500 - Std. Dev.: 3.486; 0.562; 1.389; 2.449 - Skewness: -0.056; 0.613; 0.881; 0.366 - Kurtosis: 2.777; 4.455; 2.831; 1.952 - Observations: 44; 44; 44; 44 ### Figures and series presented - Figure 1.A: The Estimated Implicit Inflation Target (series plotted 2001Q1–2011Q3; lines labeled “Baseline” and “Relaxation of the random walk assumption”); vertical axis range shown approximately 4.5–6.5. - Figure 2A: The SARB's Policy Rate, 2001q1-2011q4 (percent scale 0–16). *Source: Author's estimates; SARB.* --- _Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12177.pdf_