Inferring the Shadow Rate from Real Activity

Transcription

1 Inferring the Shadow Rate from Real Activity Benjamín García Arsenios Skaperdas June 8, 2017 PRELIMINARY Abstract We estimate a shadow rate for the effective lower bound, measured from real economic activity, that is directly comparable to the effective federal funds rate. Estimates indicate that unconventional monetary policy had significant real effects, reaching a maximum effect equivalent to a -4 percent policy rate. Each successive large-scale asset program was decreasing in effectiveness in stimulating growth. Our approach is validated by tests for parameter stability, accuracy of out-of-sample forecasts of all variables, and crossvalidation of the method when the federal funds rate is above its effective lower bound. JEL Classification: E43, E47, E52 We thank Dario Caldara, Benjamin K. Johanssen, Zeynep Senyuz, and seminar participants at the Central Bank of Ireland and Georgetown Center for Economic Research Conference. All errors are our own. We are responsible for the views expressed in this paper, and they should not be interpreted as reflecting the views of the Central Bank of Chile, the Board of Governors of the Federal Reserve System, or of anyone else associated with the Federal Reserve System. García: Central Bank of Chile, bgarcia@bcentral.cl. Skaperdas: Board of Governors of the Federal Reserve System, arsenios.skaperdas@frb.gov. 1

2 1 Introduction A natural benchmark for evaluating the effects of unconventional monetary policy is to measure how unconventional policy affects the macroeconomy in the ways that conventional monetary policy is thought to. This paper presents a relatively simple, atheoretic approach to estimate a shadow interest rate at the effective lower bound (ELB) which takes into account the effects of unconventional policy. We find that unconventional policy created a negative interest rate environment, in that it increased macroeconomic aggregates in the way that further federal funds rate decreases below zero would have. Our approach departs from current literature by inferring a shadow rate directly from observed variables capturing real economic activity. We show that a simple VAR can be used backwards. Typically, a monetary VAR is estimated using the federal funds rate, and the resulting impulse response functions are used to measure the effects of a monetary shock. Our exercise instead asks a different question: given previously measured observed effects of federal funds rate changes, what interest rate is most consistent with macroeconomic dynamics since reaching the ELB? As our benchmark specification features a VAR estimated from before the ELB, we perform a host of checks testing for structural change, parameter instability, and forecast accuracy. We find that our VAR accurately captures dynamics following the ELB. This finding mirrors other results in the time series literature (Stock and Watson, 2012) suggesting that the Great Recession was the result of large shocks rather than underlying changes in the structure of the economy. Nonetheless, we do provide upper bounds on our estimated shadow rate through a biased VAR estimation process incorporating data from the ELB. Relative to the literature, our approach departs from common estimation techniques, such as dynamic term-structure models (Krippner, 2013; Wu and Xia, 2016; Christensen et al., 2013), in that our method features no financial variables or no-arbitrage restrictions. 2

3 In a paper outside of the no-arbitrage framework, Lombardi and Zhu (2014) estimate a shadow rate with a dynamic factor model derived from interest rate series, monetary aggregates, and balance sheet data. In contrast, our underlying data includes standard series such as GDP and investment, while our approach decomposes these time series to find the interest rate most consistent with their evolution since Interestingly, our shadow rate compares closely to that of Lombardi and Zhu (2014), despite the significant differences in our data and methodology. This indicates that the stance of monetary policy instruments during the ELB, as measured by Lombardi and Zhu (2014), is mirrored by policy effects, which we measure. Another approach similar to ours in the literature is Johannsen and Mertens (2016). These authors also use time series models to better understand the effective short-term interest rate. A difference in our approach is that while Johannsen and Mertens (2016) measure the nominal interest rate that would prevail in the absence of the ELB, our results are interpreted as the likeliest effective monetary policy stance, measured in terms of the equivalent interest rate during normal times. Finally, our paper relates to state of the art non-linear DSGE models created to characterize the effective lower bound (Gust et al., 2016). While our approach is a linear approximation to macroeconomic dynamics, we find evidence that a simple VAR model forecasts accurately at the effective lower bound, as has been found in previous literature (Aastveit et al., 2016). 1 Of note, other shadow rate papers in the literature also assume a linear structure. 2 In practice, evidence of non-linearities using quarterly aggregate data has been hard to find (Ng and Wright, 2013), in part because non-linearities present at the household and firm-level may not survive aggregation. 3 1 These authors also find that constant parameter VARs are difficult to beat in out-of-sample forecasting following the crisis. 2 Wu and Xia (2016), for example, assume that their shadow rate is an affine function of factors following a linear VAR(1) process. 3 Financial market responses to monetary policy at the ELB, for example, do not show signs of nonlinearities. Swanson (2017) finds that responses to unconventional monetary policy are largely similar to those of conventional monetary policy, and that factors capturing those responses are essentially 3

4 Our paper continues as follows: we explain the data and methodology in Sections 2 and 3. Section 4 is the bulk of our paper; it presents our benchmark results, and provides a great deal of evidence that the results are valid. Section 5 explores the implications of the results for policy effectiveness. Section 6 concludes. 2 Data In order to adequately capture the relationship between interest rates and the rest of the economy, we utilize a broad set of macroeconomic variables. As in Smets and Wouters (2007) we consider output, private consumption, private investment, real wages, hours, inflation (GDP deflator), and the federal funds rate. The first four variables are per capita and used in log levels so as to contain the history of their evolution. Real wages and hours are from the non-farm business sector (Bureau of Labor Statistics). As a robustness check, we also include log per capita government consumption and investment from the National Accounts in our estimations. 3 Methodology As in Skaperdas (2016), our methodology follows from the observation that VARs can be used to measure the federal funds rate as an unobserved variable. We ask the following question: given the measured effects of a monetary shock, what is the likeliest interest rate? We estimate a VAR and extract the likeliest interest rate at the effective bound, given the VAR parameters and observed data. More explicitly, consider a bivariate VAR(1) system in structural form unchanged when estimating over the ELB separately. B 0 Y t = B 1 Y t 1 + u t (1) 4

5 Rewrite as explicit equations for the two variables with a recursive ordering, letting (i, j) denote the ith row and jth column of the preceding matrix x t = B 1 (1, 1)x t 1 + B 1 (1, 2)z t 1 + u 1,t (2) z t = B 0 (2, 1)x t + B 1 (2, 2)x t 1 + B 1 (2, 1)z t 1 + u 2,t (3) The error terms are independent, zero mean, and normally distributed. Assume that the VAR parameters have been solved for using a sample of observed values. Then in state-space form, the measurement equation is as follows, While the state equation is x t = ˆB 1 (1, 1)x t 1 + ˆB 1 (1, 2)z t 1 + u 1,t (4) }{{} deterministic z t = ˆB 0 (2, 1)x t + ˆB 1 (2, 2)x t 1 + ˆB 1 (2, 1)z t 1 + u 2,t (5) }{{} deterministic Using an observed series of values, x t, we can solve for the optimal z t using the Kalman filter for the unobserved state, both in- or out-of-sample, using VAR parameter ˆB s. Since the shocks are orthogonalized, the Kalman filter provides an optimal linear estimate of the state. In brief, the filter minimizes one step ahead forecast errors. We treat deterministic terms as exogenous forcing variables. In practice, our measurement equations and the state are set up as functions as follows: Output Consumption Investment X t = RealW age Hours Inf lation [ ] Z t = F ederalf undsrate 5

6 As shown in the two variable example, we identify the VAR using the Cholesky decomposition. Since our data consists of non-financial variables, we consider consider the Cholesky identification acceptable. 4 In support of our identification, which imposes null responses upon impact of a monetary shock to the other VAR variables, Caldara and Herbst (2016) find insignificant contemporaneous effects of monetary shocks on real variables when identifying monetary shocks through high frequency identification. In order for our methodology to capture the overall effects of unconventional monetary policy, it must be the case that unconventional policy transmits to macro variables similarly to the way a change in the federal funds rate would have. Our estimates provide a benchmark estimate of unconventional policy through this lens. 5 Other researchers have made the case that large-scale asset purchases affect the macroeconomy in a similar way to conventional interest rate changes (Gagnon et al., 2016), while forward guidance was used both during and before the ELB period (Swanson, 2017). An alternative estimation approach to that examined in this paper would be to estimate all parameters in the state-space model at once, and treat the federal funds rate as an unobserved variable at the effective lower bound. However, this would also necessitate the estimation of the Federal Reserve s reaction function during the effective bound, confounding the estimation process. The identification of the unobserved variable would come not only from the reaction of the rest of the economy, but also from the likeliest reaction of the interest rate to the economic environment. This could also bias measures of monetary shocks from before the crisis. Nonetheless, to be explained, we do address a similar type of estimation process to bound the measures of our shadow rate by a biased estimation process. 4 While incorporating financial variables, a VAR with Cholesky identification is misspecified since financial markets are known to quickly react to policy changes. 5 Note that to the extent that unconventional policy affects macroeconomic aggregates in additional channels not occurring through conventional federal funds rate changes, we would understate its beneficial effects. 6

7 4 Results 4.1 Benchmark Estimation Figure 1 presents the benchmark estimation results from this paper. We use one lag, as suggested by both the Hannan-Quinn and Schwarz Information Criteria, since our data are quarterly and less than 120 observations before the ELB (Ivanov and Kilian, 2005). In-sample, the state estimates closely follows the federal funds rate. The out-of-sample estimates directly infer the equivalent federal funds rate following the incidence of the effective lower bound. These results provide a summary statistic of the net effect of Federal Reserve policies at each quarter, to the extent that they affect other variables in the model in the way traditional monetary policy would have. Of note, the estimates are negative soon following the financial crisis. Thus, unconventional policy was able to affect macroeconomic aggregates in the way a strong and persistent reduction of the federal funds rate would have. Figure 1: Implied Monetary Stance, Benchmark Estimate q1 1990q1 1995q1 2000q1 2005q1 2010q1 2015q1 Quarter Federal Funds Rate Filtered State Estimate (+/- 1 RMSE) The fact that the shadow rate initially increases following 2008:Q3 may at first appear 7

8 unintuitive. However, the shadow rate estimated in this paper is the most likely interest rate given observed variables. Although the Federal Reserve decreased the policy rate close to its effective lower bound in 2008:Q4, it is reasonable to believe that the tightening of financial markets following the crisis caused macroeconomic aggregates to grow as if a contractionary monetary policy had been in place. Thus, the increase in the shadow rate reflects the severity of the financial shock on the real economy. As evidence of this financial shock, we present Figure 2. In it, one can see that though the federal funds rate was set to near zero, interbank rates spiked heavily, which would influence broader financing conditions, and thus the shadow rate we estimate. Figure 2: Interbank Spreads /1/2002 7/1/2004 1/1/2007 7/1/2009 1/1/2012 7/1/2014 Date 6 mth Libor- 6 mth OIS 3 mth Libor- 3 mth OIS 1 mth Libor- 1 mth OIS The confidence bands on the estimates are calculated so as to correct for the generated regressors in the first state estimation of the VAR parameters. In all figures, standard error bands are calculated using a circular bootstrap. Steps are as follows: an artificial time series is created, equal in length to the sample, using random samples of residuals drawn with replacement from a circle of the VAR residuals, initiated with 1984:Q1 values. The VAR model is estimated over the artificial time series, yielding a distribution of the 8

9 VAR parameters. This process is repeated 200 times. We then estimate the shadow rate over the original 1984:Q1-2016:Q2 data with each iteration of the VAR parameters, and report the 16th and 84th percentile intervals of the resulting distribution of state estimate root mean squared errors. 4.2 Robustness Figure 3: Implied Monetary Stance, Smoothed Estimates q1 1990q1 1995q1 2000q1 2005q1 2010q1 2015q1 Quarter Federal funds rate Smoothed State Estimate (+/- 1 RMSE) VAR coefficients estimated using 1984Q1-2008:Q3 data, 1 lag. Figure 3 present the estimation results from the (two-sided) Kalman smoother. In each quarter, this estimation process uses the entire sample of observations, rather than observations for each time t through quarter t. The smoother results in state estimates that are fairly similar to the filter. This provides evidence that the model is accurately capturing dynamics at the effective lower bound, as the state estimates, and thus the variances of the measurement equations, are not significantly changed by the addition of ELB data. Nonetheless, the filter (one-sided) estimates are our preferred specification, as they do allow some real-time fluctuation in variance estimation. 9

10 The state estimates are robust to multiple other variations on the VARs. Figure 4 charts several variations of the shadow rate. The state estimates are exactly equal with an alternative VAR ordering, where inflation is ordered before all other variables. 6 Likewise, selecting a lag length of two results in very similar estimates and dynamics. 7 Within the context of our estimations, it does not seem to be the case that government spending is an important omitted variable, in spite of the fact that the effective bound period was also characterized by unprecedented fiscal contraction. In a variety of specifications, we do not find that including government spending increases the forecast accuracy of our methodology either in- or out-of-sample. Figure 4 presents one such specification, where we add government consumption and investment, ordered first to the VAR, and omit hours. Government spending shocks seem to be associated with shocks other than monetary in the model, as the resulting state estimate is similar. Whereas a monetary shock would typically increase private investment and consumption, a government spending shock would typically do the opposite due to crowding out (Coenen et al., 2013; Coenen and Straub, 2005). The finding that forecast errors of government spending are not associated with reductions in output in the US recovery is corroborated by House et al. (2017), though those authors do find strong effects in other countries. A possible concern of the use of the state-space model is that the estimates are driven by the state equation, rather than the measurement equations. If the measurement equations do not provide a good model during the effective bound period, it could be the case that the state estimate of the federal funds rate is simply tracing out the pre-effective lower bound reaction function of the Federal Reserve. In order to address this concern, we reformulate the state equation to be a random walk as follows: 6 Any ordering of the variables before the interest rate yields equal estimates by construction, as ordering of the non-policy block does not matter. 7 Since we have a short sample our VAR does not have many lags. One could think this is problematic, as monetary policy is typically thought to have lagged effects. Note, however, that the interest rate is measured off of its level at each period, not just shocks occurring in each quarter, as would be the case for a VAR estimated in differences. 10

11 Figure 4: Implied Monetary Stance, Various Alternative VARs q1 1990q1 1995q1 2000q1 2005q1 2010q1 2015q1 Quarter Federal Funds Rate State Estimate, With Govt. Spending State Estimate, Inflation Ordered First in VAR Filtered State Estimate (+/- 1 RMSE) State Estimate, 2 Lags VAR coefficients estimated using 1984Q1-2008:Q3 data, 1 lag. z t = z t 1 + u z,t (6) The state equation now contains no estimated parameters, and agnostically treats an increase or decrease in the federal funds rate as equally likely in each quarter. The measurement equations remain the same as in the benchmark estimation. Figure 5 reveals that the state estimate dynamics are largely unchanged, meaning that the precise structure of the state equation is not crucial for estimation of the shadow rate. However, the random walk formulation does imply somewhat less accommodation overall, and especially less in level terms in 2009 and Model Cross-Validation If the VAR models are accurately capturing the dynamics of the system, one should be able to estimate them with a shorter time series, and show that the federal funds 11

12 Figure 5: Implied Monetary Stance, Other Checks q1 1990q1 1995q1 2000q1 2005q1 2010q1 2015q1 Quarter Federal Funds Rate Baseline Filtered State Estimate (+/- 1 RMSE) State Estimate with Random Walk rate is correctly predicted out-of-sample. 8 Figure 6 presents the shadow rate estimation process using a VAR estimated from 1984Q1-2002:Q1. In-sample, the estimation once again closely recovers the federal funds rate, and lies within the bounds of the benchmark estimation standard error bands. More importantly, the out-of-sample results also recover the federal funds rate during which it is observed, and in addition, the ELB period estimate is characterized by similar dynamics. While Figure 6 uses VAR parameters estimated through 2002:Q1, we can repeat this process for many endpoints. Figure 7 presents the out-of-sample states estimated with VARs from 1984:Q1 to 2001:Q2, sequentially adding one quarter to the end date at a time, until the benchmark VAR estimated through 2008:Q3. Figure 8 repeats this process, but presents the smoothed estimates. In all cases, the methodology recovers the federal funds rate closely both in- and out-of-sample. Likewise, the state estimates during the effective bound period indicate the unconventional policy gave the economy under zero 8 Note that we use out-of-sample testing to verify our approach, not for model selection, which could be problematic Hirano and Wright (2013). 12

13 Figure 6: Implied Monetary Stance, Out-of-Sample Results, VAR coefficients 1984Q1-2002:Q q1 1990q1 1995q1 2000q1 2005q1 2010q1 2015q1 Quarter Federal Funds Rate Filtered State Estimate (+/- 1 RMSE from Benchmark Estimation) Out-of-sample beginning at first red line, second red line denotes start of effective lower bound. properties. 9 9 In the Appendix, we present similar exercises where the VARs are estimated using post-1990 data, and estimated out-of-sample in the 1980s. 13

14 Figure 7: Implied Monetary Stance, Filtered Out-of-Sample Forecasts Iterated Forward q1 1990q1 1995q1 2000q1 2005q1 2010q1 2015q1 Quarter Blue line: federal funds rate. Red lines: out-of-sample state estimates. Dashed black lines: 1984Q1-2008:Q3 state estimate 1 standard error band. Figure 8: Implied Monetary Stance, Smoothed Out-of-Sample Forecasts Iterated Forward q1 1990q1 1995q1 2000q1 2005q1 2010q1 2015q1 Quarter Blue line: federal funds rate. Red lines: out-of-sample state estimates. Dashed black lines: 1984Q1-2008:Q3 state estimate 1 standard error band. 14

15 4.4 Tests of Parameter Stability A concern regarding our methodology is that structural change occurred following 2008:Q3. We have assumed that a parsimonious VAR model from before the effective lower bound period can accurately describe dynamics during the effective bound. This could be problematic if, for example, the persistence of GDP decreased following the crisis. We would then find a sequence of contractionary shocks to GDP, as GDP realizations would be lower than the VAR predictions. The filter would then erroneously attribute these contractionary GDP shocks to contractionary monetary policy, and then estimate a more positive shadow rate Explicit Tests Aastveit et al. (2016) find in a 4-variable VAR that the GDP and inflation equations do not suffer from much instability through the crisis, while the unemployment rate and interest rate parameters do seem to have shifted. The interest rate equation would be expected to have time variation at the ELB due to the ELB constraint. As an explanation for the unemployment parameters shifting, it is known that the unemployment rate has overstated the strength of the labor market during the recovery, as the labor force participation rate has been low by historical standards (Kroft et al., 2016). Since we have used data from Smets and Wouters (2007), we fortunately do not use the unemployment rate in our estimation. We test explicitly for parameter instability in our sample through the following thought experiment: if one were to randomly test if a thirty one quarter period was structurally different during the Great Moderation, what are the chances that one would get a distribution of parameters as extreme as the ones from during the effective lower bound? 10 To this end, we estimate sequential VARs over thirty one quarters from 1984:Q1 through 10 We choose thirty one quarters, as this is the length of the observed sample during the ELB. 15

16 2001:Q1. The last start date for the thirty one quarter VAR sample, 2001:Q1, is chosen such that the sample ends before the ELB. Figure 9 plots the kernel density distributions of the estimated parameters with red lines for the ELB parameter estimates. The parameters during the ELB are very similar to pre-elb parameters, and lie plausibly within the pre-elb VAR parameter distributions. The parameters that are visibly different are the estimates of lagged interest rate coefficients. In Figure 10, we present rolling estimates of the lagged interest rate parameters. The VAR coefficients using data from the ELB are very extreme in comparison to the pre-elb sample. This is intuitive: if unconventional policy had significant effects, a federal funds rate of zero does not adequately describe the stance of policy. A VAR estimated with just the federal funds rate would then find very biased effects of the federal funds rate during the zero bound, as it would be measuring a policy instrument with a much tighter stance than that realized. We thus present blue dashed lines indicating VAR parameters estimated over the ELB data using the estimated shadow rate in the kernel density estimates of Figure 9. Validating our hypothesis regarding the bias of the federal funds rate as a stance of policy, the lagged interest rate parameters using the estimated benchmark shadow rate are much more consistent with the pre-elb distribution. It is important to remember that this is not a result of the state-space model maximizing these parameters. The Kalman filter maximizes the forecasts of the measurement series. In order to get a last sense of how the distribution of parameters during the ELB compares to the in-sample thirty one quarter estimated VARs, we present Figure 11. Each line connects the standardized parameters, ordered from smallest to largest, from each VAR estimated over thirty one quarters beginning 1984:Q1 to 2001:Q1. 11 We exclude 11 The parameters are standardized with respect to equivalent parameters estimated in each VAR. For example, every estimate of the constant for output is standardized with respect to all VAR estimates of that parameter. 16

17 Figure 9: Kernel Density Estimates of VAR Parameters, from Thirty One Quarters Sampled 1984:Q1-2008:Q3, Compared to ELB Parameters output_l_output output_l_consumption output_l_investment output_l_realwage output_l_hours output_l_inflation output_l_interestrate output_constant consumption_l_output consumption_l_consumption consumption_l_investment consumption_l_realwage consumption_l_hours consumption_l_inflation consumption_l_interestrate consumption_constant investment_l_output investment_l_consumption investment_l_investment investment_l_realwage investment_l_hours investment_l_inflation investment_l_interestrate investment_constant realwage_l_output realwage_l_consumption realwage_l_investment realwage_l_realwage realwage_l_hours realwage_l_inflation realwage_l_interestrate realwage_constant hours_l_output hours_l_consumption hours_l_investment hours_l_realwage hours_l_hours hours_l_inflation hours_l_interestrate hours_constant inflation_l_output inflation_l_consumption inflation_l_investment inflation_l_realwage inflation_l_hours inflation_l_inflation inflation_l_interestrate inflation_constant interestrate_l_output interestrate_l_consumption interestrate_l_investment interestrate_l_realwage interestrate_l_hours interestrate_l_inflation interestrate_l_interestrate interestrate_constant The x-axes denote parameter estimates from sequential VARs estimated over thirty one quarters from 1984:Q1-2008:Q3. The y-axes denote density. The red lines indicate parameters from a VAR estimated over 2008:Q4-2016:Q2, while the blue dashed lines indicate the same VAR estimated with the benchmark shadow rate in place of the federal funds rate. 17

18 Figure 10: Parameter Stability of Lagged Interest Rate Coefficient: Parameter Estimates from Thirty One Quarter Rolling VARs q1 1990q1 1995q1 2000q1 2005q1 2010q1 Quarter Federal Funds Rate Hours Investment Output Inflation Real Wage Consumption This figure presents the lagged interest rate coefficient on each variable over rolling thirty one quarter VAR samples from 1984:Q1-2016:Q2. Each point indicates the parameter over a VAR beginning at that date. 18

19 lagged interest rate predictors, as these were shown to be biased. Figure 11: Distribution of Standardized Parameters: Rolling Thirty One Quarter VAR Samples Standardized Coefficient Ordinal Rank The x-axes denotes the ranking of each standardized parameter from most negative to most positive, excluding lagged interest rate parameters. Each blue line is a separate VAR estimated over a thirty one quarter period between 1984:Q1 and 2008:Q3. The dashed red line is the distribution of standardized coefficients from a VAR estimated from 2008:Q4 to 2016:Q2. Visually, the ELB parameters fall for the most part in the middle of the distribution. Closer to the tails, there do seem to be some parameters that are more extreme at the ELB, however, there are still other VAR samples that have as extreme or more extreme parameters at every point. We present parameters ranked greater than 1.65 standard deviations from the mean in the ELB sample in Table As shown, the most extreme parameters are those governing the investment, real wage, and inflation predictions. If these parameters changed significantly following the crisis, the forecasts of these 12 Note that the parameters from the thirty one quarter VARs have heavier tales than would be expected from a normal distribution. 19

20 Table 1: ELB Parameters Greater than 1.65 Standard Deviations from Respective Means Parameter Z-Score Investment, Lagged Consumption 2.36 Investment, Lagged Hours Investment, Lagged Real Wage Investment Constant 2.17 Real Wage, Lagged Consumption 2.95 Real Wage, Lagged Hours Real Wage, Lagged Inflation Inflation, Lagged Inflation 2.16 Inflation, Lagged Hours time series, and hence the measurement equations, will result in less accurate estimates of the federal funds rate as an unobserved variable when the pre-elb VAR is used for the ELB. In order to gauge the influence of these series on the shadow rate estimation, we present Figure 12. This chart plots the estimated shadow rates when we exclude these series one at a time from the measurement equations. This prevents any potentially structurally changed parameters from influencing the estimation process. The estimated shadow rate is fairly similar to the benchmark shadow rate regardless of which series is excluded. Since the shadow rate is measured from each of the six time series, the few parameters that show evidence of structural change are not important enough to change the estimated shadow rate. The one exception is that while excluding investment, the shadow rate is slightly more negative before However, the estimate is still well within the confidence bounds of the benchmark shadow rate. In conclusion, a thorough evaluation of the VAR shows that parameter stability is not an issue in the implementation of our shadow rate Forecast Evaluation One further way to test for parameter instability is to evaluate the forecast performance of a model during the time in which it is thought that the parameters have changed. If 20

21 Figure 12: Implied Monetary Stance, Excluding Series From Measurement Equations q1 1990q1 1995q1 2000q1 2005q1 2010q1 2015q1 Quarter Federal Funds Rate Excluding Inflation Benchmark State Estimate (+/- 1 RMSE) Excluding Investment Excluding Real Wages Shadow rate estimates when excluding selected series from the measurement equations one at a time. The series excluded each have at least one parameter at a tail of the distribution in Figure 11 greater than 2 standard deviations from the mean. 21

22 the model is not a good characterization of the data during that time, the out-of-sample forecast errors will be larger than in-sample forecast residuals. Even if some parameters change, it could be the case that these parameters are not influential for model dynamics, and thus that the forecast accuracy of the model could be relatively unaffected. As shown in Table 2, the in-sample one-step ahead forecast errors (Column a) are largely comparable to the one-step ahead forecast residuals from after the crisis (Columns c and d). 13 The mean absolute forecast error is within one standard deviation of forecast residuals for every variable in the VAR. 14 This means that following the crisis, the VAR predictions still forecast relatively well, meaning that the model is likely to provide similarly accurate estimates of the effective federal funds rate as before the crisis. In addition, it is notable that the mean errors of many of the series are relatively small in an economic sense. For example, the forecasts of output and consumption are on average less than 0.5 percentage points from actual values. Note that we use the estimated shadow rate in lieu of the federal funds rate following 2008:Q3, in order not to bias our forecast errors by assuming unconventional policy had no effect. We also also present a placebo test, in Column (b), where we compare out-of-sample forecast errors from before the crisis with out-of-sample forecast errors during the effective bound. This exercise is based on the following logic: using out-of-sample testing, it is clear that our methodology captures the federal funds rate well during periods when it is observed. In particular, Figure 6 showed that the federal funds rate is adequately captured out-of-sample following 2002:Q1 using VARs estimated from 1984:Q1-2002:Q1. If out-of-sample forecast errors during that time are no larger than forecast errors during the effective bound, this provides evidence that our VAR parameters are also adequately 13 We present Column (d) for reference, which omits the 2008:Q4 residual, because that quarter is characterized by unprecedented shocks due to the financial crisis that are difficult to capture with a statistical model. A regime-shifting VAR would be best for this quarter, but there exists no comparable post-war data to estimate such a model on. 14 We use mean absolute forecast errors since larger shocks during the crisis, as found by Stock and Watson (2012), would necessarily imply greater mean squared errors during the ELB sample regardless of model stability. 22

23 Table 2: VAR Mean Absolute Forecast Accuracy (a) (b) (c) (d) Forecast Residuals Forecast Errors Forecast Errors Forecast Errors Pre-2008:Q4 2002:Q2-2008:Q3 Post-2008:Q3 Post-2008:Q4 Output (0.297) (0.240) (0.437) (0.238) Consumption (0.251) (0.224) (0.376) (0.329) Investment (0.758) (0.871) (1.35) (1.033) Real Wage (0.427) (0.440) (0.824) Hours (0.260) (0.260) (0.404) (0.362) Inflation (0.114) (0.254) (0.167) (0.128) Observations Mean one-step ahead forecast errors/residuals, standard deviations in parentheses. Column (a) denotes in-sample forecast residuals from a VAR estimated from 1984:Q1-2008:Q3. Column (b) denotes out-of-sample forecast errors from a VAR estimated from 1984:Q1-2002:Q1. Column (c) denotes out-of-sample forecast errors from a VAR estimated from 1984:Q1-2008:Q3. For comparison, Column (d) repeats Column (c), but does not include the error from 2008:Q4. capturing dynamics when the effective stance of monetary policy is unobserved. Column (b) presents forecast errors from 2002:Q2-2008:Q3 using VAR predictions from 1984:Q1-2002:Q1 and using the estimated out-of-sample shadow rate from the 1984:Q1-2002:Q1 VAR predictions. Comparing Column (b) with Columns (c) and (d) shows that the effective bound forecast errors compare well to the out-of-sample forecast errors during a time which we know that the methodology adequately recovers estimates of the federal funds rate. While the output, consumption and investment equations have slightly larger forecast errors, the real wage, hours, and inflation equations have smaller forecast errors Full-Sample Estimation As a final check, a bound on our benchmark results is presented in Figure 13 where we estimate the VAR over the whole sample (1984:Q1-2016:Q2), including the effective bound 23

24 period. 15 The resulting state estimate is now a result of parameters partly estimated during the effective bound, which assuages concern about structural change. On the other hand, this also introduces a very large bias in the results: the parameters are fit in order to minimize deviations from predictions where the stance of monetary policy is set to a federal funds rate of essentially zero during the effective bound period. The VAR parameters are thus heavily biased against finding an effect of unconventional policy, and also cause extreme bias in the measures of monetary shocks from before the effective bound. Nonetheless, the full sample VAR parameters still result in a state estimate that lies within the benchmark estimate confidence interval, and still result in monetary accommodation in excess of an effective federal funds rate of zero, though less accommodation than in the benchmark estimation. The full sample state estimate also depicts similar dynamics. Figure 13 also presents a second variation of the full sample VAR state estimate using a random walk reformulation of the state equation, as in Section 4.2. By imposing a random walk rather than using estimated parameters, we do not bias the state estimates with a reaction function predicting a federal funds rate of zero at the ELB. The state equation is especially problematic at the ELB as the persistence of the federal funds rate will be biased by a period of an unchanging interest rate that is unlikely without the ELB constraint. The random walk reformulation results in a state estimate that is more accommodative than the state estimate with estimated parameters in the state equation. The shadow rate troughs just 1 percent less negative than the baseline estimate. Overall, the full sample estimation procedures provide important evidence that a federal funds rate of zero did not capture the effects of policy during the ELB, meaning that unconventional policy had significant effects. Biased estimation procedures still result in shadow rate troughing 15 It is not possible to estimate a shadow rate over only the ELB, as there are too few observations. Even if it were possible to use only ELB VAR parameters, the bias of using well-measured parameters from before the ELB could be preferable to the increase in variance from such a a small sample (see Clark and McCracken (2009)). 24

25 below a -1 percent stance of policy. Figure 13: Implied Monetary Stance, Full Sample Results q1 1990q1 1995q1 2000q1 2005q1 2010q1 2015q1 Quarter Federal Funds Rate Filtered State Estimate (+/- 1 RMSE) State Estimate with Full Sample Estimation State Estimate with Full Sample Estimation and Random Walk 5 Implications 5.1 The Large-Scale Asset Purchase Programs As the validity of the estimation has been established, the estimates can be used to answer questions about policy effectiveness. Figure 14 presents the benchmark results from Figure 1, shown following 2006, and plots vertical lines for the announcements and implementations of the three large-scale asset purchase programs (henceforth, Quantitative Easings). As evidence for the effectiveness of the QE policies, we observe that the estimated shadow rate decreases immediately during the quarter that each of those programs were implemented While we loosely attribute decreases in our shadow rate to the QE programs, the Federal Reserve also implemented many other programs during this time period, such as the Term Auction Facility during QE 1. 25

26 In our estimation, QE1 and QE2 seem to have had a large effect, while QE3 less so. QE1 seems to have immediately decreased the effective interest rate environment, which is plausible given that it helped to unfreeze credit markets. While our baseline shadow rate decreases the quarter before the announcement of QE2, that program was widely anticipated by market participants. 17 This points to important signaling effects of policy. Figure 1 presents the shadow rate of Lombardi and Zhu (2014) for comparison. Our shadow rate exhibits strong comovoment with that of Lombardi-Zhu. This is interesting given the very different methodologies and underlying data of our measures. The Lombardi-Zhu shadow rate is derived from yields, monetary aggregates, and Federal Reserve balance sheet items. The fact that the two shadow rates are characterized by such similar dynamics is comforting in that it indicates that changes in the policy instruments of the Federal Reserve, as measured by Lombardi-Zhu, are mirrored by changes in policy effects, as measured by our shadow rate. Both our shadow rate and that of Lombardi and Zhu (2014) indicate quite different effects of unconventional policy than that of Wu and Xia (2016). It would seem that Wu- Xia s use of no-arbitrage restrictions, and focus on yields, imply very different shadow rate dynamics. In contrast, our results are determined by the levels of real activity in the six VAR measurement equations. In principle, these two shadow rate methodologies could result in very different values. For example, if the transmission from financial conditions to real activity was impaired, the size of the Federal Reserve s balance sheet and the size of unconventional monetary policy s real effects would not have to be correlated. While the Wu-Xia shadow rate is correlated with the size of the Federal Reserve s balance sheet, our shadow rate depicts immediate and visible effects of the large-scale asset purchase programs, pointing to strong flow or signaling rather than stock effects. One caveat to these comparisons, however, is that our standard error bands encompass the Wu-Xia 17 See an Op-ed by Ben Bernanke in the Washington Post at that time ( 26

27 shadow rate. Figure 14: Effects of Quantitative Easing q1 2008q1 2010q1 2012q1 2014q1 2016q1 Quarter Federal Funds Rate Filtered State Estimate (+/- 1 RMSE) Wu-Xia Lombardi-Zhu The red lines denote the starts of QE1, QE2, and QE How Reasonable are the Estimates? Our shadow rate methodology indicates that the QE programs had a maximum effect equivalent to approximately a -3 to -4 percent federal funds rate during the peak effects of QEs 1 and 2. Though these estimates are fairly uncertain, are they reasonable? Previous research (Chung et al., 2012) has found that a 100 basis point change in the federal funds rate tends to be associated with about a 25 basis point change in 10-year Treasury yields. Likewise, it has been widely estimated that the QE programs lowered 10-year yields by about 60 to 100 basis points during QE 1, and a further 20 basis points during QE 2 (Chung et al., 2012; Li and Wei, 2013). Thus, our estimates of the implied stance of policy are consistent with the previous estimates of the effects of QE on long- 27

28 term yields. The basis point change in long-term yields following QE 1 would typically be associated with the magnitude of the large decrease in the monetary stance that we observe, on the order of 4 percent below zero. Likewise, our 1-2 percent change in the shadow rate following QE 2 would be within the spectrum of estimates associated with a 20 bp change in 10-year Treasuries from that program. Finally, our shadow rate also takes into account the effects of forward guidance, meaning that it is sensible that our estimates would be more accommodative than that suggested by the QE programs alone. 6 Conclusion This paper provides new estimates of unconventional monetary policy s real effects by creating a shadow rate inferred directly from real economic activity. We show that a simple VAR model indicates that monetary policy was more accommodative than a federal funds rate of zero would imply. Our relatively simple framework provides direct visual evidence of beneficial effects from the three major quantitative easing programs. We are confident that our results accurately capture the dynamics and broad magnitudes of unconventional monetary policy s real effects. Our methodology forecasts the federal funds rate correctly out-of-sample before the ELB. After the ELB, we show that our VAR accurately forecasts variables other than the federal funds rate. Finally, explicit evaluation of the ELB VAR parameters shows that they are drawn from a similar distribution as parameters from the Great Moderation. The one exception we find is that the parameters governing interest rate effects on other variables are highly biased when not incorporating the shadow rate as the stance of policy. On the technical front, our approach to estimating the effect of unconventional policy as an unobserved variable could be extended. Estimation could incorporate time-varying volatility and Bayesian estimation of parameters, which would allow for longer lag lengths 28

29 and possibly more sophisticated estimation of parameters at the effective bound. Another possibility would be to incorporate a larger number of real variables affected by the federal funds rate subject to optimal information criteria. We leave these extensions to future work. 29

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32 Appendix Figure 1: Implied Monetary Stance, Filtered Out-of-Sample Forecasts Iterated Backwards q1 1990q1 1995q1 2000q1 2005q1 2010q1 2015q1 Quarter Blue line: federal funds rate. Red lines: out-of-sample state estimates. Dashed black lines: 1984Q1-2008:Q3 state estimate 1 standard error band. 32

33 Figure 2: Implied Monetary Stance, Smoothed Out-of-Sample Forecasts Iterated Backwards q1 1990q1 1995q1 2000q1 2005q1 2010q1 2015q1 Quarter Blue line: federal funds rate. Red lines: out-of-sample state estimates. Dashed black lines: 1984Q1-2008:Q3 state estimate 1 standard error band. 33