How Robust is the VAR-Based Expectation of the Federal ......One example is a future change in the fiscal policy that is perfectly anticipated under full-model expectations but recognized

Azzam, International Journal of Applied Economics, 5(1), March 2008, 54-84 54

How Robust is the VAR-Based Expectation of the Federal Reserve Board (FRB/US) to Model-Selection Uncertainty?

Islam Azzam*

American University in Cairo

Abstract In the spring of 1996, the first working version of the FRB/US model replaced the venerable Monetary Policy System (MPS) model for forecasting and policy analysis in the US. The agents in the economy use a VAR-based expectation model, auxiliary to the large and complicated FRB/US model. This paper focuses on examining the robustness of the VAR-based expectations of the FRB/US model to the model-selection uncertainty--that is, the uncertainty about the true lag-order of the autoregressive processes. The paper estimates the degree of model-selection uncertainty in this VAR-based expectation model and examines its effect on the estimated impulse responses. It also examines the sensitivity of the monetary policy recommendations to changes in the lag order of the VAR-based expectations of the FRB/US model. The analysis found that this model is very robust to model-selection uncertainty. The analysis suggests that the agents should use VAR (1) representation as the VAR (1) forecasts better. Using the variance decompositions, we show that the policy inferences are not sensitive to the changes in the lag order of the VAR-based expectation of the FRB/US model. Keywords: model-selection uncertainty, FRB/US model, bootstrap, and impulse response functions. JEL Classification: C01, C15, E5, E37

1. Introduction The model of monetary policy used commonly by the Board of Governors of the Federal Reserve System today is referred to as the Federal Reserve Board (FRB/US) model. In the spring of 1996, the first working version of the FRB/US model replaced the venerable Monetary Policy System (MPS) model for forecasting and policy analysis. Though similarly of large scale containing some 300 equations and identities, the FRB/US model has the appealing feature that it can be simulated using expectations based on limited information. Simulations of expectations held by individuals and firms can be obtained using the forecasts of two alternative representations of the economy: (1) a small vector autoregressive (VAR) model, auxiliary to the large model and assumed to be known by the agents in the economy, and (2) the FRB/US model itself. This paper focuses on examining the robustness of the VAR-based expectations of the FRB/US model to the model-selection uncertainty--that is, the uncertainty about the true lag-order of the


autoregressive processes. The question remains, how does the uncertainty about the correct lag order affect the estimates of our VAR-based expectation of the FRB/US model? In an empirical study, I estimate the degree of model-selection uncertainty in this VAR model and examine its effect on the estimated impulse responses using Kilian’s (1998) method. Also, I examine the sensitivity of the monetary policy recommendations to changes in the lag order of the VAR-based expectations of the FRB/US model. The previous literature has focused largely on optimal rules for monetary policy when there is uncertainty about the true model. For example, in search of monetary policy rule that works well across a wide range of structural models, Levin, Wieland and Williams (1999) compare the performance of a variety of contingent monetary policy rules in terms of their implications for the variability of output and inflation in different monetary models. Tetlow and Muehlen (2000), within the context of a simple New Keynesian model, find that rules which are robust to uncertainty regarding estimated structural parameters tend to be more aggressive in response to output and inflation than optimal linear-quadratic rules. By contrast, policies designed to protect the economy against the worse-case consequences of misspecified dynamics are less aggressive. Muehlen (2001) derives mean leads, lags ad patterns of relative importance weights implied by the polynomial-adjustment-cost error correction equations that form the core of the FRB/US model. Relative importance weights measure the contributions of past and future expected changes in fundamentals on current decisions. These and the associated mean lags and leads can be considered summary measures of key dynamic properties of FRB/US. Svensson and Tetlow (2005) introduce a method that provide advice on optimal monetary policy while taking into account policy-makers’ judgment. They construct optimal policy projections (OPPs) by extracting the judgment terms that allow the FRB/US model to reproduce a forecast such as the Greenbook forecast. Given an intertemporal loss function that represents monetary policy objectives, OPPs are the projections of target variables, instruments, and other variables of interest that minimize that loss function for given judgment terms. They show that for a convention loss function, the OPPs provide significantly better performance than Taylor-rule simulations. Telow and Ironside (2006) study 30 vintages of FRB/US. They exploit archives of model code, coefficients, baseline databases and stochastic stock sets stored after each FOMC meeting from the model’s inception in July 1996 until November 2003. They document the surprisingly large changes in model properties that occurred during this period and compute Taylor-type rules for each vintage. They compare these optimal rules against plausible alternatives. Model uncertainty is shown to be a substantial problem. This paper, instead, focuses on gauging the uncertainty of the model in use. More specifically, the analysis of this paper estimates the degree of model-selection uncertainty within the VAR-based expectation model of the FRB/US framework instead of uncertainty about the optimal rule for this model. I use Kilian’s (1998) algorithm to measure this model-selection uncertainty. The question is: what is the effect of the uncertainty about the true lag order in the VAR-based


expectation models on the expectations and the model generation of the consequent policy recommendations. The importance of such uncertainty in this context is illustrated in Hafer and Sheehan (1991) who investigate the sensitivity of the policy inference derived from the VAR models to changes in the lag structure. They implement a simple four-variable VAR model with quarterly data for money, real output, prices and nominal short-term interest rates over the period 1960 through 1985. Not surprisingly, they find that the policy recommendations based on the data are quite sensitive to changes in the lag structure. More specifically, comparing policy outcomes from different lag structures of their VAR model using the variance decomposition for a twenty-quarter horizon, they conclude that, across different lag structures, the differences in M1’s effect on output, inflation and interest rates are large enough to rule out a reliable conclusion about M1’s role.

2. VAR-based Expectations of the FRB/US Model (The Historical VAR Model) Taylor (1999) summarizes most recent monetary policy models. Despite the differences in these models, there are some important common features. First, all the models are dynamic and stochastic. Second, they are general equilibrium models in the sense that they describe the behavior of the whole economy. Third, they incorporate some form of nominal rigidity, usually through some version of staggered wage or price setting. A general framework for describing the models and the methods most commonly used for evaluating monetary policy is as follows:

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tt

tttt

uyLGgLBygLAyso

yLGxuxgLBygLAy

++=

=++=

)(),(),(,

)(),(),(

(1)

where is a vector of endogenous variables, are the policy variables, is a serially uncorrelated vector of random variables with variance-covariance matrix Σ . , and are matrix or vector polynomials in the lag operator (L) while the vector g consists of all the parameters in . The first equation is a reduced form solution to the dynamic stochastic rational expectation model used for the evaluation of a monetary policy rule, while the second equation is the monetary policy rule to be considered.

ty

)(L

tx tu(LA ),(),, gLBg

G)(LG

A and B depend on the parameters (g) of the policy rule. Substitution of the policy rule into the reduced form results in a vector autoregression in and its lags. From this vector autoregression, one can easily find the steady state stochastic distribution of , characterized, for example, by the autocovariance matrix function or the spectral density of .

ty

ty

ty


The steady state stochastic distribution is a function of the parameters (g) of the policy rule along with Σ and the other parameters in A and B. So, for any choice of parameters g in the policy rule, one can evaluate any objective function that depends on the steady state distribution of .1 ty As discussed by Brayton, Mauskopf, Reifschneider, Tinsley and Williams (1997), macroeconomic models have relied on various assumptions about how individuals form expectations of future economic conditions. First, adaptive expectations depend only on past observations of the variables. The MPS model employs the adaptive expectations mechanism. Second, rational, or model-consistent, expectations are identical to the forecasts produced by the macroeconomic model in which the expectations are used, such as FRB/US model. Third, VAR-based expectations are the forecasts from a small vector autoregression (VAR) model that includes equations for only a few key economic measures, used in conjunction with a larger model such as the FRB/US model. Adaptive and VAR-based expectations would be “rational” if they were fully consistent with the macroeconomic model in which they were embedded. This discussion suggests, within the FRB/US model, two alternative assumptions regarding the degree of sophistication or rationality of individuals in their formation of expectations. One is that expectations are rational, or model consistent. In this case, household and firms are assumed to have a detailed and sophisticated understanding of how the economy functions, and expectations are identical to the forecasts of the FRB/US model. The alternative is that expectations are based on a less elaborate understanding of the economy, as represented by a small forecasting model containing a limited information set (i.e., only a few important macroeconomics variables). Because the form of the forecasting model is similar to that of a vector autoregression (VAR), such expectations are called “VAR expectations”. The VAR approach in the FRB/US model assumes that households and firms form expectations on the basis of their knowledge of the historical interactions among three variables: the federal fund rate, the cyclical state of the economy, and the rate of inflation. As mentioned in Brayton and Tinsley (1996), under the small model of VAR expectations, all sectors share a condensed description of the aggregate economy represented by a three-variable VAR. The properties of the FRB/US under full-model expectations can be similar to those under VAR expectations, if the shock or change being simulated is not unusual in a historical context. One example is a transitory change in the federal fund rate. Under either VAR or full-model expectations, output moves for a period of time in the opposite direction of the interest rate change, as does inflation, and long-term interest rates change by a fraction of the movement in the fund rate. In contrast, unusual shifts can yield different outcomes under VAR and full-model expectations. One example is a future change in the fiscal policy that is perfectly anticipated under full-model expectations but recognized only as it occurs under VAR expectations. In this case, macroeconomic variables move in advance of the fiscal change under full-model expectations, but only after the policy change under VAR expectations. The historical VAR is specified as follows:


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)()()()()()()()()()()()(

)()()()()()()()()()()()(

)()()()()()()()()()()()(

−−−−−−

−−−∞−−−

∞−−

−−−−−−

−−−∞−−−

∞−−

−−−−−−

−−−∞−−−

∞−−

Δ+Δ+Δ+Δ+Δ+Δ+Δ+Δ+Δ+−++−=Δ



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rrrxxxrrxx

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ββββββπβπβπβααππα

ββββββπβπβπβααππαπ

ββββββπβπβπβααππα

(2)

where r = Federal fund rate π = Inflation rate of personal consumption deflator (Chain weight) x = Percentage gap between actual and potential output ∞π = Expected long-run rate of inflation ∞r = Expected long–run value of the federal fund rate

trΔ = 1−− tt rr

tπΔ = 1−− tt ππ

txΔ = 1−− tt xx The historical VAR differs from the conventional VAR due to the presence of explicit endpoints for each variable in the historical VAR, ∞r , , and where = 0. Each endpoint represents the private sector perceptions of the long-run outcome of that variable.

∞π ∞x ∞x

Given the linear structure of the VAR models, VAR-based expectations can be obtained by analytical solutions rather than numerical iterations. Thus, the calculation of sectoral VAR expectations, even over infinite forecast horizons, is significantly less computationally demanding than full-model expectations. Having described what VAR-based expectations means for monetary policy, we are left with the question of what the model-selection uncertainty of the VAR-based expectations models implies for policy evaluation. This question is our focus. This paper investigates the effect of model-selection uncertainty on the policy recommendations using the VAR-based expectations for the FRB/US model. The historical VAR is a possible representation for the formation of expectations in the FRB/US model. In this historical VAR representation, the Board staff assumes a lag order of three (henceforth indicated by VAR (3)). I use Kilian’s (1998) endogenous lag order bootstrap algorithm to estimate the effect of model-selection uncertainty on the impulse response estimates from this historical VAR model.


3. Data The seasonally adjusted data, provided by Dr. David Reifschneider of the Board of Governors of the Federal Reserve System, are available at a quarterly frequency from 1965 to 2006 for the inflation rate (personal consumption expenditures, chain weighted), the federal fund rate (effective annual yield), 10-year expected inflation (Hoey/Philadelphia survey), the expected average federal fund rate (10-30 years ahead), and the output gap (for business. sector excluding. energy, housing, and farm). I found that the data used in our estimation are stationary. 4. The Effect of Model-Selection Uncertainty on the Impulse Response Estimates from the VAR-Based Expectation of the FRB/US Model I apply Kilian’s (1998) bootstrap bias correction method, which was described on section 2, on the VAR-based expectation of the FRB/US model. The analysis uses Monte Carlo simulation based on the estimated historical VAR in equation 3 to calculate the interval coverage accuracy and average length for impulse response estimates from the historical VAR model. A nominal coverage accuracy of 90% is used with 4000 Monte Carlo iterations and 400 bootstrap iterations. I have both the AIC and SC as model-selection criteria. Using the US Fed data, the estimates of VAR (3) in equation 2 are as follows where standard errors are in parentheses and estimates with asterisks are significant at 90% confidence level:

3)095.0(2)091.0(

*1)098.0(

*

3)093.0(2)093.0(1)093.0(3)081.0(2)086.0(

1)090.0(11)060.0(

*1)034.0(11)059.0(

*

3)100.0(2)101.0(1)109.0(

*

3)103.0(2)103.0(1)103.0(3)090.0(2)096.0(

*

1)100.0(

*11)067.0(1)038.0(

*11)066.0(

*

3)099.0(2)099.0(

*1)107.0(

3)101.0(2)102.0(1)101.0(

*3)089.0(2)094.0(

1)099.0(11)066.0(1)038.0(

*11)065.0(

)(095.0)(163.0)(232.0

)(034.0)(018.0)(063.0)(011.0)(024.0

)(052.0)(231.0)(036.0)(109.0

)(028.0)(022.0)(233.0

)(057.0)(131.0)(079.0)(088.0)(211.0

)(258.0)(083.0)(140.0)(140.0

)(125.0)(377.0)(064.0

)(014.0)(117.0)(260.0)(122.0)(079.0

)(066.0)(058.0)(086.0)(089.0

−−−

−−−−−

−∞−−−

∞−−

−−−

−−−−−

−∞−−−

∞−−

−−−

−−−−−

−∞−−−

∞−−

Δ+Δ−Δ+

Δ−Δ+Δ+Δ+Δ+

Δ+−−−−−=Δ

Δ−Δ−Δ+

Δ−Δ−Δ−Δ+Δ−

Δ−−−+−−=Δ

Δ+Δ−Δ+

Δ−Δ+Δ+Δ+Δ+

Δ+−−+−=Δ

ttt

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ππ

πππ

ππ

ππππ

ππ

πππ

(3) The residual correlation matrix is:


00.113.023.013.000.108.023.008.000.1

t

t

t

ttt

x

rxr

ΔΔΔ

ΔΔΔ

π

π

The estimated VAR (3) in equation 3 is stationary. The data in the Monte Carlo simulation is generated from this estimated VAR (3) which is considered to be the true model. In Table 1, although the true lag order is three, the SC selects order one while the AIC selects order two. The SC and AIC select the true lag order with probability zero and 0.36, respectively. One would recommend that the agents use an order lower than VAR (3) in their forecasting. Should the agents in the economy use the VAR (1) or VAR (2)? I will address this question in the second part of the paper. Table 2 presents the coverage accuracy and average length for endogenous and exogenous intervals for impulse response estimates in the first two forecasting periods using the AIC. The percentage difference between the average length for endogenous and exogenous intervals is the percentage increase in the variability of the impulse response estimates due to model-selection uncertainty. Table 2 shows that average length for endogenous and exogenous intervals are remarkably close to each other for all the nine impulse response functions. The model-selection uncertainty increases the variability of the impulse response estimates by no more than 7% in the first period and 9% in the second period. As the forecasting horizon increases, the variability in the impulse response estimates due to model-selection uncertainty increases. Macroeconomists are concerned more about short forecasting horizons, such as one year, because agents in the economy update their beliefs. The coverage accuracy for endogenous and exogenous intervals for impulse response estimates are close to the nominal coverage, which is 90%. The analysis also used the SC as model-selection criterion. Under the SC, the average length for the endogenous intervals is close to that for the exogenous intervals indicating that model-selection uncertainty does not increase the variability of the impulse response estimates. It is worth mentioning that I also estimated the effect of model-selection uncertainty on the variability and bias of the ijα̂ s in equation 2 which are the coefficient estimates for the adjusted variables ( 111 )()0(,) −∞−−∞( −− t− tt rrandxππ ). I examined how changing the lag order of the historical VAR affects ijα̂

ij

in equation 2. To do so, I calculate the unconditional and conditional standard deviations of α̂ . The difference between the unconditional and conditional standard deviations represents the effect of model-selection uncertainty on the variability of ijα̂ . The effect of model-selection uncertainty on the bias of ijα̂ was also calculated. The analysis found that model-selection uncertainty does not affect the variability or the bias of ijα̂ which confirms the results of Table 2. Figure 1 shows the generalized impulse response estimates for VAR (ρ) with ρ = 1,..,8 to a given shock for 25 periods using the true data. rΔ , πΔ and xΔ are defined as DR, DINF, and DX, respectively. I also included ± 1.65 the conditional standard deviation for VAR (3), which is


equivalent to 90% confidence interval. Notice that for the first four periods, none of the impulse response estimates exceed the 90% confidence interval for VAR (3) in all except one case where VAR (1) slightly exceeds the confidence interval at period three when rΔ responds to a shock in itself. At longer forecasting horizons, the variability of the impulse response estimates of rΔ and πΔ to a shock in increases and exceeds the 90% confidence interval. xΔ

Accordingly, the VAR-based expectation of the FRB/US model is very robust to model-selection uncertainty at short forecasting horizons, indicating that the policy recommendations based on it should be robust to changes in the lag order. Nevertheless, the results reported in Table 1 would suggest the possibility of improving the model’s forecasting performance with use of lower lag order than three, VAR (1) or VAR (2). 5. Should the Fed Use the VAR (1) or VAR (2) in Its Forecasting? The analysis of the previous section suggested using a lower lag order than three, which is the order used by the Fed. Which one of the VAR (1) and VAR (2) representations performs better in forecasting? Table 3 shows the mean of impulse response estimates over the eight lag orders, the standard deviations of these estimates over the eight lag orders, and the standard deviation of the impulse response estimates in VAR (1), VAR (2) and VAR (3) for five forecasting periods. In Table 3, the standard deviation of the impulse response estimates for VAR (1) are less than the standard deviations of impulse response estimates of VAR (2) and VAR (3) for all shocks after period two. Based on the results from the VAR-based expectations of the FRB/US model for forecasting longer than two periods, the VAR (1) representation would seem to exhibit lower standard deviations in impulse response estimates to all shocks. For the first two periods, the standard deviations of impulse response estimates for the three lag orders are sufficiently close to each other, so we are indifferent between the representations, VAR (1), VAR (2) and VAR (3). Thus, due to its superior performance for forecasting horizons beyond two periods, VAR (1) representation would be recommended over the VAR (2) and VAR (3) representations. Figure 2 shows the standard deviation of the impulse response estimates for the first three lag orders over 25 periods. It confirms the previous results that it is recommended to use VAR (1) rather than VAR (3). As the forecasting horizon increases, VAR (1) would be better for forecasting over the other lag orders. 6. Policy implications I use the same variance decomposition as Hafer and Sheehan (1991) to examine the sensitive of policy inferences to the change in the lag structures of VAR-based expectation of the FRB/US model, though in contrast we find the policy recommendations are not very sensitive to changes in the lag structure of the VAR model. Hafer and Sheehan (1991) calculated 90% confidence intervals for the different variance decompositions following the procedure in Runkle (1987). Runkle (1987) calculates the confidence intervals for the variance decompositions of a VAR


model that imposes equal lag lengths on each variable. Hafer and Sheehan (1991) used unequal lag length on each variable. Their confidence intervals for the variance decompositions which were not reported might not be accurate. In this paper, I use equal lag length on each variable and I calculate the standard deviation of the variance decomposition using Monte Carlo simulation. The variance decompositions for trΔ ,

tπΔ

trΔ and are calculated using the same order of variables. If the variance decomposition of

does not change over different lag orders, then we may conclude that policy inferences are not sensitive to changes in the lag structures of VAR-based expectation of the FRB/US model. In this case, the model-selection uncertainty does not affect the policy recommendations from our VAR-based expectations model. This means a different lag structure will not alter policy conclusions. For example, the variance decomposition of

xΔ

xΔ in the considered VAR model shows that shocks in , trΔ tπΔ and xΔ explain 7.5%, 0.7% and 91% of the variation in xΔ for VAR(1) in the first period, respectively. These percentage variations in xΔ due to shocks in trΔ ,

tπΔ and change very little over lag orders. For VAR (8), those percentage variations are 10.3%, 0.1% and 89.5%, respectively.

xΔ

Figure 3 shows the variance decompositions for trΔ , tπΔ and xΔ over the 8 lag orders and 1.65 of the standard deviation of the variance decompositions of trΔ , tπΔ and in the VAR (3), which is equivalent to 90% confidence interval.

xΔ

None of the variance decompositions exceeds the confidence interval for the first six periods except for the variation in tπΔ due to shocks in xΔ . For example, at period six, shocks in xΔ explain between 0.4% and 0.8%, using VAR (1) through VAR (3), and between 5.1% and 6.8%, using VAR (4) through VAR (8) in the variations in tπΔ . In this case, using VAR (1) through VAR (3) leads us to recommend that shocks in xΔ explain nothing of the variations in the tπΔ . At the same time, using VAR (4) through VAR (8), shocks in xΔ explain about 6% of the variations in tπΔ . At higher forecasting horizons than six, some variance decompositions exceed the confidence intervals. Therefore, we can conclude that for short forecasting horizons the policy inferences are robust to the lag order selection of the VAR-based expectation of the FRB/US model. The results from Figure 2 also lead us to have more confidence in the policy inferences from our historical VAR model as they show that the impulse response estimates are robust to lag order selection for short forecasting horizons. 7. Conclusion I examine the robustness of the VAR-based expectations FRB/US model, which has been used by the Board of Governors of the Federal Reserve System since 1996, to the model-selection uncertainty. The analysis, using Kilian’s (1998) method, found no significant model-selection uncertainty. That is to say this model is very robust to model-selection uncertainty.


The analysis suggests that the agents in the economy should use VAR (1) representation as the VAR (1) forecasts better. Using the variance decompositions, I show that the policy inferences are not sensitive to the changes in the lag order of the VAR-based expectation of the FRB/US model. Footnotes * Correspondence to: Islam Azzam, Assistant professor of Finance, Department of Management, American University in Cairo, 113 Kasr El Aini Street, P.O. Box 2511, Cairo 11511, Egypt; e-mail: [email protected]. 1. Examples of some of the models that use this framework for policy evaluation are Ball (1997), Ball (1999) and Svensson (1997), the time-series econometric model of Rudebusch and Svensson (1999), the Federal Reserve’s large scale rational-expectations econometric model described by Brayton, Levin, Tryon and Williams (1997), the small forward-looking models of Clarida, Gali, and Gertler (1997), Fuhrer and Moore (1995), multi-country rational expectations model of Taylor (1993), and the representative agent-optimizing models of Goodfriend and King (1997), McCallum and Nelson (1999), and Rotemberg and Woodford (1999), Svensson (1998). References Ball, L. 1997. “Efficient Rules for Monetary Policy,” NBER Working Paper No. 5952. Ball, L. 1999. “Policy Rules for Open Economies,” In: Taylor, J.B. (Ed.), Monetary Policy Rules. Chicago: University of Chicago Press. Brayton, F., A. Levin, R. W. Tryon, and J. C. Williams. 1997. “The Evolution of Macro Models at the Federal Reserve Board,” In: McCallum, B., Plosser, C. (Eds.), Carnegie-Rochester Conference Series on Public Policy, North-Holland, 47, 43-81. Brayton, F, E. Mauskopf, D. Reifschneider, P. Tinsley, and J. Williams. 1997. “The Role of Expectations in the FRB/US Macroeconomic Model,” Federal Reserve Bulletin, 83, 227-245. Brayton, F. and P. Tinsley. 1996. “A Guide to FRB/US: A Macroeconomic Model of the United States,” Finance and Economics Discussion Series: 1996-42. Braun, P. and S. Mittnik. 1993. “Misspecifications in vector Autoregressions and Their Effects on Impulse Responses and Variance Decompositions,” Journal of Econometrics, 59, 319-341. Clarida, R., J. Gali, and M. Gertler. 1997. “Monetary Policy Rules and Macroeconomic Stability: Evidence and Some Theory,” European Economic Review, 42, 1033-1067.

mailto:[email protected]


Fuhrer, J. C. and G. R. Moore, 1995. “Inflation Persistence,” Quarterly journal of Economics, 110, 127-159. Goodfriend, M. and R. King. 1997. “The New Neoclassical Synthesis and the Role of Monetary Policy,” In: Bernanke, B., Rotemberg, J. (Eds.), Macroeconomics Annual 1997. Cambridge: MIT Press, 231-282. Hafer, R. and R. Sheehan. 1991. “Policy Inference Using VAR Models,” Economic Inquiry, 24, 44- 52. Kilian, L. 1998. “Accounting for Lag Order Uncertainty in Autoregressions: The Endogenous Lag Order Bootstrap Algorithm,” Journal of Time Series Analysis, 19, 531-547. Levin, A., V. Wieland, and J. Williams. 1999. “Robustness of Simple Monetary Policy Rules under Model Uncertainty,” In: J. B. Taylor (Ed.), Monetary Policy Rule. Chicago: University of Chicago Press, 263-309. Litterman, R. 1980. “A Bayesian Procedure for Forecasting with Vector Autoregressions,” Unpublished Mimeo, Massachusetts Institute of Technology. McCallum, B. and E. Nelson. 1999. “Performance of Operational Policy Rules in an Estimated Semi-Classical Structural Model,” In: Taylor, J.B. (Ed.), Monetary Policy Rules. Chicago: University of Chicago Press. Rotemberg, J. and M. Woodford. 1999. “Interest Rate Rules in Estimated Sticky Price Models,” In: Taylor, J.B. (Ed.), Monetary Policy Rules. Chicago: University of Chicago Press. Rudebusch, G. and L. E. O. Svensson. 1999. “Policy Rules for Inflation Targeting,” In: Taylor, J.B. (Ed.), Monetary Policy Rules, Chicago: University of Chicago Press. Svensson, L. E. O. 1997. “Inflation Forecast Targeting: Implementing and Monitoring Inflation Targets,” European Economic Review, 41, 1111-1146. Svensson, L. E. O. 1998. “Open-Economy Inflation Targeting,” Institute for International Studies, Stockholm University. Svensson, L. E. O. and R. J. Tetlow. 2005. “Open-Economy Inflation Targeting,” Institute for International Studies, Stockholm University. Svensson, L E.O. and R. J. Tetlow. 2005. “Optimal Policy Projections,” International Journal of Central Banking, 1, 3, 177-207. Taylor, J. B. 1993. Macroeconomic Policy in a World Economy: From Econometric Design to Practical Operation. New York: W.W. Norton.


Taylor, J. B. 1999. “The Robustness and Efficiency of Monetary Policy Rules as a Guidelines for Interest Rate Setting by the European Central Bank,” Journal of Monetary Economics, 43, 655-679. Tetlow, R. J. and P. Muehlen. 2000. “Robust Monetary Policy with Misspecified Models: Does Model Uncertainty Always Call for Attenuated Policy?” Finance and Economics Discussion Series 2000-28. Tetlow, R. J. and B. Ironside. 2006. “Real-Time Model Uncertainty in the United States: The Fed from 1996-2003,” Finance and Economics Discussion Series 2006-08. Williams, J. 1999. “Simple Rules for Monetary Policy,” Finance and Economics Discussion Series 1999-12.

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Table 1. The Probability Distribution of the Lag Orders Selected by the AIC and SC for the Historical VAR Model

Lag order 1 2 3 4 5 6 7 8

AIC 0 0.48* 0.36 0.12 0.01 0 0 0

SC 0.58* 0.41 0 0 0 0 0 0

• * = The order that has the highest probability distribution. • The true lag order is three.

Table 2. The Coverage Accuracy and Average Length for Endogenous and Exogenous Intervals for Impulse Response Estimates from the Historical VAR Model with 90% Nominal Coverage

Impulse response function

Period

Average length for

endog. interval

Average length for

exog. interval

Model-selection uncert.

(%)

Cover. Accuracy for endog interval

Cover. Accuracy for exog. interval

1 0.290 0.275 5.2 0.87 0.84 11θ

2 0.293 0.273 7.3 0.88 0.85

1 0.286 0.273 4.6 0.87 0.85 12θ

2 0.284 0.265 7.1 0.88 0.85

1 0.247 0.243 1.7 0.81 0.81 13θ

2 0.250 0.242 3.3 0.86 0.84

1 0.311 0.294 5.8 0.90 0.88 21θ

2 0.308 0.282 9.0 0.91 0.88

1 0.293 0.281 4.2 0.88 0.88 22θ

2 0.298 0.280 6.6 0.90 0.89


1 0.254 0.250 1.6 0.84 0.83 23θ

2 0.264 0.255 3.5 0.86 0.85

1 0.269 0.256 4.8 0.85 0.82 31θ

2 0.268 0.249 7.3 0.90 0.88

1 0.258 0.248 4.0 0.88 0.87 32θ

2 0.261 0.245 6.8 0.90 0.88

1 0.225 0.224 0.4 0.86 0.84 33θ

2 0.230 0.225 2.3 0.88 0.87


Table 3. The Generalized Impulse Response Estimates over Lag Orders for the Historical VAR Model [Bold number is the minimum standard deviation over the standard deviations of VAR (1) to VAR (3).]

Responses Pe

riod

Mean of the

impulses over all lag

orders

Impulses for

VAR (3)

Stand. deviation

of the impulses

over all lag orders

Stand. deviation

of the impulse of VAR (1)

Stand. deviation


Stand. deviation


1 1.077 1.092 0.035 0.070 0.067 0.066 2 0.178 0.138 0.050 0.105 0.104 0.119 3 -0.193 -0.282 0.115 0.043 0.108 0.124 4 0.106 0.087 0.087 0.018 0.076 0.113

Response of rΔ to

generalized one std rΔ innovation 5 0.062 0.120 0.038 0.008 0.066 0.085

1 0.109 0.099 0.018 0.098 0.095 0.097 2 0.165 0.211 0.044 0.106 0.107 0.123 3 -0.156 -0.120 0.068 0.036 0.106 0.113 4 -0.114 -0.145 0.058 0.013 0.076 0.107

Response of πΔ to

generalized one std rΔ innovation 5 0.115 0.102 0.056 0.005 0.064 0.077

1 0.280 0.234 0.035 0.085 0.085 0.085 2 0.256 0.274 0.022 0.092 0.095 0.108 3 -0.149 -0.111 0.085 0.039 0.096 0.110 4 -0.039 0.006 0.031 0.016 0.049 0.099

Response of to

generalized one std

xΔ

rΔ innovation 5 -0.036 0.060 0.070 0.007 0.038 0.057

1 0.107 0.097 0.019 0.098 0.094 0.095 2 0.096 0.114 0.063 0.103 0.105 0.117 3 0.086 0.079 0.059 0.024 0.099 0.112 4 0.091 0.094 0.073 0.010 0.070 0.104

Response of rΔ to

generalized one std πΔ innovation 5 -0.066 -0.039 0.050 0.003 0.051 0.070

1 1.093 1.111 0.025 0.069 0.068 0.067 2 -0.333 -0.275 0.067 0.104 0.109 0.120 3 -0.135 -0.163 0.076 0.049 0.100 0.104 4 0.113 0.190 0.060 0.020 0.082 0.107

Response of πΔ to

generalized one std πΔ innovation 5 -0.111 -0.031 0.091 0.008 0.052 0.079

1 0.097 0.131 0.021 0.086 0.086 0.086 2 0.084 0.088 0.033 0.092 0.096 0.108 3 0.004 0.031 0.025 0.028 0.088 0.100 4 -0.032 0.005 0.028 0.010 0.041 0.094

Response of to

generalized one std

xΔ

πΔ innovation 5 -0.060 0.020 0.055 0.003 0.031 0.048


(Continue on Table 3)

Responses

Perio

d

Mean of the

impulses over all lag

orders

Impulses for

VAR (3)

Stand. deviation

of the impulses

over all lag orders

Stand. deviation


Stand. deviation


Stand. deviation


1 0.303 0.256 0.033 0.096 0.093 0.094 2 0.311 0.286 0.073 0.099 0.095 0.103 3 0.078 0.081 0.019 0.035 0.101 0.102 4 0.002 -0.049 0.053 0.015 0.051 0.103

Response of rΔ to

generalized one std xΔinnovation 5 -0.066 -0.019 0.043 0.006 0.037 0.062

1 0.107 0.146 0.025 0.098 0.093 0.096 2 -0.033 -0.057 0.015 0.102 0.100 0.109 3 -0.074 -0.096 0.052 0.032 0.101 0.102 4 -0.013 -0.023 0.027 0.012 0.051 0.105

Response of πΔ to

generalized one std xΔinnovation 5 0.173 -0.011 0.147 0.004 0.034 0.055

1 0.993 0.999 0.008 0.062 0.062 0.061 2 0.096 0.130 0.019 0.089 0.089 0.094 3 0.003 0.051 0.035 0.033 0.092 0.093 4 -0.047 -0.037 0.031 0.014 0.044 0.095

Response of to

generalized one std

xΔ

xΔinnovation 5 0.024 -0.007 0.033 0.006 0.029 0.053


Figure 1. The Impulse Response Estimates for the Historical VAR Model

-0.5

0.0

0.5

1.0

1.5

2 4 6 8 10 12 14 16 18 20 22 24

VAR1VAR2VAR3VAR4

VAR5VAR6VAR7VAR8

-1.65*SE+1.65*SE

RESONSE OF DR TO GENERALIZED ONE STD DR INNOVATION

Periods

-0.4

-0.2

0.0

0.2

0.4

0.6

2 4 6 8 10 12 14 16 18 20 22 24

VAR1VAR2VAR3VAR4

VAR5VAR6VAR7VAR8

- 1.65*SE+ 1.65*SE

RESONSE OF DINF TO GENERALIZED ONE STD DR INNOVATION


-0.4

-0.2

0.0

0.2

0.4

0.6

2 4 6 8 10 12 14 16 18 20 22 24

VAR1VAR2VAR3VAR4

VAR5VAR6VAR7VAR8

- 1.65*SE+ 1.65*SE

RESONSE OF DX TO GENERALIZED ONE STD DR INNOVATION

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

2 4 6 8 10 12 14 16 18 20 22 24

VAR1VAR2VAR3VAR4

VAR5VAR6VAR7VAR8

- 1.65*SE+ 1.65*SE

RESONSE OF DR TO GENERALIZED ONE STD DINF INNOVATION


-0.5

0.0

0.5

1.0

1.5

2 4 6 8 10 12 14 16 18 20 22 24

VAR1VAR2VAR3VAR4

VAR5VAR6VAR7VAR8

- 1.65*SE+ 1.65*SE

RESONSE OF DINF TO GENERALIZED ONE STD DINF INNOVATION

-0.2

-0.1

0.0

0.1

0.2

0.3

2 4 6 8 10 12 14 16 18 20 22 24

VAR1VAR2VAR3VAR4

VAR5VAR6VAR7VAR8

- 1.65*SE+ 1.65*SE

RESONSE OF DX TO GENERALIZED ONE STD DINF INNOVATION


-0.4

-0.2

0.0

0.2

0.4

0.6

2 4 6 8 10 12 14 16 18 20 22 24

VAR1VAR2VAR3VAR4

VAR5VAR6VAR7VAR8

- 1.65*SE+ 1.65*SE

RESONSE OF DR TO GENERALIZED ONE STD DX INNOVATION

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

2 4 6 8 10 12 14 16 18 20 22 24

VAR1VAR2VAR3VAR4

VAR5VAR6VAR7VAR8

- 1.65*SE+ 1.65*SE

RESONSE OF DINF TO GENERALIZED ONE STD DX INNOVATION


-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

2 4 6 8 10 12 14 16 18 20 22 24

VAR1VAR2VAR3VAR4

VAR5VAR6VAR7VAR8

- 1.65*SE+ 1.65*SE

RESONSE OF DX TO GENERALIZED ONE STD DX INNOVATION


Figure 2. The Standard Deviation of Impulse Response Estimates for the Historical VAR Model

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

2 4 6 8 10 12 14 16 18 20 22 24

VAR1 VAR2 VAR3

STD OF RESPONSE OF DR TO GENERALIZED ONE STD DR INNOVATION

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

2 4 6 8 10 12 14 16 18 20 22 24

VAR1 VAR2 VAR3

STD OF RESPONSE OF DINF TO GENERALIZED ONE STD DR INNOVATION


0.00

0.02

0.04

0.06

0.08

0.10

0.12

2 4 6 8 10 12 14 16 18 20 22 24

VAR1 VAR2 VAR3

STD OF RESPONSE OF DX TO GENERALIZED ONE STD DR INNOVATION

0.00

0.02

0.04

0.06

0.08

0.10

0.12

2 4 6 8 10 12 14 16 18 20 22 24

VAR1 VAR2 VAR3

STD OF RESPONSE OF DR TO GENERALIZED ONE STD DINF INNOVATION


0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

2 4 6 8 10 12 14 16 18 20 22 24

VAR1 VAR2 VAR3

STD OF RESPONSE OF DINF TO GENERALIZED ONE STD DINF INNOVATION

0.00

0.02

0.04

0.06

0.08

0.10

0.12

2 4 6 8 10 12 14 16 18 20 22 24

VAR1 VAR2 VAR3

STD OF RESPONSE OF DX TO GENERALIZED ONE STD DINF INNOVATION


0.00

0.02

0.04

0.06

0.08

0.10

0.12

2 4 6 8 10 12 14 16 18 20 22 24

VAR1 VAR2 VAR3

STD OF RESPONSE OF DR TO GENERALIZED ONE STD DX INNOVATION

0.00

0.02

0.04

0.06

0.08

0.10

0.12

2 4 6 8 10 12 14 16 18 20 22 24

VAR1 VAR2 VAR3

STD OF RESPONSE OF DINF TO GENERALIZED ONE STD DX INNOVATION


0.00

0.02

0.04

0.06

0.08

0.10

2 4 6 8 10 12 14 16 18 20 22 24

VAR1 VAR2 VAR3

STD OF RESPONSE OF DX TO GENERALIZED ONE STD DX INNOVATION


Figure 3. The Variance Decomposition for the Historical VAR Model

75

80

85

90

95

100

105

1 2 3 4 5 6 7 8 9 10

VAR1VAR2VAR3VAR4

VAR5VAR6VAR7VAR8

+1.65*SE-1.65*SE

VARIANCE DECOMPOSITION OF DR TO DR

-4

-2

0

2

4

6

8

10

1 2 3 4 5 6 7 8 9 10

VAR1VAR2VAR3VAR4

VAR5VAR6VAR7VAR8

+1.65*SE-1.65*SE

VARIANCE DECOMPOSITION OF DR TO DINF


-4

0

4

8

12

16

1 2 3 4 5 6 7 8 9 10

VAR1VAR2VAR3VAR4

VAR5VAR6VAR7VAR8

+1.65*SE-1.65*SE

VARIANCE DECOMPOSIT ION OF DR T O DX

-5

0

5

10

15

20

1 2 3 4 5 6 7 8 9 10

VAR1VAR2VAR3VAR4

VAR5VAR6VAR7VAR8

+1.65*SE-1.65*SE

VARIANCE DECOMPOSITION OF DINF TO DR


80

85

90

95

100

105

1 2 3 4 5 6 7 8 9 10

VAR1VAR2VAR3VAR4

VAR5VAR6VAR7VAR8

+1.65*SE-1.65*SE

VARIANCE DECOMPOSITION OF DINF TO DINF

-4

-2

0

2

4

6

8

10

1 2 3 4 5 6 7 8 9 10

VAR1VAR2VAR3VAR4

VAR5VAR6VAR7VAR8

+1.65*SE-1.65*SE

VARIANCE DECOMPOSIT ION OF DINF T O DX


-5

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10

VAR1VAR2VAR3VAR4

VAR5VAR6VAR7VAR8

+1.65*SE-1.65*SE

VARIANCE DECOMPOSIT ION OF DX T O DR

-4

-2

0

2

4

6

8

1 2 3 4 5 6 7 8 9 10

VAR1VAR2VAR3VAR4

VAR5VAR6VAR7VAR8

+1.65*SE-1.65*SE

VARIANCE DECOMPOSITION OF DX TO DINF


70

75

80

85

90

95

100

105

1 2 3 4 5 6 7 8 9 10

VAR1VAR2VAR3VAR4

VAR5VAR6VAR7VAR8

+1.65*SE-1.65*SE

VARIANCE DECOMPOSITION OF DX TO DX

2. VAR-based Expectations of the FRB/US Model (The Historical VAR Model)3. Data4. The Effect of Model-Selection Uncertainty on the Impulse Response Estimates from the VAR-Based Expectation of the FRB/US Model5. Should the Fed Use the VAR (1) or VAR (2) in Its Forecasting? 6. Policy implications7. ConclusionReferencesLag order

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