Evaluating Inflation Targeting Using a Macroeconometric Model
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Fair, Ray C. Working Paper Evaluating Inflation Targeting Using a Macroeconometric Model Economics Discussion Papers, No. 2007-14 Provided in Cooperation with: Kiel Institute for the World Economy – Leibniz Center for Research on Global Economic Challenges Suggested Citation: Fair, Ray C. (2007) : Evaluating Inflation Targeting Using a Macroeconometric Model, Economics Discussion Papers, No. 2007-14, Kiel Institute for the World Economy (IfW), Kiel This Version is available at: https://hdl.handle.net/10419/17937 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by-nc/2.0/de/deed.en
discussion Papers Discussion Paper 2007-14 March 26, 2007 Evaluating Inflation Targeting Using a Macroeconometric Model Ray C. Fair Yale University, New Haven Abstract: This paper uses a structurally estimated macroeconometric model, denoted the MC model, to evaluate inflation targeting in the United States. Various interest rate rules are tried with differing weights on inflation and output, and various optimal control problems are solved using differing weights on inflation and output targets. Price-level targeting is also considered. The results show that 1) there are output costs to inflation targeting, especially for price shocks, 2) price-level targeting is dominated by inflation targeting, 3) the estimated interest rate rule of the Fed (in Table 4) is consistent with the Fed placing equal weights on inflation and unemployment in a loss function, 4) the estimated interest rate rule does a fairly good job at lowering variability, and 5) considerable economic variability is left after the Fed has done its best. Overall, the results suggest that the Fed should continue to behave as it has in the past. JEL: E52 Keywords: inflation targeting, interest rate rules, optimal control Correspondence: Cowles Foundation and International Center for Finance, Yale University, New Haven, CT 06520-8281. Voice: 203-432-3715; Fax: 203-432-6167; e-mail: ray.fa[email protected]; website: fairmodel.econ.yale.edu. I am indebted to William Brainard and William Nordhaus for helpful comments. http://www.economics-ejournal.org/economics/discussionpapers © Author(s) 2007. This work is licensed under a Creative Commons License - Attribution-NonCommercial 2.0 Germany
1 Introduction There has been much discussion in the recent literature on whether ination targeting (IT) by a monetary authority is a good idea. One approach is to look at performances of IT versus non IT countries. Using this approach, Ball and Sheridan (2005) nd no evidence that IT improves a country's performance. There are, however, a number of endogeneity problems associated with this approach, as the authors are aware, and it is not clear what to make of the results. In comparisons across countries it is hard to hold other things constant, especially initial conditions. In addition, a country that is not formally an IT country may behave roughly like one, and a country that is formally an IT country may behave somewhat more exibly. An alternative approach is to examine how an IT policy performs relative to other types of monetary policies in an economic model. Different rules can be examined, or formal optimal control problems can be solved. This is the approach taken in this paper. The model used is a version of the multicountry (MC) macroeconometric model in Fair (2004). The MC model is quite different from the macro model that is primarily used in the current literature, namely the New Keynesian (NK) model, and some justication is needed for using a different model. The NK and MC models are briey compared in Section 2. Section 3 then examines ination targeting using the MC model. Various interest rate rules are tried with differing weights onination andoutput, and various optimalcontrol problemsare solvedusing differing weightson ination and output targets. Price-leveltargeting is also considered. 2
2 The NK and MC Models 2.1 NK Model Goodfriend and King (1997) lay out what they call the New Neoclassical Synthesis, whichis representedby theNK model. The fourfeatures ofthis synthesisare: 1) intertemporal optimization, 2) rational expectations, 3) imperfect competition, and 4) costly price adjustment. The NK model plays a prominent role in Clarida, Galí, and Gertler (1999) in their review of recent research in monetary policy, as it does in Woodford (2003). Virtually all the papers in Taylor (1999a) use some version of this model. Ireland (2004c, p. 923) states that The development of the forward-looking, microfounded New Keynesian model stands, in the eyes of many observers, as one of the past decade's most exciting and signicant achievements in macroeconomics. 1 Woodford (2006, p. 17) suggests that NK models have sufcient claim to quantitative realism to be of interest to policy-making institutions. In the NK model an innitely lived, representative household maximizes the discounted value of expected future utility. An intertemporal optimality condition relates current consumption to expected future consumption and the real inter1 OtherrecentexamplesoftheuseofthebasicNKmodelareAmatoandLaubach(2004),Andrés, López-Salido, and Nelson (2005), Belaygorod and Dueker (2005), Benigno (2004), Bouakez, Cardia, and Ruge-Murcia (2005), Chari, Kehoe, and McGrattan (2000), Christiano, Eichenbaum, and Evans (2005), Clarida, Galí, and Gertler (2001), Coenen and Wieland (2005), Corsetti and Pesenti (2005), Giannoni and Woodford (2005), Iacoviello (2005), Ireland (2004a), Gürkaynak, Sack, and Swanson (2005), Keen (2004), Kim and Henderson (2005), King and Wolman (2004), Ludec and Sill (2004), Leith and Malley (2005), Levin, Wieland, and Williams (2003), Levin and Williams (2003), Lindé (2005), Lubik and Schorfheide (2004), Pappa (2004), Rabanal and Rubio-Ramírez (2005), Ravenna and Walsh (2006), Rudebusch (2005), Steinsson (2003), and Yun (2005). 3
est rate. Equating consumption to output yields an aggregate demand equation in which current output depends on expected future output and the real interest rate. The price equation, which has come to be called the new-Keynesian Phillips curve, is a forward-looking Phillips curve in which current ination depends on expected future ination and an output gap. It is derived from the optimizing behavior of monopolistically competitive rms, where rms change prices randomly as discussed in Calvo (1983) or face some kind of adjustment costs. 2 An interest rate rule is then sometimes added as a third equation in which the nominal interest rate depends on ination and the output gap. Data on output (usually real GDP), ination (usually the percentage change in the GDP deator), and the federal funds rate or the three-month Treasury bill rate aretypically used for the model. Sometimesdata ona fewother variables areused, depending on the setup. In particular, the labor income share is sometimes used in the price equation in place of the output gap, as in Galí and Gertler (1999) and Sbordone (2002), if the price equation is being analyzed separately. Sometimes all the parameters are calibrated and sometimes some parameters are calibrated and some are estimated. Estimation includes maximum likelihood and matching the model's impulse responses to those of an estimated VAR. This work is all done under the assumption of rational expectations. The parameters that are calibrated or estimated are usually the structural parameters of the theoretical model, and so 2 Recent studies dealing with the New Keynesian Phillips curve but not the entire NK model are Batini, Jackson, and Nickell (2005), Galí, Gertler, and López-Salido (2005), which is a defense of the earlier widely cited Galí and Gertler (1999) paper, Kurmann (2005), Mankiw and Reis (2002), who propose an alternative price equation, Mavroeidis (2005), Nessen and Vestin (2005), Rudd and Whelan (2005), Sahuc (2005), and Sbordone (2005), which is a defense of the earlier widely cited Sbordone (2002) paper. 4
this analysis is not subject to the Lucas (1976) critique. 2.2 MC Model ThetheoreticalmodeluponwhichtheMCmodelisbasedwasrstpresentedinFair (1974a). An easier-to-read presentation is in Fair (1984). The following is a brief outline of the model. It has two of the four features of the New Neoclassical Synthesis, namely intertemporal optimization and imperfect competition. Households maximizeexpectedfutureutilityandrmsmaximizeexpectedfutureafter-taxcash ow. The horizons for the maximization problems are nite. The choice variables for a household are consumption, leisure, and money holdings. The main choice variables for a rm are its price, wage rate, production, and investment. Expectationsoffuture valuesby householdsandrms arebasedoncurrent andpastvalues; they are not assumed to be rational. Disequilibrium is allowed for, and it takes the form of rms telling households the maximum amount of labor they will hire in the period and of actual sales differing from expected sales. A household takes as given its initial values of money and bonds and the current values of the price, wage rate, interest rate, personal income tax rate, transfer payments, and the labor constraint from rms. It forms expectations of the future values of these variables and solves it optimization problem given a terminal condition on the value of its money plus bonds. A rm faces a putty-clay technology. Adjustment costs are postulated for changesinlaborandthecapitalstock. Firmssetpricesandwagesinamonopolistic competitive setting. The demand for a rm's product depends on its price relative 5
to the prices of the other rms. A rm expects that other rms' prices are affected by the price that it sets. In other words, a rm expects that other rms will raise (lower) their prices if the rm raises (lowers) its own price. Similarly, the supply of labor to a rm depends on its wage rate relative to the wage rates of the other rms, and a rm expects that other rms' wage rates are affected by the wage rate that it sets. 3 A rm takes as given all the initial values, including the initial values of other rms' prices and wage rates and the current values of the interest rate and the prot tax rate. It forms expectations of the relevant future values, where again its expectations of other rms' prices and wage rates depend on its own behavior, and solves its optimization problem. It chooses its price, wage rate, amount of each type of machine to purchase, and production. Given its price and wage rate decisions, a rm has an expectation of its sales and of the amount of labor that will be supplied to it. If actual sales turn out to be different from expected, this results in an unexpected change in inventories. If actual labor supply exceeds expected labor supply, the rm is assumed to hire only the expected amount. In fact, the model is set up so that rms communicate to households the amount of labor they are willingtohire (namely, the rms'expectedamounts), and householdsoptimize under this constraint, as noted above. Regardingtheexpectationsofhouseholdsandrmsinthetheoreticalmodel,for anumber of variables equations are postulatedspecifying howthe expectationsare 3 No adjustment costs are postulated for price changes and wage rate changes, and all rms can change their prices and wage rates each period. This is contrary to the fourth feature of the NewNeoclassical Synthesis mentionedabove, namelycostly price adjustment. Thisassumption of costlypriceadjustmentis, ofcourse,controversial,anditisnotnecessarilyadesirablefeatureofthe synthesis. Bils and Klenow (2004) is a recent study casting doubt on the sticky price assumption. 6
formed. FortheoverallmodelinFair(1974a)itisalsospeciedthathouseholdsand rms estimate the parameters of these equations based on past data. In this sense the expectations are sophisticated. The key point about expectations, however, is that they are not specied to be rational or converge to being rational. Because expectations are not rational, disequilibrium can occur, which drives many of the properties of the model. Households and rms never learn the true model; they grope around in a complex world, never quite understanding everything. Government scal policy decisions are exogenous. The government chooses the two tax rates, transfer payments, the amount of goods to purchase, and the amount of labor to hire. On the monetary policy side, an interest rate rule is postulated in which the interest rate depends on ination and unemployment. Unemploymentinthemodelisthedifferencebetweenthelaborthathouseholdswould supplyif the labor constraintwere not bindingand the amount theyactually supply taking into account the labor constraint in their optimizing problem. Allowsof fundsand balancesheet constraintsare accountedfor inthe model. Onesector'ssavingissomeothersector'sdissaving. Onesector'snancialliability is some other sector's nancial asset. The model in Fair (1974a) was a closed-economy model, but a two-country model was introduced in Fair (1984). Again, all ows of funds and balance sheet constraints among the sectors of the countries are accounted for. The choice of a household now includes how much to purchase of the foreign good, which is affected by the price of the foreign good relative to the price of the home good. The exchange rate is determined by a reaction function of one of the country's monetary authorities. 7
The model is solved by numerical techniques, given chosen parameter values and initial conditions. In a model in which disequilibrium is possible, the order of transactions matters, and the order chosen is 1) the government, 2) rms, and then 3) households. Transactions take place after households have optimized. Because rms don't have complete knowledge of the model, their price and wage settingbehaviormayresultinsalesdifferingfromexpectedsalesandlabordemand differing from the unconstrained labor supply. The numerical work consists of running various experiments for the individual optimization problems and then running experiments using the entire model. The experiments are designed to explore the properties of the theoretical model. Returning to the expectational assumptions used in the model, Mankiw and Reis (2002, 2006) in recent work have modied the standard NK model by adding the assumption of sticky information. Households and rms are inattentive and base their decisions on outdated information sets. This work is essentially incorporating ideas from behavioral economics into the NK model. This assumption of sticky information is to some extent in the spirit of the expectational assumptions described above. Agents do not know the true model and therefore do not form rational expectations. They have limited information. Contrary to the case in the present model, however, in the Mankiw and Reis (2006) model, when agents update, they do know everything. For example, if there is no sticky information, the Mankiw and Reis model is just a standard classical exible-price model. In the model above, on the other hand, agents never know everything. But there are similarities, and in general the above expectational assumptions are in the spirit of the assumptions of behavioral economics in that there is a lot that agents don't 8
Table 1 Outside Sample RMSEs (percentage points) Real GDP GDP Deator No. Obs. Qtrs ahead Qtrs ahead Qtrs ahead Model 4 8 4 8 4 8 1. US 1.02 1.46 0.78 1.39 76 72 2. US+ 1.33 1.84 0.87 1.52 76 72 3. Hybrid RBC 3.45 70 4. Diagonal RBC 2.16 70 5. NK 2.62 6.05 0.88 1.70 55 51 • Rows 1 and 2 rows from Fair (2004), Table 14.1, p. 166. • Rows 3 and 4 from Ireland (2004b), Table 5, p. 1218. • Row 5 computed from Del Negro et al. (2006), Table 2, p. 36. • Basic prediction periods: 1983.12002.3 for rows 1 and 2; 1985.12002.2 for rows 3 and 4; 1985.42000.1 for row 5. US+ models, 70 for the RBC models, and 55 for the NK model. Table 1 shows that the NK model does poorly regarding real GDP. The fourquarter-aheadRMSEisabouttwicealargeasthosefortheUSandUS+models,and the eight-quarter-ahead RMSE is over three times as large. For the four-quarterahead results, the NK model is better than the hybrid RBC model, but worse than thediagonalRBCmodel. TheNKmodelismuchclosertotheUSandUS+models for the GDP deator. These results thus suggest that the NK aggregate demand equation is not well specied, a point argued above. In light of these results the quote from Woodford (2006) at the beginning of this section seems premature. Another way of testing the NK model is to test the assumption of rational expectations, which play a large role in the model. Although it is hard to test 15
this assumption, results have generally not been supportivesee, for example, Fair (2004), Fuhrer and Rudebusch (2004), and Rudd and Whelan (2006). The results in Rudd and Whelan (2006) are particularly strong against the assumption of rational expectations in the new-Keynesian Phillips curve. Given the results to date, a useful working hypothesis would appear to be that expectations are not rational rather than rational. Returning to methodology, early examples of the estimation of equations that are meant to approximate the decision rules of economic agents are Tinbergen (1939) and Klein (1950). There is considerable economic theory involved in this work, andinfactnearly half ofKlein's bookisdevotedto intertemporal optimizing models of households and rms. 11 But none of this early empirical work directly estimated the parameters of the theoretical models. Theory was only used to guide the choice of left hand side and right hand side variables. This approach dominated macro model building through the 1960s. The Lucas (1976) critique in the early 1970s changed the macro research landscape, and it eventually led to the DSGE approach that is currently popular. Whether this was a positive change for macro is an open question. Given the heterogeneity of agents, the complexity of the actual decision making processes, the complexity of the interactions among agents, and the quality of the macro data, it may be too much to expect that a good approximation of the economy can be obtained by directly estimating the parameters of a representative-agent theoretical model like that of the NK model. It may be better to settle for estimated approximations to decision rules. And if expectations are not rational, the Lucas critique is not likely to be a problem. The 11 For an interesting discussion of this, see Solow (1991). 16
basic NK model does not appear trustworthy for analyzing monetary policy issues, including ination targeting. Models more tied to the data are needed, and the MC model is one alternative. It is used for the work in the next section. Table 2 summarizes the comparison of the basic NK and the MC model discussed in this section. 3 Estimated Effects of Ination Targeting 3.1 Interest Rate Channels It will rst be useful to outline the various channels through which interest rates affect output in the U.S. part of the MC model. Consider a decrease in the U.S. short term interest rate, say a policy change by the Fed. This decreases long term interest rates through estimated term structure equations. Interest rates appear as explanatory variables in the consumption, residential investment, and nonresidential xed investment equations, all with negative coefcient estimates. In addition, decreases in interest rates have a positive effect on the change in stock prices through an estimated capital gains and losses equation, which has a positive effect on household wealth. This in turn has a positive effect on consumption because wealth appears as an explanatory variable in the consumption equations. Also, a decrease in U.S. interest rates (relative to other countries' interest rates) leads to a depreciation of the U.S. dollar through estimated exchange rate equations. 12 12 A relative interest rate variable appears in the exchange rate equations for Canada, Japan, the United Kingdom, and Germany (Euroland after 1999). (All exchange rate equations are relative to the U.S. dollar.) 17
Table 2 The Basic NK Model versus the MC Model Property NK Model MC Model Intertemporal optimization? Yes. Yes. Rational expectations? Yes. No. Imperfect competition? Yes. Yes. Costly price adjustment? Yes. No. Estimation. Parameters of the theoretical model are calibrated or estimated. The theoretical model is used to guide the specication of the econometric model, which is then estimated. No calibration for econometric model. Demand disaggregation. One aggregate demand equation. Three consumption equations: services, nondurables, durables; three investment equations: nonresidential xed, residential, inventory; import demand equation. Government sector? Usually not. Yes. Foreign sector? Usually not. Yes. Stock effects? No. Yes,ondurableconsumption, residential investment, nonresidential xed investment, inventory investment. Wealth effects? No. Yes, on the three categories of consumption. Wage equation? Usually not. Yes, separately estimated wage and price equations. Real versus nominal interest rate effects. Real effects imposed. Tested, where nominal interest rates generally dominate. Effects of a positive price shock with the nominal interest rate held constant. Explosive or indeterminate. Contractionary. Lucas critique a problem? No. Not under the assumptions about expectations. Long run tradeoff between ination and output? No. Lack of tradeoff not tested because of limited data; see last paragraph in Section 3.2. Relationship likely to be nonlinear. Accuracy. See Table 1. See Table 1. 18
Other things being equal, this depreciation is expansionary because U.S. exports rise and U.S. imports fall. A decrease in interest rates thus has a positive effect on aggregate demand through these channels. 13 3.2 The U.S. Price Equation It will next be useful to outline the main price equation in the U.S. part of the MC model. In this equation the log of the price level (the private nonfarm price deator) is regressed on a constant, the lagged logged price level, the log of the wage rate, the log of the import price deator, the unemployment rate, and the time trend. The coefcient estimates are presented in Table 3. The cost variables are the wage rate and the import price deator, and the demand variable is the unemployment rate. The time trend is added to pick up trend effects on the price level not captured by the other variables. Adding the time trend to this equation is like adding a constant term to an equation specied using the ination rate rather than the price level. This equationdoeswell invarious chi-squaredtestsreported inTableA10, p. 206, in Fair (2004), with updated results on the website. No signicant improvement in t occurs when 1) the logged price level lagged twice, the log of the wage rate lagged once, the log of the import price deator lagged once, and the unemployment rate lagged once are added as explanatory variables, 2) the equation is estimated under the assumption of fourth order serial correlation of the error term, 13 There is one effect that works in the opposite direction. An decrease in interest rates decreases household interest income, which has a negative effect on household expenditures through a disposable income variable in the household expenditure equations. This effect is, however, smaller than the positive effects, and so the net effect of an interest rate decrease is positive. 19
Table 3 U.S. Price Equation LHS Variable is log PF RHS Variable Coef. t-stat. cnst -0.036 -3.21 log PF−1 0.881 92.56 log W 0.040 3.36 log PIM 0.050 21.23 UR/100 -0.177 -7.40 time trend 0.00032 9.88 SE 0.00343 •PF = private nonfarm price deator. •W = nominal wage rate adjusted for labor productivity. •PIM = import price deator. •UR = unemployment rate. • Estimation period: 1954.12006.1. • Estimation method: 2SLS. 3) the log of the wage rate led once is added, 4) the log of the wage rate led four times is added, 5) the log of the wage rate led eight times is added, and 6) an output gap variable is added. When the output gap variable is added, the unemployment rate retains its signicance, and so it dominates the output gap as an explanatory variable. If the wage rate variable were dropped from the equation in Table 3 and the equation were specied as an ination equation rather than a price-level equation, the coefcient on log PF−1 would be one. In addition, if lagged ination were added as an explanatory variable to the ination equation, this would introduce log PF−2 with restrictions on the coefcients of both log PF−1 and log PF−2 . These restrictions were tested in Fair (2000) and updated to other countries in 20
Chapter 4 in Fair (2004). They were rejected for the United States and generally rejected for the other countries. They suggest that the price equation should be specied in terms of price levels rather than ination rates or changes in ination rates. Using changes in ination rates is off by two derivatives! The wage equation in the U.S. part of the MC model has log W on the left hand side and on the right hand side: the constant, log W−1 , log PF , log PF−1 , and the time trend. The price and wage equations are identied because log PIM and UR are excluded from the wage equation, and log W−1 is excluded from the price equation. Intheestimationofthewageequationalongrunrestrictionwasimposed regarding the real wage, which is that the derived real wage equation does not have on the right hand side the price level separately or the wage rate separately. This restriction is not rejected by the data. The price and wage equations were tested in Fair (2000) and (2004, Chapter 4) against standard NAIRU equations, and they lead to considerably more accurate price level and ination predictions. This is consistent with the rejection of the NAIRU dynamics mentioned above. A long run property of the price and wage equations is the following. If, say, theunemploymentrateispermanentlydecreasedbyonepercentagepoint,theprice levelis permanentlyhigher, but theination rateconvergesback toits initial value. There is no permanent effect on the ination rate. The evidence in favor of this property is the lack of rejection of the restrictions discussed above. Regarding this long run property, it is obviously not sensible to think that the unemployment rate can be driven to zero with no permanent effect on the ination rate. The problem in my view with the specication in Table 3 (or with specicationsintermsofinationratesor changesininationrates)isthelinearity 21
assumptionregardingtheeffectoftheunemploymentrateormeasuresoftheoutput gap on the price level (or the ination rate or the change in the ination rate). At low levels of the unemployment rate, this effect is likely to be nonlinear. I have tried for both the United States and other countries to pick up nonlinear effects, but there appear to be too few times in which the unemployment rate is very low (or the output gap very small) to allow sensible estimates to be obtained. This does not mean, however, that the true functional form is linear, only that the data are insufcient for estimating the true functional form. What this means regarding the MC model is that one should not run experiments in which unemployment rates or output gaps are driven to historically low levels. Price-level or ination-rate equations are unlikely to be reliable in these cases. Because of this, an effort has been made in the experiments below to stay around historical values. 3.3 The U.S. Interest Rate Rule The nal equation to discuss is the U.S. estimated interest rate rule. This rule was rst estimated and added to my U.S. model in 1978Fair (1978). This is the rst instance that such as rule was added to a model, but the rules themselves go back to Dewald and Johnson (1963). This was long before the rules came to be called Taylor rules; they should really be called Dewald-Johnson rules. The estimated rule is presented in Table 4. The left hand side variable is the three-month Treasury bill rate ( RS ), which is taken as the control variable of the Fed. 14 The Fed is estimated to respond to 14 The actualcontrol variable is thefederal fundsrate, butthis rateand RS aresohighly correlated that it makes little difference which is used. 22
Table 4 U.S. Interest Rate Rule LHS Variable is RS RHS Variable Coef. t-stat. cnst 0.774 5.23 RS−1 0.922 53.30 ˙ PD 0.071 4.18 UR -0.125 -4.22 ∆UR -0.761 -6.02 ˙ M1−1 0.012 2.30 D794823 ·˙ M1−1 0.215 9.73 ∆RS−1 0.228 4.24 ∆RS−2 -0.332 -6.76 SE 0.463 Stability test, 1954.1-1979.3 versus 1982.4-2006.1: Wald statistic is 15.33 (8 degrees of freedom, p -value = .0531.) •RS = three-month Treasury bill rate. •PD = price deator for domestic sales. •UR = unemployment rate. •M1 = money supply. •D794823 = dummy variable that is 1 between 1979:4 and 1982:3 and 0 otherwise. • A dot over a variable means percentage change at an annual rate. • Estimation period: 1954.12006.1. • Estimation method: 2SLS. ination, 15 the unemployment rate, the change in the unemployment rate, and the lagged growth of the money supply. The lagged values of RS are meant to soak 15 NoteinTable 4thattheFedistakentorespond tochanges in PD , the pricedeatorfor domestic sales, not PF , the private nonfarm price deator. PD , contrary to PF , includes import prices and excludes export prices. It is close in concept to the consumer price index. Better results are obtained using PD rather than PF in the interest rate rule. The exact denitions of PD and PF are in Fair (2004) and on the website. 23
up the dynamics, which are estimated to be fairly complicated. Between 1979:4 and 1982:3 (to be called the early Volcker period) the Fed, according to its own announcements, operated under a procedure that focused more on monetary aggregates than was the case before (or that was the case subsequently). This behavioral change was handled in the specication by adding a variable that is the lagged growth of the money supply multiplied by a dummy variable that is one in the early Volcker period and zero otherwise. As can be seen in Table 4, the coefcient estimate for the lagged money supply growth is about 20 times larger in the early Volcker period than otherwise. This way of accounting for the Fed policy shift does not, of course, capture the richness of the change in behavior, but at least it seems to capture some of the change. The equation in Table 4 does well in various chi-squared tests (reported in Table A30, p. 216, in Fair (2004), with updated results on the website). No signicant improvement in t occurs when 1) RS lagged four times, the ination ratelaggedonce,theunemploymentratelaggedtwice,andthepercentagegrowthin the money supply lagged twice are added as explanatory variables, 2) the equation isestimatedundertheassumptionoffourthorderserialcorrelationoftheerrorterm, 3) the ination rate and the unemployment rate led once are added, 4) the ination rate and the unemployment rate led four times are added, 5) the ination rate and the unemployment rate led eight times are added, and 6) and 7) two measures of expected future ination are added. The stability test listed at the bottom of Table 4 is of the hypothesis that the coefcients of the rule are the same before the early Volcker period as after. Much of the literature is of the view that the Fed behaved differently in the two periods, 24
Table 7 Effects of a Positive Demand Shock Changes from Base Values Quarters Ahead Variable 1 4 8 12 16 20 Sum Case 1: RS Exogenous PD .03 .25 .72 1.17 1.50 1.67 ˙ PD .13 .38 .46 .39 .23 .10 Y .30 1.25 1.78 1.76 1.56 1.37 1.41 UR -.06 -.44 -.74 -.73 -.59 -.42 -.56 RS 000000 Case 2: Estimated Rule PD .02 .21 .51 .72 .84 .87 ˙ PD .10 .30 .29 .19 .07 .01 Y .29 1.17 1.49 1.33 1.13 1.02 1.12 UR -.06 -.42 -.63 -.54 -.39 -.27 -.43 RS .06 .46 .73 .75 .67 .56 Case 3: Ination Rule .2838 PD .03 .23 .59 .86 1.01 1.04 ˙ PD .11 .34 .34 .23 .08 .01 Y .30 1.21 1.59 1.44 1.20 1.06 1.19 UR -.06 -.43 -.67 -.59 -.43 -.28 -.46 RS .03 .25 .52 .63 .58 .46 Case 4: Ination Rule .5676 PD .03 .21 .49 .67 .73 .73 ˙ PD .10 .30 .26 .15 .03 -.01 Y .29 1.17 1.46 1.25 1.03 .95 1.07 UR -.06 -.42 -.62 -.51 -.34 -.24 -.41 RS .06 .45 .85 .94 .80 .59 Case 5: Price-level Rule PD .03 .24 .63 .87 .83 .58 ˙ PD .12 .36 .36 .16 -.11 -.27 Y .30 1.24 1.65 1.38 .91 .53 1.07 UR -.06 -.44 -.70 -.58 -.31 -.06 -.42 RS .01 .13 .53 1.05 1.46 1.61 • Simulation period is 1994.11998.4. • For notation see notes to Table 5. For this experiment the estimated rule and the ination rules respond similarlycases 2, 3, and 4. The interest rate is increased, which lowers both 31
output and the price level relative to case 1 of no interest rate change. Comparing the estimated rule to say, ination rule .2858, the differences are small, and one could conclude that moving from current Fed behavior to behavior in which only ination is in the rule makes little difference. This, of course, is not true for the price shock in Table 6, and so the consequences of changing rules depends on the type of shock. Regarding case 5, the price-level rule, there is a slower initial response and then larger effects at the end. 3.6 Stochastic Simulation Results The shocks in Tables 6 and 7 are just made up shocks. A more general way of examining the consequences of using different interest rate rules is to use historically estimated residuals and stochastic simulation. There are 328 stochastic equations in the MC model, 182 quarterly and 146 annual. There is an estimated error term for each of these equations for each period. Although the equations do not all have the same estimation period, the period 19772004 is common to all equations. There are thus available 28 vectors of annual error terms and 112 vectors of quarterly error terms. These vectors are taken as estimates of the economic shocks, and they are drawn in the manner discussed below. Since these vectors are vectors of the historical shocks, they pick up the historical correlations of the error terms. If, for example, shocks in two consumption equations are highly positively correlated, the error terms in the two equations will tend to be high together or low together. Thebasepathisagaintakentobethehistoricalpath,whichisobtainedbyadded 32
theestimated residualsto allthe equationsand taking themto beexogenous. Thus, for all the stochastic simulations the estimated residuals are added to the model and the draws are around these residuals. Each trial for the stochastic simulation is a dynamic deterministic simulation for 1994:11998:4 using a particular draw of the error terms. For each of the ve years for a given trial an integer is drawn between 1 and 28 with probability 1/28 for each integer. This draw determines which of the 28 vectors of annual error terms is used for that year. The four vectors of quarterly error terms used are the four that correspond to that year. Each trial is thus based on drawing ve integers. The solution of the model for this trial is an estimateof whatthe worldeconomywouldhave beenlikehad theparticular drawn error terms actually occurred. (Remember that the drawn error terms are on top of the historical residuals for 1994:11998:4, which are always used.) The number of trials taken is 1000, so 1000 world economic outcomes for 1994:11998:4 are available for analysis. The historical residuals are added to whatever interest rate rule is used, but no errors are drawn for it. Adding the historical residuals means that when the model inclusive of the rule is solved with no errors for any equation drawn, a perfect tracking solution results. 19 Not drawing errors for the rule means that the Fed does not behave randomly but simply follows the rule. Let yj t be the predicted value of endogenous variable y for quarter t on trial j , and let y∗ t be the base (actual) value. How best to summarize the 1000 ×20 values of yj t ? One possibility for a variability measure is to compute the variability of yj t 19 Eachoftheruleshasadifferentsetofestimatedresidualsassociatedwithitbecausethepredicted values from the rules differ. 33
around y∗ t for each t : (1/J)∑J j=1(yj t−y∗ t)2 , where J is the total number of trials. 7 The problem with this measure, however, is that there are 20 values per variable, which makes summary difcult. A more useful measure is the following. Let Lj be: Lj=1 T T ∑ i=1 (yj t−y∗ t)2 (1) where T is the length of the simulation period (20). Then the measure is L=1 J J ∑ j=1 Lj (2) L is a measure of the deviation of the variable from its base values over the whole period. It is not an estimated variance, just a summary measure of variability. Selected results are presented in the rst ve rows in Table 8: values of L are presented for ve variables. Comparing rows 1 and 2, the estimated rule does a fairly good job in lowering the values of L . L for PD falls from 4.69 to 3.08, and L for Y falls from 2.88 to 2.22. L increases for RS from zero to .97. 20 In row 3 ination rule .2838 lowers L for PD more (to 2.53) at a cost of higher values of L for Y (2.55) and RS (1.59) compared to the estimated rule. The results in rows 2 and 3, for the estimated rule and ination rule .2858, are not as similar as they are for the demand shock in Table 7, but they are more similar than for the price shock in Table 6. This is as expected since the errors used for the stochastic-simulation 20 Whentheexperimentinrow2isdoneforthe2000:12004:4period(insteadof1994:11998:4), the values of L are: 3.86 for PD , 2.90 for ˙ PD , 2.08 for Y , .58 for UR , and .64 for RS . In this later period the actual values of RS are on average smaller, and in the experiment there are more times when the 0.5 constraint for the minimum value of RS is binding. This is the main reason that L for RS is lower in the later period: the Fed has less room to maneuver. (Remember that the base path for an experiment is just the historical path.) This constraint results in somewhat higher values of L for PD and ˙ PD in the later period, but in general the results are fairly close. 34
Table 8 Variability Estimates: Values of L MC Model PD ˙ PD Y UR RS 1. No rule ( RS exogenous) 4.69 2.85 2.88 .80 0 2. Estimated rule 3.08 2.45 2.22 .59 .97 3. Ination Rule .2838 2.53 2.33 2.55 .67 1.59 4. Ination Rule .5676 1.63 2.03 2.56 .63 4.04 5. Price-level Rule 2.60 2.83 3.21 .78 3.61 US(EX,PIM) Model 6. No rule ( RS exogenous) 4.48 2.58 3.29 .97 0 7. Estimated rule 3.66 2.44 2.50 .72 .87 8. Optimal ( λ1= 1.5, λ2= 1.5 ) 3.72 2.40 2.46 .72 .98 9. Optimal ( λ1= 0.0, λ2= 3.0 ) 3.04 2.25 2.97 .86 .99 10. Optimal ( λ1= 0.0, λ2= 1.0 ) a 2.29 2.64 4.32 1.15 3.33 a Price-level loss function; see text • Simulation period: 1994:11998:4. • See notes to Tables 5 and 6. draws consist of both demand and price shocks. The draws are, of course, more representative of actual shocks than are the shocks used in Tables 6 and 7. The second ination rule is more extreme than the rst (row 4 versus row 3). L for PD falls to 1.63, but L for RS is now 4.04. It is interesting in this case that the cost of lowering L for PD is added variability of RS , not of Y and UR . Other things being equal, lowering the variability of PD in the MC model lowers the variability of Y , and this affect dampens the effects that work in the opposite direction, which arise from the higher variability of RS . Again, the price-level rule (row 5) is not very good. Comparing rows 3 and 5, the price level rule has about the same value of L for PD , but much larger values 35
for Y and RS . 3.7 Optimal Control Results Optimal control techniques are the obvious ones to use in evaluating ination targeting, and so the most weight should probably be placed on the following results. Theoptimalcontrolmethodologyrequiresthatalossfunctionbepostulated for the Fed. Assume that the loss for quarter t is: Ht=λ1(URt−UR∗ t)2+λ2(˙ PDt−˙ PD∗ t)2+ 9.0(∆RSt−∆RS∗ t)2 +1.0/(RSt−0.499) + 1.0/(16.001 −RSt) (3) where ∗ denotes a base value. λ1 is the weight on unemployment deviations, and λ2 is the weight on ination deviations. The last two terms in (3) insure that the optimal values of RS will be between 0.5 and 16.0. The middle term penalizes changes in RS ; more will be said about it below. As was done for the other experiments, the estimated residuals are rst added to the stochastic equations and taken to be exogenous. The base path is then the historical path, and the target values in (3) are the historical values. Assume that the control period of interest is 1 through T , where in the present case 1 is 1994:1 and T is 1998:4. Although this is the control period of interest, in order not to have to assume that life ends in T , the control problem should be thought of as one of minimizing the expected value of ∑T+n t=1 Ht , where n is chosen to be large enough to avoid unusual end-of-horizon effects near T . The overall control problem should thus be thought of as choosing values of RS that minimize the expected value of ∑T+n t=1 Ht subject to the model used. 36
If the model used is linear and the loss function quadratic, it is possible to derive analytically optimal feedback equations for the control variables. In general, however, optimal feedback equations cannot be derived for nonlinear models or for loss functions with nonlinear constraints on the instruments, and a numerical procedure must be used. The following procedure was used for the present results. It is based on a sequence of solutions of deterministic control problems, one sequence per trial. Recallwhat a trial for thestochastic simulationis. A trial isa setof drawsof20 vectors of error terms, one vector per quarter. Given this set, the model is solved dynamically for the 20 quarters using an interest rate rule (or no rule). This entire procedure is then repeated the chosen number of trials, at which time the summary statisticsarecomputed. As willnowbediscussed, eachtrialfortheoptimalcontrol procedure requires that 20 deterministic control problems be solved. For purposes of solving the control problems, the Fed is assumed to know the model (its structure and coefcient estimates) and the exogenous variables, both past and future. The Fed is assumed not to know the future values of any endogenous variable or any error draw when solving the control problems. The Fed is assumed to know the error draws for the rst quarter for each solution. This is consistent with the use of the above rules, where the error draws for the quarter are used when solving the model with the rule. The procedure for solving the overall control problem is as follows. 1. Draw a vector of errors for quarter 1, and add these errors to the equations. Take the errors for quarters 2 through k to be their historical values (no draws), where k is dened shortly. Choose values of RS for quarters 1 through k that minimize ∑k t=1 Ht subject to the model as just described. 37
This isjust adeterministic optimalcontrol problem, whichcan besolved, for example, by themethod inFair(1974b). 21 Let RS∗ 1 denotethe optimal value of RS for quarter 1 that results from this solution. The value of k should be chosen to be large enough so that making it larger has a negligible effect on RS∗ 1 . (This value can be chosen ahead of time by experimentation.) RS∗ 1 is a value that the Fed could have computed at the beginning of quarter 1 (assuming the model and exogenous variables were known) having knowledge of the error draws for quarter 1, but not for future quarters. 2. Record the solution values from the model for quarter 1 using RS∗ 1 and the error draws. These solution values are what the model estimates would have occurred in quarter 1 had the Fed chosen RS∗ 1 and had the error terms been as drawn. 3. Repeat steps 1 and 2 for the control problem beginning in quarter 2, then for the control problem beginning in quarter 3, and so on through the control problem beginning in quarter T . For an arbitrary beginning quarter s , use the solution values of all endogenous variables for quarters s−1 and back, as well as the values of RS∗ s−1 and back. 4. Steps 1 through 3 constitute one trial, i.e., one set of T drawn vectors of errors. Do these steps again for another set of T drawn vectors. Keep doing this until the specied number of trials has been completed. The solution values of the endogenous variables carried along for a given trial from quarter to quarter in the above procedure are estimates of what the economy would have been like had the Fed chosen RS∗ 1 ,..., RS∗ T and the error terms been as drawn. The optimal control procedure is too costly in terms of computer time to be able to be used for the MC model, and for this work the U.S. subset of the model was used, denoted US(EX,PIM). This model is exactly the same as the model for the United States in the MC model except for the treatment of U.S. exports 21 This method sets up the problem as an unconstrained nonlinear optimization problem and uses an optimization algorithm like DFP to nd the optimum. 38
( EX ) and the U.S. price of imports ( PIM ). These two variables change when RS changesprimarily because the value of the dollar changesand the effects of RS on EX and PIM were approximated in the following way. First, log EXt−α1RSt was regressed on a constant, t , log EXt−1 , log EXt−2 , log EXt−3 , and log EXt−4 , and log PIMt−α2RSt was regressed on a constant, t , log PIMt−1 , log PIMt−2 , log PIMt−3 , and log PIMt−4 . Second, these two equations were added to the US(EX,PIM) model for particular values of α1 and α2 , and an experiment was run in which the estimated interest rate rule of the Fed was dropped and RS was decreased by one percentage point. This was done many times for different values of α1 and α2 . The nal values of α1 and α2 chosen were ones whose experimental results most closely matched the results for the same experiment using the complete MC model. The nal values chosen were -.0004 and -.0007 respectively. Third, the experiment in row 2 of Table 8 was run for the US(EX,PIM) model (with the EX and PIM equations added) and with the estimated errors from the EX and PIM equations being used in the drawing of the errors. When an error for the EX equation was drawn, it was multiplied by β1 , and when an error for the PIM equation was drawn, it was multiplied by β2 . The experiment was run many times for different values of β1 and β2 , and the nal values chosen were ones that led to results similar to those in the row 2 of Table 8. The values were β1=.4 and β2=.75 . The results using these values are in row 7 of Table 8. The chosen values of α1 , α2 , β1 , and β2 were then used for the experiments in rows 810. Because of computational costs, 100 rather than 1000 trials were used for the optimal control experiments. The results are presented in rows 6-10 in Table 8. 39
Eachexperimentinarowusesthesamesetsoferrordraws,whichlessensstochastic simulationerroracrossexperiments,althoughthesesetsoferrordrawsaredifferent from those used for the experiments in rows 15. Rows 6 and 7 are equivalent to rows 1 and 2: no rule and the estimated rule, respectively. Comparing these rows, the same pattern holds for both the overall MC model and the US(EX,PIM) model, namely that the estimated rule substantially lowers the variability of both PD and Y . Row 8 uses equal weights on unemployment and ination in the loss function. The values of λ1 and λ2 of 1.5 were chosen after some experimentationusing the coefcient of 9 on the middle term in equation (3)to have the value of L for RS to be similar to its value when the estimated rule is used. The value of L is .98, which is close to .87 for the estimated rule. The aim is to constrain the optimal control procedure from variations in RS much different from what the Fed is estimated to have done historically (aside from the early Volcker period). The results using the estimated rule in row 7 and the equally-weighted optimal control procedure in row 8 are quite similar. In other words, the estimated rule is consistent with the Fed solving an optimal control problem with equal weights on unemployment and ination. Row 9 uses a zero weight on unemployment and a weight of 3.0 on ination. Again, the weight of 3.0 led the value of L for RS (.99) being close to the value using the estimated rule. Comparing row 9 to row 7, L for PD has fallen from 3.66 to 3.04 at a cost of L rising for Y from 2.50 to 2.97. For the ination rate ( ˙ PD ) L falls from 2.44 to 2.25, and for the unemployment rate L rises from .72 to .86. 40
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