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Optimal simple rules in RE models with risk sensitive preferences
Economics Letters 97 (2007) 260 – 266 www.elsevier.com/locate/econbase
Optimal simple rules in RE models with risk sensitive preferences ☆ Mingjun Zhao ⁎ Ohio State University, Arps Hall 410, 1945 North High Street, Columbus, OH 43210, United States Received 22 March 2006; received in revised form 16 March 2007; accepted 19 March 2007 Available online 5 July 2007
Abstract This paper provides a useful method to solve optimal simple rules under risk sensitive preference in macromodels with forward looking behavior. An application to a new Keynesian model with lagged dynamics is offered and risk sensitive preference is found to amplify the policy responses. © 2007 Elsevier B.V. All rights reserved. Keywords: Risk sensitive control; Commitment; Simple rules JEL classification: C61; E52; E58
1. Introduction In dynamic economic applications, it is very useful to incorporate forward looking rational expectations elements. This paper extends the version of Hansen and Sargent (1995) risk sensitive control in backward looking system and solves simple optimal rules. Solution of optimal policies in a linear-quadratic framework for a class of RE macromodels is extensively discussed in Soderlind (1999) and Hansen and Sargent (2003). The jump variables in this class of models do not respond to contemporaneous exogenous shocks. The economies we considered are much broader and the unconstrained optimal commitment rules are generally not easy to derive. However, it's straightforward to pursue some kind of simple rules.
☆
I would like to thank Bill Dupor genuinely for his intellectual tutoring and warm encouragement. ⁎ Tel.: +1 614 292 2069; fax: +1 614 292 3906. E-mail address: [email protected].
0165-1765/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2007.03.004
M. Zhao / Economics Letters 97 (2007) 260–266
261
The paper illustrates the extension with an application to the optimal monetary policy in a DSGE model. The central bank is assumed to have a risk sensitive preference and commit to a nominal income growth rule. Compared to the standard intertemporally time-consistent preference, risk sensitive preference makes the central bank undertake a precautionary policy stance. It will respond more aggressively resulting in higher current period loss in exchange for tighter control of future uncertainty. 2. Problem formulation Many dynamic linear rational expectations models evolve according to AEt xtþ1 ¼ Bxt þ Cut þ Det
ð1Þ
where xt is an n × 1 vector of endogenous variables, ut is a k × 1 vector of policy instruments, and et is an m × 1 Gaussian noise with mean zero and covariance matrix Ω. Some of the endogenous variables are predetermined (backward looking), and we assume that they are ordered first in the vector xt. For ′ , x2t ′ )′ where x1t is n1 × 1 and x2t is notational convenience, partition the vector xt accordingly into (x1t n2 × 1 with n1 + n2 = n and x10 given. The policy maker chooses policy instruments ut to minimize the fixed point L0 of Lt ¼ xtVQxt þ utVRut
h r i 2b logE exp Ltþ1 jt r 2
ð2Þ
where β ∈ (0, 1) is a discount factor and σ b 0 is the risk sensitive parameter. Lt+1 indicates the continuation value of the loss. When σ = 0, we obtain the standard specification because in that case Lt = x′Qx t t + u′Ru t t + βE[Lt+1|t]. So here σ b 0 reflects an additional aversion to the continuation loss risk beyond that represented by the expectation operator. 1 The current formulation of discounted, risk-adjusted losses is similar to Hansen and Sargent (1995), and I generalize the stable dynamic system they considered to incorporate rational expectations. This additional risk adjustment can be viewed as a special case of Epstein and Zin's (1989) recursive preference specification. More generalized policy rules are considered as well. In the standard control theory, we anchor the policy to the predetermined variables, something we know at the current period. Recently, policy rules targeting on expected variables draw a lot of attention as well, such as Clarida et al. (2000). 3. Optimal simple rules The policy maker is assumed to be able to commit to a simple decision rule of the form ut = −F1xt − F2Etxt+1. There may be restrictions on the elements in F1 and F2. Augmenting Eq. (1) with the policy rule gives A 0nk B C xt D 0nk et xtþ1 E ¼ þ ð3Þ F2 0kk t utþ1 0km 0kk 0k1 F1 Ikk ut 1
By the convexity of the exponential function, E exp r2 L jt zexp r2 E½Ljt .
262
M. Zhao / Economics Letters 97 (2007) 260–266
which in terms of yt = (x′,t u′)′ can be written as t A⁎ Et ytþ1 ¼ B⁎ yt þ D⁎ vt
ð4Þ
where vt = (e′,t 0k×1 ′ )′. This can be solved using the generalized Schur decomposition as in Klein (2000). Given the square matrices A⁎ and B⁎, the decomposition gives the unitary square complex matrices Q and Z 2 such that A⁎ ¼ Q H SZ H and B⁎ ¼ Q H TZ H
ð5Þ
where QH and ZH are the transpose of the complex conjugates of Q and Z, and S and T are upper triangular. Reorder the decomposition so that the stable generalized eigenvalues come first. 3 Count the number of the stable generalized eigenvalues ns. When ns = n1, there is a unique solution to Eq. (3). If there are more stable eigenvalues than there are predetermined variables (ns N n1), we can follow Blanchard and Kahn to select smallest eigenvalues needed. Following Klein (2000), the solution to Eq. (3) can be written as y2t ¼ M1 y1t þ M2⁎ vt
ð6Þ
y1;tþ1 ¼ N1 y1t þ N2⁎ vt
ð7Þ
4 where y1t = x1t, y2t = (x′2t , u′)′ and t 1 M1 ¼ Z21 Z11
ð8Þ
1 1 M2⁎ ¼ ðZ22 Z21 Z11 Z12 ÞT22 Q2 D⁎
ð9Þ
1 1 N1 ¼ Z11 S11 T11 Z11
ð10Þ
1 1 1 1 1 T11 Z11 Z12 T22 Q2 D⁎ þ Z11 S11 ½T12 T22 Q2 D⁎ þ Q1 D⁎ N2⁎ ¼ Z11 S11
ð11Þ
and Si , j , Ti , j , Z i , j and Qi (i, j = 1, 2) are conformable partitioned matrices of S, T, Z and Q. As in HS (1995), it is desirable to represent losses in terms of the true state variables at each period recursively. Suppose the problem has been solved for time t + 1 and future periods, and the This is also called ‘qz’ decomposition and this Q is different from the one in Eq. (2). Generalized eigenvalues are defined as tii / sii, i = 1, … , n where tii and sii are diagonal elements of T and S, and those with modulus less than one are called stable. 4 Equivalently, the solution can be expressed as y2t = M1y1t + M2et and y1,t+1 = N1y1t + N2et where M2 and N2 are the first m columns of M2⁎ and N2⁎. 2 3
M. Zhao / Economics Letters 97 (2007) 260–266
263
loss at time t + 1 can be written as a quadratic form in the true state variables at t + 1 as Lt+1 = x1,t+1 ′ 5 Vt+1x1,t+1 + dt+1. Then using the lemma in Jacobson (1973) h r i 2 V Vtþ1 x1;tþ1 þ dtþ1 jt qðLtþ1 jtÞu log E exp x1;tþ1 r 2 ð12Þ 1 f ¼ dtþ1 þ log detðX1 þ rN2VVtþ1 N2 Þ þ x1tV N1VV tþ1 N1 x1t r ~ where V t+1 = Vt+1 − σVt+1N2(Ω− 1 + σN2′Vt+1N2)− 1 N2′Vt+1.
Given
yt ¼
y1t y2t
¼
In1 0 Q f f y1t þ n1 m et ¼ M 1 y1t þ M 2 et and W ¼ M1 M2 0kn
0nk R
, we can further compute the cur-
rent loss in terms of true state variables x1t xt ¼ E½ ytVWyt jt þ bqðLtþ1 jtÞ ¼ x1tVVt x1t þ dt
ð13Þ
where f f f Vt ¼ M 1VW M 1 þ bN1VV tþ1 N1
ð14Þ
b f f dt ¼ traceðM 2VW M 2 XÞ þ bdtþ1 þ logdetðX1 þ rN2VVtþ1 N2 Þ r
ð15Þ
and
Note since the jump variables depend on the Gaussian noise, we should take the expectations conditional on information at period t for the first term in the current loss. Thus, we successfully map a translated quadratic loss measure for next period's losses into a translated quadratic loss measure today, both expressed in terms of true state variables at respective periods. Accordingly the initial period loss is V V ⁎ x10 þ d ⁎ x10
1 b fV f where V⁎ is the fixed point of Eq. (14) and d ⁎ ¼ traceðM 2 W M 2 XÞ þ logdetðX1 þ rN2VV ⁎ N2 Þ . 1b r
ð16Þ
By choosing the elements in the committed rule F1, F2 to minimize Eq. (16), we can find the optimal simple rule. Numerical non-linear optimization algorithms such as Nelder–Mead or simulated annealing can be employed. Sometimes, they are also termed “handcrafted” feedback rules.6 4. An application To illustrate the properties of this rational expectations-augmented risk sensitive preference and proposed solution approach, I apply this to a well discussed monetary model. This is a modified version of The formula works only when (Ω− 1 + σN2′Vt+1N2) N 0. The key assumption for this lemma to be held is the Gaussian distribution of error terms. 6 In a broad sense, “handcrafted” feedback rules mean all feedback rules that are not computed from formal optimizing procedures such as dynamic programming and Riccati equations. 5
264
M. Zhao / Economics Letters 97 (2007) 260–266
new Keynesian model with lagged dynamics. With the presence of habit formation, consumption Euler equation takes the form yt ¼ ð1 dÞyt1 þ dEt ytþ1 /ðRt Et ptþ1 Þ þ eyt
ð17Þ
where yt is the current output gap, πt is the inflation rate and Rt is the nominal interest rate. The coefficients δ and ϕ are closely related to the index of the importance of habit formation in the utility function. Habit formation has recently been studied by Fuhrer (2000) and McCallum and Nelson (1999) in models for the analysis of monetary policy. An alternative specification for lagged output dynamics is the introduction of rule-of-thumb consumers, such as Campbell and Mankiw (1989) and Amato and Laubach (2003). Analogous to output, inflation also features lagged dynamics. Dynamic indexation as in Christiano et al. (2005) and rule-of-thumb pricing setting as in Gali and Gertler (1999) are two alternatives to obtain lagged inflation in the formation of current inflation. pt ¼ jðyt þ yt1 Þ þ ð1 gÞpt1 þ gEt ptþ1 þ ept
ð18Þ
Both the output shock eyt and the inflation shock eπt are assumed iid normal with mean 0 and variance σy2 and σπ2 respectively. The central bank sets short interest rates to minimize the loss h r i 2b Lt ¼ Et fp2t þ ky y2t þ kR R2t g log E exp Ltþ1 jt ð19Þ r 2 As emphasized earlier, the central banker adjusts continuation losses to reflect an additional sensitivity to risk. It is then straightforward to rewrite the model in the forms (1) and (2), with x1t = ( yt−1, πt−1), x2t = ( yt, πt), and ut = Rt. We solve for optimal nominal GDP growth rule Rt ¼ a½ yt yt1 þ pt
ð20Þ
that is to say, the commitment rule subjects to the restriction F1 = [α, 0, −α, −α]′ and F2 = [0, 0, 0, 0]′. As shown in McCallum and Nelson (1999), nominal income growth rules describe US monetary policy practice since 1979 as well as, if not better than the influential Taylor type rules. In addition, not like the Taylor type rules, nominal income growth rules avoid the need to measure capacity or potential output. We are going to look at the effects of risk sensitive preference on the choice of policy rule. The parameters in the model are set as follows: β = 0.98, δ = 0.75, ϕ = 0.2, κ = 0.3, γ = 0.5, λy = 0.5, λR = 0.2 and σπ = σy = 1. Initially the economy is in the steady state. Risk sensitive preference leads to stronger reaction. The policy coefficient α = 2.65 for the standard preference (i.e. σ = 0), and 2.82 for risk sensitive preference with σ = −0.01. As −σ increases, the policy becomes more responsive.7 So here the risk sensitive central bank engages in a precautionary policy. Risk sensitive preferences imply that the central bank is not indifferent to the question of when uncertainty is resolved. It is willing to take a policy action today that tries to remove the possibility of a future bad outcome. By responding more aggressively to the nominal income growth, the current change of interest rate is larger, inducing higher current loss. On the other hand, future paths of the economy are under better control. Just as a precautionary consumer would sacrifice part of current consumption and save more to guard against future uncertainty, the risk sensitive central bank chooses to vary interest rate more in hope 7
The condition for Jacobson lemma will fail as −σ gets too large for given variances of shocks. See Footnote 5.
M. Zhao / Economics Letters 97 (2007) 260–266
265
that the future economy is in good hands. In this sense, the risk sensitive preference and robust control deliver similar messages. Hansen and Sargent (2004) demonstrate the equivalence between these two approaches for the backward looking economic system. Although there is a lack of formal proof of this equivalence or disequivalence in forward looking models, as hinted by Walsh (2003), risk sensitive preference approach makes more sense than model uncertainty aversion argument in practice, especially with rational expectations in the model. In the robust control argument, the central banker has standard loss preference and is uncertain about the economy structure. Robust policies are designed from a distorted model. It obviously eliminates the separation between the forecasters and the decision makers. Different worst case scenarios result in different forecasts about forward looking variables. Then the staff needs to incorporate the policy maker's uncertainty aversion into the forecasting exercise. While with risk sensitive preference, the policy decision and forecasting exercises will be based on one structural economy that the decision maker is confident about. The staff can provide fairly accurate and timely forecasts and the choice among the alternative paths for the decision makers. The risk sensitive preference determines which instrument path is eventually chosen. In the current setting, the relative variances of shocks that hit the economy also matter in the design of monetary policy. First, they are useful for private agents to form rational expectations and thus have impact on the path of the economy. More importantly, the relative variances are a good measure of additional risk that risk sensitive central bank cares about beyond that represented by expectation operators, which basically reflects first moment risk. For example, if σπ = 1 and σy = 1.5, α = 2.53 for the standard preference and α = 2.84 when σ = −0.01. The same increase of degree of risk sensitivity now prompts larger amplification of response coefficient, since different variabilities of shocks implicate more uncertainty to the central bank. References Amato, J.D., Laubach, T., 2003. Rule-of-thumb behavior and monetary policy. European Economic Review 47, 791–831. Campbell, J.Y., Mankiw, N.G., 1989. Consumption, income, and interest rates: reinterpreting the time series evidence. In: Blanchard, O.J., Fischer, S. (Eds.), NBER Macroeconomics Annual 1989. MIT Press, pp. 185–216. Clarida, R., Gali, J., Gertler, M., 2000. Monetary policy rules and macroeconomic stability: evidence and some theory. Quarterly Journal of Economics 115, 147–180. Christiano, L.J., Eichenbaum, M., Evans, C.L., 2005. Nominal rigidities and the dynamic effects of a shock to monetary policy. Journal of Political Economy 113 (1), 1–45 (FEB). Epstein, L.G., Zin, S.E., 1989. Substitution, risk aversion, and the temporal behavior of consumption and asset returns: a theoretical framework. Econometrica 57 (4), 937–969. Fuhrer, J.C., 2000. Habit formation in consumption and its implications for monetary-policy models. American Economic Review 90, 367–389. Gali, J., Gertler, M., 1999. Inflation dynamics: a structural econometric analysis. Journal of Monetary Economics 44, 195–222. Hansen, L.P., Sargent, T.J., 1995. Discounted linear exponential quadratic Gaussian control. IEEE Transactions on Automatic Control 40 (5), 968–971. Hansen, L.P., Sargent, T.J., 2003. Robust control of forward-looking models. Journal of Monetary Economics 50, 581–604. Hansen, L.P., Sargent, T.J., 2004. Misspecification in Recursive Macroeconomic Theory. Unpublished manuscript. Jacobson, David H., 1973. Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games. IEEE Transactions on Automatic Control 18 (2), 124–131. Klein, P., 2000. Using the generalized Schur form to solve a multivariate linear rational expectations model. Journal of Economic Dynamics & Control 24, 1405–1423.
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McCallum, B.T., Nelson, E., 1999. Nominal income targeting in an open-economy optimising model. Journal of Monetary Economics 43, 553–578. Soderlind, Paul, 1999. Solution and estimation of RE macromodels with optimal policy. European Economic Review 43, 813–823. Walsh, Carl E., 2003. Implications of a changing economic structure for the strategy of monetary policy. Monetary Policy and Uncertainty: Adapting to a Changing Economy, pp. 297–348.
Optimal simple rules in RE models with risk sensitive preferences ☆ Mingjun Zhao ⁎ Ohio State University, Arps Hall 410, 1945 North High Street, Columbus, OH 43210, United States Received 22 March 2006; received in revised form 16 March 2007; accepted 19 March 2007 Available online 5 July 2007
Abstract This paper provides a useful method to solve optimal simple rules under risk sensitive preference in macromodels with forward looking behavior. An application to a new Keynesian model with lagged dynamics is offered and risk sensitive preference is found to amplify the policy responses. © 2007 Elsevier B.V. All rights reserved. Keywords: Risk sensitive control; Commitment; Simple rules JEL classification: C61; E52; E58
1. Introduction In dynamic economic applications, it is very useful to incorporate forward looking rational expectations elements. This paper extends the version of Hansen and Sargent (1995) risk sensitive control in backward looking system and solves simple optimal rules. Solution of optimal policies in a linear-quadratic framework for a class of RE macromodels is extensively discussed in Soderlind (1999) and Hansen and Sargent (2003). The jump variables in this class of models do not respond to contemporaneous exogenous shocks. The economies we considered are much broader and the unconstrained optimal commitment rules are generally not easy to derive. However, it's straightforward to pursue some kind of simple rules.
☆
I would like to thank Bill Dupor genuinely for his intellectual tutoring and warm encouragement. ⁎ Tel.: +1 614 292 2069; fax: +1 614 292 3906. E-mail address: [email protected].
0165-1765/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2007.03.004
M. Zhao / Economics Letters 97 (2007) 260–266
261
The paper illustrates the extension with an application to the optimal monetary policy in a DSGE model. The central bank is assumed to have a risk sensitive preference and commit to a nominal income growth rule. Compared to the standard intertemporally time-consistent preference, risk sensitive preference makes the central bank undertake a precautionary policy stance. It will respond more aggressively resulting in higher current period loss in exchange for tighter control of future uncertainty. 2. Problem formulation Many dynamic linear rational expectations models evolve according to AEt xtþ1 ¼ Bxt þ Cut þ Det
ð1Þ
where xt is an n × 1 vector of endogenous variables, ut is a k × 1 vector of policy instruments, and et is an m × 1 Gaussian noise with mean zero and covariance matrix Ω. Some of the endogenous variables are predetermined (backward looking), and we assume that they are ordered first in the vector xt. For ′ , x2t ′ )′ where x1t is n1 × 1 and x2t is notational convenience, partition the vector xt accordingly into (x1t n2 × 1 with n1 + n2 = n and x10 given. The policy maker chooses policy instruments ut to minimize the fixed point L0 of Lt ¼ xtVQxt þ utVRut
h r i 2b logE exp Ltþ1 jt r 2
ð2Þ
where β ∈ (0, 1) is a discount factor and σ b 0 is the risk sensitive parameter. Lt+1 indicates the continuation value of the loss. When σ = 0, we obtain the standard specification because in that case Lt = x′Qx t t + u′Ru t t + βE[Lt+1|t]. So here σ b 0 reflects an additional aversion to the continuation loss risk beyond that represented by the expectation operator. 1 The current formulation of discounted, risk-adjusted losses is similar to Hansen and Sargent (1995), and I generalize the stable dynamic system they considered to incorporate rational expectations. This additional risk adjustment can be viewed as a special case of Epstein and Zin's (1989) recursive preference specification. More generalized policy rules are considered as well. In the standard control theory, we anchor the policy to the predetermined variables, something we know at the current period. Recently, policy rules targeting on expected variables draw a lot of attention as well, such as Clarida et al. (2000). 3. Optimal simple rules The policy maker is assumed to be able to commit to a simple decision rule of the form ut = −F1xt − F2Etxt+1. There may be restrictions on the elements in F1 and F2. Augmenting Eq. (1) with the policy rule gives A 0nk B C xt D 0nk et xtþ1 E ¼ þ ð3Þ F2 0kk t utþ1 0km 0kk 0k1 F1 Ikk ut 1
By the convexity of the exponential function, E exp r2 L jt zexp r2 E½Ljt .
262
M. Zhao / Economics Letters 97 (2007) 260–266
which in terms of yt = (x′,t u′)′ can be written as t A⁎ Et ytþ1 ¼ B⁎ yt þ D⁎ vt
ð4Þ
where vt = (e′,t 0k×1 ′ )′. This can be solved using the generalized Schur decomposition as in Klein (2000). Given the square matrices A⁎ and B⁎, the decomposition gives the unitary square complex matrices Q and Z 2 such that A⁎ ¼ Q H SZ H and B⁎ ¼ Q H TZ H
ð5Þ
where QH and ZH are the transpose of the complex conjugates of Q and Z, and S and T are upper triangular. Reorder the decomposition so that the stable generalized eigenvalues come first. 3 Count the number of the stable generalized eigenvalues ns. When ns = n1, there is a unique solution to Eq. (3). If there are more stable eigenvalues than there are predetermined variables (ns N n1), we can follow Blanchard and Kahn to select smallest eigenvalues needed. Following Klein (2000), the solution to Eq. (3) can be written as y2t ¼ M1 y1t þ M2⁎ vt
ð6Þ
y1;tþ1 ¼ N1 y1t þ N2⁎ vt
ð7Þ
4 where y1t = x1t, y2t = (x′2t , u′)′ and t 1 M1 ¼ Z21 Z11
ð8Þ
1 1 M2⁎ ¼ ðZ22 Z21 Z11 Z12 ÞT22 Q2 D⁎
ð9Þ
1 1 N1 ¼ Z11 S11 T11 Z11
ð10Þ
1 1 1 1 1 T11 Z11 Z12 T22 Q2 D⁎ þ Z11 S11 ½T12 T22 Q2 D⁎ þ Q1 D⁎ N2⁎ ¼ Z11 S11
ð11Þ
and Si , j , Ti , j , Z i , j and Qi (i, j = 1, 2) are conformable partitioned matrices of S, T, Z and Q. As in HS (1995), it is desirable to represent losses in terms of the true state variables at each period recursively. Suppose the problem has been solved for time t + 1 and future periods, and the This is also called ‘qz’ decomposition and this Q is different from the one in Eq. (2). Generalized eigenvalues are defined as tii / sii, i = 1, … , n where tii and sii are diagonal elements of T and S, and those with modulus less than one are called stable. 4 Equivalently, the solution can be expressed as y2t = M1y1t + M2et and y1,t+1 = N1y1t + N2et where M2 and N2 are the first m columns of M2⁎ and N2⁎. 2 3
M. Zhao / Economics Letters 97 (2007) 260–266
263
loss at time t + 1 can be written as a quadratic form in the true state variables at t + 1 as Lt+1 = x1,t+1 ′ 5 Vt+1x1,t+1 + dt+1. Then using the lemma in Jacobson (1973) h r i 2 V Vtþ1 x1;tþ1 þ dtþ1 jt qðLtþ1 jtÞu log E exp x1;tþ1 r 2 ð12Þ 1 f ¼ dtþ1 þ log detðX1 þ rN2VVtþ1 N2 Þ þ x1tV N1VV tþ1 N1 x1t r ~ where V t+1 = Vt+1 − σVt+1N2(Ω− 1 + σN2′Vt+1N2)− 1 N2′Vt+1.
Given
yt ¼
y1t y2t
¼
In1 0 Q f f y1t þ n1 m et ¼ M 1 y1t þ M 2 et and W ¼ M1 M2 0kn
0nk R
, we can further compute the cur-
rent loss in terms of true state variables x1t xt ¼ E½ ytVWyt jt þ bqðLtþ1 jtÞ ¼ x1tVVt x1t þ dt
ð13Þ
where f f f Vt ¼ M 1VW M 1 þ bN1VV tþ1 N1
ð14Þ
b f f dt ¼ traceðM 2VW M 2 XÞ þ bdtþ1 þ logdetðX1 þ rN2VVtþ1 N2 Þ r
ð15Þ
and
Note since the jump variables depend on the Gaussian noise, we should take the expectations conditional on information at period t for the first term in the current loss. Thus, we successfully map a translated quadratic loss measure for next period's losses into a translated quadratic loss measure today, both expressed in terms of true state variables at respective periods. Accordingly the initial period loss is V V ⁎ x10 þ d ⁎ x10
1 b fV f where V⁎ is the fixed point of Eq. (14) and d ⁎ ¼ traceðM 2 W M 2 XÞ þ logdetðX1 þ rN2VV ⁎ N2 Þ . 1b r
ð16Þ
By choosing the elements in the committed rule F1, F2 to minimize Eq. (16), we can find the optimal simple rule. Numerical non-linear optimization algorithms such as Nelder–Mead or simulated annealing can be employed. Sometimes, they are also termed “handcrafted” feedback rules.6 4. An application To illustrate the properties of this rational expectations-augmented risk sensitive preference and proposed solution approach, I apply this to a well discussed monetary model. This is a modified version of The formula works only when (Ω− 1 + σN2′Vt+1N2) N 0. The key assumption for this lemma to be held is the Gaussian distribution of error terms. 6 In a broad sense, “handcrafted” feedback rules mean all feedback rules that are not computed from formal optimizing procedures such as dynamic programming and Riccati equations. 5
264
M. Zhao / Economics Letters 97 (2007) 260–266
new Keynesian model with lagged dynamics. With the presence of habit formation, consumption Euler equation takes the form yt ¼ ð1 dÞyt1 þ dEt ytþ1 /ðRt Et ptþ1 Þ þ eyt
ð17Þ
where yt is the current output gap, πt is the inflation rate and Rt is the nominal interest rate. The coefficients δ and ϕ are closely related to the index of the importance of habit formation in the utility function. Habit formation has recently been studied by Fuhrer (2000) and McCallum and Nelson (1999) in models for the analysis of monetary policy. An alternative specification for lagged output dynamics is the introduction of rule-of-thumb consumers, such as Campbell and Mankiw (1989) and Amato and Laubach (2003). Analogous to output, inflation also features lagged dynamics. Dynamic indexation as in Christiano et al. (2005) and rule-of-thumb pricing setting as in Gali and Gertler (1999) are two alternatives to obtain lagged inflation in the formation of current inflation. pt ¼ jðyt þ yt1 Þ þ ð1 gÞpt1 þ gEt ptþ1 þ ept
ð18Þ
Both the output shock eyt and the inflation shock eπt are assumed iid normal with mean 0 and variance σy2 and σπ2 respectively. The central bank sets short interest rates to minimize the loss h r i 2b Lt ¼ Et fp2t þ ky y2t þ kR R2t g log E exp Ltþ1 jt ð19Þ r 2 As emphasized earlier, the central banker adjusts continuation losses to reflect an additional sensitivity to risk. It is then straightforward to rewrite the model in the forms (1) and (2), with x1t = ( yt−1, πt−1), x2t = ( yt, πt), and ut = Rt. We solve for optimal nominal GDP growth rule Rt ¼ a½ yt yt1 þ pt
ð20Þ
that is to say, the commitment rule subjects to the restriction F1 = [α, 0, −α, −α]′ and F2 = [0, 0, 0, 0]′. As shown in McCallum and Nelson (1999), nominal income growth rules describe US monetary policy practice since 1979 as well as, if not better than the influential Taylor type rules. In addition, not like the Taylor type rules, nominal income growth rules avoid the need to measure capacity or potential output. We are going to look at the effects of risk sensitive preference on the choice of policy rule. The parameters in the model are set as follows: β = 0.98, δ = 0.75, ϕ = 0.2, κ = 0.3, γ = 0.5, λy = 0.5, λR = 0.2 and σπ = σy = 1. Initially the economy is in the steady state. Risk sensitive preference leads to stronger reaction. The policy coefficient α = 2.65 for the standard preference (i.e. σ = 0), and 2.82 for risk sensitive preference with σ = −0.01. As −σ increases, the policy becomes more responsive.7 So here the risk sensitive central bank engages in a precautionary policy. Risk sensitive preferences imply that the central bank is not indifferent to the question of when uncertainty is resolved. It is willing to take a policy action today that tries to remove the possibility of a future bad outcome. By responding more aggressively to the nominal income growth, the current change of interest rate is larger, inducing higher current loss. On the other hand, future paths of the economy are under better control. Just as a precautionary consumer would sacrifice part of current consumption and save more to guard against future uncertainty, the risk sensitive central bank chooses to vary interest rate more in hope 7
The condition for Jacobson lemma will fail as −σ gets too large for given variances of shocks. See Footnote 5.
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that the future economy is in good hands. In this sense, the risk sensitive preference and robust control deliver similar messages. Hansen and Sargent (2004) demonstrate the equivalence between these two approaches for the backward looking economic system. Although there is a lack of formal proof of this equivalence or disequivalence in forward looking models, as hinted by Walsh (2003), risk sensitive preference approach makes more sense than model uncertainty aversion argument in practice, especially with rational expectations in the model. In the robust control argument, the central banker has standard loss preference and is uncertain about the economy structure. Robust policies are designed from a distorted model. It obviously eliminates the separation between the forecasters and the decision makers. Different worst case scenarios result in different forecasts about forward looking variables. Then the staff needs to incorporate the policy maker's uncertainty aversion into the forecasting exercise. While with risk sensitive preference, the policy decision and forecasting exercises will be based on one structural economy that the decision maker is confident about. The staff can provide fairly accurate and timely forecasts and the choice among the alternative paths for the decision makers. The risk sensitive preference determines which instrument path is eventually chosen. In the current setting, the relative variances of shocks that hit the economy also matter in the design of monetary policy. First, they are useful for private agents to form rational expectations and thus have impact on the path of the economy. More importantly, the relative variances are a good measure of additional risk that risk sensitive central bank cares about beyond that represented by expectation operators, which basically reflects first moment risk. For example, if σπ = 1 and σy = 1.5, α = 2.53 for the standard preference and α = 2.84 when σ = −0.01. The same increase of degree of risk sensitivity now prompts larger amplification of response coefficient, since different variabilities of shocks implicate more uncertainty to the central bank. References Amato, J.D., Laubach, T., 2003. Rule-of-thumb behavior and monetary policy. European Economic Review 47, 791–831. Campbell, J.Y., Mankiw, N.G., 1989. Consumption, income, and interest rates: reinterpreting the time series evidence. In: Blanchard, O.J., Fischer, S. (Eds.), NBER Macroeconomics Annual 1989. MIT Press, pp. 185–216. Clarida, R., Gali, J., Gertler, M., 2000. Monetary policy rules and macroeconomic stability: evidence and some theory. Quarterly Journal of Economics 115, 147–180. Christiano, L.J., Eichenbaum, M., Evans, C.L., 2005. Nominal rigidities and the dynamic effects of a shock to monetary policy. Journal of Political Economy 113 (1), 1–45 (FEB). Epstein, L.G., Zin, S.E., 1989. Substitution, risk aversion, and the temporal behavior of consumption and asset returns: a theoretical framework. Econometrica 57 (4), 937–969. Fuhrer, J.C., 2000. Habit formation in consumption and its implications for monetary-policy models. American Economic Review 90, 367–389. Gali, J., Gertler, M., 1999. Inflation dynamics: a structural econometric analysis. Journal of Monetary Economics 44, 195–222. Hansen, L.P., Sargent, T.J., 1995. Discounted linear exponential quadratic Gaussian control. IEEE Transactions on Automatic Control 40 (5), 968–971. Hansen, L.P., Sargent, T.J., 2003. Robust control of forward-looking models. Journal of Monetary Economics 50, 581–604. Hansen, L.P., Sargent, T.J., 2004. Misspecification in Recursive Macroeconomic Theory. Unpublished manuscript. Jacobson, David H., 1973. Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games. IEEE Transactions on Automatic Control 18 (2), 124–131. Klein, P., 2000. Using the generalized Schur form to solve a multivariate linear rational expectations model. Journal of Economic Dynamics & Control 24, 1405–1423.
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McCallum, B.T., Nelson, E., 1999. Nominal income targeting in an open-economy optimising model. Journal of Monetary Economics 43, 553–578. Soderlind, Paul, 1999. Solution and estimation of RE macromodels with optimal policy. European Economic Review 43, 813–823. Walsh, Carl E., 2003. Implications of a changing economic structure for the strategy of monetary policy. Monetary Policy and Uncertainty: Adapting to a Changing Economy, pp. 297–348.