Nonlinear Phillips curves, mixing feedback rules and the distribution of inflation and output

Journal of Economic Dynamics & Control 28 (2003) 467 – 492 www.elsevier.com/locate/econbase

Nonlinear Phillips curves, mixing feedback rules and the distribution of in-ation and output Luisa Corradoa;∗ , Sean Hollyb a Department

of Economics and CEIS University of Rome ‘Tor Vergata’, via Columbia 2, Rome 00133, Italy b Department of Applied Economics, University of Cambridge, Sidgwick Site, Cambridge, UK

Abstract Optimal nominal interest rate rules are usually set assuming that the underlying world is linear. In this paper, we consider the performance of ‘optimal’ rules when the underlying relationship between in-ation and the output gap may be nonlinear. In particular if the in-ation–output trade-o6 exhibits nonlinearities this will impart a bias to in-ation when a linear rule is used. By deriving some analytical results for the higher moments and in particular the skewness of the distribution of output and in-ation, we show that the sign of the skewness of the distribution of in-ation and output depends upon the nature of the nonlinearity. For the convex modi9ed hyperbolic function used by Chadha et al. (IMF Sta6 Papers 39(2) (1992) 395) and Schaling (Bank of England Working Paper Series, 1999) in-ation is positively and output negatively skewed. Whereas, if a concave–convex form is used the skewness of in-ation and output is reversed. To correct this bias we propose a piecewise linear rule, which can be thought of as an approximation to the nonlinear rule of Schaling (1999). In order to evaluate the relevance of these results, we turn to some illustrative empirical results for the US and the UK. We show that this reduces the bias, but at the expense of an increase in the volatility of the nominal interest rate. c 2003 Published by Elsevier B.V.  JEL classi*cation: C30; E31; E61 Keywords: Optimal control; Feedback rules; Nonlinear models

∗

Corresponding author. Tel.: +39-06-72595639. E-mail addresses: [email protected] (L. Corrado), [email protected] (S. Holly).

c 2003 Published by Elsevier B.V. 0165-1889/03/$ - see front matter
doi:10.1016/S0165-1889(02)00184-7

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1. Introduction The widespread use of delegated bodies such as central banks to operate monetary policy and the increasing use of short-term interest rates rather than monetary aggregates as instruments, has stimulated considerable research into how interest rates should be set in a stochastic environment. Two general approaches have been proposed. They can be either rules aG la Taylor where nominal interest rates respond to current deviations of in-ation and output from target; or as Svensson (1997) has suggested, a 2 year ahead in-ation forecast should be targeted, since there is an inherent lag in setting the interest rate and it showing up in in-ation. Both rules assume that the underlying world is linear, which implies that the rule used is symmetric about the natural or NAIRU level of output. 1 In this paper we consider the performance of rules, and in particular ‘optimal’ rules, when the underlying relationship between in-ation and the output gap may be nonlinear. Schaling (1999) has explicitly examined the implications for in-ation–forecast targeting of a convex Phillips curve. Convexity in the in-ation–output relationship implies that, under conditions of full employment, in-ation responds strongly to excess demand, while in a recession it can be insensitive to the change in the level of activity. Several other recent studies (Laxton et al., 1995; Turner, 1995; Clark et al., 1995; Debelle and Laxton, 1997) argue that the level of economic activity has a nonlinear e6ect on in-ation. 2 We add to this body of work by deriving some analytical results for the higher moments and in particular the skewness of the distribution of output and in-ation. We also show that the sign of the skewness of the distribution of in-ation and output depends upon the nature of the nonlinearity. For the convex modi9ed hyperbolic function used by Chadha et al. (1992) and Schaling (1999) in-ation is positively and output negatively skewed. Whereas, if a concave–convex form is used the skewness of in-ation and output is reversed. In order to evaluate the relevance of these results, we turn to some illustrative empirical results for the US and the UK. These are not meant to be de9nitive since a very simple formulation is being used. As an approximation to the nonlinear Phillips curve used to derive the analytical results, we estimate a two-threshold or piecewise linear model for the in-ation–output relationship. The results suggest that while a modi9ed hyperbolic function seems to capture the curvature of the US Phillips curve, the relationship for the UK is better captured by a concave–convex functional form.

1 At the same time this emphasis on the rules that underpin monetary policy has re-awakened interest in the applicability of control theory to economics, that blossomed during the 1970s (see Chow, 1975; Kendrick, 1981; Holly and Hughes Hallett, 1989). As is well known the seminal paper by Kydland and Prescott (1977) appeared to undermine the application of control theory to economics. However, it was the method of dynamic programming rather than control theory, per se, that was undermined because the time separability of the objective function was lost. For an early application of non-recursive control methods to rational expectations models see Holly and Zarrop (1983). 2 It should be emphasised that in Phillip’s original 1958 article he speci9cally considered a nonlinear relationship between wage in-ation and unemployment.

L. Corrado, S. Holly / Journal of Economic Dynamics & Control 28 (2003) 467 – 492

469

Using the linear estimates from each segment, we can then calculate the optimal feedback rule using the standard approach (Chow, 1975) appropriate for that segment of the Phillips curve. When we stochastically simulate this model for output, in-ation and a feedback rule for the interest rate, we 9nd empirically that the analytical results are borne out. As a benchmark, a linear Phillips curve with a linear feedback rule generates, as expected, normally distributed, certainty equivalent outcomes about the desired targets for in-ation and output. If we then consider the case in which a linear feedback rule computed as if the underlying model was linear, is applied in the nonlinear case, we 9nd that there is a bias in mean in-ation and output. Moreover there is signi9cant skewness. To overcome the mean bias and asymmetry we propose a piecewise linear, or mixing feedback rule in which the response of interest rates depends on the slope of the Phillips curve. In this case much of the bias and asymmetry, at least in the case of the US can be ameliorated. 2. A benchmark model of output and ination An increasingly common framework for organizing the discussion of theoretical and empirical issues concerning monetary policy uses a simple IS relationship for deviations, in the short run, of output from trend (the output gap) combined with a Phillips curve and a loss function for the policymaker that provides a metric for the conduct of policy. A very general form of this model that encompasses a wide variety of theoretical perspectives and a linear or nonlinear Phillips curve is given below: q


yt+1 = 1 yt + 2 Et+1 yt+2 −

(i+s+1 it−i − i+s+1 Et+1 t−i+1 ) + t+1 ;

(1)

i=−s

t+1 = 0 t + (1 − 0 )Et+1 O t+2 + 1 O t−1 + min Et {it }

q


i=−s ∞
−t

i+s+1 yt+i (1





=t

!1



−

i+s+1 ’yt+i )

n

;

     ( +2 | t − ∗ )2 (y )2 (Oi )2 + !3 ; + !2 2 2 2

t+1 = t + vt+1 ;

(2) (3) (4)

y is the output gap, i is the short-term nominal interest rate, is the in-ation rate and E the expectations operator;  is an exogenous autoregressive demand shock with vt+1 having mean zero and unit variance. If i+s+1 = i+s+1 the real interest rate a6ects output; if i+s+1 = 0 only the nominal interest rate enters the speci9cation. The relationship for in-ation captures a range of nonlinear functions. The degree of nonlinearity is captured by ’. For ’=0 and n=1, we have the standard linear relationship between the output gap and in-ation. The third relationship in the benchmark model is the quadratic loss function usually employed in the literature. 3 The last term allows weights to be 3

For a recent discussion of the quadratic case see Chadha and Schellekens (1999).

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attached to the change in the nominal interest rate. The consequence is that the interest rate is smoothed relative to the frequency of in-ation and the output gap. For a full discussion of interest rate smoothing see Clarida et al. (1999). Let us assume in our benchmark model that 0 = 1, 1 = 0 and s = q = 0. We consider two di6erent nonlinear forms of the Phillips curve. If n=−1 the relationship describing the in-ation process is that proposed by Chadha et al. (1992) and Laxton et al. (1995). This modi9ed hyperbola can be written as O t+1 =

1 yt

(1 −

1 ’yt )

:

(5)

Alternatively for n = 2, we have a modi9ed cubic function. The modi9ed hyperbola and the cubic functions are shown in Fig. 1. As Chadha et al. (1992) show, there are a number of limiting cases: lim

’→0

@

= @y

1;

(6)

which is the linear trade-o6 between in-ation and the output gap lim

y→1=

@

= ∞; 1 ’ @y

(7)

where 1= 1 ’ is the upper bound to the short run increase in output. While, lim

y→−∞

@

= 0: @y

(8)

At this lower limit, even though output is collapsing, in-ation is completely unresponsive to output. It is clear from the 9rst limiting condition that ’ captures the degree of curvature of the nonlinear function. If n = 2 we obtain the cubic function: O t+1 =

1 yt (1

−

1 ’yt )

2

;

(9)

which is asymmetric around the origin. The properties of the function can be derived by analysing the behaviour in some limiting cases: lim

y→1=3

1 ’;1=

1’

@

= 0; @y

(10)

where 1=3 1 ’ and 1= 1 ’ identify the value of the output gap where the 9rst derivative of the in-ation–output relationship is equal to zero. Also the in-ection point, where the change of curvature occurs is given by lim

y→2=3

@2

= 0: 2 1 ’ @y

(11)

As for the modi9ed hyperbola we can de9ne the limiting behaviour when the degree of nonlinearity, ’, tends to zero: lim

’→0

@

= @y

1;

(12)

L. Corrado, S. Holly / Journal of Economic Dynamics & Control 28 (2003) 467 – 492

Fig. 1. The modi9ed hyperbola and the cubic function.

471

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L. Corrado, S. Holly / Journal of Economic Dynamics & Control 28 (2003) 467 – 492

which is again the linear trade-o6 between in-ation and output and @

= −∞; lim y→−∞ @y @

= ∞; lim y→∞ @y

(13) (14)

so in-ation has no upper or lower bounds. This modi9ed cubic di6ers from a standard cubic by being asymmetric with respect to the origin. It is also quadratic over the range (0; 1= 1 ’), and the in-ation–output gap slope is negative over the (narrower) range (1=3 1 ’; 1= 1 ’). As well as the nonlinear relationship for in-ation and output, a variety of di6erent versions of this framework are nested in the benchmark model. A dynamic general equilibrium approach to macroeconomics has been used, for example, by Goodfriend and King (1997) and McCallum (1999) in order to derive a forward looking version where ’ = 0; 1 = 0; 2 = 1; 0 = 1 = 0, s = −1, q = 1. The current output gap depends on the real interest rate, now and in the future, through the expected output gap term. In-ation also depends on the current output gap and its future evolution. Alternatively, the backward looking approach of Taylor (1994), Svensson (1997) and Bean (1998) emphasises that, empirically, there are pure delays in the impact of interest rates on output, and output on in-ation. In this case ’ = 0, 2 = 0; 0 = 1; 1 = 0 and s = q = 0. This means that there is a pure delay of two periods between a change in interest rates and it showing up in the rate of in-ation. In both classes of model the solution to the policy problem, with an assumption about the formation of expectations, can be expressed as a feedback rule. This can be derived explicitly or written as a Taylor type rule. 4 2.1. A feedback rule for the nonlinear Phillips curve We now derive a feedback rule for the nominal interest rate from a simpli9ed model nested in (1)–(3), where we assume s = q = 0, 0 = 1, 1 = 0 and 1 = 1 = 1. 5 The model simpli9es to yt+1 = 1 yt − (it − t ) + t+1 ; O t+1 = f(•) = t+1 = t + vt+1 ;

1 yt (1

−

1 ’yt )

(15) n

for n = −1;

(16) (17)

where 1 ¿ 0; 0 6 ’ ¡ 1; 0 ¡ 1 ¡ 1 and 0 ¡  ¡ 1. As in Svensson (1997) we assume that the interest rate in period t a6ects in-ation only starting in period t + 2. Therefore the dynamic programming problem can be solved by assigning the interest rate in period t to the in-ation rate two periods ahead. 4

For a recent survey of this class of model see Clarida et al. (1999). Note that in this simpli9ed model current and not expected in-ation a6ects the real interest rate in the output equation. Moreover, we ignore interest rate smoothing so !3 = 0. 5

L. Corrado, S. Holly / Journal of Economic Dynamics & Control 28 (2003) 467 – 492

473

Following Schaling (1999) let V ( t ; yt ) represent the value function. The minimization problem becomes:

 !1 ( t+2 − ∗ )2 + !2 yt2 V ( t ; yt ) = min Et 2 {it }  + Et [V ( t + f(•); yt+1 )] : (18) Expected in-ation two periods ahead is 1 !2 yt (1 − 1 ’yt )2 : Et t+2 = ∗ − 2  1 !1

(19)

Hence, the in-ation forecast 2 years ahead equals the in-ation target plus a correction factor which depends upon the degree of nonlinearity of the in-ation–output relationship and on the relative weights placed upon the deviation of in-ation and output from their target in the loss function. Considering the limiting value for expected in-ation if the degree of nonlinearity tends to zero we get: 1 !2 yt ; (20) lim Et t+2 = ∗ − 2 ’→0  1 !1 which is the result in Svensson (1997). That is, the two year in-ation forecast should equal the in-ation target only if output equals the natural rate. Otherwise it should deviate from the in-ation target in proportion to how much output deviates from the natural level. Let us now assume that there is no output stabilisation, which implies !2 = 0. Also let us assume that the weight on in-ation, !1 equals one. In this case expected in-ation two years ahead is Et t+2 = ∗

(21)

and taking into account that:

6

Et t+2 = t + f(•) + Et f(•)t+1 the feedback rule for the nominal interest rate, as in Schaling (1999), is given by: 1 1 ( t − ∗ ) + f(•) + 1 yt ; (22) it = t + t + 1

1

where  = (1=1 − ’[( t − ∗ ) + f(•)]). The relationships in (22) are Taylor type rules with state-contingent weights. Since lim’→0  = 1 and lim’→0 f(•) = 1 yt the feedback rule (22) reduces to 1 ( t − ∗ ) + (1 + 1 )yt ; (23) ii = t + t + 1

which is the simple Taylor rule for the linear model where the nominal interest rate is equal to in-ation plus the real rate plus two disequilibrium terms. One of the disequilibrium terms is the discrepancy of actual in-ation from the target; the other disequilibrium term is the output gap, y. On the other hand, if there are nonlinearities 6

Note that Et f(•)t+1 =

1 (1 yt

− (it − t ) + t )=(1 −

1 ’(1 yt

− (it − t ) + t )) = f(Et yt+1 ).

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L. Corrado, S. Holly / Journal of Economic Dynamics & Control 28 (2003) 467 – 492

in the in-ation–output relationship, the Taylor rule becomes state-contingent and the weights on the two disequilibrium terms increase with the degree of nonlinearity in the in-ation–output relationship. To evaluate the e6ect that a linear feedback rule has when the world is nonlinear we apply, in line with the simulation reported in the empirical section, the linear rule (23) in the output relationship and we then use it to evaluate the distribution of in-ation. We analyse 4 di6erent cases that depend upon whether the Phillips curve is a modi9ed hyperbola (n = −1) or our modi9ed cubic function (n = 2) and on whether the nominal interest rate appears in the output relationship or not. 2.2. The e;ect of nonlinearity on the distribution of in
E[{f(•) − E[f(•)]}3 ] : [E{(f(•) − E[f(•)])2 }]1:5

(24)

To evaluate the in-ation bias when the optimal linear rule is applied in a nonlinear setting, we 9rst replace the output level at time t, as de9ned in (15), in the in-ation–output relationship (16). When 1 = 1 =1 the real interest rate enters the output speci9cation, whereas if 1 =1 and 1 = 0 only the nominal interest rate matters. In both cases we replace the nominal interest rate, it , with the rule for the nominal interest rate reported in (23) in order to evaluate the e6ect that a linear feedback rule has on the in-ation distribution when the in-ation–output relationship is nonlinear (see Appendix A). The deviation of in-ation from target is now a nonlinear function of output and in-ation at time t−1 and of the current output shock: f(•) = h(x); where the vector x=( t−1 ; yt−1 ; vt ) has mean x∗ and standard deviation we can use a second order Taylor approximation around x∗ :

(25) x.

7

Therefore,

h(x) ∼ = h(x∗ ) + (x − x∗ )h (x∗ ) + 0:5(x − x∗ )2 h (x∗ );

(26)

which implies E[h(x)] ∼ = h(x∗ ) + 0:5(x − x∗ )2 h (x∗ ):

(27)

Hence it is possible to demonstrate (Aizenman and Hausmann, 1994) that: 3 =

7

1:5[E(x − x∗ )4 − ( x )4 ]h (x∗ ) | h (x0 ) | ( x )2

We consider a variance–covariance matrix where the covariances are set to zero.

(28)

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475

from which we conclude that sign(3 ) = sign h (x∗ ):

(29)

As Appendix A shows, evaluating the sign of the leading minors of the Hessian of the second-order partial derivatives related to (25) at x∗ , one can easily see that for n = −1 the Hessian is positive semide9nite, so f is convex, whereas for n = 2 the Hessian is negative semide9nite, so f is concave. The sign of the Hessian in both cases is determined by its 9rst leading minor: 1 = 1 = 1

1 = 0

n = −1

2’ ¿ 0

2’(1 +

n=2

−4’ ¡ 0

−4’(1 +

2 1) ¿ 0 2 1)

(30)

¡0

The degree of skewness clearly depends on the degree of nonlinearity, ’. Moreover, it is greater when nominal rather than real interest rates matter for the output gap, and is increasing in the slope of the Phillips curve. But more critically, the sign of the asymmetry depends on the form of the nonlinear function. If we have the modi9ed hyperbola, skewness is positive because the function is convex everywhere. 8 Whereas, with the modi9ed cubic which is not centred at the origin skewness is negative. The function is concave in the range (−∞; 2=3 1 ’), and convex over the smaller segment (2=3 1 ’; ∞); the total e6ect is to make skewness negative. 3. Some illustrative empirics for the UK and US In this section, we turn to an illustrative exploration of nonlinearities and asymmetries in the Phillips curve using data for the US and the UK and consider what role feedback rules might play in o6setting nonlinearities. This is not intended to be a de9nitive empirical contribution but instead is intended to explore what the consequences might be if in fact nonlinearities were an important aspect of the relationship between in-ation and the output gap. We use quarterly data from 1966 to 1997. In Table 1 we report estimates of a number of di6erent functional forms for the Phillips curve. In column one we show a linear version of the Phillips curve. This is a very parsimonious formulation. In line with other studies, current deviations of output from trend do not have a signi9cant e6ect on in-ation, but the lagged output gap is signi9cant. 9 We seek to approximate the hyperbolic and cubic functions used in the previous section using a piecewise linear functional form 10 (see Corrado et al. (2002) for more 8 It is obvious that for the standard cubic which is a concave–convex function with the point of in-ection at the origin that skewness will be zero. 9 We estimate the output gap using a Hodrick–Prescott 9lter. 10 Of course we could estimate the modi9ed hyperbolic and modi9ed cubic functions directly. However, because we want to explore the possibility of using a mixing feedback rule later in the paper we prefer the method of this section.

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L. Corrado, S. Holly / Journal of Economic Dynamics & Control 28 (2003) 467 – 492

Table 1 Phillips curve estimates Linear

One threshold

Two thresholds

US

UK

US

UK

1

0.472 (6.28)

0.345 (4.91)

0.470 (6.24)

0.346 (4.90)

1

0.077 (3.79)

0.225 (4.24)

0.065 (2.25) 0.088 (3.31)

0.212 (2.71) 0.23 (3.35)

1+ 1− 1 2 3 !1 !2

US 0.463 (6.45)

0.332 (4.75)

0.20 (3.00) 0.02 (0.78) 0.46 (3.97)

0.58 (1.25) 0.15 (2.17) 0.75 (2.31)

−0:02 0.02

R2 SEE LM (2) ARCH (1)

0.42 0.0034 0.182 0.474

0.26 0.0106 0.453 0.213

0.42 0.0035 0.213 0.552

0.26 0.0106 0.512 0.223

UK

0.48 0.0031 0.182 0.474

−0:02 0.03 0.26 0.0106 0.231 0.260

discussion). In column two we report a kinked functional form, with the discontinuity occurring at the zero output gap. This single threshold model can be written as − + O t = 1 O t−1 + 1+ yt−1 + 1− yt−1 ;

(31)

− + yt−1 and yt−1 refer to observations of the output gap lying above and below zero, respectively. Finally in column three we report estimates of a two-threshold, or piecewise linear formulation. The general form of this model can be written as

O t = 1 O t−1 +  1 yt−1 I1 +  2 yt−1 I2 +  3 yt−1 I3 :

(32)

The terms Ii with i = 1; 2; 3 are de9ned as follows: I1 = {1 for yt−1 ¡ !1 ; 0 otherwise}; I2 = {1 for !1 ¡ yt−1 ¡ !2 ; 0 otherwise}; I3 = {1 for yt−1 ¿ !2 ; 0 otherwise};

(33)

L. Corrado, S. Holly / Journal of Economic Dynamics & Control 28 (2003) 467 – 492

477

where !1 and !2 are the threshold values. Continuity of this function requires the imposition of the following constraints:  1 !1 =  2 !1 ;  2 !2 =  3 !2 :

(34)

Expanding (32) and substituting in the continuity constraints yields:

t = t−1 + 1 O t−1 + !1 ( 2 −  1 )I1 + !2 ( 2 −  3 )I3 +  1 yt−1 I1 +  2 yt−1 I2 +  3 yt−1 I3 + ut :

(35)

Eq. (35) embodies a piecewise linear function with a kink at the point where the output gap begins to exert upward (above !2 ) and downward (below !1 ) pressure on in-ation. It can be thought of as an approximation of any hyperbolic functional form, which is convex whenever 3 ¿ 2 ¿ 1 , concave whenever 3 6 2 6 1 , and concave–convex if 3 6 2 ¿ 1 . Its advantage is that it is possible to test for asymmetry in a very direct way. When the economy is overheated, above !2 , the Phillips curve is steeper, with a given increase in output having a larger e6ect on in-ation. Below !1 , when the economy is depressed, the e6ect of output on in-ation weakens. This can also be thought of as a piecewise linear approximation to the functional forms considered earlier. A special case of these piecewise functional forms is the kinked function of Laxton et al. (1993), where the kink is at the point where the output gap is zero. This is similar to the non-symmetric error correction model of Granger and Lee (1989). 11 The slopes in the three regimes are derived by a two stage procedure: we 9rst estimate (35) by least squares for given !1 and !2 and conduct a grid search over the ! parameters for those values which minimize the residual sum of squares. Given !1 and !2 , the slopes in the three regimes, 1 , 2 and 3 are estimated using a structural model which comprises in-ation and output, as de9ned in the following section. If the estimates of the general model indicate either 1 = 2 or 2 = 3 then we adopt a speci9cation with two piecewise linear segments and if both restrictions hold then we adopt a simple linear function. A similar approach has been applied by Escribano and Pfann (1998). The third column in Table 1 contains the two threshold speci9cation of the Phillips curve. Using quarterly data for the US, we 9nd signi9cant nonlinearities similar to Filardo (1998). By using a grid search routine to de9ne the threshold levels and then estimating the model for in-ation and output by nonlinear least squares, he 9nds that the slope in the weak regime is 0.2, in the balanced regime −0:02 and in the overheated regime 0.49. Our results con9rm that in the weak regime, the slope is 0.2. In the balanced regime the slope is essentially -at whereas in the overheated regime the slope is 0.46, more than twice as steep as in the weak-economy regime. For the UK we 9nd that a concave– convex relationship 9ts more or less as well as the linear model, though a test of this against the null of a linear model gave a *2 =2.99 (p = 0:223). Since a J -test against 11

See Escribano and Pfann (1998) for an extended discussion of nonsymmetric error correction models.

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L. Corrado, S. Holly / Journal of Economic Dynamics & Control 28 (2003) 467 – 492

US

UK

πt - πt-1

πt - πt-1

D

D L’

C

B L

φ 1 = −0.02

L’

φ 2 = 0.02

C

φ1 = −0.02

B

Output Gap

A

φ 2 = 0.03 Output Gap

L A

Fig. 2. The linear and piecewise phillips curves for the US and the UK.

the alternatives is inconclusive, we can reject neither speci9cation. 12 The slope of the Phillips curve is substantially less in the weak than in the over-heated regime and with an almost -at slope in the balanced economy, as in the case of the US, suggesting that in-ation does not respond to the output gap in this regime. In the overheated regime the in-ation–output relationship is 30% steeper than in the weak regime, suggesting that in-ation is much more sensitive in this region. The piecewise estimates of the slopes, labelled ABCD, are shown in Fig. 2 against the linear versions of the Phillips curve, LL . The US estimates seem to approximate more the modi9ed hyperbola while the UK is closer to the modi9ed cubic described in Section 2. 3.1. Modelling output In this section we augment the relationship for in-ation with an empirical equation for the output gap. For the US we use a backward-looking version of the IS curve, which contains the lagged real interest rate on the right-hand side. Notwithstanding the arguments of Fair et al. (2002) referred in the next paragraph, other research (Roberts, 1995; McCallum and Nelson, 1998), 9nd that this speci9cation for the IS curve seems to 9t the data better. As in Estrella and Fuhrer (2000) we use the federal funds rate for the nominal interest rate. In-ation is de9ned in terms of the chain-weighted gross domestic product (GDP) de-ator, rather than the consumer price index (CPI).

12 There is also some Monte Carlo evidence to suggest that the power of tests of a null of linearity against the alternative of nonlinearity is very low. See Cook et al. (1999).

L. Corrado, S. Holly / Journal of Economic Dynamics & Control 28 (2003) 467 – 492

479

The IS equation is the following: yt = 0:0018 − 0:075(it−1 − t−1 ) + 0:8975yt−1 (−2:45)

(19:9)

2

R =0:76; SEE =0:008; LM (2)=6:71; ARCH (1)=1:13, sample period: 1966:4 –1998:2. Because this is a model of the output gap we estimated the equation so that the long run output gap is zero by imposing the restriction that the intercept o6sets approximately 2.5% the average real interest rate over the sample period. Note, that while the IS model for the US is in terms of the real interest rate, it proved impossible with UK data to 9nd a signi9cant e6ect except when the nominal interest rate was used. This is what we report and use in the simulations later in the chapter. Following Bean (1998) we allow the interest rate to have a di6erent e6ect on output depending upon whether a 9xed exchange rate regime is in operation or not yt = −0:0004D1 +0:011D2 −0:013D1 (it−3 +it−4 )−0:053D2 (it−3 +it−4 )+0:79yt−1 ; (−0:07)

(−3:67)

(15:55)

R2 = 0:72; SEE = 0:009; LM (2) = 0:251; ARCH (1) = 0:276, sample period: 1969: 1–1997:4. D1 and D2 are dummy variables; D1 takes the value 1 from prior to 1972 and 0 thereafter while D2 takes the value 0 prior to 1972 and 1 thereafter. 3.2. Real versus nominal interest rates As Fair (2002) has recently pointed out, the appearance of the real interest rate in the relationship for aggregate demand has an important policy implication. If there is a positive shock to in-ation and the nominal interest rate is unchanged, the real interest falls and stimulates demand which in turn raises in-ation and lowers the real interest rate further. Without changes in nominal interest rates this is explosive. To ensure that the real interest rate does not fall in response to a shock to in-ation, the rule determining the nominal interest rate must imply that interest rates rise by more than the rise in in-ation. There are a number of attractive theoretical reasons why the real interest rate should appear in expenditure equations, since non-neutralities will generate implausible (long run) properties in macroeconomic models. Nevertheless, Fair points to a large body of empirical evidence that positive in-ation shocks (holding nominal interest rates constant) has a negative impact on expenditure though real wealth e6ects (a Pigou e6ect) as well as real wage e6ects (a Keynes e6ect). This implies, empirically that the e6ect of a positive in-ation shock on output is not the same, in the short run, as a negative nominal interest shock of the same magnitude. In a recent study on the e6ects of monetary policy on the countries of the Euro area, Angeloni et al. (2002) survey the results from structural econometric models and VARs. In structural VARs nominal interest rates are used explicitly and it is rare for analysis of the e6ects of monetary shocks to concentrate on the e6ects of real interest rates on aggregate demand. Overall there does appear to be a considerable body of empirical work to suggest that nominal interest rates (rather than real) play an

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L. Corrado, S. Holly / Journal of Economic Dynamics & Control 28 (2003) 467 – 492

important role in the (short run) transmission of monetary policy. It does, therefore, seem sensible to explore the consequences of a nominal interest e6ect compared to a real interest rate e6ect (where the e6ect of in-ation is constrained to have a coeTcient of equal and opposite sign to the nominal interest rate). Essentially, for a null of a real interest rate e6ect we are examining the consequences of a type 2 error. As we have emphasised earlier we do not o6er the empirical results of this section as evidence for either nonlinearities in the Phillips curve or for a nominal rather than a real interest rate e6ect in the equation for output. However, we o6er them as a vehicle to explore numerically some of the analytical results that we have obtained in Section 2.

4. The optimal control rule In this section we brie-y reprise the standard LQG (Linear Quadratic Gaussian) model of optimal control (see Chow, 1975). First we de9ne a loss function for the monetary authorities in terms of a state variable z: n

Lt =

1
(zt; j − zt;dj ) Qi (zt; j − zt;dj ) 2

(36)

t=0

for j = US, UK where: 

yt



   t       t−1    zt; US =  ;  it       it−1   

  0   0     0   d zt; US =   ; 0     0   0

Oit 

zt; UK

yt



   t       t−1       it    = ;  it−1       it−2       it−3    Oit

zt;dUK

  0   0     0      = 0;   0     0   0

       QUS =      

!1

0

0

0

0

!2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0



 0    0   ; 0    0   !3

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L. Corrado, S. Holly / Journal of Economic Dynamics & Control 28 (2003) 467 – 492



QUK

         =        

!1

0

0

0

0

0

0

0

!2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

481



 0    0    0   : 0    0    0   !3

(38)

The vector of state variables z is chosen to make the model easy to write in state-space form. The state-space form of a model relates the current state to its one period lag, the current value of the control variable and a set of (possibly time varying) intercept terms as shown in (39): zt; j = Aj zt−1; j + Bj it; j + et; j ; where



1

  1    0  AUS =   0    0  0 

AUK

1   1    0    0  =  0    0    0  0

(39)

0



−1

0

1

0

1 + 1

− 1

0

0

1

0

0

0

0

0

0

0

0

0

1

0

0

0

−1

0

 0   0  ; 0   0  0

0

0

0

0

3

3

1 + 1

− 1

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

1

0

0

0

0

0

0

1

0

0

0

−1

0

0

0

  0   0     0   BUS =   ; 1     0   1    0 0    0 0       0 0        1 0    ; BUK =   : 0 0        0 0      0 0    1 0

(40)

(41)

Note that the state vector is de9ned in order to include the control variable i, and its 9rst di6erence Oi, as part of the state. This enables us to place penalties on either deviations of the interest rate from some target value or on changes in interest rates.

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L. Corrado, S. Holly / Journal of Economic Dynamics & Control 28 (2003) 467 – 492

The general solution for this form of the control problem can be written as: it; j = Kt; j zt−1; j + kt; j ;

(42)

where Kt; j (t=1; : : : ; T ) are a sequence of feedback control matrices and kt; j (t=1; : : : ; T ) represents what is known as the tracking gain in the control literature. These are solved recursively by 9rst solving the period T problem to obtain a solution for iT; j conditional on iT −1; j . This is used to write a value function for period T which depends on iT −1; j which in turn forms part of the objective function for the period T − 1 problem. Using d this procedure, along with the terminal conditions HT; j =Qj and kT; j = hT; j = Qj zT; j we can solve for the sequence of feedback control matrices and tracking gains as KT; j = −(Bj HT; j Bj )−1 (Bj HT; j Aj ); kT; j = −(Bj HT; j Bj )−1 Bj (HT; j eT; j − hT; j ); HT −1; j = Qj + (Aj + Bj KT; j ) HT; j (Aj + Bj KT; j ); hT −1; j = kT −1; j + (Aj + Bj KT; j ) (hT; j − HT; j eT; j ):

(43)

These are the well-known discrete time Ricatti equations. Since we have assumed that the Bj ; Aj and Qj matrices are constant, it is also possible to show that the feedback control matrices should converge to a constant matrix for some t ¡ t ∗ . The tracking gain will converge only if both the targets in the loss function and the intercepts in the state-space representation of the model are constant. 5. Some model simulations In this section we turn to a stochastic analysis of the illustrative models for the US and the UK in order to draw out numerically some of the analytical results of Section 2. But as well as this, our intention is to evaluate how well an optimal rule derived on the assumption that the underlying Phillips curve is linear, behaves when in fact the Phillips curve is nonlinear and how this a6ects the distribution of outcomes for in-ation, output and interest rates. We want, also, to determine empirically, how important the distinction between real and nominal interest rates is for the nature of the results. The optimal rule is the equilibrium solution for the optimal control problem. This is obtained by iterating the Ricatti equations in the linear model until the feedback control rule converges. The solution takes the form 13 itUS = k0 + K1 yt−1 + K2 t−1 + K3 t−2 + K4 it−1 ;

(44)

itUK = k0 + K1 yt−1 + K2 t−1 + K3 t−2 + K4 it−1 + K5 it−2 + K6 it−3 + K7 it−4 : (45) In order to make the rules comparable we impose equal weights of 1 on output, in-ation and nominal interest rate deviations from a target of zero, both for US and UK. 13

The extra terms in the UK rule re-ect the extra lagged interest rate terms in the IS curve.

L. Corrado, S. Holly / Journal of Economic Dynamics & Control 28 (2003) 467 – 492

483

Table 2 Feedback rule coeTcients

k0 K1 K2 K3 K4 K5 K6 K7

Linear

Piecewise

itL

it1

it2

it3

yt−1 ¡ !1

!1 ¡ yt−1 ¡ !2

yt−1 ¿ !2

US

UK

US

UK

US

UK

US

UK

0.010 1.105 2.215 −0:978 0.564

0.074 1.221 1.162 −0:376 0.525 −0:095 −0:097 −0:049

0.014 2.036 2.301 −0:977 0.423

0.119 2.595 1.055 −0:341 0.337 −0:186 −0:198 −0:101

0.008 0.586 1.719 −0:769 0.667

0.062 0.895 1.195 −0:386 0.581 −0:072 −0:072 −0:036

0.019 3.641 2.404 −0:962 0.238

0.137 3.199 1.018 −0:329 0.270 −0:225 −0:241 −0:124

The 9rst column of Table 2 shows the weights in the optimal linear feedback rule for both the US and the UK. Given the weights used, the feedback on output is very strong in both countries and a shock to output brings a countervailing change in interest rates larger than the magnitude of the shock. In response to an in-ation shock in the US nominal interest rates are changed by more than the size of the shock in line with the standard result in Taylor type rules, so that the real interest rate has the right e6ect on output. Given the weights on in-ation lagged one and two periods, we can rewrite this as a proportional/integral rule with the weights on the lagged change in in-ation and lagged level of in-ation as 2:215O t−1 + 1:237 t−2 . In contrast to the US, the UK response to in-ation shocks is more muted because there is less need to o6set the e6ects of in-ation on the real interest rate. The implicit proportional/integral rule is 1:162O t−1 + 0:786 t−2 . In Table 3 we show the steady-state results of 2000 stochastic simulations 14 of the model. In conducting our stochastic simulations we allow the interest rate to respond to lagged stochastic disturbances to the in-ation and output equation over the simulation period, using the feedback control coeTcients determined through dynamic programming. Column 1 shows the results of simulating the linear model using the feedback rule calculated above. The error covariances we use are those provided by the estimated equations. This generates a steady-state solution for the linear model with the output gap and in-ation at zero. This provides the benchmark for the subsequent simulations. As it would be expected, given that the random shocks are drawn from a normal distribution, the distribution of output and in-ation outcomes are also normally distributed about the target. However, given the seeds that we have used for the stochastic simulations the standard error on the nominal interest rate is very large. If the target for in-ation was indeed 2.5% for the UK and around 2% for the US, this would imply negative 14

The stochastic simulations use the antithetics suggested by Bianchi et al. (1978).

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L. Corrado, S. Holly / Journal of Economic Dynamics & Control 28 (2003) 467 – 492

Table 3 Steady-state results from the stochastic simulations Number of solutions 2000—simulation period 1998:4 –2006:4 LM/LFR Means of series Deviation of in-ation from target Deviation of output from target Standard deviation In-ation Output Interest rate

NLM/LFR

US

UK

0 0

0 0

NLM/PLFR

US

UK

US

UK

1.9 −0:81

−2:56 0.56

0.92 −0:15

−0:73 0.32 9.99 3.68 14.94

3.88 3.28 11.43

7.73 3.50 9.78

6.44 3.95 18.51

11.46 4.09 15.02

10.25 2.62 16.94

Skewness In-ation Output Interest rate

0.00 0.00 0.00

0.00 0.00 0.00

0.88 −0:23 0.73

−0:017 0.094 0.035

−0:007 −0:231 0.044

0.062 0.030 0.131

Kurtosis In-ation Output Interest rate

3.17 2.86 3.19

3.08 2.82 3.24

5.75 3.25 5.47

4.96 3.01 5.37

2.93 2.50 3.08

3.70 2.92 2.77

Bowman–Shenton (*2 (2)) In-ation Output Interest rate

2.63 1.57 3.15

0.60 2.43 5.15

892.72 24.53 692.28

323.09 3.0177 470.14

0.321 38.23 1.27

42.24 0.776 9.843

LM: linear model. LFR: linear feedback rule. NLM: nonlinear model. PLFR: piecewise linear feedback rule.

interest rates. Let us assume that the target in-ation rate is suTciently high for this not to be a problem. 5.1. The e;ect of a linear rule when the Phillips curve is nonlinear The question we want to ask is how well does a rule calculated on the assumption that the Phillips curve is linear, perform in a world in which the Phillips curve actually takes on the piecewise linear form of (35). The second column of Table 3 shows that using the linear rule in a nonlinear world imparts a bias to the steady-state in-ation rate which is positive for the US and negative for the UK. Both output and in-ation are more widely dispersed compared with the linear model and the US distribution is skewed positively for in-ation and negatively for output. In the case of the UK the results are less clear-cut. There is signi9cant mean bias, skewed in the opposite direction to the results for the US though there is positive excess kurtosis in in-ation. Output is normally distributed. In fact empirically, the nonlinear curve for the UK appears to be much more asymmetric

L. Corrado, S. Holly / Journal of Economic Dynamics & Control 28 (2003) 467 – 492

485

(see the slope and threshold values estimates in Table 1) compared to the US. In the case of the UK for an output gap between 2% and 3% in-ation falls in the balanced regime, where the slope of the Phillips curve is 0.15, while if the output gap is between −2% and −3% (equal in absolute value but of a negative sign) in-ation falls in the weak regime where the slope of the Phillips curve is 0.58, almost four times as steep. So for negative output gap between −2% and −3% in-ation falls more than for a positive output gap in the same range. This explains why with a concave– convex relationship between in-ation and output which is asymmetric around the origin and shifted to the right, as for the UK, in-ation tends to be negatively skewed. Finally, as with the US there is signi9cant positive excess kurtosis for both in-ation and output. 15 We can also see this visually in Figs. 3 and 4. Here we have plotted kernel density estimates of the steady-state distributions. The dotted line represents the linear model with the linear feedback rule, itL , reported in the 9rst two columns of Table 3. The solid line is the distribution when the Phillips curve is nonlinear but the feedback rule linear. In Table 3, as well as reporting measures of dispersion, skewness and kurtosis we also report the Bowman and Shenton (1975) omnibus test for normality which is distributed as a *2 with two degrees of freedom. 5.2. A piecewise or mixing feedback rule In this section we propose the use of a piecewise or mixing feedback rule that uses the slope estimates of the two threshold model of Section 3. Given these slope coeTcients reported in column 3 of Table 1, for the di6erent segments of the convex function, we calculated three rules it1 ; it2 ; it3 . Table 2 shows the value of the feedback rule in the di6erent regimes for the US and the UK. Note that in the middle regime, where in-ation is almost unresponsive to output, the feedback on output deviations is moderate, whereas it becomes extremely responsive in the over-heated regime, where the slope of the Phillips curve is higher, in particular for the US where we have found the slope in the upper output regime to be more than twice as steep as it is in the lowest regime. The mixing feedback rule is de9ned as itm = it1 I1 + it2 I1 + it3 I3 ; where as before I1 = {1 for yt−1 ¡ !1 ; 0 otherwise}; I2 = {1 for !1 ¡ yt−1 ¡ !2 ; 0 otherwise}; I3 = {1 for yt−1 ¿ !2 ; 0 otherwise}:

15

We have not been able to provide any analytical results in Section 2 for kurtosis.

(46)

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L. Corrado, S. Holly / Journal of Economic Dynamics & Control 28 (2003) 467 – 492

Fig. 3. UK-kernel distribution: comparison between the linear model with a linear feedback rule (LM/LFR) and the nonlinear model with a linear feedback rule (NLM/LFR).

At each threshold there is a discrete switch to the linear rule appropriate to the di6erently slopped Phillips curve. In Table 3 column 2 and Figs. 5 and 6 we show that the mixing feedback rule in the US reduces, though it does not entirely eliminate, the mean bias in in-ation and output arising from the nonlinear Phillips curve. Skewness in in-ation is eliminated, though skewness in output is hardly a6ected. Overall the highly non-normal distribution for in-ation that arises when the linear feedback rule is used is entirely ameliorated by the mixing rule. However, the distribution of output is hardly a6ected. For the UK, the mixing rule reduces the in-ation bias, lowers the standard deviation in in-ation, output and interest rates, and eliminates the kurtosis, but at the expense of increasing the skewness of the interest rate.

L. Corrado, S. Holly / Journal of Economic Dynamics & Control 28 (2003) 467 – 492

487

Fig. 4. USA-kernel distribution: comparison between the linear model with a linear feedback rule (LM/LFR) and the nonlinear model with a linear feedback rule (NLM/LFR).

6. Conclusions We have explored a number of issues concerned with the design of monetary policy in a world in which the Phillips curve might be signi9cantly nonlinear. We have investigated not only the behaviour of the 9rst two moments of the distribution of outcomes but also skewness and kurtosis. Using small empirical models of the US and the UK, we have been able to demonstrate with stochastic simulation techniques that a mixing rule, which involves piecewise linear feedback rules corresponding to di6erently slopped parts of the Phillips curve, goes some way towards correcting the mean biases that are induced by the nonlinearity and restoring much of the normal distribution of outcomes in the US, at least for in-ation. We have deliberately carried out the analysis over a wide range of stochastic outcomes in order to excite those regions in which the nonlinearity is important. It may

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L. Corrado, S. Holly / Journal of Economic Dynamics & Control 28 (2003) 467 – 492

Fig. 5. UK-kernel distribution: comparison between the linear model with a linear feedback rule (LM/LFR) and the nonlinear model with piecewise linear feedback rule (NLM/PLFR).

be that this has little relevance for contemporary policymaking, when recent history has delivered relatively small shocks to in-ation and output, and the monetary regimes that have been in place in the US and the UK for the last decade, have been very successful in keeping in-ation low and stable and the volatility of output at historically low levels. Within the bands that current policy has delivered, the relationship between output and in-ation is likely to be more or less linear. So the results that we have for regions well outside recent experience may be of esoteric interest. Nevertheless, the results may have some relevance for the debate on the merits of gradualism versus cold turkey responses to shocks. A large shock that temporarily moved the economy outside the comfort zone, may require swift and decisive action in order to bring the economy back under control before the nonlinear processes have a chance to take e6ect. The US Fed response to the events of 2001 may be a case in point.

L. Corrado, S. Holly / Journal of Economic Dynamics & Control 28 (2003) 467 – 492

489

Fig. 6. USA-kernel distribution: comparison between the linear model with a linear feedback rule (LM/LFR) and the nonlinear model with piecewise linear feedback rule (NLM/PLFR).

Acknowledgements We are grateful to Ken Wallis, to all the participants of the Macromodelling Bureau Seminar held in Warwick (July 2000) and to two anonymous referees for their comments on the work. The usual disclaimer applies.

Appendix A. We now study the curvature of the distribution for in-ation when a linear feedback rule is applied to the nonlinear model. This gives f(•) =

1 yt (1

−

1 ’yt )

n

for n = −1; 2;

(A.1)

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L. Corrado, S. Holly / Journal of Economic Dynamics & Control 28 (2003) 467 – 492

yt = 1 yt−1 − (it−1 − t−1 ) + t−1 + vt ; ii−1 = t−1 + t−1 +

1 1

(A.2)

( t−1 − ∗ ) + (1 + 1 )yt−1 ;

(A.3)

where (A.1) is the nonlinear in-ation–output trade-o6, (A.2) is the output equation with 1 = 1 = 1 and (A.3) is the linear feedback rule. By replacing (A.3) in (A.2) and then (A.2) in (A.1) we are able to evaluate the performance of optimal linear rules when the underlying relationship between in-ation and the output gap is nonlinear. To assess whether the resulting in-ation distribution is positively or negatively skewed, we have to evaluate the curvature of the resulting function, f, near equilibrium (see condition (29) in the text). Since f is a function of the vector of variables x = ( t−1 ; yt−1 ; vt ) with continuous partial derivatives of 9rst and second order, we study the sign of the Hessian:    f

f y f v      (A.4) H (x) =  fy fyy fyv  ;    fv fvy fvv  where H (x) accounts for the symmetry of the cross-derivatives f y = fy ; f v = fv

and fyv = fvy . Note that in the de9nition of H (x) we have omitted the time subscript for convenience. The second-order partial derivatives are: n = −1

n=2

1 = 1 =1

1 =0

f

2’g(x)−3

fyy

2’

2 −3 1 g(x)

2’

2 −3 1 (g(x)+’ 1 t−1 )

fvv

2’

2 −3 1 g(x)

2’

2 −3 1 (g(x)+’ 1 t−1 )

f y

2’

1 g(x)

−3

2’

f v

−2’

1 g(x)

−3

fyv

−2’

2 −3 1 g(x)

2’(1+

1)

2

(g(x)+’

1 t−1 )

−3

1 = 1 =1

1 =0

−2’c(x)

−2’(1+ 1 )2 (c(x) +3’ 1 t−1 )

−2’

2 1 c(x)

−2’

2 1 (c(x)+3’ 1 t−1 ) 2 1 (c(x)+3’ 1 t−1 )

−2’

2 1 c(x)

−2’

1 (1+ 1 )(g(x)+’ 1 t−1 )

−3

−2’

1 c(x)

−2’ 1 (1+ 1 )(c(x) +3’ 1 t−1 )

−2’

1 (1+ 1 )(g(x)+’ 1 t−1 )

−3

2’

1 c(x)

2’ 1 (1+ 1 )(c(x) +3’ 1 t−1 )

−2’

2 −3 1 (g(x)+’ 1 t−1 )

2’

2 1 c(x)

2’

2 1 (c(x)+3’ 1 t−1 )

(A.5) where g(x) = (1 + ’(( t−1 − ∗ ) + c(x) = (2 + 3’(( t−1 − ∗ ) +

1 (yt−1

− vt )));

1 (yt−1

− vt ))):

(A.6)

L. Corrado, S. Holly / Journal of Economic Dynamics & Control 28 (2003) 467 – 492

491

To assess whether f is convex/concave, we have to evaluate the sign of the leading minors of H (x). By substituting the relevant partial derivatives reported in (A.5) in the Hessian and evaluating the sign of its leading minors one can easily see that for n = −1, the Hessian, H (x), is positive semide9nite, so f is convex, whereas for n = 2 the Hessian, H (x), is negative semide9nite, so f is concave. The sign of H (x) in both cases is determined by the 9rst leading minor: 1 = 1 = 1 n = −1 n=2

2’g(x)−3 −2’c(x)

1 = 0 2

(g(x) + ’ 1 t−1 )−3

2

(c(x) + 3’ 1 t−1 )

2’(1 +

1)

−2’(1 +

1)

(A.7)

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