A. The Economy

The economic behavior of the public is summarized by an expectations-augmented Phillips curve:²

where λ is positive coefficient, u_t is the rate of unemployment, $u_t^n$ is the natural rate of unemployment, π_t is the rate of inflation, ${\pi }_t^e$ is the public’s forecast of inflation at time t constructed at time t − 1, and η_t is a supply disturbance. The choice of unemployment as measure of real economic activity has no effect on the analytical derivations below and allows the comparison of empirical results in section 3 with theoretical predictions in earlier literature.

Under the assumption of rational expectations,

where E_t−1 is the expectation conditional on all information available at time t − 1. The public’s information set is denoted by I_t and is assumed to contain all model parameters, current and past realizations of the variables, and current and next-period inflation targets. The assumption that future inflation targets form part of I_t follows from the fact that inflation targets are preannounced.

The natural rate of unemployment is assumed to evolve over time according to:

where ζ_t denotes the unpredictable component of the natural rate. This specification includes as special cases the stationary (0 ≤ δ < 1) and unit root (δ = 1) models of the natural rate employed in earlier literature. For example, the model in Barro and Gordon (1983a) can be obtained by setting θ_i = 0 for i = 1,…, q − 1, the one in Ireland (1999) by setting δ = 1, ψ = 0, and θ_i = 0 for i = 2,…, q − 1, and the one in Cukierman (2000) by setting δ = 0, and θ_i = 0 for i = 1,…,q − 1. The intention here is to adopt a general time series specification for the natural rate and then examine to what extent results are robust to the use of different forecasting models for $u_t^n.$³ The constant term in (3) could be written as, ψ(1 − δ) in order to rule out the presence of a drift when unemployment is I(1). However, for the estimation of the model, it is desirable to allow a nonzero intercept. Standard procedures can then be used to test whether this term is significantly different from zero or not. In order to exclude the possibility of $u_t^n.$ being an I(2) process, all roots of the polynomial $1-{\displaystyle \sum _{i=1}^{q-1}{\theta }_i{z}^i}$ are assumed to lay outside the unit circle.

Modeling the natural rate of unemployment as time-varying is important for at least two reasons. First, it seems plausible that changes in technology, labor force demographics, unionization rates, and welfare benefits could affect the labor market and generate movements in the natural rate. For the United States, Weiner (1993), Tootell (1994), and Staiger, Stock, and Watson (1997) report evidence that the natural rate has changed during the postwar period. Shimer (1998) argues that in the absence of the baby boom, the rate of unemployment would have neither increased in the 1960’s and 1970’s, nor fallen afterwards.

Second, although the assumption that $u_t^n$ is constant can be innocuous for some theoretical results, it leads to the empirical prediction that realized unemployment is white noise. However, for most countries and sample periods the rate of unemployment is serially correlated and substantially persistent. As an example, Table 1 presents the first 10 autocorrelations of the quarterly rate of unemployment in three inflation-targeting countries, namely Canada, Sweden, and the United Kingdom. In all cases, the correlation between current and previous unemployment (i.e., the first autocorrelation) is above 0.90, and the correlation between current unemployment and its realization two years before (i.e., the eight autocorrelation) is above 0.55.

Table 1.

Autocorrelations of the Unemployment Rate

Notes: The statistics were calculated using quarterly, seasonally-adjusted data for the periods 1990:01 to 2000:06 (Canada), 1993:01 to 2000:06 (Sweden), and 1991:01 to 2000:06 (United Kingdom).

Table 1.

Autocorrelations of the Unemployment Rate

Autocorrelation	Canada	Sweden	United Kingdom
1	0.966	0.935	0.987
2	0.930	0.875	0.971
3	0.892	0.823	0.954
4	0.849	0.773	0.937
5	0.803	0.724	0.917
6	0.751	0.685	0.896
7	0.702	0.639	0.875
8	0.657	0.598	0.851
9	0.614	0.556	0.827
10	0.577	0.512	0.802

Notes: The statistics were calculated using quarterly, seasonally-adjusted data for the periods 1990:01 to 2000:06 (Canada), 1993:01 to 2000:06 (Sweden), and 1991:01 to 2000:06 (United Kingdom).

View Table

Table 1.

Autocorrelations of the Unemployment Rate

Autocorrelation	Canada	Sweden	United Kingdom
1	0.966	0.935	0.987
2	0.930	0.875	0.971
3	0.892	0.823	0.954
4	0.849	0.773	0.937
5	0.803	0.724	0.917
6	0.751	0.685	0.896
7	0.702	0.639	0.875
8	0.657	0.598	0.851
9	0.614	0.556	0.827
10	0.577	0.512	0.802

Notes: The statistics were calculated using quarterly, seasonally-adjusted data for the periods 1990:01 to 2000:06 (Canada), 1993:01 to 2000:06 (Sweden), and 1991:01 to 2000:06 (United Kingdom).

As in Barro and Gordon (1983a) and Ireland (1999), the process of $u_t^n$ is not affected by the monetary policy instrument (see below) or lagged unemployment rates. This assumption reflects the view that the natural rate is primarily determined by factors outside the scope of monetary policy (for example, exogenous demographic or technological variables). In this sense, this paper adopts Friedman’s interpretation of the natural unemployment rate as” the level that would be ground out by the Walrasian system of general equilibrium equations, provided there is imbedded in them the actual structural characteristics of the labor and commodity markets, including market imperfections, stochastic variability in demands and supplies, the cost of gathering information about job vacancies and labor availability, the costs of mobility, and so on” [Friedman (1968, p. 8)].

An alternative process for the natural rate is proposed by Lockwood and Philippopoulos (1994). Their specification also seeks to capture the persistence observed in the unemployment rate, and consists of a strictly stationary process for the natural rate where the first unemployment lag is one of the regressors. Unfortunately, when coupled with the more general preference specification employed here, the model has no closed-form solution, the value function needs to be approximated by a finite-order polynomial, and the uniqueness of the Nash equilibrium cannot be insured. Since (3) also captures the serial correlation in unemployment but in a more tractable manner, and permits both stationary and nonstationary specifications, I did not pursue their approach further.

The policy maker is assumed to affect the rate of inflation through a policy instrument (as in Walsh(1995,1998 ch. 8)). We can think of this instrument as a monetary aggregate or a short-term nominal interest rate. The instrument is imperfect in the sense that it cannot determine inflation completely and its effect takes place with a (one-period) lag:

where$f(·)$ is a monotonic, continuous, and differentiable function, i_t is the policy instrument, and ε_t is a control error that represents imperfections in the conduct of monetary policy. Since i_t is chosen at time t − 1, it follows that i_t ∈ I_t−1 This simple specification serves two purposes. First, it relaxes the usual assumption that the monetary authority chooses directly the rate of inflation after observing (before the public does) the random shocks. The policy maker here has no informational advantage over the public since neither of them observe at time t − 1 the realization of the disturbances at time t. Because there is no private information, the central banker’s information set coincides with the public’s and is also given by I_t. Second, (4) introduces an additional structural disturbance to the model and permits the derivation of time-series predictions on the joint probability distribution function of inflation and unemployment.⁴

Finally, to complete the description of the economy, define ξ_t to be the 3x1 vector that contains all the model’s structural disturbances at time t. It is assumed that ξ_t is serially uncorrelated and normally distributed with zero mean and constant conditional variance-covariance matrix:

where Ω is a 3 x 3 positive-definite matrix. Evidence supporting the assumption that ξ_t is conditionally homoskedastic is presented in section 3.4. The assumption of normality is crucial for obtaining an analytical solution, but section 3.5 examines the robustness of empirical results to shocks drawn from a different distribution. For the moment, no restrictions are imposed on the off-diagonal elements of Ω.

B. The Central Banker

The central banker is assumed to have preferences over inflation and unemployment. Preferences are described by the function:

where C:ℜ² —> ℜ, ${\tilde{\pi }}_t$ is the inflation target, ${\tilde{u}}_t$ is the unemployment target, and ϕ is a positive coefficient that measures the relative importance of unemployment stabilization. In the institutional setup considered here, the inflation target is assigned by the government to the monetary authority before time t. Since the target is preannounced, it follows that ${\tilde{\pi }}_t$∈ I_t−1 A general model for u_t is presented below.

In contrast to most of the previous literature, where both components of the policy maker’s loss function are quadratic, the inflation component in (5) is given by the linex function g(x) = [exp (αx) − αx −1]/α² [Varian (1974)].⁵ Zellner (1986), Granger and Pesaran (1996), and Christoffersen and Diebold (1997) study this function in the context of optimal forecasting. In order to develop some intuition, the linex function is plotted in Figure 1(a) for the special case where α > 0. For rates of inflation above the target, the exponential term eventually dominates and the loss associated with a positive deviation rises exponentially. For rates of inflation below the target, it is the linear term that becomes progressively more important as inflation decreases and, consequently, the loss rises linearly. This asymmetry can be easily seen by considering, for example, the loss associated with a ±1 inflation deviation from ${\tilde{\pi }}_t$. It is apparent that even though their magnitudes are the same, the −1 deviation delivers a smaller loss than the +1 deviation. Hence, positive inflation deviations are weighted more severely than negative ones in the central banker’s loss function. The converse holds when α < 0.

Preferences

Citation: IMF Working Papers 2001, 161; 10.5089/9781451857818.001.A001

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Preferences

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Preferences

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It is useful to compare the linex function with the very-familiar quadratic loss function in Figure 1(b). The quadratic function is symmetric around zero. This means that positive and negative deviations of the same size yield exactly the same loss. Hence, only the magnitude, and not the sign, of the deviation is important for the policy maker. Applying L’Hôpital’s rule, it is possible to show that the quadratic function can be obtained as a special case of the linex function when the asymmetry parameter (α) tends to zero.⁶ This result is important because it proves that the preference specification employed here nests the one assumed in earlier literature as a special case. It also suggests that the hypothesis that the central banker’s preferences are quadratic over inflation could be tested econometrically by evaluating whether α is significantly different from zero.

The functional form in (5) is attractive because it is analytically tractable, yields a closed-form solution, and generates clear empirical predictions. In principle, one could extend it to allow asymmetries regarding unemployment. This possibility is discussed in section 3.5, where it is shown that the basic predictions of the model are unchanged by this extension. In related research, Ruge-Murcia (2001) estimates a model with asymmetric unemployment preferences for the G7 countries. Reduced-form estimates do not support the hypothesis of asymmetric unemployment preferences for Canada and the United Kingdom.

The targeted rate of unemployment is assumed proportional to the expected natural rate:

Previous literature usually assumes that ${\tilde{u}}_t$ is strictly less than the natural rate (i.e., 0 < k < 1). This assumption is based on the notion that distortions in goods and labor markets render the natural rate of unemployment higher than socially optimal. Persson and Tabellini (2000) note that this assumption is crucial in generating an inflation bias in the linear-quadratic framework of Kydland and Prescott (1977) and Barro and Gordon (1983a). On the other hand, King (1996) and Blinder (1998) suggest on the basis of institutional evidence, that central bankers target the expected natural rate of unemployment (i.e., k = 1). Both views are accommodated here by allowing 0 < k ≤ 1. Analytical results obtained under both assumptions are compared below.

C. Nash Equilibrium

Consider the problem of a central banker who must choose the sequence of instruments that minimizes the present discounted value of her loss function:

where β ∈ (0,1) is the discount rate. The optimization is made subject to the expectations-augmented Phillips curve [eq. (1)], and takes as given the public’s inflation forecasts and the inflation targets. Recall that the natural rate of unemployment is assumed to be determined primarily by factors outside the scope of monetary policy. Hence, the policy instrument does not affect the path of $u_t^n$ and the central banker’s objective function can be break down into a sequence of one-period problems. This decomposition simplifies the solution of the model and, as it will be shown below, delivers a unique Nash equilibrium.⁷

The first-order necessary condition is

where $k{E}_{t-1}{u}_t^n$ substitutes u˜_t. Since the loss function is globally convex, the solution to (7) delivers a unique minimum. In order to find the conditional expectation in (7), two intermediate results are useful. First, as shown below, the assumption that shocks are normal implies that, conditional on the information set, inflation is normally distributed. Then, $\text{exp}\left[\alpha \left({\pi }_t-{\tilde{\pi }}_t\right)\right]$ is log normal with mean $\text{exp}\left[\alpha \left(E_{t-1}{\pi }_t-{\tilde{\pi }}_t+\left(\alpha /2\right){\sigma }_{\pi }^2\right)\right],$ where ${\sigma }_{\pi }^2$ denotes the conditional variance of the inflation rate. (See below for the derivation of ${\sigma }_{\pi }^2$ in terms of the elements of ξ_t.) Second, the conditional expectation of unemployment can be found by taking E_t−1 in both sides of (1) to obtain $E_{t-1}{u}_t=E_{t-1}{u}_t^n-\lambda \left(E_{t-1}{\pi }_t-{\pi }_t^e\right).$ With these results, the first-order condition can be rewritten as

In the quadratic model, the first-order condition of the central banker’s minimization problem is linear and can be solved explicitly to obtain her reaction function in terms of the public’s inflation forecast, ${\pi }_t^e$.⁸ To see this, take the limit of (8) as $\alpha \to 0$and rearrange to obtain

The central banker’s reaction is linear and monotonically increasing on the public’s inflation forecast (note that 0 < λ²ϕ/(1 + λ²ϕ) < 1). It is trivial to show that in this case, the Nash equilibrium always exists and is unique.

In contrast, under asymmetric preferences, the first-order condition only defines the reaction function implicitly:

Using the implicit function theorem, it is possible to show that

for all values of α. Hence, as in the quadratic model, the central banker’s reaction is a monotonically increasing function of the public’s inflation forecast. Also

that is less than zero for α > 0, equal to zero for α → 0, and larger than zero for α < 0. In other words, for α > 0 (α < 0) the central banker’s reaction is a concave (convex) function of ${\pi }_t^e$

In order to develop further the reader’s intuition and to illustrate future theoretical results, it is useful to plot the central banker’s reaction function for different values of the preference parameter α. This is done in Figure 2 under the assumption that the inflation target is $\tilde{\pi }=0$ and remaining parameters are λ = 2, ϕ = 0.5, k = 0.8, uⁿ = 5, and ${\sigma }_{\pi }^2={2.5}^2$. Treating all parameters as fixed, the central banker’s reaction was computed by solving numerically the implicit function (10) for given values of π^e The figure also includes the reaction function of the quadratic central banker, and the public’s reaction function that is summarized by the rational expectations relation (2). Graphically, the Nash equilibrium is the point where (10) and (2) intersect.

Reaction Functions

Citation: IMF Working Papers 2001, 161; 10.5089/9781451857818.001.A001

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Reaction Functions

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Although in all cases the central banker’s reaction is an increasing function of π^e, her willingness to accommodate the public’s inflation forecast depends on the preference parameter α. Consider the case where α < 0. The central banker responds to ${\pi }_t^e$ at an increasing rate and the inflation rate will be larger than under quadratic preferences. For values of $\alpha \le -1/\left(\lambda \varphi \left(1-k\right)E_{t-1}{u}_t^n\right)$, there is no finite rate of inflation at which (10) and (2) intersect, and the Nash equilibrium will not exist [see proposition 1 below]. Consider now the case where α > 0. The central banker accommodates the public’s inflation forecast at a decreasing rate and inflation will be always smaller than under quadratic preferences. For large-enough values of α, a deflationary bias, whereby inflation systematically undershoots its targeted value, can arise in equilibrium. This result is important because it shows that asymmetric preferences can provide a theoretical foundation for Stanley Fischer’s observation that a deflationary bias is a possible outcome in the practice of monetary policy [Fischer (1994)].⁹

Conditions for the existence and uniqueness of the Nash equilibrium are presented in the following lowing proposition:

From this proposition follows:

Notice in (11) that depending on the sign of α, inflation is an increasing or decreasing function of its conditional variance. Recall that when the loss function is quadratic, certainty equivalence holds and the model solution is the same regardless of whether there is uncertainty or not. In contrast, when the loss function is asymmetric on inflation, the marginal cost of departing from ${\tilde{\pi }}_t$ is not linear in inflation, but convex (when α > 0) or concave (when α < 0). When α > 0, an increase in uncertainty rises the expected marginal cost of deviating from ${\tilde{\pi }}_t$. Then, uncertainty induces a prudent behavior on the part of the central banker. A comparable result can be found in the literature on precautionary savings. When the assumption of quadratic utility is relaxed and labor-income risk in nondiversifiable, uncertainty increases the expected marginal utility of future consumption. To satisfy the Euler condition, prudent households decrease current consumption compared to future consumption and increase their savings.

I now derive the stochastic processes of the rates of inflation and unemployment and show that, conditional on the information set, both are normally distributed. Taking conditional expectations of both sides of (4), noting that i_t forms part of the public’s information set, substituting back into (4), and using (11) yields

The notation π_t(α) makes explicit the dependence of the inflation process on the parameter that measures the asymmetry in the central banker’s preferences. Since ${\tilde{\pi }}_t$ and $E_{t-1}{u}_t^n$ are included in the public’s information set at time t − 1:E(π_t |I_t−1) = E_t−1 π_t and $\mathit{\text{Var}}\left({\pi }_t|I_{t-1}\right)={\sigma }_{\pi }^2={\sigma }_{\epsilon }^2=\mathbf{A}{\Omega }{\mathbf{A}}^{\prime },$, where A = (0,0,1). Since the linear combination Aξ_t is normally distributed, then

as claimed above.

Turning now to the rate of unemployment, plug (12) into (1) and use the assumption of rational expectations to obtain: $u_t=u_t^n-\lambda {ϵ}_t+{\eta }_t$. As usual, the rate of unemployment realized in equilibrium does not differ systematically from the natural rate. Since $u_t^n$ is serially correlated, u_t will be serially correlated too. Decompose the process of the natural rate as $u_t^n=E_{t-1}{u}_t^n+{\zeta }_t$ to write

Since $E_{t-1}{u}_t^n$ forms part of $I_{t-1}:E\left(u_t|I_{t-1}\right)=E_{t-1}{u}_t^n\text{and}\mathit{\text{Var}}\left(u_t|I_{t-1}\right)={\sigma }_u^2=\mathbf{B}\Omega {\mathbf{B}}^{\prime }$, where B=(1,1, −λ). Since the linear combination Bξ_t is normally distributed, then

In contrast to models where the supply disturbance is the only shock, the error term here is a linear combination of the structural disturbances, rather than only a scaled version of the supply shock.

Under current inflation targeting arrangements, the targets are publicly announced and, consequently, are observable by the econometrician. This means that (12) generates testable empirical implications regardless of whether ${\tilde{\pi }}_t$ is socially optimal or not.¹⁰ Readers interested on these predictions are invited to go directly to section 2.5. However, from the normative point of view, it is still important to derive the optimal inflation target in the more general setup where preferences are asymmetric and compare results with the ones obtained using a quadratic loss function. This task is undertaken in the following section.

D. Digression: The Optimal Inflation Target

In order to make results comparable with the ones is earlier literature, assume that society’s preferences are described by the quadratic loss function:

where ${\pi }_t^{*}$ and $u_t^{*}$ denote the socially-optimal rates of inflation and unemployment, and remaining notation is as previously defined. Consider the problem of a government that chooses the sequence ${\left\{{\tilde{\pi }}_t\right\}}_{t=0}^{\infty }$ minimizes the present discounted value of the social loss function. It is important to note that in practice, targets are sometimes jointly agreed by the government and the monetary authority. For example, section 9 of the Reserve Bank of New Zealand Act requires the Finance Minister and the Governor of the Reserve Bank to negotiate and make public the Policy Target Agreements (PTA’s) that state the inflation targets [Bernanke et al. (1999, p. 88)]. Similarly, in Canada the targets are jointly determined and announced by the government and central bank. However, since in this model there is no private information and preferences are observable prior to delegation, the simplifying assumption that the choice of targets rests with the government is innocuous.¹¹

The government’s problem is

where the minimization is made subject to (12). Because the choice of the inflation target for time t does not affect the inflation process beyond time t, the government’s objective function can also be decomposed into a sequence of one-period problems. Noting that the unemployment rate does not depend on ${\tilde{\pi }}_t$ and that $\partial {\pi }_t/\partial {\tilde{\pi }}_t=1$, the first-order condition is $E_{t-1}\left[\left(\partial G/\partial {\pi }_t\right)\left(\partial {\pi }_t/\partial {\tilde{\pi }}_t\right)\right]=E_{t-1}{\pi }_t-{\pi }_t^{*}=0$ Using (11) and solving for the optimal inflation target yields

The notation ${\tilde{\pi }}_t\left(\alpha \right)$ makes explicit the dependence of the optimal inflation target on the parameter that measures the asymmetry in the central banker’s preferences. As in Svensson (1997), the optimal target is time-varying, rather than the constant value typically observed in practice. However, the optimal target here is nonlinearly related to unemployment, depends on the conditional variance of inflation, and can be smaller or larger than the optimal inflation rate.

The optimal inflation target under quadratic preferences can be obtained by taking the limit of (15) as α→0 to obtain

Notice that when 0 < k < 1, the optimal target is strictly lower than optimal inflation. The result ${\tilde{\pi }}_t\left(0\right)<{\pi }_t^{*}$ means that the government entrusts the quadratic central banker with a target that is low enough vis a vis ${\pi }_t^{*}$ that, even if the latter is subject to an inflation bias, the marginal costs and benefits of inflation to the central banker are equalized at the socially-optimal inflation rate. This result was first derived by Svensson (1997). In the special case where the central banker targets the natural rate of unemployment (k = 1), the optimal inflation target corresponds exactly to the socially-optimal inflation rate.

The optimal inflation target can deliver the optimal monetary policy. To see this, replace (15) into (12) to obtain

Inflation realizations differ from ${\pi }_t^{*}$ only by a mean-zero and serially uncorrelated random term. The unconditional mean of inflation is the socially-optimal rate. Since the disturbance ϵ_t is outside the control of the monetary authority, the model highlights the fact that inflation is not perfectly controllable by the central banker. Consequently, announced inflation targets are likely to be specified in terms of a tolerance range rather than as a sole numerical value. This implication of the model appears consistent with empirical evidence: of the eight inflation targeting countries surveyed by Bernanke and Mishkin (1997), only Finland defines its target as a number alone. Israel has defined its inflation target as range in 1993 and from 1996 onwards, and a as number in 1994 and 1995. Since June 1995, the United Kingdom defines its target as a point rather than as a range. However, after June 1997 there is a 1% threshold around the target rate that, if breached by the inflation rate, prompts and explanatory letter from the Monetary Policy Committee to the Chancellor of the Exchequer.

E. Implications

As noted above, the fact that inflation targets are observable means that (12) generates testable predictions regardless of whether the targets are optimal or not. These predictions can be formulated in terms of either the rate of inflation or its deviation from the announced target. For the general case where the loss function is asymmetric, it is trivial to rewrite (12) as:

Notice that the term $\left(1/\alpha \right)\text{ln}\left(1+\alpha \lambda \varphi \left(1-k\right)E_{t-1}{u}_t^n\right)$ is nonnegative but $-\left(\alpha /2\right){\sigma }_{\pi }^2$ can be positive or negative depending on the sign of the preference parameter α.

Consider first the case α < 0, meaning that the central banker weights positive deviations from the target less severely than negative ones. Since $-\left(\alpha /2\right){\sigma }_{\pi }^2$ is larger than zero, the average inflation deviation from the target is unambiguously positive. Consider now the case α > 0, meaning that the central banker weights positive deviations from the target more severely than negative ones. Then, $\left(\alpha /2\right){\sigma }_{\pi }^2>0$ and for certain parameter values, it is possible that its magnitude be large enough that ${\pi }_t\left(\alpha \right)-{\tilde{\pi }}_t<0$ Hence, inflation could be on average below the announced target. In summary, under asymmetric preferences, inflation could be systematically above or below its target depending on the central banker’s preference parameter α. It is shown below that the quadratic model predicts that average inflation is no smaller than the announced target. In the special case examined by Green (1996) and Svensson (1997) where 0 < k < 1, average inflation is always larger than the target.

The asymmetric model predicts that the conditional variance of inflation is helpful in predicting its mean. In a cross section, this means, for example, that if α > 0, countries with more volatile inflation should have a lower average inflation deviation from target. Some limited evidence to this effect is presented in section 3.4. In a time-series, the preference parameter is statistically identified from the coefficient of the conditional variance of inflation and from the nonlinear term on unemployment.

As a comparison, it is useful to derive the implications of the model with quadratic preferences. Take the limit of (17) as α → 0 to obtain

Since the term $\lambda \varphi \left(1-k\right)E_{t-1}{u}_t^n$ is nonnegative, the average realization of ${\pi }_t\left(0\right)-{\tilde{\pi }}_t$ is also nonnegative. When 0 < k < 1, this model predicts that inflation is systematically above the announced target. Because this overshooting is rationally anticipated by the public, it follows that the targeting policy can only be imperfectly credible [see Svensson (1997)]. Quadratic preferences also imply that the relation between inflation and unemployment is linear and that the conditional variance has no explanatory power on the current deviation from the target. If one generalizes Svensson’s model to allow the case where the central banker targets the natural rate of unemployment (k = 1), the model predicts that the average inflation deviation from the target is zero.

A more direct comparison of both models can be obtained by examining the variable

This variable compares explicitly the deviations predicted by the quadratic and asymmetric models. It is plotted in Figure 3 as a function of the preference parameter α and for three different values of $E_{t-1}{u}_t^n,$ namely 2%, 4%, and 6%. In constructing this graph, I have set λ = 2, ${\sigma }_{\pi }^2=1$, ϕ = 0.5, and k = 0.8. Notice that when α → 0 both models predict exactly the same deviation from the target and, consequently, their difference is zero. For negative values of α, the asymmetric model predicts positive deviations that are larger than the (also positive) deviations predicted by the quadratic model. The nonlinear relation between unemployment and inflation can be quite relevant for certain negative values of α. For positive values of α, the asymmetric model predicts negative deviations, or positive deviations that are smaller than the ones predicted by the quadratic model. Consequently, their difference is negative. Note that for α > 0, the relationship between the variables is approximately linear.

Difference in Predicted Inflation Deviation from Target

Citation: IMF Working Papers 2001, 161; 10.5089/9781451857818.001.A001

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Difference in Predicted Inflation Deviation from Target

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Difference in Predicted Inflation Deviation from Target

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As previous game-theoretical models of monetary policy [see, among others, Barro and Gordon (1983a, b), Rogoff (1985), and Svensson (1997)], the rates of inflation and unemployment are predicted to be positively correlated when 0 < k < 1. However, under asymmetric preferences, the relationship is nonlinear. The analysis of the data in section 3.2 suggests that the nonlinear model yields a more accurate fit of the inflation/unemployment observations than the linear specification.

In the special case where the central banker targets the natural rate of unemployment (k = 1), the model predicts no systematic relationship between inflation and unemployment. However, an inflation bias can still arise in the case where α < 0. This suggests that the result that the inflation bias is zero when k = 1 [see McCallum (1995, 1997) and Blinder (1998)] is not robust to the generalization of the policy maker’s loss function.¹² In the more plausible case where α > 0, a pure deflationary bias arises and inflation is on average below its target for any positive α.

Finally, consider the effect on the inflation deviation from target of an innovation that increases the public’s forecast of the natural rate. In the quadratic case:

that is positive, constant and independent of the expected rate of unemployment when the innovation takes place. In the asymmetric case:

that is also positive but depends on the expected unemployment rate. For example, when α > 0, the innovation always has a smaller effect than in the quadratic model and decreases with the rate of unemployment. The empirical predictions of the models with quadratic and asymmetric preferences are summarized in Table 2.

Table 2.

Summary of Predictions

Table 2.

Summary of Predictions

Prediction	Model
	Quadratic	Asymmetric
Positive relation between ${\pi }_t-{\tilde{\pi }}_t$ and ut	Yes	Yes
Nonlinear relation between ${\pi }_t-{\tilde{\pi }}_t$ and ut	No	Yes
Average deviation from target is nonnegative	Yes	Not necessarily
Conditional variance helps forecast ${\pi }_t-{\tilde{\pi }}_t$	No	Yes

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Table 2.

Summary of Predictions

Prediction	Model
	Quadratic	Asymmetric
Positive relation between ${\pi }_t-{\tilde{\pi }}_t$ and ut	Yes	Yes
Nonlinear relation between ${\pi }_t-{\tilde{\pi }}_t$ and ut	No	Yes
Average deviation from target is nonnegative	Yes	Not necessarily
Conditional variance helps forecast ${\pi }_t-{\tilde{\pi }}_t$	No	Yes

A. The Data

The empirical predictions of the model are examined using monthly observations of the rates of inflation and unemployment for Canada, Sweden, and the United Kingdom. These inflation-targeting countries feature time series that are sufficiently long to allow the meaningful estimation and testing of the model. In addition, their monetary policy objectives appear not to involve (at least explicitly) exchange rate considerations. Thus, despite very occasional interventions in foreign exchange markets, their exchange rate regime could be characterized as floating [see Bernanke et al. (1999)]. This is important because modeling the objectives of a central banker explicitly concerned about the nominal exchange rate, would require a more general model with the loss function appropriately augmented to include this institutional feature. For example, Finland had an inflation target of 2% per year between February 1993 and December 1998, and was simultaneously a member of the Exchange Rate Mechanism (ERM). Under the ERM, the nominal exchange rate was to be kept in a ±15% range around a target value and later fixed at an “irrevocable” rate prior to monetary union. Because of this, some authors argue that Finland’s monetary policy might not correspond to one of pure inflation targeting [see Gerlach (1999, p. 1263) for a discussion].

Australia, New Zealand, and Israel also employ inflation targets in the conduct of their monetary policy. Unfortunately, their statistical offices publish data on inflation and/or unemployment only on a quarterly basis. The Australian Bureau of Statistics collects price data every quarter, pricing most articles in the first two months of each quarter. Only certain items (for example, milk, bread, fresh meat, fresh fruit and vegetables, holiday travel and accommodation) are surveyed each month. Similarly, Statistics New Zealand prices most items at the mid-point of each quarter. Food is the only commodity group of the Consumer Price Index (CPI) for which an index is prepared each month. Although monthly CPI data is available for Israel, the labor force survey of the Central Bureau of Statistics takes place only once per quarter. All this means that, since inflation targeting is a relatively recent policy, the number of observations available to estimate and test the model for these countries is too small to yield reliable results.¹³

The Canadian inflation targets were announced in February 1991 and originally envisaged a reduction in the annual rate of inflation to 3% by the end of 1992, 2.5% by mid-1994, and 2% by the end of 1995. The inflation target of 2% per year has been renewed in December 1993 (to the end of 1998), in February 1998 (to December 2001) and in May 2001 (to December 2006). I follow the literature [for example, Lafrance (1997)], in interpreting the Canadian targets during the period December 1992 to December 1995 as the linear interpolation of the announced values. The targets are regarded as mid-points in a band of plus or minus 1 percentage point and apply to a measure of “core” inflation that excludes the volatile food and energy components and the first-round effect of changes in indirect taxes. The annual percentage change of the CPI and Core CPI were obtained from the Weekly Financial Statistics published by the Bank of Canada (http://www.bankofcanada.ca).

In Sweden, the Governing Board of the Riksbank announced in January 1993 an inflation target of 2% per year from January 1995 onwards. The target applies to the annual change in the CPI with no exclusions and involves a tolerance range of ±1% around the target.¹⁴ The raw CPI series was taken from the Web Site of the Riksbank (http://www.riksbank.se).

In the United Kingdom, the Chancellor of the Exchequer announced an inflation target of between 1% and 4% per year in October 1992. Initially, targets were meant to apply “through the end of the present parliament” alone. In June 1995, the government reinterpreted the inflation target as a numerical value of 2.5% per year. Since June 1997 there has been a 1% threshold around the target rate that, if breached by the inflation rate, prompts an explanatory letter from the Monetary Policy Committee to the Chancellor of the Exchequer. Throughout, the targeted measure of inflation is the annual change in the Retail Price Index excluding mortgage interest payments (or RPIX). Both the RPI and the RPIX were taken from the OECD Main Economic Indicators. Plots of the rates of inflation and announced targets for Canada, Sweden, and the United Kingdom are presented in Figure 4.

Inflation Rates

Citation: IMF Working Papers 2001, 161; 10.5089/9781451857818.001.A001

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Inflation Rates

Citation: IMF Working Papers 2001, 161; 10.5089/9781451857818.001.A001

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Inflation Rates

Citation: IMF Working Papers 2001, 161; 10.5089/9781451857818.001.A001

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• Download figure as PowerPoint slide

Note that by design, inflation targets are intended to apply to annual, rather than to monthly, inflation. However, because (i) annual inflation is the sum of the twelve most recent observations of monthly inflation,¹⁵ (ii) the policy holds continuously, and (iii) past rates of monthly inflation are predetermined at time t, it is possible estimate the model using data observed at a higher frequency than the time horizon for which the policy is defined. See Ruge-Murcia (2000) for a discussion.

Unemployment is measured by the survey-based, seasonally-adjusted rate of unemployment published by the OECD Main Economic Indicators.¹⁶ The sample periods are as follows: Canada from 1992:12 to 2000:6, Sweden from 1995:1 to 2000:6, and the United Kingdom from 1992:10 to 2000:6. The different sample periods start at the time when the inflation targeting policy took effect in each of country and end with the latest available observation of the variables at the time the data was collected.

B. A Rough Look at the Data

Prior to estimation, this section uses simple summary statistics and plots to examine to what extend the model predictions are (or are not) borne by the data. Consider first the predictions of the quadratic and asymmetric models regarding the average inflation deviation from the target. Under quadratic preferences, inflation should be on average above or at the target. Under asymmetric preferences, inflation can be on average above or below the target depending on the sign and magnitude of the preference parameter α. A negative average inflation deviation from the announced target is suggestive of a positive value of α.

Table 3 contains the sample mean of ${\pi }_t-{\tilde{\pi }}_t$ and the percentage of inflation observations above and below the target. For Canada and Sweden, the average deviation from target is negative and quantitatively important (−0.61 and −1.23, respectively), and the proportion of observations below target is high (91.2% and 81.2%, respectively). In the case of Sweden, the average deviation is below the lower limit of its inflation target zone. For the United Kingdom, the average inflation deviation from the target is small and positive (0.15) and 69.9% of the observations are above ${\tilde{\pi }}_t$.¹⁷

Table 3.

Inflation Deviation from Target

Notes: The statistics were calculated using monthly observations for the periods 1992:12 to 2000:06 (Canada), 1995:01 to 2000:06 (Sweden), and 1992:10 to 2000:06 (United Kingdom).

Table 3.

Inflation Deviation from Target

			Canada	Sweden	United Kingdom
		Average Deviation from Target	−0.61	−1.23	0.15
		Observations Above Target (in %)	8.79	18.18	69.89
		Observations Below Target (in %)	91.21	81.82	30.11

Notes: The statistics were calculated using monthly observations for the periods 1992:12 to 2000:06 (Canada), 1995:01 to 2000:06 (Sweden), and 1992:10 to 2000:06 (United Kingdom).

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Table 3.

Inflation Deviation from Target

			Canada	Sweden	United Kingdom
		Average Deviation from Target	−0.61	−1.23	0.15
		Observations Above Target (in %)	8.79	18.18	69.89
		Observations Below Target (in %)	91.21	81.82	30.11

Notes: The statistics were calculated using monthly observations for the periods 1992:12 to 2000:06 (Canada), 1995:01 to 2000:06 (Sweden), and 1992:10 to 2000:06 (United Kingdom).