The Phillips curve and the role of monetary policy in Chile

ABSTRACT In this paper, the empirical analysis finds that the dynamics of inflation and unemployment can be described by a Phillips curve when allowing for a positive co-movement between trend-adjusted productivity and unemployment. This suggests that improvements in productivity have been achieved by laying off the least productive part of the labor force. Furthermore, the natural rate of unemployment is a function of the long-term interest rate, indicating that monetary policy is not completely neutral in the long run. This result rejects the natural rate hypothesis and, at the same time, provides empirical support for the structural slump theory in a world of imperfect knowledge. The recent theory of imperfect knowledge economics (IKE) seems to address the problem that many economic models lack: persistence in the observed data. By combining IKE and the structural slumps theory it is possible to obtain predictions that are theoretically and empirically consistent.


Introduction
Since Phillips (1958) observed a negative relationship between wage inflation and the unemployment rate, known as the Phillips curve, numerous studies have analyzed this relationship empirically and theoretically. Over time, the relationship between the inflation rate and some measure of the economic cycle has been analyzed, giving rise to different formulations of the Phillips curve that have been generally used for the design of economic policies and forecasting inflation.
Models that analyze inflation dynamics are often classified either as standard Phillips curve models or New Keynesian Phillips curve models (see Karanassou, Sala, & Snower, 2010 for a thorough review of the historical development and formulations of the Phillips curve).
Standard Phillips curve models from the early 1960s find empirical support for a negative relationship between the inflation rate and unemployment as formulated by the traditional Phillips curve: where Δp t is the inflation rate, u t is the unemployment rate, c > 0 and b > 0 are constants, and v 1t is a stationary error term. Phillips (1958) estimated Equation (1) CONTACT Leonardo Salazar lsalazar@ubiobio.cl for the UK as did Samuelson and Solow (1960) for the United States with the expected coefficients signs. This relationship, however, broke down in the 1970s when a positive relationship between inflation and unemployment was observed rather than a negative one; this was called stagflation. It caused Friedman (1968) and Phelps (1968) to develop a new form of the Phillips curve, the expectations-augmented Phillips curve. The idea was that the curve shifts over time as a function of changes in inflationary expectations. Thus, there is a natural rate of unemployment acting as a long-run attractor for the unemployment rate, and, in the long run, unemployment is independent of the rate of inflation. This formulation is given by where Δp e t is the expected inflation rate, u Ã is the natural rate of unemployment that is generally assumed to be constant, and v 2t is a stationary error term. Thus, under the expectations-augmented Phillips curve, there is only a short-run trade-off between inflation and unemployment rate. The New Keynesian Phillips curve models relate actual and expected inflation to some measure of aggregate marginal cost instead of to unemployment.
The expectations augmented Phillips curve assumes that the rate of unemployment should converge to its natural rate after a supply shock. This assumption is known as the natural rate hypothesis (NRH) and posits that the Phillips curve is vertical in the long run. However, the NRH is not well supported empirically (see Gomes &Da Silva, 2008, for Chilean data, andFarmer, 2013, for US data), that is the unemployment rate does not converge to a unique constant value in the long run. This might potentially explain this variable's persistence and also suggests that the natural rate of unemployment is time varying. Phelps (1994) argues, in his structural slumps theory, 1 that the long swings observed in European unemployment rates can be explained by fluctuations in exchange rates and real interest rates. Specifically, domestic real interest rates likely influence the natural rate of unemployment. In this case, the natural rate of unemployment is time varying, and its fluctuations are associated with the movements observed in real interest rates. The structural slumps theory provides two reasons for explaining the positive comovement between the natural rate of unemployment and the interest rate. First, higher real interest rates increase the natural rate of unemployment by discouraging investment. Examples would be an investment in the retention of workers (high interest rate reduces the probability of paying higher wages) or investment that could increase the productivity of the firm's workforce. Second, equilibrium employment will decrease with higher interest rates when government actions reduce firms' labor demand. For example, actions that affect the wealth of the working-age population will raise the real wage that workers demand (Aghion, Frydman, Stiglitz, & Woodford, 2003).
The structural slumps theory assumes a world where the unemployment rate and the real interest rate are stationary. Empirical studies, however, often find real interest rates to be indistinguishable from a unit root process. Such persistence in real interest rates is consistent with the theory of imperfect knowledge economics (IKE), developed by Frydman and Goldberg (2007, 2011), and Juselius and Juselius (2012 argue, based on Finish data, that the structural slumps theory based on IKE is better able to explain the persistent swings observed in the data. Juselius and Juselius (2012) give the following rationale. Under IKE, nominal interest rates exhibit strong persistence due to a nonstationary uncertainty premium, whereas inflation rates are more stable over time due to international competitiveness. This implies that the Fisher effect does not hold as a stationary condition. 2 The uncertainty premium is generally related to the concept of a "gap effect" which, in the foreign currency market, can be measured by the deviation of the real exchange rate from its long-run purchasing power parity value. In an IKE world, due to speculative behavior in the currency market, nominal exchange rates tend to move away from relative prices for long periods of time, causing the real exchange rate to behave like a near I 2 ð Þ process and persistent deviations of the real exchange rate from its long-run benchmark value will be reflected in the uncertainty premium and hence in domestic and foreign interest rates.
Then an increase (decrease) in nominal interest rates, in general, is not likely to be followed by an increase (decrease) in consumer price inflation and the real interest rate will tend to rise (drop). This is likely to result in inflows (outflows) of speculative capital, causing an appreciation (depreciation) of the real exchange rate and worsening (improving) domestic competitiveness. Under this situation, domestic firms in the tradable sector cannot count on exchange rates to restore competitiveness after a shock to relative costs (e.g., a large wage rise). In this case, domestic firms will be prone to adjust profits rather than prices and the former can be adjusted through improvements in labor productivity by laying off the least productive workers. Thus, an increase in both labor productivity and unemployment might be expected in periods of real appreciation and rising real interest rates.
Then a more adequate and general representation of the Phillips curve is given by where v 3t and v 4t are stationary error terms and the time-varying natural rate of unemployment, u Ã t , can be expressed as a function of the real interest rate, i t , and z 0 > 0. In this article, the Phillips curve is empirically analyzed and the role of monetary policy is assessed for the Chilean economy. The results, based on a cointegrated vector autoregressive (CVAR) model, show that the natural rate of unemployment is positively co-moving with the interest rate. Therefore, the monetary policy in Chile is not completely neutral in the long run. Evidence of a natural rate of unemployment as function of the interest rate has also be found in Juselius and Ordóñez (2009) for Spanish data, in Juselius (2006) for Danish data, and in Berentsen et al. (2011) for United States data. Indeed, Berentsen et al. (2011) conclude that monetary policy may have been responsible for a sizable part of movements in trend unemployment over the last half century.
Additionally, the results provide empirical support for the structural slump theory in a world of imperfect knowledge. That is, while a positive co-movement is found between unemployment and productivity, there is a negative co-movement between productivity and the real exchange rate, indicating that international competitiveness might affect the unemployment rate.
The results in this paper also suggest that there is a long-run trade-off between inflation and unemployment when allowing for a positive co-movement between trend-adjusted productivity and unemployment. This is consistent with empirical evidence that finds a long-run trade-off between inflation and unemployment in the United States (Fair, 2000;Karanassou, Sala, & Snower, 2005;King & Watson, 1994), Spain (Dolado, López-Salido, & Juan, 2000), Finland (Juselius & Juselius, 2012), and the European Union (Karanassou et al., 2005). This paper, however, includes variables that might explain the high persistence in unemployment and inflation rate: the interest rate which is positively co-moving with the natural rate of unemployment, as in Juselius and Juselius (2012), and the real exchange rate which is negatively co-moving with productivity over time.
The fact that the interest rate co-moves with the natural rate of unemployment in the long run challenges the classical dichotomy that nominal variables do not affect real variables and that inflation and unemployment can be separately analyzed if there is no trade-off between them in the long run. This result, however, is in consonance with Blanchard (2003), which implicitly states that monetary policy may have real effects on the economy: . . .if we accept the fact that monetary policy can affect the real interest rate for a decade and perhaps more, then, we must accept, as a matter of logic, that it can affect activity, be it output or unemployment, for a roughly equal time (Blanchard, 2003, p. 3).

Theoretical framework
This section discusses the expectations-augmented Phillips curve in a framework developed by Hoover (2011), assuming an economy with imperfect competition and imperfect information about the current price level. In this framework, supply shocks are also allowed to shift the relationship between the unemployment rate and inflation. 3

Price setting
A firm sets its price based on its expectation of the price level prevailing during the current period, taking demand and supply conditions into account. The price setting is written as where Δ is the first difference operator, p j;t ¼ ln P j;t À Á and P j;t is the price set by firm j, p e j;t ¼ ln P e j;t and P e j;t is the expected level of price prevailing during the current period. This price is set by firm j at the end of period t À 1 based on an information set available at the end of the same period, Z j;tÀ1 . The expected price can be written as P e j;t ¼ E j;tÀ1 P t jZ j;tÀ1 Â Ã where E Á ½ is the expectation operator.
Functions f Á ð Þ and g Á ð Þ determine how demand and supply factors affect the pricing decision of firm j. Equation (4) represents a single firm's price behavior. Taking the average of all firms in the economy and assuming that firm-specific supply and demand factors average out, the economy's price behavior is written as Δp t ¼Δp e t þ f aggregate demand factors ð Þ þ g aggregate supply factors ð Þ ; where Δp t is the current inflation rate, Δp e t is the average expectation of general price inflation for all firms, f Á ð Þ is reflecting demand-pull inflation, and g Á ð Þ captures cost-push inflation.

The Phillips curve and the natural rate of unemployment
Functions f Á ð Þ and g Á ð Þ must be explicitly defined to apply Equation (5) to actual data. Since Phillips (1958), an accepted and usual measure of the aggregate demand, has been the unemployment rate: where u t is the unemployment rate, b is assumed to be a positive constant given the countercyclical behavior of the unemployment rate, and a > 0. Now, assuming for the moment that aggregate supply factors can be ignored, an unemployment rate that sets actual inflation equal to expected inflation is obtained by inserting Equation (6) into Equation (5): where u Ã t is the natural rate of unemployment. If a and b are stable over time, the natural rate can be expressed without subscript t. Equation (5) can then be rewritten as (8) is the extension of the Phillips curve that allows for supply shocks. When the unemployment rate is below (over) its natural rate, inflation tends to increase faster (slower) than expected. Equations (7) and (8) show two classical results. First, the natural rate of unemployment is constant, and second, the Phillips curve is vertical at this level. That is, when expectations are fulfilled and the aggregate supply factors are set at their "natural" levels, there is no long-run trade-off between unemployment and inflation.

Discussion
According to the theoretical framework, the unemployment rate should converge to its natural rate after a supply shock. This assumption is known as the NRH and posits that the Phillips curve is vertical in the long run. Farmer (2013) proposes that the NRH can be tested in the following way. Under the assumption that expectations are rational, the number of periods (e.g., quarters) where actual inflation is above its expected value should be roughly equal to the number of periods in which the opposite situation is observed. Then, over a decade, the average inflation rate should be approximately equal to the average expected inflation. If the inflation rate over long periods is plotted together with the unemployment rate, a vertical line at the natural rate of unemployment should be visible, supporting the NRH and rational expectations. Figure 1 shows Chile's average inflation and unemployment rates by decade. 4 The plotted points are not vertically aligned, and there is no tendency for them to lie around a vertical line. Farmer (2013) obtains a similar result for United States data and categorically concludes that because expectations are unlikely to be systematically biased over decades, the NRH is false. However, such a strong conclusion should not rely on a simple graph, and further tests must be provided.
Most of the Phillips curve literature on Chile has dealt with the evaluation of the transmission mechanism of monetary policy (Cabrera & Lagos, 2000), the forecast of inflation (Nadal De Simone, 2001), and the estimation of NAIRU, or the non-accelerating inflation rate of unemployment (Restrepo, 2008). But some studies also analyze hysteresis as an explanation for the persistence of the unemployment rate (Gomes & Da Silva, 2008;Solimano and Larraín 2002).
These studies find that there is a significant and negative relationship between inflation and cyclical unemployment, as well as between cyclical unemployment and the inflationary gap. This is interpreted as evidence for a short-run Phillips curve (Restrepo, 2008). The literature also finds that the Phillips curve seems to provide better forecasting of inflation when the explicit inflation target is used as an independent variable instead of the output gap (Nadal De Simone, 2001). Since the GDP gap as well as measures of the business cycle do not show a significant response to an increase Figure 1. Average inflation and unemployment by decade in Chile. Note: 1980s includes 1986Note: 1980s includes -1989Note: 1980s includes , and 2010s includes 2010Note: 1980s includes -2016 in the monetary policy interest rate, Cabrera and Lagos (2000) argue that the Phillips curve is not a proper tool to analyze the transmission mechanism of monetary policy.
The hysteresis hypothesis has been offered as a way to explain the persistence observed in the rate of unemployment in Chile. Solimano and Larraıń (2002) show that this hypothesis cannot be rejected, and Gomes and Da Silva (2008) argue that hysteresis is a better explanation of the actual unemployment than the NAIRU. Restrepo (2008) shows that the latter rate is time varying but does not offer an explanation for its cause.
Before 1990, the monetary policy in Chile was mostly subordinated to fiscal deficit financing (Nicolás, 1998). This situation changed when a new institutional framework established the independence of the Chilean Central Bank in December of 1989 and few weeks after taking office, the new authorities implemented a sharp increase in the longterm interest rate of Central Bank's internal debt (Rosende & Tapia, 2012). Since 1991, explicit annual inflation targets have been used and the Central Bank aims to achieve a level of interest rate, expenditure, and output consistent with the targets.
The Central Bank of Chile influences markets rates, which are freely determined by supply and demand conditions, through its policy instruments. Until the middle of 1995, monetary policy was implemented through open market operations using 90-day real instruments and since 2001 the Central Bank decided to nominalize the monetary policy and the monetary policy interest rate was set at an annual 6.5%. Figure 2 shows the monetary policy interest rate, as determined by the Central Bank of Chile, together with the long-and short-term interest rates. The latter two are calculated as the average long (1 year) and short (3 months) deposit rates in the Chilean financial system. The three interest rates are co-moving over time, suggesting that market interest rates follow the policy rate. Note: Data on monetary policy rate is only available since 1995:2.All graphs have been scaled by mean and range.
Regardless of the specification of the Phillips curve, most of the available studies are based on a single-equation approach where the rate of inflation is explained by some indicator of the economic cycle (e.g., unemployment, production, marginal cost). Such an approach can be justified by the classical dichotomy that nominal variables do not affect real variables and that inflation and unemployment can be separately analyzed if there is no trade-off between them in the long run. However, the empirical evidence for this dichotomy is quite weak. For instance, Fisher and Seater (1993), King and Watson (1994), Fair (2000), and Karanassou et al. (2005) present evidence of a significant longrun trade-off between inflation and the unemployment rate. Even if the classical dichotomy holds, a single-equation approach is silent about any feedback mechanisms embedded in the data.
Furthermore, most empirical and theoretical studies assume the natural rate of unemployment to be an exogenous variable. This can also be justified by the classical dichotomy. Generally, in its first stage the Phillips curve is estimated under the assumption of a constant natural rate, and from the residuals of this equation, a time-varying natural rate of unemployment is then derived (Mankiw & Ball, 2002). Most Phillips curve studies focused on industrialized economies, particularly in European countries, and only a small proportion on developing countries. The latter is probably due to the scarceness of homogeneous datasets that cover sufficiently long periods, which may be the result of economic and political instability in these economies. Whatever the reason, the lack of research on developing economies is likely to hinder a suitable design of economic policies. Figure 3 Panel (a) shows the quarterly inflation rate, which exhibits a gradual decrease over the sample. This might be associated with the implementation of inflation targeting in 1990. By mandate, the Central Bank of Chile is compelled to keep the annual inflation rate centered on 3% with fluctuations in the range of 2-4%. This has been more or less the case since 2000. Figure 3 Panel (b) shows the evolution of the unemployment rate, which exhibits an important increase in 1998, possibly as a result of the Asian crisis. 5 The unemployment rate exhibits a higher mean after that crisis, suggesting that the mean of the natural rate of unemployment might have increased as well. Indeed, the rate of unemployment rose from an average rate of 6.9-8.4% after the Asian crisis. The unemployment rate exhibits another significant increase in 2009 when the global financial crisis hit the Chilean economy. This increase, however, seems transitory. 6 The unemployment rate is not the only variable seemingly affected by the Asian crisis. Figure 3 Panel (c) shows a deceleration in the growth of real productivity around 1998, which again might indicate a slowdown in economic activity as a result of the Asian crisis.

Stylized facts
The unemployment rate and productivity exhibit seasonality caused by the agricultural activity in Chile, which is higher during the last and first quarter of each year. Figure 3 Panel (d) shows the real exchange rate, and since 2000, it exhibits a higher mean. This may be related to the free floating exchange rate regime implemented by the Central Bank of Chile in September 1999. The graph suggests that the real exchange rate may be fluctuating around a constant level, though with persistent deviations.
Two major reforms were introduced in the Chilean financial market between 2000 and 2001. While the first reform, promulgated in 2000, gave greater protection to both domestic and foreign investors, the second reform, enacted in 2001, liberalized the financial system, implying, among other things, capital account deregulation. These reforms seemed to affect the real interest rate which had exhibited a lower mean since 2001 (Figure 3 Panel (e)). At the same time, Figure 3 Panel (f) shows a large volatility reduction in the interest rate spread between the long-and short-term rates.

Baseline model
The quarterly data cover the period 1990:4-2017:1, and the information set 7 is given by where Δp t is the inflation rate; u t is the unemployment rate; ri t ¼ i L t À Δp t is the long-term interest rate and i L t is the long-term nominal interest rate; sp t ¼ i L t À i S t is the interest rate spread and i S t is the short-term nominal interest rate (following Juselius & Juselius, 2012, the spread is used as a proxy for expected inflation); c t is the real labor productivity; and q t is the real exchange rate (for further details, see Table A1). The baseline CVAR model for vector x t is whereχ t ¼ x t ; d s00:4;t ; t 98:3;t ; t Â Ã 0 , d s00:4;t is a step dummy that takes the value of 1 since 2000:4, 0 otherwise; t 98:3;t is a broken linear trend taking values of 1,2,. . .,62 since 1998:3, 0 otherwise; and t is a linear trend. d s98:3;t is a shift dummy that takes the value of 1 since 1998:3, 0 otherwise, corresponding to the first difference of the broken linear trend. 1 is a vector of constant terms, D p;t contains 14 permanent dummies (0,. . .,0,1,0,. . .,0), 8 and S t is a vector of centered seasonal dummies (for further details see Table A3). Finally, ε t is an i.i.d. vector of normally distributed stochastic errors.
The step dummy, d s00:4;t , accounts for deregulation in the Chilean financial markets, and the broken linear trend, t 98:3;t , accounts for the productivity slowdown in 1998. Both deterministic variables, together with the linear trend, are restricted to lie in the cointegrating space.
The software CATS 3 for Ox Metrics was used to estimate model (9). Table 1 reports the residual misspecification tests of model (9). 9 The upper part shows that the model is, in general, well behaved. The hypotheses of non-autocorrelation and non-ARCH cannot be rejected, and there are only weak signs of non-normality. The univariate tests in the lower part of Table 1 show that only residual ARCH and signs of non-normality are present in the interest rate spread. The ARCH problem is evident from Figure 3 Panel (f) and the normality problem is associated with excess of kurtosis rather than skewness. Except for these minor misspecification signs, the model seems reasonably well specified. The upper part of Table 2 reports the I 1 ð Þ trace tests and shows the eigenvalues, λ i , for the null hypothesis of r ¼ 0; 1; 2; 3; 4; 5 together with simulated p-values of the trace test. The hypothesis r ¼ 4 cannot be rejected based on a simulated p-value of 0.31. To check the adequacy of this choice, the lower part of Table 2 reports the four largest characteristics roots for r ¼ 6; 5; 4. The unrestricted model, r ¼ 6 has only one reasonably large root, 0.85, suggesting that the model contains one unit root at most. r ¼ 4 introduces additional persistence in the system since two unit roots are set to one. When r ¼ 5, only one root is set to one, leaving a complex pair of roots with a modulus of 0.70 as the largest roots in the system. Thus, the following analysis is based on r ¼ 5 cointegrating vectors.

Identification of the long-run structure
To identify plausible economic relationships among the variables, a set of restrictions, H e β : H 1 φ 1 ; H 2 φ 2 ; . . . ; H r φ r À Á , must be imposed onβ where H i is a restriction matrix of dimension p 1 Â p 1 À m i ð Þ, p 1 is the dimension ofχ t , m i is the number of restrictions imposed onβ, p 1 À m i is the number of freely varying parameters, and φ i is a p 1 À m i ð ÞÂ1 vector of unknown parameters. This test is asymptotically χ 2 distributed with degrees of freedom equal to P r i¼1 m i À r À 1 ð Þ ð Þ (Johansen, 1996).
Model (9) may be written for the variable x i;t as Δx it ¼ P r j¼1 α ij P p m¼1 β mj x m;tÀ1 , i ¼ 1; 2; . . . ; p. Then, for α ij Þ0, the sign of α ij β ij determines whether the variable x i;t is error increasing or error correcting according to the following rule: if α ij β ij < 0, then Δx it is equilibrium error correcting to β 0 j x tÀ1 . Otherwise, the variable is equilibrium error increasing. However, when the system is stableall characteristic roots are either inside or on the unit circleany error-increasing behavior is sooner or later compensated by errorcorrecting behavior.   Table 3 reports an identified structure onβ with eleven overidentified restrictions that could not be rejected based on χ 2 11 ð Þ ¼ 8.85 with a p-value of 0.64. 10 To facilitate interpretation, an α ij coefficient in boldface means that the cointegrating relation i is equilibrium correcting, whereas an error increasing coefficient is in italics. The results in Table 2 show that all eigenvalues are inside of the unit circle. Thus, the system is stable, and any error-increasing behavior is compensated by error-correcting behavior.
The first cointegrating vector,β 0 1χt , is interpreted as a Phillips curve when allowing for trend-adjusted productivity 11 and is expressed as wherec t ¼ c t þ 0:013t 98:3;t À 0:02t stands for trend-adjusted productivity andυ 1;t ,I 0 ð Þ measures the equilibrium error. Equation (10) shows that unemployment in excess of trend-adjusted productivity would lead to downward pressure on the inflation rate. This finding is consistent with the reasoning above that in a world of imperfect knowledge, firms exposed to international competition facing a positive shock to relative costs cannot, in general, count on exchange rates to restore competitiveness. Accordingly, firms will be prone to adjust profits rather than prices (Juselius & Juselius, 2012). Profits can be adjusted through improvements in productivity by laying off the least productive workers. Evidence of a positive co-movement between unemployment and productivity has also been found in Juselius (2006) for Danish data.
The adjustment coefficients show that, while the inflation rate and productivity are equilibrium error correcting to the Phillips curve, the unemployment rate is equilibrium error increasing. Thus if unemployment is above its long-run equilibrium value, the trend-adjusted productivity will tend to increase and, at the same time, the Note: Á ð Þ is the t-value. "−" is a zero restriction. A coefficient in boldface stands for an equilibrium error correcting behavior. A coefficient in italics stands for an equilibrium error increasing behavior. unemployment rate will tend to increase as long as enterprises improve labor productivity over its long-run trend. This introduces further increases in the equilibrium error. The inflation rate will start decreasing, which tends to restore equilibrium. Moreover, the adjustment coefficients show that long-term interest rate and interest rate spread have been positively affected by the equilibrium error in the Phillips curve and that the real exchange rate has appreciated when the unemployment rate has been above its long-run equilibrium value.
The second vector,β 0 2χt , is interpreted as an unemployment relationship and is expressed as u t ¼ ri t þ 0:01d s00:4;t À 0:001t þυ 2;t ; (11) whereυ 2;t ,I 0 ð Þ is the equilibrium error. The linear trend in (11) can be understood from Figure (3) Panel (e). The graph shows that the real interest rate has shown a tendency to decrease over time, 12 which is approximated by the deterministic trend in Equation (11).
The adjustment coefficients show that, while the unemployment rate is equilibrium error correcting to Equation (11), the real interest rate is equilibrium error increasing. Then, even if the interest rate is above its long-run equilibrium value, it will continue to rise. At the same time, the unemployment rate will also increase, restoring equilibrium. This seems to be greatly aggravated by the fact that the inflation rate has been positively affected by the equilibrium errorυ 2;t . That is, the inflation rate shows a tendency to decrease when the real interest rate is increasing, generating further increases in the unemployment rate. In addition, the adjustment coefficients show that the real exchange rate has appreciated when the real interest rate has been above its long-run trend.
The natural rate of unemployment can be derived from Equation (11), 13 and is expressed as where the natural rate of unemployment, u Ã t , is time varying and positively associated with the long-term nominal interest rate. This result indicates that monetary policy in Chile is not neutral over time. That is, the Central Bank of Chile conducts its monetary policy using the interest rate as the main tool to keep the inflation rate close to its target, and Equation (12) suggests that fluctuations in the nominal interest rate are positively associated with the natural rate of unemployment. In addition to the structural slump theory's explanation, the positive co-movement between the interest rate and the natural rate of unemployment might be justified using a general equilibrium model of unemployment and money demand developed in Berentsen et al. (2011). This model is based on frictions in both labor and goods markets where the costs and hence the benefits of holding money are explicitly modeled. Furthermore, there are two kinds of agents: firms and household. Then, a higher interest rate increases the cost of holding money which in turn leads households to economize on real balances, reducing profits, retail trades and, ultimately, employment.
Equation (12) shows that the natural rate is co-moving with the nominal rather than the real interest rate, which can be explained by the fact that the rate of inflation has been fairly stable over time, as shown in Figure 3 Panel (a).
In an IKE world, this finding provides empirical support for Phelps's hypothesis of a Phillips curve where the natural rate of unemployment is positively co-moving with the interest rate. A time-varying natural rate of unemployment is consistent with the nonvertical scatter of the inflation and unemployment rates shown in Figure 1.
Equation (11) also shows that unemployment is nonstationary per se and needs to be combined with another variable to obtain stationarity. This result provides an explanation of the persistence of the unemployment rate other than the hysteresis hypothesis of previous studies in Chile.
The third vector,β 0 3χt , is interpreted as a short-term real interest rate relationship and is expressed as ri t ¼ sp t À 0:01d 00:4;t þυ 3;t ; or, equivalently i S À Δp À Á t ¼ À0:01d s00:4;t þυ 3;t (13) whereυ 3;t ,I 0 ð Þ measures the equilibrium error. It describes a stationary short-term real interest rate after controlling for an equilibrium mean shift in the last quarter of 2000. The latter is likely to reflect the effect of reforms introduced in the Chilean financial system. The adjustment coefficients show that, while the long-term real interest rate is equilibrium error correcting to Equation (13), the spread is equilibrium error increasing. When the real interest rate is above its long-run equilibrium value, the spread will tend to decrease which increases the equilibrium error. At the same time, the real interest rate will tend to decrease, restoring the equilibrium. The inflation rate has been positively affected by the equilibrium error, which also helps to reduce the real interest rate.
The fourth relation,β 0 4χt , is interpreted as a central bank's reaction rule and is expressed as sp t ¼ À0:07Δp t þ 0:0003t 90:4À98:3 ð Þ ;t þυ 4;t ; whereυ 4;t ,I 0 ð Þ measures the equilibrium error. It indicates that the central bank tends to increase the short-term interest rate to counteract inflationary pressures, thereby reducing the spread. This is consistent with the countercyclical policy of the Central Bank of Chile. The latter tends to increase its interest rate when there are signs of inflationary pressure. The term t 90:4À98:3 ð Þ ;t , 14 that stands for a trend between 1990:4 and 1998:3, is needed because before 1998:3 the inflation rate behaved as a trending variable, which is shown in Figure 3 Panel (a), but since 1998:3 the inflation rate is fluctuating around a roughly constant average, which is consistent with the full implementation of inflation targeting in 1999.
The adjustment coefficients show that while the spread is equilibrium error correcting to the central bank's reaction rule, the inflation rate is equilibrium error increasing. When the spread is above its long-run equilibrium value, the inflation rate will tend to increase, generating further increases in the equilibrium error. At the same time, the spread will tend 14 The term t 90:4À98:3 ð Þ ;t is equivalent to t À t 98:3;t .
to narrow, restoring the equilibrium. The long-term interest rate has been negatively affected by the equilibrium error, which helps to narrow the interest rate spread.
Finally, the fifth relation,β 0 5χt , is interpreted as a productivity relationship and is expressed as where ĉ t ¼ c t À 0:002t stands for trend-adjusted productivity, q t ¼ q t À 0:17d s00:4;t stands for the real exchange rate after controlling for an equilibrium mean shift, and υ 5;t ,I 0 ð Þ measures the equilibrium error. It shows that the trend-adjusted productivity has been negatively co-moving with the real exchange rate. This finding provides empirical support for the predictions of Phelps's structural slumps theory in an IKE world. That is, firms exposed to international competition that face a positive shock to relative costs will improve productivity rather than increase prices to adjust profits in periods of real appreciation. Figure 4 shows the cointegrating relationships and despite some signs of persistent deviations, they seem mean-reverting.

Policy implications
According to the IKE theory, the long swings observed in the nominal exchange rate around relative prices will be associated with similar swings in the interest rate spread between the domestic and foreign interest rates. Furthermore, the structural slumps theory predicts that the domestic real interest rate may influence the unemployment rate. Then, the latter is likely to be affected by persistent swings in the real exchange rate and by the monetary policy of central banks that impact interest rates.
The main objective of the Central Bank of Chile is "safeguarding the stability of the currency and the normal functioning of the internal and external payment systems" (BCCh, 1989; Section III). To achieve its objective, the Central Bank conducts its monetary policy based on inflation targeting in a free-floating exchange rate regime. The main instrument to keep inflation close to its target is the monetary policy interest rate. Changes in the latter are passed to the interbank interest rate through open-market operations, interest-bearing reserves, and discount-window policy. Commercial banks, in turn, pass these fluctuations to lending and/or deposit rates, 15 which changes consumption, savings, and investments, affecting the aggregate demand and, hence, the price level in the economy.
The results in this paper indicate that the Central Bank, in pursuing its mandate, may affect not only the unemployment rate but also the natural rate of unemployment. Moreover, the adjustment coefficients show that when the real interest rate is above its long-run equilibrium, it will tend to rise as real interest exhibits an equilibrium errorincreasing behavior. Whatever the case, it seems that high and persistent unemployment might be caused by fighting inflation. Thus, Chilean monetary policy must be efficiently combined with other economic policies (e.g., fiscal policy) that stimulate employment in periods in which contractionary monetary policy is at work.
The fact that the natural rate of unemployment is co-moving with the market interest rate, which in turn follows the monetary policy rate, indicates that the Central Bank is 15 Figure 2 shows the co-movement between the monetary policy interest rate and the deposit rates.
driving the natural rate of unemployment. This, however, does not prevent that other variables may affect the natural rate of unemployment. Among others, price of raw materials, financial wealth, and labor market institutions might play a significant role in determining the natural rate of unemployment. The lack of accurate and historical data hinders their inclusion in the present model and interactions between these variables and interest rates are well beyond the scope of the current article.
The results also suggest that firms exposed to international competition in a world of imperfect knowledge adjust profits through productivity improvements by laying off the least productive workers. Because Chile has a small and open economy that is fairly and increasingly exposed to international competition, then in periods of real exchange appreciation, policies aimed to reduce unemployment should focus on the tradable sector.

Conclusion
This paper has analyzed the dynamics of inflation and unemployment in Chile using a CVAR model. The results indicate that these dynamics may be described by a Phillips curve when allowing for a positive co-movement between trend-adjusted productivity and unemployment rate. That is, unemployment in excess of trend-adjusted productivity would lead to downward pressure on inflation. Moreover, the natural rate of unemployment is time varying and positively associated with the long-term interest rate, suggesting that monetary policy is not completely neutral in the long run. Note: The graphs correspond to the cointegrating relationships in model (9) where the effects of the short-run dynamics, Γ 1 Δx t , have been concentrated out. For further details, see Chapter 7 in Juselius (2006).
The results also show that the trend-adjusted productivity is negatively co-moving with the real exchange rate, implying that during periods of real exchange rate appreciation competitiveness is negatively affected. Consequently, firms adjust profits by increasing productivity rather than prices.
Altogether, the empirical analysis provides support for the Phelps's (1994) structural slumps theory in a world of imperfect knowledge. Phelps states that fluctuations in exchange rates and the real interest rate may explain the fluctuations in unemployment rate and that the natural rate of unemployment is a function of the domestic real interest rate in the economy. The imperfect knowledge theory states that pronounced persistence in real exchange rates are associated with a similar persistence in the real interest rate. Therefore, these theories predict that unemployment and labor productivity increase in periods of rising real interest rates and real exchange appreciation.
As previously mentioned, the fact that the natural rate of unemployment is co-moving with the market interest rate does not prevent that other variables (e.g., the price of raw materials, financial wealth, and labor market institutions) may affect the natural rate of unemployment. The analysis of these potential interactions is well beyond the scope of this article and is suggested here as challenging research areas for future studies.

Appendix B. Unit root test
The unit root test reported in Table A2 is a test of variable stationarity. Specifically, the hypothesis of stationarity of the variable x i can be expressed as where e i is a known vector of dimension p 1 Â 1 ð Þ, p 1 is the dimension of e x t in (9) and β 0 is a freely-varying vector of dimension p 1 Â r À 1 ð Þ. The test is χ 2 distributed with p 1 À r À n d degrees of freedom and n d is the number of exogenous and deterministic variables that is maintained in the test (Doornik & Juselius, 2017).
The results in Table A2 show that inflation rate, productivity, and unemployment are not stationary; the latter less significantly though. Additionally, the long-term real interest rate, interest rate spread, and real exchange rate behave as stationary variables. 16