Estimating the output gap in a changing economy.

Univariate filters used in output gap estimation are subject to criticism as being purely statistical and having no economic content. The information content of the output gap measures estimated by standard multivariate filtering techniques, on the other hand, can be distorted because of the possibly unrealistic restriction that system parameters stay constant over time. In this study, we seek to address these shortcomings by proposing an output gap estimation method that takes into account changing economic relations. We employ a nonlinear time series framework along with the extended Kalman filter, in which economic content is used by inflation and output gap dynamics and the parameters are allowed to be time varying. We use the Turkish economy as a laboratory to show that our method provides useful results, both in terms of the properties of output gap estimates and for the assessment of change in macroeconomic dynamics.

JEL Classification: C32, C63, E30

1. Introduction

The emergence of price stability as the overriding goal of monetary policy in recent decades has led central banks to use all available information in the economy to foresee the future course of price dynamics. One important variable that is closely monitored by the central banks in this context has been the output gap--output in excess of its "neutral" or "potential" level. A positive output gap, for example, is often perceived as a signal of "excess demand," which might require tightening to prevent the economy from overheating.

Output gap, by definition, cannot be directly observed and thus has to be estimated. However, the information content of conventional output gap measures might be limited. (1) Not surprisingly, studies on developing proper alternative techniques have intensified in recent years, especially after the shortcomings of the Hodrick--Prescott (HP) filter--the most commonly used methodology--were realized. (2) In this context, multivariate filtering techniques utilizing information from macroeconomic relationships have been proposed as alternatives. (3)

One appropriate estimation technique in such a setting is the Kalman filter, which is a recursive algorithm for optimally forecasting the unobserved component given the observed variables and the imposed economic structure. However, existent multivariate techniques often impose strong restrictions in defining the relationships between key macroeconomic variables. For example, in these settings, the relationships that govern the dynamics of the economy stay intact over the sample period. This restriction might not hold for a constantly changing economy--at least for economies experiencing massive structural changes, as emerging markets do. Adopting different monetary policy regimes and experiencing frequent fiscal and financial restructuring periods affect the behavior of economic agents over time, changing the reduced-form relations between macroeconomic variables in these countries. (4) Therefore, although standard multivariate filters address and hence improve upon--the criticism regarding both the HP filter and other univariate statistical procedures, there may be room for improvement if one can relax the typical restriction that system parameters stay constant over time. This is the main motivation of our paper.

In this study, we present a multivariate unobserved components model to estimate both the output gap and the time-varying system parameters. We use the Turkish economy as a laboratory to demonstrate this technique. As mentioned above, the estimation of output gap by this method allows us to address changing relationships in the economy, leaving room for assessing the effect of different policy regimes on the relationship between key macroeconomic variables such as inflation, interest rates, exchange rates, and output gap.

When the state variables (including output gap, potential output, or both) and the system parameters are to be estimated simultaneously in a time-varying fashion, the model takes a nonlinear characteristic, and the standard Kalman filter (SKF) needs to be modified. In this case, the extended Kalman filter (EKF) emerges as an estimation procedure. Such a methodology has been previously used in Ozbek and Ozlale (2005) in a simple parametric form of output. However, their study does not allow the exogenous variables to be incorporated into both the state and the measurement equation. Nor do they present a structural background, which would be necessary to interpret the time-varying effects of the relevant macroeconomic variables on the estimated output gap.

We believe that the findings in this paper will serve as a reference in two distinct ways. The major departure of this study from the existing literature of unobserved component methodology of output gap estimates is the joint estimation of unobserved variables and time-varying parameters in a simultaneous system. As a by-product, this methodology permits us to interpret the resulting output gap series and its time-varying relationship with other variables, allowing us to conduct an ex post evaluation of the changing economic structure across different policy regimes.

The rest of the paper is organized as follows. In the next section, we present and discuss the model along with a state space representation. The estimation methodology is also introduced in this section. In section 3, we apply the methodology to Turkish data and present both the estimated output gap series and the estimated time varying parameters, interpreting them in view of developments in the Turkish economy in the last decade. A model sensitivity analysis is performed in section 4, where we analyze whether the results remain robust to different specifications about inflation and the output gap dynamics. This section also contrasts the estimated gap series with the ones obtained from the HP filter and the standard Kalman filter procedure. Finally, section 5 concludes the paper.

2. The Model

The general form of the model, in which the time-varying parameters and the gap series are estimated, is shown in Equations 1-5.

Inflation Output Gap Dynamics

[[pi].sub.t] = [[alpha].sub.1,t], [[pi].sub.t-1] + [[alpha].sub.2,t] [[pi].sub.t-2] + [[alpha].sub.3,t] [gap.sub.t-1] + [[alpha].sub.4,t] [reer.sub.t] + [v.sub.t] (1)

Actual Output Decomposition 

[y.sub.t] = [y.sup.*.sub.t] + [gap.sub.t] (2)

Potential Output Equation

[y.sup.*.sub.t] [y.sup.*.sub.t-1] + [[mu].sub.t-1] + [[eta].sub.t] (3)

Potential Output Growth Rate Equation

[[mu].sub.t] = (1 - [[rho].sub.t]) [[mu].sub.0] + [[rho].sub.t] [[mu].sub.t-1] + [[epsilon].sub.t] (4)

Output Gap Dynamics

[gap.sub.t] = [[gamma].sub.1,t] [gap.sub.t-1] + [[gamma].sub.2,t]] [r.sub.t] + [[gamma].sub.3,t] [DI.sub.t] + [[gamma].sub.4,t] [reer.sub.t] + [[zeta].sub.t]

where [[pi].sub.t], is the inflation rate defined as the logarithmic difference of the quarterly seasonally adjusted consumer price index (CPI), [gap.sub.t], is the unobserved output gap, [reer.sub.1], is the logarithmic difference of the real effective exchange rate (positive means appreciation), [y.sub.t] is the logarithmic seasonally adjusted real gross domestic product, [y.sup.*.sub.t] is the unobserved potential output, [[mu].sub.t] is the potential output growth rate, [r.sub.t] is the ex post real interest rate on the basis of government securities, and [DI.sub.t] is the demand index, which is constructed from the Business Tendency Survey of the Central Bank of the Republic of Turkey. The derivation of the demand index along with other data descriptions is presented in Appendix A. Finally, [v.sub.t], [[eta].sub.t], [[epsilon].sub.t] and [[zeta].sub.t] represent shocks to the system, which are assumed to be independent and identically distributed with zero mean and constant variances.

It is important to point out that the parameters of the system are time varying. Therefore, one has to make a time series specification for the evolution of these parameters. Following the fairly standard procedure in the literature, we have assumed that each time-varying parameter follows a random walk. Thus, the system includes nine more equations, in which each time-varying parameter follows a random walk process.

Equation 1 is a fairly standard reduced-form Phillips curve specification including lagged inflation terms, lagged output gap, and the change in the real effective exchange rate. Accordingly, persistence in inflation is reflected in inertial terms up to two lags. Output gap is assumed to affect inflation with a lag since it takes time for the pressure on production costs to be revealed and for prices to be adjusted in response to a demand shock. Changes in the real effective exchange rate capture the effects of the exchange rate dynamics on inflation through both the "cost of production channel" and the prices of imported final goods. (5)

Equation 2 is the identity defining output as the sum of the potential output (trend component) and the output gap (transitory component). Equation 3 defines potential output as a random walk with a drift model, implying that shocks to trend output are permanent. Moreover, the drift term, trend growth, is allowed to vary over time, and the persistence can be shaped with respect to different values of [rho]. Needless to say, trend growth may change over time along with productivity, labor force, or technology developments. Moreover, in a recent study, Aguiar and Gopinath (2004) state that emerging markets are subject to extremely volatile shocks to the stochastic trend and provide evidence that emerging market business cycles are driven by shocks to the trend growth rate, which could result from extreme and relatively frequent changes in economic policies. Taking these factors into account, potential growth rate is modeled as time-variant. In this respect, Equation 4 defines potential growth as a first-order autoregressive process with a long-run average growth rate of and the autoregressive coefficient 0 < [rho] < 1 representing the persistence in trend growth. The magnitude of P shows the persistence of the deviations from the long-run growth rate [[mu].sub.0]. (6) The system allows for a time-varying estimation of the persistence parameter. An estimated [rho] close to 1 means the potential output can deviate from the steady state for substantially long periods. In this respect, the setting also provides a framework to test the "cycle is the trend" hypothesis in emerging market economies, which is discussed in Aguiar and Gopinath (2004).

Equation 5 specifies the output gap dynamics. Rather than modeling the output gap by a purely stochastic process as most models do--we include variables that can provide extra information on the evolution of the output gap. We choose the variables employed in the equation so as to characterize a set of broad macroeconomic variables that can affect the actual output, but not the potential output. The main variables included are the lagged output gap, the real interest rate, the expectations of the business sector participants in the economy, and the changes in the real effective exchange rate. These factors are viewed to be more effective on the demand side, and thus neutral for the behavior of potential output in the short run. Therefore, these variables appear in the output gap equation to account for the deviation of actual output from its potential level.

In this respect, the real interest rate undertakes its traditional role as affecting the consumption and the investment behavior in the economy. The inclusion of [DI.sub.t] captures the information content that is embedded in the private sector expectations. In other words, the expectations index is supposed to capture the firms' prospects about demand conditions, which could be a major determinant of the output gap along with the real interest rate.

Although the roles of expectations and real interest rates in the whole system are clear, the role of the exchange rate dynamics on the output gap is less certain and thus should be discussed in detail. As mentioned in the first section, the seemingly positive relationship between inflation and output gap can partly be driven by another factor, which could affect both of the variables contemporaneously. In this context, because the real exchange rate can play such a role, it is included in both the first and the fifth equations. An appreciation can be associated with capital inflows and thus could lead to a temporary hike in actual demand, increasing the gap between the actual and the potential. Such an appreciation will also lead to a decrease in the cost of imported intermediate goods, which could increase supply in the economy. Taking these factors into account, the final effect of the changes in the exchange rate on the output gap is not clear. Various factors such as the exchange rate elasticity of net exports, the magnitude of the exchange rate pass-through, and the importance of imported capital goods in overall production should be taken into account to reach a robust conclusion.

It can also be argued that several other exchange rate measures could be used in both the inflation specification and the output gap equation. For example, import price inflation could be a better candidate for explaining the exchange rate pass-through to prices, whereas deviations of the real exchange rate from its long-run trend could be used in the output gap specification. (7) However, we believe that a common variable, which can affect both inflation and output gap dynamics, could be more appropriate to identify the links among the dynamics of output, inflation, and the exchange rates. Therefore, changes in the real effective exchange rate have been used for inflation as well as the output gap specification.

State Space Representation of the Model

State space modeling has been used extensively for the estimation of potential output in recent years. It not only provides the opportunity for building encompassing models but also simplifies the formulation of rather complicated problems. Moreover, once the model is written in a state space form, the Kalman filter can easily be utilized to obtain the required estimates. The general form of the state space formulation can be represented as (8)

x(t) = Fx(t - 1) + Gu(t) + [e.sub.1](t)

z(t) = Hx(t) + [e.sub.2](t),

where [e.sub.1] and [e.sub.2] denote vectors of serially uncorrelated disturbances with mean zero and covariance matrices [R.sub.1] and [R.sub.2], respectively. Furthermore, u(t) is the vector of exogenous variables. In this respect, our measurement equation, wherein the evolution of the observed variables (inflation and output) is described as a function of the unobserved state variables, is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The second measurement equation is the identity specified in Equation 2 of the model, which states that the actual output is equal to the sum of the potential output and the output gap. The unobserved variables (potential output, potential output growth rate, and the output gap) and inflation rate evolve according to the following transition equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [[eta].sub.t], [[epsilon].sub.t], [[zeta].sub.t], and [v.sub.t] are assumed to be white noise processes.

Nonlinearity in State Space Models and EKF

The transition matrix contains time-varying parameters to be estimated. Moreover, both these parameters and the state variables, which are to be estimated simultaneously, are presented in multiplicative form. Such a representation causes the state space model to take a nonlinear form, wherein it would be inappropriate to employ the standard Kalman filter. In this context, the EKF emerges as the estimation methodology, which consists of the use of the SKF equations to the first-order approximation of the nonlinear model about the last estimate. In this case, the time-varying parameters, which were all assumed to follow a random walk, are treated as a new state vector and added to the initial transition equation. Then, the EKF procedure can be applied to estimate the new state vector, which contains the parameter vector as one of its components. The EKF procedure, its application, and the smoothing algorithm are described more comprehensively in Appendices B and C.

3. Empirical Findings

On the basis of the discussion above, first we form the parameter vector, which contains nine equations. Next, we merge the state vector and the parameter vector to produce the new enhanced transition equation. Finally, we apply the extended Kalman filter to obtain the estimates for both the state variables and the time-varying parameters in the model (see Appendix B). In this section, we display and interpret these estimation results. Initially, we start with the estimated potential output and the output gap, which are presented in Figure 1. The output gap estimate confirms that the Turkish economy has been subject to severe shocks economic crises ending in rapid contractions in economic activity--several times in the sample period. As a consequence, output gap estimates exhibit sharp movements. Such an observation illustrates the excessive boom-bust cycle peculiar to emerging markets. Moreover, estimated potential output is far less smooth than a potential output that could be obtained with a standard filtering technique. These findings are also in line with Aguiar and Gopinath (2004), who argue that shocks to potential output are vital in explaining business cycles in emerging markets.

[FIGURE 1 OMITTED]

Interpretation of Output Gap Estimates: A Brief Account of Turkish "Business Cycles"

In 1994:II, because of the deep financial crisis, output declined drastically in Turkey. Our measures point to a negative output gap, which reached almost 8% during this period. Following the trough in 1994:II, the economy started to recover and the output gap closed after approximately seven quarters. Between 1996 and 1998, actual output remained above its potential level, with 1998:III being the peak for the economy. This was a period in which high interest rates attracted short-term capital, fiscal policy was expansionary, and the central bank was conducting real exchange rate targeting with an accommodative monetary policy. However, it was obvious that such an expansionary period increased the fragility of the economy and was thus destined to be rather short lived. During the period starting in 1998:III, several internal and external shocks hit the Turkish economy. The Russian crisis in 1998:III and the devastating earthquake in 1999:III were the major shocks driving the economy into a recession, which manifested itself as a trough in 1999:III.

One common feature of the 1994 and 1998 crises was the short-lived negative output gap (less than four quarters) and relatively fast recovery of economic activity after the crisis. Perhaps this explains why those contraction periods did not lead to a decline in inflation. To permanently solve the prolonged problem of high inflation and unsustainable growth, Turkey announced an IMF-supported exchange rate based stabilization program in the context of a crawling peg regime to be adopted by the beginning of 2000. It did not take a long time for the economy to recover. Indeed, the revival was quick, characterized by a rapid expansion in output and thus in the output gap. The early stages of the program witnessed a sizeable drop in interest rates, rapid credit growth, and the appreciation of the Turkish lira in real terms, eventually boosting domestic demand. Consistent with the underlying story, our measures indicate a positive output gap by the beginning of 2000:1.

In 2000, the high-rated increases in imports, because of expanding domestic demand and production, caused concerns about the sustainability of the current account. Moreover, because of the fragile structure of the banking sector, having potential problems such as maturity mismatches and open foreign exchange positions, the economy was even more vulnerable to speculative attacks. In addition to the anxiety associated with economic dynamics, political problems such as the reluctance of the government to introduce structural reforms became evident. As perceptions of the vulnerability of the economy became more pronounced, economic agents were already questioning the prospects of the macroeconomic program. The first signal of the breakdown of the program came in November 2000. The collapse of the crawling peg regime in February 2001 dragged the economy into the deepest recession that Turkey experienced in the sample period. The crisis had been extremely detrimental to the economic activity, which led to a trough in 2001:IV.

In the postcrisis period, a new economic program was implemented. The main pillars of the program were banking sector reforms, central bank independence, and fiscal discipline. These macroeconomic policies induced significant achievements on the way to economic stability. In this respect, along with declining inflation, high growth rates were attained after 2002. As a consequence, actual output converged to its potential while output gap remained at a disinflationary level in 2005:II. One exception was 2004:II, during which the actual output exceeded its potential level. This was a period with rapid credit expansion triggered mainly by the decline in interest rates to historically low levels, real appreciation of the Turkish lira against foreign currencies, and sector-specific consumption incentives, particularly in motor vehicles. Extensive credit use during this period was mostly directed toward deferred consumption of durables and machinery and equipment investment. Accordingly, domestic demand grew by 21.4% in the first half of 2004 compared with the same period of the previous year. However, to curb domestic demand, restrictive measures such as the elimination of consumption incentives and the rise in the cost of credit use were put into effect in the following period. As a result, the positive level of the output gap in 2004:II did not last for long and turned out to be transitory.

The figures reveal that, although it seems to have closed as of 2005:II, the output gap's slow convergence contributed dramatically to the disinflation process after the February 2001 economic crisis. Indeed, one major difference between the 2001-2004 period and the 1994 and 1998-1999 periods was definitely the gradual closing of the gap in the former.

Time-Varying Parameter Estimates

The evolution of the time-varying estimates of the system parameters are displayed in Appendix D. The common emphasis of the recursive estimates is that economic crises had a significant influence on the parameters. The 1994 crisis, especially, seems to have led to drastic changes in most of the parameter estimates.

The autoregressive coefficient of potential growth is estimated as 0.78 at the end of the sample, implying that potential growth is fairly persistent. This means that in the absence of shocks, output growth would converge to within 1% of the steady-state rate in just about five years. On the other hand, the sum of the coefficients for [[pi].sub.t-1] and [[pi].sub.t-2] ([[alpha].sub.1] and [[alpha].sub.2]) is high, pointing to a significant degree of persistence in inflation (Table 1).

The parameter estimates in the inflation specification reveal that a positive output gap is associated with higher inflation ([[alpha].sub.3]) as expected, whereas the real exchange rate coefficient takes on an unexpected estimate ([alpha].sub.4]), albeit the effect being very close to zero is economically insignificant. In fact, the real exchange rate has a considerable share in the total cost of production and a real depreciation is expected to raise inflation both by an increase in the prices of imported final goods and through the higher cost of production. Historical evidence indicates that a period of improved confidence in the economy coupled with sustained real appreciation leads to a drop in inflation through tradable goods in the CPI basket and weakening indexation behavior. Therefore, the unexpected estimate of [[alpha].sub.4] might have been caused by an identification problem resulting from the tight relationship between the real exchange rate and the output gap, which might capture the whole effect of a linear combination of real marginal cost components (i.e., real unit labor cost and real exchange rate). It is also worth noting that the time-varying estimates of [[alpha].sub.4] in the first alternative model follow a similar path across time with those of the baseline model but with a negative end-point estimate of -0.03 (Appendix Figure D1). The positive relationship between inflation and output gap is consistent with the standard Phillips curve notion. (9)

The role of the real exchange rate on output gap measures, which is shown by the parameter [[gamma].sub.4], deserves close attention. Our estimates imply that appreciation periods are associated with positive gaps. In other words, appreciation (depreciation) of domestic currency is mostly associated with periods of economic expansion (contraction). This unconventional finding suggests that a third factor might drive both output and exchange rates (such as economic crises or capital flows) in most of the sample period. Moreover, such a finding could suggest that the cost channel of the exchange rate on output gap dominates the demand channel. That is to say, an appreciation of domestic currency leads to a decline in the cost of imported capital goods, thereby creating an incentive for increasing production given favorable demand conditions. Such an increase outweighs a possible worsening in net exports driven by the real appreciation of domestic currency.

The real interest rate has the expected sign to explain output gap dynamics, except for the beginning of the sample period. Moreover, the effect of the real interest rate on the output gap as of 2005:11 is estimated as -0.09. Real interest rates and output gap have a positive relationship until 1996. It is interesting to observe that after 1996, the relationship between real interest rate and output gap turns out to be negative. The unstable empirical coefficient pertaining to the relationship between the interest rate and the output gap in Turkey can be attributed to several factors. First of all, interest rates had never been used as a policy instrument to attain the inflation target before the floating regime. Instead, most of the sample period is dominated by crawling peg/fixed exchange rate regimes, in which exchange rate developments were the primary determinant of inflation and output dynamics in Turkey because of high indexation. Second, in an unstable economy with high and persistent inflation and uncertainty, real interest rates generally remain at excessively high levels. This, in turn, renders the economic agents' decisions as insensitive to intertemporal shifts in real interest rates, weakening the link between the interest rates and spending decisions. In this respect, the interest rate sensitivity of aggregate demand was limited and the conventional monetary transmission mechanism was not evident in Turkey during the 1980s and 1990s.

The discussion on the time-varying behavior of the real interest rate coefficient can also be extended to another field, namely the literature on "central bank credibility." Following Rotemberg and Woodford (1999), as long as aggregate demand depends on long and short rates, the monetary authority will not need to move interest rates too much, provided that agents are rational and believe in the policymakers' commitment. In other words, under the aforementioned conditions, a commitment to raise interest rates is sufficient to bring about a contraction in aggregate demand in response to a shock that is expected to lead to inflationary pressures. Therefore, regarding the period after the February 2001 crisis, the sustained decline in the time-varying estimates of the real interest rate (increased sensitivity of output gap to real interest rate) can also be attributed to the ongoing improvement in the credibility of policymakers, as well as a probable increase in the share of forward-looking agents subsequent to the successful implementation of macroeconomic policies during this period.

Time-varying parameter estimates suggest that transition to a financial restructuring period and a new policy regime--implicit inflation targeting along with the floating exchange rate regime--in February 2001 had significant effects on the underlying economic dynamics in Turkey. Estimates are consistent with the view that interest rates have been the main policy instrument in recent years to attain announced targets for the inflation rate. Moreover, there is evidence of a weakened exchange rate pass-through on consumer prices. (10) All in all, the parameter estimates and their historical evolution are in line with the main characteristics of the Turkish economy.

4. Sensitivity Analysis

To analyze whether the results presented in the previous section are sensitive to the specification of the model (and to the magnitude of the shocks), we estimate both the output gap and the system parameters with two alternative specifications of output gap dynamics. We also compare our output gap estimates with the standard Kalman filter. Finally, we contrast the results of our methodology with those of the HP and standard Kalman filters by investigating the robustness properties of the output gap estimates.

Alternative Models

As mentioned above, in view of the inherent uncertainty regarding the main characteristics of the Turkish economy, we will estimate the same system with two alternative models and contrast the results with our baseline setting. As a first exercise, we change the output gap dynamics. Given the theoretically ambiguous sign of the relationship between the output gap and the real exchange rate, as well as the ad hoc nature of the demand index, in the first alternative model, we drop the demand index and the real exchange rate from the output gap dynamics. In this case. only the lagged value of the output gap and the real interest rate appear in the output gap specification. Our second alternative drops the real exchange rate in the inflation equation, specifying inflation as a function of only its lagged values and the output gap, whereas the remaining equations appear exactly the same as in the baseline model.

Figure 2 displays the output gap estimates obtained under our baseline model and two alternative models. The output gap derived with the second alternative model (with different inflation dynamics) is close to the gap computed by our baseline model. In other words, as long as the output gap appears as a right-hand side variable in the inflation equation, our methodology is quite robust to changes in the inflation dynamics. The figure also depicts the first alternative model (with different output gap dynamics) as exhibiting some quantitative discrepancy with the other two and thus revealing some degree of sensitivity to the output gap specification. However, the differences are not sizeable in quantitative terms. Moreover, these discrepancies mostly occur in the peak or the trough of the cycle. All three output gap measures are qualitatively similar, so the periods of contraction and expansion are evidently the same.

[FIGURE 2 OMITTED]

The end-sample coefficient estimate of the output gap in the inflation specification for the first alternative model is 0.09. In the second alternative model, when the coefficient of the real exchange rate is restricted to zero, the estimated coefficient of the output gap in the inflation equation goes up to 0.24 at the end of the sample (Table 2). Therefore, all three models imply that a positive output gap is associated with rising inflation. All in all, the models point out a similar output gap path and the results are fairly robust to different specifications for output gap and inflation dynamics.

Extended Compared with Standard Kalman Fiher

We have already argued that model coefficients are not expected to stay constant in a country that is subject to severe shocks and regime shifts. This was our main motivation for use of the extended Kalman filter. However, it still remains to check whether the output gap estimates derived with the assumption of time-varying parameters differ significantly from those of time-invariant parameters, which is the task of this subsection. To do so, we estimate our baseline model with the standard Kalman filter, restricting the coefficients to be constant over time and compare the results with those obtained from the extended Kalman filter (Figure 3).

[FIGURE 3 OMITTED]

The alternative output gap figures are similar in terms of their turning points. However, they differ qualitatively, especially in the sharp contraction and expansion periods--the 1994 and 2001 crises and the expansion period in 2000. The most striking difference concerns the expansion in 2000. Contrary to the extended Kalman filter estimates, the standard Kalman filter suggests that the output gap was barely closed at the end of 2000. This is at odds with our intuition, in the sense that this period was characterized by a sizeable drop in interest rates, large capital inflows, and rapid credit growth, eventually boosting the domestic demand. We believe that the output gap measure derived by the extended Kalman filter is more consistent with the underlying history, pointing to a significant positive output gap in the year 2000. On the other hand, the end-sample information revealed by the two estimates is similar, with the output gap just closing.

After the comparison of estimated figures from standard and extended Kalman filters, it would be helpful to evaluate the uncertainty surrounding the estimated output gaps for the baseline model across the methods. A comparison of the estimated average standard error of the output gaps for the extended and standard Kalman filter methods ([[bar.[sigma]].sub.gap_EKF]/[[bar.[sigma]].sub.gap_SKF]) implies that the introduction of time variation in the model parameters increases the uncertainty by 32.1%. Hence, the results indicate that a higher uncertainty is associated with extended Kalman filter estimates. (11)

Revision Properties of Estimates (Comparison with SKF and HP Filter)

Ideally, the output gap estimates should neither be revised drastically nor exhibit large swings when new observations are added to the analysis. One major drawback of typical output gap estimating techniques such as the HP filter, however, is their sensitivity to new data releases. Because the HP filter is a two-sided optimization procedure that uses both lead and lagged information, the accuracy of the filter diminishes at the end of the sample because of missing lead information. In other words, because filters use both past and future data, problems can arise at the end of the sample because of the lack of future data. This problem is generally called the "end-point" problem. In fact, this is unfortunate considering that the last observations are often the most vital ones for monetary policy analysis in the sense that these are the ones used for forecasting inflation. To assess whether our results suffer from the same syndrome and to see how sensitive they are to the inclusion of new data or revisions, we compare our estimates with the results of a more standard technique, such as the HP filter and the standard Kalman filter. (12)

Initially, we evaluate the revision properties of the HP and extended Kalman filter estimates at certain cut-off points. To this end, Figure 4 compares the results from two filtering methods when the sample is cut at 2002:I and at 2003:I along with the full sample estimates. The resulting HP filter estimates exhibit major differences. For example, when the sample is cut at 2003:I, the output gap is positive with a magnitude of 3.1%, whereas the full sample estimates point to a negative gap with a magnitude of 1.6% during the same period. On the other hand, carrying out the same exercise with our multivariate filter shows that the corresponding gap estimates are close to each other and fairly insensitive to the inclusion of new data. In other words, our output gap estimates are more robust than those of the HP filter.

[FIGURE 4 OMITTED]

In the following, the alternative filtering methods, namely extended Kalman, standard Kalman, and HP filter, are compared by running the filters at every possible cut-off point for the last five years. The comparison of the estimated figures with the final estimates at every data point may provide a course to evaluate the success of filters in terms of revision properties. At this point, it is important to keep in mind that, for each model, the output gap series that is calculated by using the whole sample is assumed to be the "best" output gap measure. In the assessment of the relative performance of the methods, the mean absolute deviation (MAD) statistic is employed following Graft (2004), in which MAD is defined as (1/t) [summation][absolute value of [P.sub.t] - [F.sub.t]], where [P.sub.t] and [F.sub.t] stand for the provisional and the final estimated figures, respectively. Figure 5 plots the MAD values across the alternative filters starting from 2001:I. For example, values at 2001:I denote the mean absolute deviations between the output gap figures estimated using the whole sample and the corresponding ones obtained when the sample is cut at 2001:I. In this respect, Figure 5 summarizes the average revisions in absolute terms by running a number of rolling output gap estimates at every possible point for the last five years.

[FIGURE 5 OMITTED]

A comparison of MAD statistics over time seems to indicate that the extended Kalman filter delivers better revision properties compared to its rivals. Additionally, revisions of HP filter estimates (in absolute terms) are smaller than those of standard Kalman filter estimates. When parameters are fixed at their final maximum likelihood estimates, the standard Kalman filter outperforms the HP filter. However, given that it is not possible to know the full sample parameter estimates in real life, the results of this exercise point out that standard Kalman filter estimates are prone to comparatively large revisions.

These findings reveal that estimating the output gap with HP and standard Kalman filters can be quite misleading, especially when the economy is subject to large swings, since the revisions in the output gap can be substantial. Hence, it would be fair to conclude that multivariate filters relying on time-varying parameters may perform better than their constant parameter counterparts.

5. Concluding Remarks

We have presented a time-varying parameter methodology for constructing an estimate of output gap by exploiting the extended Kalman filter technique in a multivariate setting, incorporating economic content by the inclusion of inflation and output gap dynamics. Our findings suggest that estimating the output gap in a multivariate time-varying setting may have many advantages over univariate techniques such as the HP filter. Sensitivity analysis, for example, shows that using a purely statistical filter can lead to serious policy mistakes. Some reverse engineering exercise can be useful for understanding the latter statement. Suppose that the Central Bank of Turkey had employed a typical Taylor rule where output gap is one of the parameters against which the monetary authority reacts. Had the bank measured the output gap by HP filter, ceteris paribus, the monetary stance would have been tightened in Turkey after 2002:II. In contrast, had the bank used the extended Kalman filter to measure the output gap, the Taylor rule would have commanded a loosening in the monetary policy. Indeed, with no explicit tightening in monetary policy, inflation fell drastically after 2002. This ex post evaluation suggests that our multivariate time-varying filter might yield superior policy advice compared to statistical counterparts such as the HP filter.

Nevertheless, these findings do not imply that the caution associated with the output gap measures is unwarranted. First of all, data revisions and lags in the announcement of data give rise to criticism about the real-time accuracy of the output gap estimates, since the information available to the policymaker at the time policy decisions are taken may be subject to significant changes. Second, and more importantly, since output gap is unobservable by definition, its measures are not any more reliable than the model they are based on. (13) In that sense, establishing the appropriate model to represent the structure of the economy is of great importance. For example, one possible extension worth exploring could be the joint estimation of output gap with unemployment gap.

Notwithstanding these shortcomings and practical problems, output gaps still appear--and are likely to continue appearing--in the inflation reports of many central banks. Nonetheless, being aware of the numerous shortcomings of the output gap estimation techniques, central banks also acknowledge that output gap is only one of the many variables in the information set that policymakers use for forecasting future inflation and interpretation of the output gap measures along with the information provided by other indicators is critical to improving the overall quality of forecasting inflation.

Appendix A: Data Description

The sample period covers quarterly data between 1988:II and 2005:II. The issue of seasonality is handled with the commonly used program TRAMO/SEATS (Gomez and Maravall 1998).

* [y.sub.t] is the logarithmic seasonally adjusted gross domestic product at 1987 constant prices.

* [[pi].sub.t] is the log difference of the quarterly seasonally adjusted consumer price index (1994 = 100 and 2003 = 100).

* [reer.sub.t] is the log difference of the CPI-based real effective exchange rate (1995 = 100).

* [r.sub.t] is the ex post real interest rate on government securities, quarterly average of compound interest rates for discounted Treasury auctions, weighted by the share of the respective amounts sold.

* [DI.sub.t] is the demand index constructed from the Business Tendency Survey (BTS) of the Central Bank of the Republic of Turkey. Components of the index are BTS question 9 (total amount of orders received this month), question 16 ([trend of the next three months] volume of goods sold in the domestic market), and question 18 ([trend of next three months] volume of raw-material stocks).

Appendix B: Extended Kalman Filter Algorithm (14)

The following algorithm presents the results when the extended Kalman filter is applied to the estimation of the parameter vector of [theta] = ([[alpha].sub.1], [[alpha].sub.3], [rho], [[gamma].sub.1], [[alpha].sub.2], [[alpha].sub.4], [[gamma].sub.2], [[gamma].sub.3], [gamma].sub.4]). In addition to the usual Kalman filter algorithm, here we also have random parameters, which are assumed to be evolving according to the random walk process. Simply, in the EKF case, because of the nonlinear relationship, we linearize the process and measurement functions at the current state estimate with the use of partial derivatives and then apply the usual Kalman filter algorithm.

The unobserved state vector X can be seen as partitioned into two parts: One is the usual unobserved state variables and the other is the unknown parameter vector

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (B1)

where [bar.K] and [bar.P] are the Kalman gain and covariance matrices, respectively, for the extended state. The general algorithm is shown in Equations B2 B9.

[S.sub.t] = [H.sub.t][P.sub.1](t)[H.sup.T.sub.t] + [H.sub.t][P.sub.2](t)[D.sup.T.sub.t] + [D.sub.t][P.sup.T.sub.2](t)[H.sup.T.sub.t] + [D.sub.t][P.sub.3](t)[D.sup.T.sub.t] + [R.sub.2] (B2)

L(t) = [[P.sup.T.sub.2](t)[H.sup.T.sub.t] + [P.sub.3](t)[D.sup.T.sub.t]][S.sup.-1.sub.t] (B3)

K(t) = [[F.sub.t][P.sub.1](t)[H.sup.T.sub.t] + [M.sub.t][P.sup.T.sub.2](t)[H.sup.T.sub.t] + [F.sub.t][P.sub.2](t)[D.sup.T.sub.t] + [M.sub.t][P.sub.2](t)[D.sup.T.sub.t] + [R.sub.12]][S.sup.-1.sub.t] (B4)

[P.sub.1](t + 1) = [F.sub.t][P.sub.1](t)[F.sup.T.sub.t] + [F.sub.t][P.sub.2](t)[M.sup.T.sub.t] + [M.sub.t][P.sup.T.sub.2](t)[F.sup.F.sub.t] + [M.sub.t][P.sub.3](t)[M.sup.T.sub.t] - K(t)[S.sub.t][K.sup.T](t) + [R.sub.1] (B5)

[P.sub.2](t + 1) = [F.sub.t][P.sub.2](t) + [M.sub.t][P.sub.3](t) - K(t)[S.sub.t][L.sup.T](t) (B6)

[P.sub.3](t + 1) = [P.sub.3](t) - L(t)[S.sub.t][L.sup.T](t) + Q (B7)

[??](t + 1) = [F.sub.t][??](t) + [G.sub.t]u(t) + K(t)[z(t) - [H.sub.t][??](t)] (B8)

[??](t) = [??](t - 1) + L(t)[z(t) - [H.sub.t-1][??](t)] (B9)

Here,

[F.sub.t] = F([??](t),[G.sub.t] = G([??](t)), [H.sub.t] = H([??](t)), (B10)

[M.sub.t] = M([??](t),[??](t),u(t)), with M([??],x,u) = [partial derivative]/[partial derivative][theta][F([theta])x + G([theta]u]|[sub.[theta]] = [??]], (B11)

and

[D.sub.t] = D([??](t- l),[??](t)) with D([??], x) = [partial derivative]/[partial derivative][theta][H([theta])x]|[sub.[theta]] = [??]]. (B12)

Finally, regarding the initial values,

[P.sub.1](O) = [P.sub.1,0],[P.sub.2](0) = 0, [P.sub.3](0) = [P.sub.3,0], [??](0) = [x.sub.0],[??](0) = [[theta].sub.0] (B13)

Appendix C: The Smoothing Algorithm

The potential output can be estimated in two ways depending on what information is used. The filtered estimate at time t is one-sided and it uses information to time t([y.sup.*.sub.t|t]). Therefore, the Kalman filter (used as a real- time or online algorithm) estimates the state vectors exploiting current and past information. On the other hand, a smoothed value is two-sided and uses information from the whole sample, up to time T([y.sup.*.sub.t|T]), where 0 [less than or equal to] t [less than or equal to] T. In this way, the smoothing algorithm allows for considering future information as well in the estimation of potential output. Unless there is some immediate real-time constraint, state estimates can be improved with the smoothing algorithms. Referring to the fixed- interval smoothing (Ranch-Tung Striebel two-pass smoother) algorithm, the smoothed estimator can be defined as

X(t - 1|T) = X(t - 1|t - 1) + B(t)[X(t|T) - X(t|t - 1)], (C1)

with its corresponding covariance matrix 

[bar.P](t - 1|T) = [bar.P](t - 1|t - 1) + B(t)[[bar.P](t|T) - [bar.P](t|t - 1)]B[(t).sup.t], (C2)

where B(t) = [bar.P](t - 1|t - 1)f(t)' [[bar.P].sup.-1](t|t - 1) for t = T ... 1 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

with k representing the number of the time-varying parameters. Because the smoother is based on more information than the filtered estimator, it will have an mean square error that is smaller than that of the filtered estimator.

Appendix D: Time-Varying Parameter Estimate

[GRAPHICS OMITTED]

We thank Levent Ozbek for his valuable assistance and suggestions. We are also grateful to Dilara Ece, Arzu Cetinkaya, and Ayse Tatar Curl for their contributions throughout this project. The views expressed are those of the authors and should not be attributed to the Central Bank of the Republic of Turkey.

Received March 2006; accepted November 2006.

References

Aguiar, M., and G. Gopinath. 2004. Emerging market business cycles: The cycle is the trend. Mimeo. Chicago, IL: University of Chicago.

Anderson, B. D. O., and J. B. Moore. 1979. Optimal filtering. Englewood Cliffs, NJ: Prentice Hall.

Benes, J., and P. N'Diaye. 2003. A multivariate filter for measuring potential output and the NAIRU. The Czech National Bank's Forecasting and Policy Analysis System, Chapter 5, pp. 99-118.

Billmeier, A. 2004. Ghostbusting: Which output gap measure really matters? IME Working Paper No. 04/146.

Boone, L., M. Juillard, D. Laxton, and P. N'Diaye. 2001. How well do alternative time-varying parameter models of the NAIRU help policy makers forecast unemployment and inflation in the OECD countries? IMF Working Paper, Draft Version.

Butler, L. 1996. A semi-structural method to estimate potential output: Combining economic theory with a time-series filter. The Bank of Canada's New Quarterly Projection Model (QPM), Part 4.

Edwards, S., and M. A. Savastano. 1999. Exchange rates in emerging economies: What do we know? What do we need to know? NBER Working Paper No. 7228.

Gomez, V., and A. Maravall. 1998. Seasonal adjustment and signal extraction in economic time series. Banco de Espana No. 9809.

Graft, M. 2004. Estimates of the output gap in real time: How well have we been doing? Reserve Bank of New Zealand Discussion Paper Series DP2004/04.

Harvey, A. C. 1990. Forecasting, structural time series models and the Kalman filter. Cambridge, UK: Cambridge University Press.

Kara, H., H. Kucuk-Tuger, B. Tuger, U. Ozlale, and E. Yucel. 2005. Exchange rate pass-through in Turkey: Has it changed, and to what extent? Central Bank of the Republic of Turkey (CBRT) Working Paper No. 05/04.

Karagedikli, O., and L. C. Plantier. 2007. Do multivariate filters have better revision properties? An empirical analysis. Reserve Bank of New Zealand Discussion Paper Series, forthcoming.

Kuttner. K. N. 1994. Estimating potential output as a latent variable. Journal of Business and Economic Statistics 12:361-68.

Leigh, D., and M. Rossi. 2002. Exchange rate pass-through in Turkey. IMF Working Paper No. 02/204.

Ljung, L., and T. Soderstrom. 1983. Theory and practice of recursive identification. Cambridge, MA: The MIT Press.

Meese, R., and K. Rogoff. 1983. The out-of sample failure of empirical exchange rate models. In Exchange rate and international macroeconomics, edited by J. Frenkel. Chicago: University of Chicago Press.

Ogunc, F., and D. Ece. 2004. Estimating the output gap for Turkey: An unobserved components approach. Applied Economics Letters 11:177-82.

Orphanides, A. 1999. The reliability of output gap estimates in real time. Finance and Economics Discussion Series 199- 938, August 1999.

Ozbek, L., and U. Ozlale. 2005. Employing extended Kalman filter in measuring output gap and potential output for the Turkish economy. Journal of Economic Dynamics and Control 29:1611-22.

Ozlale. U, and K. Metin-Ozcan. 2003. Does time inconsistency problem apply for Turkish monetary policy. Unpublished paper, Bilkent University, Ankara, Turkey,

Rotemberg, J. J., and M. Woodford. 1999. Interest-rate rules in an estimated sticky-price model. In Monetary policy rules, edited by J. B. Taylor. Chicago: University of Chicago Press.

(1) See Billmeier (2004) and Orphanides (1999) for an account of the information content of the output gap estimates widely used in the literature.

(2) The HP filter is subject to criticism as being purely statistical and having no economic content because it does not exploit any information other than the series to be detrended. Moreover, the cyclical component always sums to zero in the HP filter, which does not necessarily hold, especially when the sample size is small. The criticism also focuses on the view that the values suggested by Hodrick and Prescott are specific to U.S. data and could lead to misspecification of the underlying economic structure of other economies. The weighting parameter in the objective function, which controls the degree of smoothness of the trend component, is subjectively determined by the user. Economic theory provides little information regarding the value of this smoothing parameter. As stated by Butler (1996), rather than imposing the weighting parameter, it is possible to estimate this parameter in a multivariate setting. Boone et al. (2001) show that the optimization procedure used by the HP filter can be specified as a univariate unobserved components model in which the smoothing parameter represents the relative variances of a shock to the cycle component and a shock to the trend growth rate component (in our notation corresponds to [lambda] = [[sigma].sup.2.sub.[zeta]]/[[sigma].sup.2.sub.[epsilon]]), which can be estimated by prediction error decomposition via the Kalman filter.

(3) Kuttner (1994), for example, proposes a filter that estimates the output gap consistent with the observed inflation using a Phillips curve relation. See Ogunc and Ece (2004) for an application to Turkish economy. These authors find that incorporating the supply side into the output gap system reduces the parameter uncertainty and the total standard error, hence improving output gap estimate.

(4) For example, the effects of the transition to a floating exchange rate regime along with an inflation-targeting framework after the collapse of the exchange rate-based stabilization program cannot be adequately captured in a model in which the system parameters are assumed to be constant over time, especially if the models are based on reduced-form ad hoc relationships that do not take forward-looking elements into account.

(5) See Leigh and Rossi (2002) for more on exchange rate pass-through in Turkey.

(6) The sustainable steady-state real growth rate for the economy is assumed to be 4.5% on an annual basis, which corresponds to 1.106% per quarter.

(7) By employing the Prior Consistent filter, Benes and N'Diaye (2003) use such a measure. However, following Meese and Rogoff (1983) and Edwards and Savastano (1999), there is no commonly agreed methodology in determining the equilibrium real exchange rate and, thus, the deviations from such equilibrium.

(8) For a detailed description of the slate space models and the Kalman filter, see Anderson and Moore { 19791 and Harvey (1990).

(9) In a previous study where the standard Kalman filter was employed, Ogunc and Ece (2004) also found a positive relationship between inflation and output gap for the Turkish economy with a parameter estimate of 0.24.

(10) See Kara et al. (2005) for more evidence on reduced exchange rate pass-through in Turkey.

(11) This should not be surprising because more parameters are estimated in the extended Kalman filter.

(12) We could also contrast the performance of our methodology with a multivariate HP filter. However, on the basis of the results of several recent studies, which find counter evidence against the superiority of the multivariate filters over a simple HP filter in terms of real-time properties, we prefer to use the simple standard HP filter in our comparison. Graff (2004) shows that during the recorded history of the multivariate HP filter in the Reserve Bank of New Zealand, the revisions to real-time output gap have been no smaller than if a standard HP filter had been used. Using New Zealand data, Karagedikli and Plantier (2007) also find that the real-time properties of multivariate HP filters do not improve over the simple HP filter because of uncertainty in the trends and the equilibrium values of augmenting variables, which are included in the system to alleviate the effect of revisions on the output gap estimates.

(13) Some recent studies cast doubt on the positive relationship between inflation and output gap implied by the Phillips curve notion. See, for example, Ozlale and Metin-Ozcan (2003), who document a negative relationship between inflation and output gap. Moreover, recent studies based on micro-founded models indicate that what drives inflation is the real marginal cost rather than some ad hoc output gap measure, implying that reduced-form traditional Phillips curve relationships might not be the best way to model inflation dynamics.

(14) The algorithm is discussed in detail in Theory and practice of recursive identification (Ljung and Soderstrom 1983).

Hakan Kara, Central Bank of the Republic of Turkey, Research and Monetary Policy Department, Istiklal Caddesi No. 10, Ulus, Ankara 06100, Turkey; E-mail [e-mail].

Fethi Ogunc, Central Bank of the Republic of Turkey, Research and Monetary Policy Department, Istiklal Caddesi No. 10, Ulus, Ankara 06100, Turkey; E-mail [e-mail].

Umit Ozlale, TOBB University of Economics and Technology, Department of Economics, Sogutozu Caddesi No. 43, Sogutozu Ankara 06560, Turkey; E-mail [e-mail]; corresponding author.

Cagri Sarikaya, Central Bank of the Republic of Turkey, Research and Monetary Policy Department, Istiklal Caddesi No. 10, Ulus, Ankara 06100, Turkey; E-mail [e-mail].

Table 1. End-Sample Estimates for the Model Parameters

Model Parameters   Baseline Model

[[alpha].sub.1]         0.38
[[alpha].sub.2]         0.56
[[alpha].sub.3]         0.35
[[alpha].sub.4]         0.01
[rho]                   0.78
[[gamma].sub.1]         0.21
[[gamma].sub.2]        -0.09
[[gamma].sub.3]         0.20
[[gamma].sub.4]         0.04

Table 2. End-Sample Estimates for the Alternative Model Parameters

                  Baseline   Alternative   Alternative
Model Parameters   Model       Model 1       Model 2

[[alpha].sub.1]     0.38        0.11          0.57
[[alpha].sub.2]     0.56        0.76          0.34
[[alpha].sub.3]     0.35        0.09          0.24
[[alpha].sub.4]     0.01       -0.03           --
[rho]               0.78        0.71          0.79
[[gamma].sub.1]     0.21        0.49          0.23
[[gamma].sub.2]    -0.09       -0.03         -0.08
[[gamma].sub.3]     0.20         --           0.21
[[gamma].sub.4]     0.04         --           0.04