Nonlinear Relation of Inflation and Nominal Interest Rates A Local Nonparametric Investigation

Transcription

1 Nonlinear Relation of Inflation and Nominal Interest Rates A Local Nonparametric Investigation Marcelle Chauvet and Heather L. R. Tierney * Abstract This paper investigates the relationship between nominal interest rates and inflation in the U.S. in the last 50 years. In particular, we use a recursive framework in which all model parameters are time varying, not only across sub-samples, but at any point in time. This allows examination of potential structural breaks in inflation response to monetary shocks as well as systematic changes over the business cycle. Our goal is to study not only if monetary policy and inflation coefficients have changed over time, but also if they change across the state of the economy, or when inflation is in a high or low growth phase. We propose a nonparametric tool to investigate local dynamic impulse response functions in a VAR system. The advantage of this tool is that it allows analysis at each iteration of the local average effects of a shock to any given variable in the VAR system. In addition, potential nonlinearities, nonstationarities, and asymmetric behavior can be examined without the need for specifying a functional form, and information in the tail regions can be incorporated in the model. We use this framework to shed light into the potential nonlinear dynamics of inflation and interest rates. KEY WORDS: Inflation, nominal interest rates, vector autoregression, nonparametric, monetary policy. JEL Classification: E40, E52, E58 Marcelle Chauvet, Department of Economics, University of California, Riverside, CA ; chauvet@ucr.edu. Heather L.R. Tierney, Department of Economics and Finance, College of Charleston, 5 Liberty Street, Charleston, SC, 29424; tierneyh@cofc.edu.

2 . Introduction In recent years, several factors have revitalized interest in monetary policy in the U.S. The unusual price stability achieved in the early 2000s, with a declining inflation rate after the 200 recession raised concerns regarding the risks of potential deflation. This possibility instigated public debates and several academic contributions over the costs and benefits of the preemptive decrease in nominal interest by the Fed to its lowest level in many decades in order to stimulate economic growth. More recently, the resurge in core prices have triggered new debates on monetary policy conduct. This paper investigates the relationship between nominal interest rates and inflation in the U.S. in the last 50 years. In particular, we use a recursive framework in which all model parameters are time varying, not only across sub-samples, but at any point in time. This allows examination of potential structural breaks in inflation response to monetary shocks as well as systematic changes over the business cycle. Our goal is to study not only if monetary policy and inflation coefficients have changed over time, but also if they change across the state of the economy, or when inflation is in a high or low growth phase. This framework can shed light into the potential nonlinear dynamics of inflation and interest rates. We propose a nonparametric tool to investigate local dynamic impulse response functions in a VAR system. In general, nonparametric estimation requires aggregation of coefficients for the purpose of statistical inference. This paper, in contrast, constructs nonparametric local orthogonalized impulse response functions that do not involve aggregation. The advantage of this tool is that it allows analysis at each iteration of the local average effects of a shock to any given variable in the VAR system. In addition, potential nonlinearities, nonstationarities, and asymmetric behavior can be examined without the need for specifying a functional form, and information in the tail regions can be incorporated in the model. The methodology involves first the estimation of a vector autoregression model (VAR), which is undertaken using local linear least squares non-parametric algorithm (LLLS). Next, we obtain the nonparametric conditional orthogonalized impulse response functions. In particular, the functions are formed using Choleski decomposition in the factorization of the nonparametric conditional variance-covariance matrix for each t th iteration of the VAR for t =,, T. As a result, T sets of nonparametric conditional orthogonalized impulse response functions are obtained. These functions measure the short and medium run effects from any t th iteration of the 2

3 resulting VAR, and provide information on the strength or weakness of the current relationship between inflation and nominal interest rates. A large body of research has been devoted to the examination of the relationship between inflation and nominal interest rates using vector autoregression models. One of the findings is that sudden rises in interest rates have been followed by an initial positive response in inflation, which is known and the price puzzle. Sims (992) suggests that this might be caused by an insufficient response by the Federal Reserve to expectations of higher future inflation. However, some authors find that the puzzle is stronger or weaker across sub-samples. Others find evidence of breaks in the relationship between inflation and nominal interest rates, especially around 980 (see e.g. Tierney 2005). Finally, there is a large recent literature examining the reduced variability of inflation and nominal interest rates since the mid 980s. Some authors attribute this to changes in the real side of the economy that have transmitted to inflation, whereas others reckon that the culprit is the more transparent and effective monetary policy conduct, which has reduced the oscillations in inflation and interest rates. The method proposed can shed light on the price puzzle, potential nonlinear relationship between inflation and nominal interest rates, and changes in their variance over time. The local nonparametric VAR is applied to inflation and nominal interest rates, and the estimated nonparametric coefficients, variances and impulse response functions are used as tools to investigate the impact of monetary policy in the economy around periods of high and low inflation growth, taking into account the possibility of parameter changes across business cycle phases and structural changes. We find substantial evidence of nonlinearities and nonstationarities in the relationship between inflation and interest rates. For example, during low inflation periods the response of inflation to shocks in interest rates is much more effective than during periods of high inflation. In addition, we find that the coefficients linking these two variables display breaks and reversal correlation around business cycle turning points. The estimated time-varying coefficients also display nonstationarities in several dates, particularly during high inflation phases. Finally, we also find that both the variance of innovations in inflation and in interest rates have decreased over time, although this has taken place a lot earlier for inflation than for interest rates. 3

4 The structure of this paper will be as follows: Section 2 presents the nonparametric VAR model and the proposed methodology for estimation of the parameters and impulse response functions. The empirical model and results are presented in Section 3, and Section 4 concludes. 2. Theoretical Model The theoretical model involves a local and a global component, which can be obtained from the nonparametric regression coefficients and variance-covariance matrices. More specifically, in order to form the nonparametric VAR(p) the local linear least squares non-parametric method (LLLS) is used. The LLLS method fits a local least squares line within an interval as specified by the window width of the empirical kernel density function. The empirical kernel density function places more weight on the observations closest to the conditioning observation and increasingly less weight as the distance between the measured observation and conditioning observation increases. This conditioning process is done for each and every observation of each regressor, and hence is better able to capture the information in the tails regions. As with a parametric VAR(p), the nonparametric VAR can be estimated equation-byequation provided that each equation has the same set of regressors, and the error terms of each equation are not correlated with the set of regressors. Let the regressand for each iteration of the VAR be denoted as y = ( y L y L y y ), which is an ( n ) t t rt n t nt column vector with t =,,T where T equals the total number of observations, n is the total number of equations in each iteration of the VAR, and r represents the r th regressand or, equivalently, the r th equation of the VAR with r =,, n. Furthermore, let the complete matrix of regressands comprised of n column vectors and T number of observations be denoted as Y ( Y Y Y Y ) ( ) Y = y L y L y y r r rj rt rt = L L with r n n, where j =,,T. Hence, the transpose of each row of the matrix Y represents an iteration of the VAR. Regarding the set of regressors for each t th iteration of the VAR, the set of regressors is denoted as x = ( z x L x L x x ), which is a ( n q) matrix with q = ( k+ ) t t t mt k t kt including an intercept term for each equation. In our framework, the n rows of the regressor This notation is useful when the VAR is estimated by the equation-by-equation method. 4

5 matrix are identical for each t th iteration of the VAR. z t is the column vector of constant terms with dimension ( n ), and x mt is a column vector with dimension ( n ), for m =,,k. Similar to the regressand matrix, the regressor matrix is composed of q column vectors with a total of T number of observations, which is denoted as X = ( Z X L X L X X ) with Xm ( xm xmj xmt xmt) m k k = L L where j =,,T. Thus, each row of the matrix X represents an iteration of the VAR. We use Gaussian as the empirical kernel density function. For each iteration of the VAR(p), the Gaussian kernel density is of the form: where K( ψ j ) T j= ( j ) K = K ψ, () X xj Xm xmj Xk xkj = exp k + L + L + 2 h 2 hm h k ( 2π ) X xj Xm xmj Xk x kj with ψ j = h L h L m h and j =,, T. p The window width represents the interval in which the local least squares line is to be fitted. In order to prevent over-or-under-smoothing of the data, the window width, h m, which represents the window width of the m th regressor X m, is chosen through the method of cross validation. In this method, the optimal window width minimizes the sum of squared errors of each equation of the VAR (see Pagan and Ullah, 999). The j th iteration of the kernel density function, K, can be represented as a ( T T) matrix of zeroes with the only non-zero element being in the j th row and j th column, which represents the j th conditional non-parametric coefficients based upon the j th observation of the set of regressors, for the r th regressand of the VAR(p) that consists of a ( q ) column vector denoted as: rj ( X KX ) β = X KY (2) r Using the transpose of Equation (2), β rj is a ( q) row vector. Calculating all T iterations for each r th regressand, the compilation of the VAR(p) results in a ( T q) matrix of nonparametric, 5

6 coefficients, which is denoted as β r. The local conditional orthogonalized impulse response functions will later be formed through the use of β r. 2 Once the nonparametric coefficients for the r th regressand are obtained, the VAR(p) can be re-written as a linear combination since the LLLS nonparametric method fits a line within the window width, namely: y = x β + u, (3) rj j rj rj 2 with rj ~ ( 0, rj ) u σ. y rj, β rj, and u rj refer to the j th observation of the r th regressand, the set of nonparametric coefficients, and the error term, respectively. 3 Since the nonparametric VAR provides local estimation, the local estimation for each j th iteration of the nonparametric VAR can be exploited to form the local conditional orthogonalized impulse response functions through the use of the conditional nonparametric variance-covariance matrix. In order to produce the nonparametric conditional orthogonalized impulse response functions, the diagonal elements of the conditional nonparametric variance-covariance matrix, Σ ( x j ), are formed for each rj th iteration of the VAR(p) by 2 σ ( u x ) T j= rj j = T K j= ( Ψ j) K u ( Ψ j ) 2 rj (4) where r refers to the r th equation of the VAR and j refers to the j th iteration of the VAR. The offdiagonal elements of the conditional nonparametric variance-covariance matrix, Σ ( x j ), are formed for each rj th iteration of the VAR(p) by cov u j, u2j, L, urj xj = T j= ( Ψ j)( j 2 j L rj) K u u u T j= K ( Ψ j ) 2 When using demeaned or standardized data, the dimensions of β r will be (T x k) due to the exclusion of the constant term. 3 x j refers to the j th observations of the regressor set, which are identical for each set of n-equations in the VAR, and hence is not denoted with a subscript of r (which denotes the r th equation in the VAR). 6

7 T T K j ( Ψ j)( uj) K( Ψ j)( urj) j= j= L (5) T T K( Ψ j) K( Ψ j) j= j= Typically, the VAR is dependent on p number of lags, and it can be written in terms of the p th order polynomial in the lag operator as: Φ (L) y = u, (6) j j j where Φ ( L) j represents the nonparametric AR coefficients of the j th iteration of the VAR(p), which contains n-number of equations. y j refers to all n-equations of the j th iteration of the VAR. In this instance, the nonparametric estimation of the VAR will produce T sets of Equation (6). The nonparametric coefficients of the VAR(p) from Equation (6) are used to form its MA( ) representation, which are orthogonalized through the application of the conditional Choleski decomposition based upon the conditional nonparametric variance-covariance matrix, Σ ( x j ). Specifically, the j th iteration of the VAR with n-equations is of the form: y = M(L) u (7) j j j where M ( L ) ( L Φ ) P ( x j ), P( xj ) j j is the lower triangular Choleski matrix that satisfies ( x j) P( xj) P( x = j) and w( x ) ( ) j = P xj uj, with E w( x j ) = 0, and V w( xj) Σ = In. The orthogonalized MA( ) coefficients are used to form the nonparametric conditional orthogonalized impulse response functions. At time j + s, the matrix form of the local nonparametric dynamic multipliers caused by a shock at time j, is of the form ( j+ s) ( j ) d y dw x = M. (8) js Due to the local properties of nonparametrics, a set of local nonparametric conditional orthogonalized impulse response functions for each j th iteration of the VAR, can be produced thereby generating a total of T sets of local nonparametric conditional orthogonalized impulse response functions with n 2 individual local nonparametric conditional orthogonalized impulse response functions in each set. 7

8 In summary, the local benefits of the theoretical model of the nonparametric VAR are that the model is able to produce T sets of AR coefficients as well as T sets of conditional orthogonalized MA( ) coefficients that are used to form the local nonparametric conditional orthogonalized impulse response functions. Aside from the local qualities of the nonparametric VAR, it is also able to produce global coefficients through the use of aggregation. In order to obtain the global coefficients, the nonparametric regression coefficients for each r th equation and each m th regressor is aggregated by taking the average. Thus, a global set of AR coefficients is produced, which is then used to form the global MA( ) coefficients that are needed for the formation of the impulse response functions. In order to orthogonalize the global MA( ) coefficients, the variance-covariance matrix needs to be obtained, which is dependent upon the error terms. The error terms for the r th equation of the VAR are obtained from the following equation: U = Y Xβ, (9) r r r, np where U r is a column vector of error terms with dimensions of ( T ) from the r th equation of the VAR. 4 The complete matrix of error terms that are formed from the global nonparametric, i.e., the aggregated nonparametric coefficients for the VAR is denoted as the matrix ( ) U = U L U L U U. Once the aggregation process has been completed, and the r n n error terms have been obtained, the formation of the impulse response functions is formed akin to the impulse response functions of the parametric VAR. 3. Empirical Results 3. The Data The log of the first difference of the seasonally adjusted Consumer Price Index (CPI) is used as a measure of inflation, and the three-month Treasury bill rate (secondary market) is used as the nominal interest rate. 5 The detrended series of inflation and nominal interest rates are used, 4 It should be noted that r,np ( k ) if a constant term is excluded. β has the dimensions of ( q ) if a constant term is included and the dimensions of 5 The data are obtained from the St. Louis F.R.E.D. For CPI (CPIAUCSL) and the 3-month T-Bill rates (TB3MS), monthly data are converted to quarterly data. 8

9 which is important to warrant stability of the VAR (Tierney 2005). The sample period is from the second quarter of 960 to the second quarter of Before forming the VAR(p), we investigate stationarity and structural breaks in inflation and nominal interest rates. First, we find that the Augmented Dickey-Fuller (ADF) test fails to reject the null of non-stationarity for both inflation and nominal interest rates. As Tierney (2005), we find a structural break in 98:Q3 in the inflation series and in 980:Q2 in nominal interest rates, based on the log likelihood ratio of the Chow breakpoint test and the Chow forecast test. These results are confirmed by applying Andrews and Ploberger (992) endogenous break tests. Due to the weak power of the ADF test, Perron s test for nonstationarity (989) is applied to inflation and nominal interest rates with the inclusion of a break in the mean and the drift, which is of the form: x = a + a t+ µ D + µ D + x (0) a t L 3 T t where x t is either inflation or nominal interest rates, t refers to the time trend, and x a t are the residuals of the detrended series. D L is a dummy variable that takes a value of 0 for t < T B and a value of for t T B, where T B is the time of the structural break. D T is a dummy variable multiplied by the time trend, which takes a value of 0 for t < T B, and the value of the time trend, t, for any t T B. 3.2 Results As discussed in section 2, from a dataset with T observations, T sets of localized nonparametric conditional orthogonalized impulse response functions can be obtained where T refers to the number of observations used to estimate the VAR(p) once the p lags are taken into account. 6 This permits one to study the localized effects of a shock to the system at any given point in time. We trace the ratio of responses of inflation caused by a shock to interest rates, which is referred to as Γ k' with k' representing a finite horizon. The localized nonparametric conditional orthogonalized dynamic multipliers of inflation caused by a shock to interest rates will be examined using several different methods. In particular, the local nonparametric VARs are 6 The lag length of the VAR is determined by AIC and SBC. 9

10 estimated in three distinct ways. First, we estimate the nonparametric VARs for the full sample from 960: 2 to 2004:2. The nonparametric impulse response functions are obtained from the average nonparametric coefficients for each regressor, which are used to form the variancecovariance matrix. Second, we estimate the VAR for sub samples and for periods in which inflation is increasing and decreasing. Third, we estimate recursively the VARs(4) for each observation in the sample from 960:2 to 2004: Method - Full Sample Estimation Figure plots inflation and nominal interest rates. In the first part of the sample these series move close together up to 980, from which point nominal interest rates stay above inflation until Overall inflation and nominal interest rates display a positive correlation, which is known as the price puzzle. That is, increases in interest rates have historically been associated with subsequent increases in inflation instead of falls as predicted by monetarist theory. The impulse response function (irf) of inflation to one standard deviation in nominal interest rate innovation obtained from method is plotted in Figure 2, which also compares the parametric and the nonparametric irf versions using the full sample. Both irf are very similar, increasing in the first five quarters, and declining thereafter. The response of inflation to interest rate shocks is consistent with the price puzzle found in the literature (Bernanke and Blinder 992; Christiano, Eichenbaum, and Evans 994, Sims, 992, Balke and Emery 994). Sims (992) also shows that the price puzzle is consistent across several countries such as France, Germany, Japan, the U.K., and the U.S Method 2 - Sub-Sample and Inflation Phases Sub-Samples Some authors have shown that the price puzzle is sensitive to the sample used. In particular, Balke and Emery (994) find that that the price puzzle is stronger for the period pre-980 than for post-982. Using method 2, we estimate the nonparametric irfs for four subperiods to investigate possible changes in the relationship between inflation and nominal interest rates: 960:2 to 979:3, 980: to 984:, 980: to 2004:2, and 984: to 2004:2. The irfs for these subperiods are shown in Figure 3. The irf for 960:2 to 979:3 and 984:2 to 2004:2 exclude the turbulent period in which monetary policy targeted reserves instead of interest rates. Their shape 0

11 is very similar, with inflation response peaking in the second period and decreasing slowly until it reaches zero around nine quarters. However, the positive correlation between these two series is stronger for the first period compared to the second. The subperiod from 980: to 984: shows different dynamics: inflation innovation increases substantially more for two quarters in response to a shock in nominal interest rates, but has a steep fall and become negative after five quarters. These differences in the irfs across samples illustrate the difficulty of evaluating the average relationship between inflation and nominal interest rates. The local nonparametric method proposed in this paper allows us to investigate this further, since we can estimate recursive irf for each period in time and for small subperiods, without losing degrees of freedom. High and Low Inflation Phases We study the response of inflation innovations to unexpected changes in interest rates across high and low inflation periods. We are mostly interested in identify times in which there is a persistent change in inflation we classify a high inflation phase when inflation increases persistently for several quarters until it reaches a peak. By the same token, low inflation phases start when inflation falls for several quarters until it reaches a trough. This is in contrast with the NBER method to classify classical business cycle phases, in which a recession starts when economic growth falls below a long run trend and ends when it reaches the previous peak. Our classification for inflation phases and the NBER s are illustrated in the diagrams below: Classification of High and Low Inflation Phases

12 Classification of NBER Expansions and Recessions The difference between these two methods is that a high (low) inflation phase includes periods in which inflation is still relatively low (high) but is increasing (decreasing) steadily. In addition, the level of inflation is not as relevant as its gradient. That is, inflation can be historically low as in the early 2000s, but its gradient is positive indicating the beginning of a high inflation phase. This classification seems more appropriate to capture the effectiveness of monetary policy in light of the persistence of inflation once it starts increasing (decreasing). The metric proposed to determine inflation phases is as follows: a high inflation phase starts in quarter t if inflation π t- was in a low phase in quarter t- and πt + 2 πt+ πt πt. That is, inflation grows for three consecutive quarters. A low inflation phase starts in quarter t if inflation π t- was in a high phase in quarter t- and πt + < πt < πt. That is, inflation falls for two consecutive quarters. Figure 4 plots inflation, inflation phases, and the NBER recessions. As it can be seen, when inflation starts increasing it does so slowly and steadily. However, when inflation falls, it drops abruptly, which makes it easier to identify the beginning of a low inflation phase than the start of a high inflation phase. Notice that inflation phases are associated with NBER recessions all NBER recessions begin around the end of high inflation phases. There were only two high 2

13 inflation phases, in and 2002, in which a recession did not follow. However, the economy entered a slowdown in Figures 5 and 6 show the nonparametric irfs of inflation to nominal interest rate innovations during high and low inflation phases, respectively. Although in both cases inflation innovations increase to a shock in interest rates reflecting the price puzzle, during low inflation phases inflation innovation starts decreasing in the fourth quarter and becomes negative already in the six quarter. On the other hand, during high inflation phases an unexpected increase in interest rates raises inflation innovation for six quarters, from which point it starts decreasing very slowly. In addition, the magnitude of increase of inflation innovation to a shock in interest rates is substantially smaller during low inflation phases compared to high inflation phases Method 3 Recursive Estimation In this section we report the results for the nonparametric VARs estimated for each point in the sample from 960:2 to 2004:2. This yields 73 VARs after adjusting for the lags and, consequently, the same number of time varying coefficients, variances, and impulse response functions for each regressor. Time-Varying Coefficients Figure 7 shows the time-varying coefficients for lagged inflation in the inflation equation, which measure inflation persistence. The dynamics of inflation as captured by these coefficients can be divided in two distinct behaviors. Before the recession and after the recession, inflation dynamics show very low persistence with the coefficients averaging On the other hand, between 970 and 990 the inflation coefficients are substantially higher and show oscillatory movements. The coefficients indicate nonstationarities, especially in 974 and 986, but also with several instances in the mid 970s. In particular, the coefficients were large and positive during the period in which inflation was increasing substantially between 975 and 980. Around or at NBER recessions, inflation coefficients become large and negative. This is consistent with a peak reversal of inflation around economic recessions. A similar pattern is observed for the dynamics of nominal interest rates (Figure 8), although the coefficients show smaller oscillations overall and nonstationarities in only a few cases. The period between 974 and 984 was marked by larger oscillations. As with inflation dynamics, 3

14 the coefficients of nominal interest rates show a large decline around the beginning of economic recessions, reflecting peak reversal changes in monetary policy from tight to loose in face of weaker economic prospect. Figures 9 and 0 show the time-varying relationship between inflation and nominal interest rates. Figure 9 plots the lagged inflation coefficients in the nominal interest equation, which indicate how nominal interest rates respond to past inflation. There is a positive relationship between lagged inflation and nominal interest rates, which increases right before or during recessions, and decreases at their end. Since the beginning of recessions are generally times in which inflation is at its peak, these coefficients show the sensitivity of monetary policy to changes in inflation when inflation is historically high. Interestingly, during expansions monetary policy reaction reverts with nominal interest rates responding the least to inflation at those times compared to the rest of the sample. Figure 0 shows the lagged nominal interest rate coefficients in the inflation equation. Inflation is particularly responsive to lagged interest rates around economic recessions. In fact, in all recessions the modulus of the nominal interest rate coefficients increased to above unity. It is interesting to notice that some times the inflation has a positive relationship with lagged nominal interest rates, and some others a negative one. The most noticeable period in which inflation decreased following an increase in nominal interest rates was during the 200 recession, when there was a large negative drop in the interest rate coefficients. On the other hand for most of the previous recessions, there was a large and positive increased in these coefficients. Time-Varying Variance and Covariance Figures and 2 plot the variance of inflation and interest rate innovations, respectively. An interesting feature that emerges is that the variances of both innovations tend to increase a couple of quarters before a recession. In particular, the variance of inflation innovation increased substantially before all but the 974 and 98 recessions. The only time in which the variance of inflation innovation rose and a recession did not follow was in 984. However, the economy experienced a low growth between 984 and 986. The variance of nominal interest rate innovations display a similar pattern, but it tends to slightly lead inflation variance as shown in Figure 3. Both variances show some clustering, especially between 960 and 970, and between 995 and In addition, the variance of inflation innovation decreases substantially 4

15 since 986, whereas the variance of interest rate innovations decreases only after 990. Figure 4 shows the covariance between inflation and nominal interest rate innovations. For most of the sample the correlation is positive or slightly negative. However, it turns substantially negative between 997 and 999. Impulse Response Functions (IRF) The individual response functions for each data point can be used as valuable tool to evaluate the nonlinear responses of inflation to nominal interest over time. For example, the irf in 979:4 would indicate how the monetary policy shock in that date affected inflation innovation. In addition, the individual irf can be averaged for small sub-periods or for high and low inflation phases without running into the trade off of low degrees of freedom. Figures 5 and 6 plot the nonparametric irf of inflation to nominal interest rate innovations during each individual high and low inflation phases, respectively. During two high inflation periods a shock to interest rate had as a counterpart an increase in inflation for a couple of quarters followed by subsequent decreases, in 960:2-964:2 and 998:-200:. The shape of the irf for the former is the closest to the dynamics of irf for the full sample as shown in Figure 2. Only during the high inflation period in 983:2-984: is monetary policy effective in reducing inflation, with a very minor increase in the first two quarters followed by a substantial decrease from the third to the tenth quarter, during which inflation response is negative. The irf for the other high inflation phases share in common a very small first response of inflation innovation to a shock to interest rate, with an abrupt fall or rise in quarter ten or eleven. Figure 6 compares the irfs for low inflation phases. The low inflation phase in 960:2-964:2 and 984:2-986: show an oscillatory response of inflation innovation to shocks to interest rates, with an initial increase followed cyclical movements, which were not yet dampening after eleven quarters. The periods in which inflation innovations were most responsive to shocks to monetary policy were during the low inflation phases in 980:2-98: and 982:3-983:, when inflation innovation had just a very mild initial increase, followed by a sharp decrease in quarter six to eight, from which point inflation response is negative. These periods are followed by the high inflation period in 983:2-984: in which monetary policy was also the most effective. Thus, the individual irfs indicate that the period between 980:2 and 984: monetary policy shock achieved the expected negative response in inflation innovation. 5

16 The irfs for the remaining low inflation phases had very low initial response of inflation and a sudden increase around quarter ten or eleven. Figure 7 compares the time-varying coefficient of lagged inflation from the inflation equation and lagged nominal interest rate from the interest rate equation across high inflation phases. The time-varying coefficients estimated using our local nonparametric method show evidence of important nonstationarities in the dynamics of inflation and nominal interest rates, and nonlinearities in their relationships. As it can be observed, the coefficients are close to a unit root or show explosive behavior in all high inflation phases from 974 to 99. The only high inflation phases in which the coefficients are stationary throughout the period are in 964:- 970:, and in the two most recent ones, in 998:-200: and in 2002:-2002:4. Notice that not only the data used in the VAR are stationary, the estimated coefficients from full sample nonparametric VAR (Figure 2), from the high inflation phase VAR (Figure 3), and from the low inflation phase VAR (Figure 4) were stationary. However, the dynamics of the VAR when estimated at each point in time (irfs in Figures 5 and 6 and corresponding timevarying coefficients in Figure 7) are sometimes nonstationary, which compromise interpretation of the impulse response functions in the long run. According to Phillips (998), when irfs are computed from unrestricted VARs with roots near unity, the long run responses are inconsistent. In fact, as the horizon of the impulse response function increases, the dynamic multipliers become random variables since the shock does not necessarily dies out in the presence of nonstationarities. Since the impulse response functions become more unstable as the horizon increases, the long-term response to shocks is not reliable, although the short term multipliers are unbiased. Philips (998) also shows that the forecast error variances at long horizons are inconsistent. However, the largest differences in performance when the estimated VAR coefficients are unit roots or near unit roots is with respect to policy analysis. The local nonparametric VAR method proposed in this paper enables evaluation of nonstationarities at each point in time, which allows assessment on whether the impulse response functions could be used for policy analysis. As illustrated above, there are several instances in which this tool could be very misleading. This is especially the case during high inflation phases, although much less so during low inflation phases. During high inflation phases there are not only nonstationarities but also breaks and reversals in the relationship between inflation and nominal interest rates, especially right before and during economic recessions. On the other 6

17 hand, the coefficients are stationary in most low inflation phases. The exceptions are for 970:2-972:, 975:-976: and in 980:2-98:. 4. Conclusion This paper proposes a local nonparametric VAR method to investigate nonlinearties, nonstationarities and potential breaks in the relationship between inflation and nominal interest rates. We find substantial evidence of nonlinearities and nonstationarities in the relationship between inflation and interest rates that can shed light on the effectiveness of monetary policy across business cycle phases and high or low inflation phases. This tool might also prove particularly important for real time applications, as it allows recursive analysis of the non-parametric relationship of the variables in the VAR system at each point in time. We are investigating this in an on-going project. 7

18 References Balke, N. and K. M. Emery, 994, Understanding the Price Puzzle, Economic Review, Fourth Quarter, Dallas Fed. Bernanke, Ben S., and Alan S. Blinder (992), The Federal Funds Rate and the Channels of Monetary Transmission, American Economic Review 82 (September): Boivin, J. and M. Giannoni, 2002, Assessing Changes in the Monetary Transmission Mechanism: A VAR Approach, : FRBNY Economic Policy Review, May, 97-. Christiano, Lawrence J., Martin Eichenbaum, and Charles Evans (994), Identification and the Effects of Monetary Policy Shocks, in M. Blejer, Z. Eckstein, Z. Hercowitz, and L. Leiderman, eds., Financial Factors in Economic Stabilization and Growth (Cambridge: Cambridge University Press). Leeper, Eric M., Christopher A. Sims, and Tao Zha What Does Monetary Policy Do? Brookings Papers on Economic Activity, no. 2: -63. Pagan, A. and Ullah, A, 999. Nonparametric Econometrics, Cambridge University Press, Cambridge, p , 50-54, 95. Perron, P., 989, The Great Crash, the Oil Price Shock, and the Unit Root Hypothesis, Econometrica, 6, Phillips, P, 998. Impulse Response and Forecast Error Variance Asymptotics in Nonstationary VARs, Journal of Econometrics, 83, Tierney, H., 2005, The Nonparametric Time Detrended Fisher Effect, Ph.D. Dissertation, University of California, Riverside. 8

19 Figure Inflation ( ), Nominal Interest Rates ( ) and NBER Recessions Figure 2 Nonparametric ( ) and Parametric ( ) Impulse Response Functions of Inflation to One S.D Nominal Interest Rate Innovation Full Sample

20 Figure 3 Nonparametric Impulse Response Functions of Inflation to One S.D Nominal Interest Rate Innovation Sub-Samples.0 960:2-979: : : : - 984:.0 984: :

21 Figure 4 Inflation, High Inflation Phases (Shaded Area), and NBER Recessions (Dotted line)

22 Figure 5 Nonparametric Impulse Response Function of Inflation to One S.D Nominal Interest Rate Innovation High Inflation Phases Figure 6 Nonparametric Impulse Response Function of Inflation to One S.D Nominal Interest Rate Innovation Low Inflation Phases

23 Figure 7 Aggregate Inflation Coefficients from Inflation Equation and NBER Recessions Figure 8 Aggregate Nominal Interest Rate Coefficients from Nominal Interest Rate Equation and NBER Recessions

24 Figure 9 Aggregate Lagged Inflation Coefficients from Nominal Interest Equation and NBER Recessions Figure 0 Aggregate Lagged Nominal Interest Rate Coefficients from Inflation Equation and NBER Recessions

25 Figure Inflation Innovation Variance and NBER Recessions Figure 2 Nominal Interest Rate Innovation Variance and NBER Recessions

26 Figure 3 Inflation ( ) and Nominal Interest Rate ( ) Innovation Variances and NBER Recessions Figure 4 Covariance Inflation and Nominal Interest Rate and NBER Recessions

27 Figure 5 Nonparametric Impulse Response Function of Inflation to One S.D Nominal Interest Rate Innovation Individual High Inflation Phases.0 964:3-970: 972:2-974:4 976:2-980: :2-982:2 983:2-984: 986:2-990: : - 200: 2002: :

28 Figure 6 Nonparametric Impulse Response Function of Inflation to One S.D Nominal Interest Rate Innovation Individual Low Inflation Phases.2 960:2-964: :2-972: 975: - 976: :2-98: 982:3-983: 984:2-986: :4-997:4 200:2-200:

29 Figure 7 Nonparametric Time-Varying Coefficients for Lagged Inflation on Inflation Equation ( ), Lagged Nominal Interest Rates on Nominal Interest Rates Equation ( ), and High Inflation Phases (Shaded Area)