A competing risks analysis of the duration of federal target funds rates

Computers & Operations Research 39 (2012) 785–791

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Computers & Operations Research journal homepage: www.elsevier.com/locate/caor

A competing risks analysis of the duration of federal target funds rates Ester Gutie´rrez , Sebastia´n Lozano Dept. of Industrial Management, University of Seville, Camino de los Descubrimientos, s/n, 41092-Sevilla, Spain

a r t i c l e in f o

abstract

Available online 10 October 2010

The behaviour of short term interest rates has been the focus of extensive studies in economics and finance. This paper analyses the duration between changes in the Intended Federal Funds Rate (IFFR) taking into account different factors and the possibility of two outcomes (rate increase and rate decrease). Due to the lack of independence between these outcomes, the use of traditional survival analysis was ruled out and a novel approach based on competing risks is used. The estimation results show the influence of the current interest rate level, US GDP growth rate, US inflation rate, inflation rate differential with Euro-zone and the sign of the previous interest rate change. Test results also confirm that the duration previous to small changes (of a quarter point) and to large changes (above a quarter point) are statistically different for both rate hikes and rate cuts. Accurate confidence intervals for the Cumulative Incidence Function (CIF) are provided even with small sample size under non-normality. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Interest rates Federal Reserve Survival analysis Competing risks

1. Introduction Although there are similarities in the structure of Central Bankers (CB), there are also some key differences in terms of monetary policy. Many CBs, such as the Eurosystem of CBs or the Bank of Japan, assign overriding importance to price stability in contrast to the Federal Reserve’s (‘‘the Fed’’ hereafter) multiobjective mandate to effectively promote the goals of maximum employment, stable prices and moderate long-term interest [1]. One of the ways the Fed implements US monetary policy is through its control over the Federal Funds Rate—the rate at which depository institutions trade balances at the Fed. The Federal Funds Rate is the interest rate paid on overnight loans made between depository institutions. The Fed Funds Rate is a benchmark for home equity lines of credit, credit cards and other consumer loans as well as the prime rate used for short-term business loans. Since the late 1980s, the Fed has implemented monetary policy by using open-market operations to target an Intended Federal Funds Rate (IFFR). In this way, the Federal Funds Rate is not freely determined by market forces but is, in fact, administered by the Fed. The Federal Open Market Committee (FOMC) sets the IFFR at a level that it believes will foster financial and monetary conditions consistent with achieving its monetary policy objectives, and it adjusts that target in line with evolving economic developments. A change in the IFFR, or even a change in expectations about the IFFRs future level, can set off a chain of events that will affect other short-term interest rates, longer-term interest rates, the foreign

 Corresponding author. Tel.: +34 954486198; fax: + 34 954487329.

E-mail address: [email protected] (E. Gutie´rrez). 0305-0548/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2010.10.002

exchange value of the dollar and stock prices. In turn, changes in these variables will affect households’ and businesses’ spending decisions, thereby affecting growth in aggregate demand and the economy. Although the federal funds rate may vary day-to-day in response to uncontrollable market factors, Fed actions are generally successful in achieving the FOMC’s IFFR target on average. The aim of this paper is twofold: on the one hand, to empirically analyse the timing and determinants of the IFFR changes; and, on the other hand, given that Fed can raise or lower the IFFR, to explore the differences in the determinants of these two outcomes. The methodology used is based on competing risks (CR) survival analysis, in which an individual is exposed to two or more alternative event causes. In these cases, the occurrence of one type of event hinders the occurrence of any other event [2,3]. CR analyses are encountered frequently in medical research. In economics, the most common application concerns individual unemployment durations [4–6], strike durations [7–9] and mortgage and loan defaulting [10,11]. In our case, we define two types of events: rate hike versus rate cut. The main benefits of the proposed approach versus existing conditional time series approaches of interest rates durations is its nonparametric nature, which does not require any distributional assumption. It is well known that the FOMC reacts systematically (though not exclusively) to information about inflation and economic activity in a way first identified by Taylor [12]. Taylor [13] estimates the original Taylor rule, in which the Fed is assumed to set the federal funds rate in response to the current inflation rate and output gap. Some criticisms of the Taylor rule have been expressed by, among others, Meltzer [14] and Orphanides [15]. The research literature on interest rate modelling is quite extensive. Among the prominent models offered as explanations

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E. Gutie´rrez, S. Lozano / Computers & Operations Research 39 (2012) 785–791

for interest rates behaviour are Brennan and Schwartz [16–18], Vasicek [19], Cox et al. [20,21], Hull and White [22], Heath et al. [23] and Brace et al. [24]. Many researchers have also investigated the relationship between variables and the term structure of interest rates. Modelling and forecasting of interest rates has traditionally proceeded in the framework of linear stationary methods such as Autoregressive Moving Average, Vector Autoregressions (VAR), Bayesian VAR, etc., but only with moderate success. Nonlinear methods have also been applied in the context of modelling interest rates: (i) the Autoregressive Conditionally Heteroscedasticy (ARCH) model [25] and the Generalized ARCH model [26]; (ii) the stochastic models of [27]; (iii) approaches in which the type of nonlinearity is spelt out explicitly, e.g., bilinear, generalized autoregressive, smooth threshold autoregressive, Markov switching, etc. [28]. Most research on FOMC decision-making describes the IFFR with a parametric statistical model. Thus, Robertson and Tallman [29] use a series of VAR models to forecast the one- and two-month ahead federal funds rate. Hamilton and Jorda [30] estimate a model in which the FOMC decides whether to change the target rate according to an Autoregressive Conditional Hazard (ACH) duration specification. Once a change ‘‘trigger’’ has been reached, the FOMC makes an ordered-probit decision that determines the change in the target rate, according to 25 b.p. increments. A similar marked point process approach is used in Dolado and Marı´a-Dolores [31] for evaluating changes in the interest rate target of the Bank of Spain. Hu and Phillips [32] discrete choice model to estimate not only the magnitude of a target rate change, but also the timing of the change. More recently, Grammig and Kehrle [33] propose a model for the IFFR combining the ACH model proposed by Hamilton and Jorda [30] and an autoregressive conditional multinomial model introduced by Russell and Engle [34]. The remainder of the paper is organized as follows. In Section 2, the methodology, based on a CR model, is presented. Section 3 describes the dataset used. In Section 4, the results of the proposed approach are presented and discussed. Finally, Section 5 summarizes and concludes.

2. Methodology The empirical analysis carried out in this research allows for different interest rate outcomes by using a CR survival analysis model. Kalbfleisch and Prentice [35] describe CR as the general situation in which an individual can experience more than one type of event so that a CR model is a model for multiple durations that start at the same point time. The subject is observed until one of the multiple durations is completed. In our case, two event risks are considered (i.e., up and down changes of the interest rate) and the duration of the first type of event that occurs is successively recorded. In contrast to the conditional times series approach, CR analysis does not require any underlying probabilistic distribution assumption, can incorporate censured data and does not require any structure on the dependencies between the events considered. However, a remarkable weakness of CR analysis is related to the difficulty of interpretation of the results when the number of causes increases. Several approaches have been proposed that extend the Cox Proportional Hazards (PH) model to correlated survival data [36,37]. Pintilie [38] shows, through simulations, that the Cox PH model can be used to test the effect of a covariate on the hazard of failure from the cause of interest in the presence of CR. In our case, this means that the significance of the factors that influence the occurrence of the two possible interest rate change outcomes can be tested and assessed.

Note that, in the presence of CR the traditional survival methods should be applied with caution. Thus, the traditional Kaplan and Meier [39] method (KM) for summarizing time-to-event data will produce, when CR are present, an overestimate of the probability of being at risk of failure at any time, i.e. of the probability of any of type of interest rate change event at any time [40–45]. Kalbfleisch and Prentice [35] suggested an alternative approach that accounts for CR. This method is labelled Cumulative Incidence Function (CIF). The CIF quantifies the cumulative probability of cause-specific failure (i.e. rate hike or rate cut in our case) in the presence of competing events without assumptions about the dependence between the events [41,42,46]. The starting point for the analysis are the CR data (T, K), where T is the time to the occurrence of the first cause-specific event (i.e. a rate hike or a rate cut) and K A ð1, . . . , nK Þ is the event type, where nK is the number of event types. In our case, nK ¼2 corresponding to the two types of events considered (e.g. K ¼1 for interest rate rise and K ¼2 for interest rate cut). The basic results that can be obtained from these CR data are the cause-specific hazard and CIF. The cause-specific hazard function for an event K ¼k at time t is

lk ðtÞ ¼ lim

Dtk0

Prðt oT r t þ Dt,K ¼ k=T ZtÞ Dt

ð2:1Þ

and represents the rate of occurrence of the kth cause-specific event (for example, a rate hike). In other words, the hazard function gives the instantaneous potential per unit time for the kth event (for example, a rate hike) to occur, given that no changes have taken place in interest rates up to time t. From this, for small Dt, we have that Prðt oT r t þ Dt,K ¼ kÞ  PrðT Z tÞlk ðtÞ Dt

ð2:2Þ

and, as noted in Jeong and Fine [58], the left-hand side in (2.2) approximates the probability density function for the kth causespecific event as Dt approaches 0. This implies that the corresponding probability distribution function (i.e. the CIF) for the kth cause-specific event is Z t Fk ðtÞ ¼ SðuÞdLk ðuÞ ð2:3Þ 0

where S(t)¼Pr(T>t) is the survivor function (i.e. the probability that at a given time t no interest rate change has occurred yet) and R Lk ðtÞ ¼ 0t lk ðuÞdu is the cumulative hazard function for a rate hike or a rate cut, which describes how the risk of each particular outcome changes with time. In practical applications, S(t), Lk ðtÞ and Fk(t) are estimated from CR data using nonparametric or semiparametric methods, with the advantage that there is no need to assume an underlying distributional form for the CIF. In practice, the random variable T is typically subject to independent right censoring. To estimate the overall survival ^ function S(t) it is valid to use the KM estimator SðtÞ, which is obtained by calculating the proportion of times that an interest rate change has occurred by time t. Similarly, the cause-specific cumulative hazard function Lk ðtÞ may be estimated by the ^ ðtÞ [47,48]. Hence, an estimate of the Nelson–Aalen estimator L k CIF for kth cause-specific event (i.e. rate hike or rate cut) is given by Z t ^ ðuÞ ^ F^ k ðtÞ ¼ L ð2:4Þ SðuÞd k 0

The linear 100(1  a) per cent pointwise confidence interval for the CIF F^ k ðtÞ, for a fixed time t for cause kth corresponds to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F^ k ðtÞ 7 za=2 V^ ðF^ k ðtÞÞ ð2:5Þ where za/2 is the upper a/2 quantile of the standard normal distribution and V^ is the variance estimator of the corresponding

E. Gutie´rrez, S. Lozano / Computers & Operations Research 39 (2012) 785–791

CIF, which can be obtained using Aalen [48] estimator  X dkj ½F^ k ðtÞF^ k ðtj Þ2 V^ ðF^ k ðtÞÞ ¼ ðnj 1Þðnj dj Þ t rt

Table 1 Variable definitions and data sources (data source is http://www.federalreserve.gov unless indicated otherwise).

j

X

^ j1 Þ2 dkj ðnj dkj Þ þ Sðt n2j ðnj 1Þ tj r t 2

X

½F^ k ðtÞF^ k ðtj Þ

^ Sðt j1 Þ

tj r t

dkj ðnj dkj Þ nj ðnj dj Þðnj 1Þ

787

ð2:6Þ

Note that in the above expression dkj is the number of events of type k (rate hike or rate cut) that have occurred at time tj, while dj ¼d1j + d2j is the total number of events up to time tj and nj the number of times that no interest rate change has occurred prior to tj. Note that, alternatively, Dinse and Larson [59] estimator can be used although Choudhury [49] finds that the difference between both estimators is very small. In investigations of the KM estimator, Kalbfleisch and Prentice [35] and Borgan and Liestøl [50] have shown that confidence intervals for survival functions based on the log(  log) and arcsine transformations perform better than the usual linear confidence intervals. Similarly, Choudhury [49] has shown, based on simulations, that confidence intervals for survival functions based on the log( log) transformation perform better than the usual linear confidence intervals. In that case, the 100(1  a) per cent confidence interval for F^ k ðtÞ would be given by pffiffiffiffiffiffiffiffiffiffiffiffiffi ^ ^ ð2:7Þ F^ k ðtÞexpf 7 za=2 V ðF k ðtÞÞ g

Variable of interest IFFRD Intended Federal Funds Rates Duration Covariates IFFR Intended Federal Funds Rate IR US Inflation rate Source: http://www.bls.gov/ IR-EUI Difference between US inflation rate and European Union (EU) inflation rate Source: http://epp.eurostat.ec.europa.eu/ ER Exchange rate (US $/Euro) http://www.jeico.com/jeifront.html (data up to 2000) http://www.federalreserve.gov (data from 2000) UGDP Quarterly growth rate of real US GDP (percent change based on chained 2000 dollars) Source: http://www.bea.gov/ UGDP-EGDP Difference between quarterly growth rate of real US GDP and quarterly growth rate of real European GDP Source: http://epp.eurostat.ec.europa.eu/ IFFRD(t  1) Duration of the previous IFFR sIFFRD(t  1) Indicator variable coded one if the difference between IFFRD(t  1) and the IFFRD(t  2) is positive and zero otherwise. This variable takes into account the acceleration effect in the duration of the previous move of the IFFR.

25 b.p. This not only ensures that the boundaries of the confidence intervals for F^ k ðtÞ are contained in [0,1] but may also improve the coverage accuracy in small samples. When analyzing cause-specific event patterns, investigators may be interested in the effects of covariates on the event-specific event probabilities, in our case on the probabilities of the two types of interest rate changes. Fine and Gray [51] and Fine [52] adapted the PH model [53] to the CIF case and proposed inferences for the effects of binary and continuous covariates. Thus, Fine and Gray [51] considered the PH model to directly infer the effects of covariates on the cumulative incidence of type k events. Their model was originally posited in terms of the subdistribution hazard function [54]

lk ðt; ZÞ ¼ lk0 ðtÞexpðZ T bk Þ

ð2:8Þ

where lk0 ðtÞ is the cause-specific baseline hazard function, Z is a time-independent P-dimensional covariate vector and bk is the corresponding P-dimensional regression coefficients vector. The cumulative probability of a type k event is given by Fk ðt; ZÞ ¼ 1expfexpðZ T bk Þuk ðtÞg

ð2:9Þ

where uk ðtÞ ¼ logk

Z

t 0



lk0 ðsÞds

The explanatory variables (i.e., covariates) in our empirical specification (see Table 1 for variables definition) include main macroeconomic aggregates for the US economy. The economy in relation to full employment is not considered because of data lags (and difficulties in estimating the full-employment level of output). In order to make the analysis richer and to test for empirical implications of theoretical models we enlarge our information set with additional variables related to Euro-zone macroeconomic aggregates.

IFFR increase More than 25 b.p. FOMC IFFR change decisions 25 b.p. IFFR decrease More than 25 b.p. Fig. 1. The CR situation in a FOMC decision about IFFR.

3. Descriptive duration analysis The empirical analysis has been carried out using data from the Federal Reserve Statistical Release gathered by the Board of Governors of the Federal Reserve System, for the period 1999–2008 (till December 16, 2008). It includes all changes of IFFR taking place during that period. In monetary policy studies on interest rates sensitivity, the duration of IFFR could be a key quantity of interest. During the course of a FOMC meeting, the decision about IFFR change may be an increase (to contain inflation pressures) or a rate cut (to stimulate economic growth). The dependent variable in this study is the duration (expressed in days) between a change in the IFFR by the FOMC and the subsequent change (rate hike or rate cut). In the dataset used, approximately 50% of the interest rate changes were rate rises and 50% were rate cuts. Typically, as depicted in Fig. 1, FOMC sets the federal funds rate at a level it believes will foster financial and monetary conditions consistent with achieving its monetary policy objectives. It should be noted that since 1990 the FOMC has always changed the IFFR in multiples of 25 b.p. during the 46 changes occurred. The magnitude of change was of 25 b.p. in approximately 70% of changes. In contrast, 30% were changes of 50 b.p. or above (75 b.p.). Fig. 2 displays the pattern of interest rates at the Fed over 1999–2008 (till December 16, 2008). The plotted points exhibit

E. Gutie´rrez, S. Lozano / Computers & Operations Research 39 (2012) 785–791

788

specific observation, the midpoint of the range has been considered in the analysis. Fig. 3 shows a positive skewness and a large kurtosis in the distribution of the IFFR duration, suggesting a leptokurtic shape. The histogram indicates that the IFFR duration distribution has a fat tails when compared to a Gaussian distribution. However, it is significant that the IFFR duration distribution exhibits non-normality due to leptokurtosis, as evidenced by the Jarque–Bera test (w22;exp ¼ 80:15; p-valueo0.01).

4. Estimation results for the CR model The CIF curves that represent the probability of an event of a certain type (rate hike or rate cut) computed as per Eq. (2.4) are plotted in Fig. 4. The estimated probability that an IFFR rise will be experienced before three months (90 days) is 0.4572. Similarly, the probability of an IFFR cutting down before three months is 0.3904. After 55 days after the last change in the IFFR, the general trend of an IFFR rise is more likely a fall. Computing confidence intervals provides useful information about uncertainty related to parameter estimates. Table 3 gives the 95% confidence intervals for the CIF for interest rate hikes and interest rate cuts at 60, 120 and 240 days using Eqs. (2.6) and (2.7). According to the magnitude of the difference between two successive IFFR observations a factor with two levels or groups is defined. Thus, when the difference between two consecutive IFFRs is less than or equal to 25 b.p. it is labelled small; otherwise it is labelled large. Table 4 shows the results of the Gray [54] test and the

7

35

6

30

5

25 Frequency

IFFR

some pattern of cyclic behaviour. The interest rates had a long run down and run up from 2000 to 2002 and from 2004 to 2006, respectively. On average each run lasts around two years. During 2003 until early spring of 2004, the IFFR was reduced to 1%, as a consequence of an unusually slow employment growth and to combat fears of deflation. Since 2006, there has been a slowdown trend in IFFR which still persists, due to the US housing market correction and the subprime mortgage crisis. Table 2 shows mean duration of changes on interest rates (IFFRD). The average duration is 80 days and the median is 48 days. On average, decreasing changes on interest rates show longer durations (88.50 days versus 71.56 days for increasing changes) and the median duration is longer for increasing interest rates than decreasing interest rates (49 days versus 43 days for decreasing changes in the interest rates). This is due to the negative skew in the case of increasing interest distribution and the positive skew in case decreasing interest rates distribution. The maximum duration (408 days) was reached with an IFFR equal to 5.25% (from 18/09/2006 to 31/10/2007), and the minimum duration 8 days with an IFFR equal to 4.25% (from 22/01/2008 to 30/ 01/2008). The largest rise in IFFR (50 b.p.) occurred only once (on 16 May 2000) and the largest drops (75 b.p.) have occurred three times (on January 22nd 2008, March 18th 2008 and April 30th 2008). In Fig. 2 it can be seen that these three large drops are almost consecutive and recent. Actually, no other 75 b.p. drops have occurred in recent Fed history (i.e., since 1990). Note also that, on December 16th 2008, the Federal Reserve cut the IFFR to an all-time low, actually a range between 0% and 0.25%. For this

4 3 2

20 15 10

1

5

16/12/08 Last IFFR change

0

0 0

01/01/2000 01/01/2002 01/01/2004 01/01/2006 01/01/2008

Date Fig. 2. IFFR during 1999–2008.

50

100 150 200 250 300 350 400 Days

Fig. 3. Histogram of frequency of IFFR durations.

Table 2 Number of IFFR changes/year and their associated IFFR duration. Year

All interest rates changes No. of changes in IFFR

1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 (till 16/12/ 08) Total

3 3 11 1 1 5 8 5 2 7

%

6.52 6.52 23.91 2.17 2.17 10.87 17.39 10.87 4.35 15.22

Rate hikes Mean IFFR duration 121.33 60.67 52.18 330.00 231.00 107.60 45.50 55.80 224.50 53.00

46 100 80.00 Median ¼ 48 Minimum ¼ 8; Maximum¼ 408

No. of changes in IFFR 3 3 0 0 0 5 8 4 0 0

Rate cuts %

13.04 13.04 0.00 0.00 0.00 21.74 34.78 17.39 0.00 0.00

Mean IFFR duration

No. of changes in IFFR

121.33 60.67 – – – 107.60 45.50 49.50 – –

0 0 11 1 1 0 0 1 2 7

23 100 71.56 Median ¼ 49 Minimum ¼34; Maximum¼371

%

0.00 0.00 47.83 4.35 4.35 0.00 0.00 4.35 8.70 30.43

Mean IFFR duration – – 52.18 330 231 – – 81.00 224.50 53.00

23 100 88.50 Median ¼43 Minimum¼ 8; Maximum¼ 408

E. Gutie´rrez, S. Lozano / Computers & Operations Research 39 (2012) 785–791

0.6

Rate hike Rate cut

0.5

CIF

CIF

0.4 0.3 0.2

789

1.0

Large Rate Cut

0.8

Small Rate Hike

0.6 0.4

Small Rate Cut

0.2

0.1

Large Rate Hike 0.0

0.0 0

100

200 Days

300

400

Fig. 4. Estimated CIF for interest rate hikes and interest rate cuts.

Table 3 95% confidence intervals for the CIF estimates for rate hikes and rate cuts. Variance method

Aalen [48] Choudhury [49]

Time

Rate Rate Rate Rate

Confidence intervals

hike cut hike cut

60 days

120 days

240 days

(0.269, (0.227, (0.258, (0.224,

(0.311, (0.247, (0.311, (0.244,

(0.332, (0.308, (0.322, (0.307,

0.558) 0.510) 0.531) 0.503)

0.604) 0.533) 0.603) 0.536)

0.626) 0.603) 0.616) 0.601)

Table 4 Tests for equality of CIF across small and large changes in IFFR. Test

Statistic

Gray [54]

Rate hike Rate cut

Pepe and Mori [42]

p-value 10.42806 22.77651 11.48244

1.24115e  03 1.81976e  06 7.02569e  04

Pepe and Mori [42] test for the equality of CIF of these two groups. These tests confirm that CIF curves for small and large FFTR are statistically different for rate hikes and rate cuts. Hence, the hazards rates are quite different for these two groups of events. A plot of estimated CIF for each combination of the type and amount of IFFR change is shown in Fig. 5. For rate cuts, the CIF for large changes rises very sharply from 30 to about 80 days and afterwards rises slowly until about 330 days. This behaviour is similar for small rate hikes although these usually take a little longer to occur. There is also certain similarity between the CIF of small rate cuts and large rate hikes, both of which take longer than the other two cases. The cumulative incidence of rate hikes (small and large) at 60 days is calculated to be 58.12% and 6.6%, respectively. The cumulative incidence of rate cuts (small and large) at 60 days is calculated to be 19.27% and 73.33%, respectively. In order to identify the impact of the macroeconomic magnitudes described in Table 1 on the time duration of IFFR, two types of modelling have been carried out, namely cause-specific hazards using the Cox PH model [53] and hazards of the subdistributions using the CR regression model of Eqs. (2.8) and (2.9). The corresponding estimated regression coefficients are presented in Table 5 so that the proportional impact of each variable on the rate hike or rate cut hazard rate can be calculated by taking the exponent of the corresponding coefficient. It can be noted that the number of significant covariates is much higher in the CR Regression than in the Cox PH and that both models find that the current value IFFR (i.e., a current high IFFR) reduces the risk of rate hikes at any time by a factor of exp(  0.643) or exp( 0.765) depending on the model used. Also, both models

0

100

200 Days

300

400

Fig. 5. Estimated CIF curves for IFFR hikes and cuts as competing events for each type of magnitude of change (small and large changes).

identify sIFFRD(t  1) to be a significant covariate and UGDP-EGDP as an insignificant covariate. Specifically, with respect to sIFFRD(t 1), both models give negative coefficients for the two competing risks. Thus, if the previous rate change was a hike (sIFFRD(t  1)¼ 1), then the risks of any immediate change of the IFFR, either up or down, is reduced by a factor that, for a rate hike, is estimated by the Cox PH model as exp( 1.547) and by CR regression as exp(  1.010) and for a rate cut as exp( 1.634) by the Cox PH model and exp(  1.125) by CR regression. Both models indicate that the IFFRD(t  1), i.e., the duration of the previous IFFR, does not have an appreciable influence on the duration of the current IFFR. On the contrary, both models indicate that a high US GDP growth reduces the risk of a rate cut by a factor of exp( 0.345) according to the Cox PH and exp(  0.320) according to CR regression. With respect to the effects of US GDP growth on the risk of a rate hike, although the Cox PH does not find it significant, CR regression estimates an increased risk of exp(0.321). In the CR regression model, Dollar–Euro exchange rate is positively associated with the probability of interest rates hikes. Since exchange rate movements are an important channel through which monetary policy affects the economy, these results are consistent with empirical evidence [55]. Although both models agree in finding the influence of the gap between US and EU GDP growth rates not significant, both models indicate that the difference between US and EU inflation rates increases the risk of an IFFR increase by as much as exp(4.230), according to the Cox PH, and exp(4.531), according to CR regression. This is clearly related to the objective of curbing inflation to avoid losing competitiveness. The effect of the gap in inflation rates on the risk of an IFFR cut is negative in both models although only significant in the case of CR regression. Thus, when US IR is higher than Euro IR, the risk of an IFFR cut is reduced by exp(  3.064) according to the Cox PH and by exp(  4.422) according to CR regression. Finally, IR appears associated, in both the Cox PH and CR regression models, with a reduced risk of interest rates hikes and positively associated with interest rates cuts. Thus, a high IR increases the risk of a rate cut by a factor exp(1.419) according to the Cox PH and exp(1.894) according to the CR regression model. The effect on the risk of a rate hike is the opposite: a high IR reduces the risk of a rate hike by a factor of exp( 1.171) according to the Cox PH and exp( 1.716) according to the CR regression model. These results may seem to contradict the expected direct linkage between IR and IFFR. However, note that in the duration analysis we have carried out, the temporal dimension is crucial and, although IR and IFFR are obviously dependent, there are complex delays that confound this dependence. Thus, Fig. 6, which shows the evolution of IFFR and IR over the period under study

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Table 5 Cox PH and CR regression estimatesa. Dependent variable: IFFRD. Variable

Cause-specific Cox PH

CR regression

Rate hike Coefficient IFFR IR IR-EUI ER UGDP UGDP-EGDP IFFRD(t  1) sIFFRD(t  1)

 0.643  1.171 4.230 1.512 0.104 0.032  0.006  1.547

Rate cut Coefficient

(0.237)* (0.586)** (3.730)* (2.622) (0.530) (0.077) (0.004) (0.576)*

0.985 1.419  3.064  2.071  0.345  0.370 0.006  1.634

Rate hike Coefficient

(0.344)* (0.556)** (1.247)** (1.755) (0.275) (0.169) (0.003) (0.678)*

 0.765  1.716 4.531 2.676 0.321 0.046  0.003  1.010

Rate cut Coefficient

(0.151)* (0.422)* (0.929)* (1.490)** (0.141)** (0.051) (0.003) (0.439)**

1.141 1.894  4.422  2.164  0.320  0.391 0.008  1.125

(0.302)* (0.550)* (1.379)* (1.330) (0.1431)* (0.186) (0.002)* (0.451)*

a

Standard errors in parentheses. Significant at 0.01 (two-sided). ** Significant at 0.05 (two sided).

7

Variable IFFR IR

6

Variable

5 4 3 2 1 0

Probability of rate hike conditioned on small and large rate hike

*

1.0

Small

0.8

Large

0.6 0.4 0.2 0.0 0

Date Fig. 6. Intended Federal Funds Rates and US inflation rate during 1999–2008.

(1999–2008), indicates that the IFFR has, in general, been close to the IR. This shows the importance of Fed operations in the aim of steering interest rates. It can be noted that the IR dispersion has remained within a narrower range than the IFFR. Looking at developments in both macroeconomic variables, it is remarkable that the US has often exhibited simultaneous declining (respectively increasing) IR and rising (respectively falling) IFFR since 1999. Hence, it is not surprising to find that not only there is no empirical evidence that an increase (respectively decrease) in IR is followed up immediately by a rate hike (respectively rate cut) in IFFR. According to our CR analysis, quite the opposite seems to happen. Mishkin [56] explanation of this involves the Fed forwardlooking behaviour and pre-emptive monetary policy. On the other hand, it has also been claimed that short-term velocity movements, due to implicitly accommodated money demand shocks, blur the short-term relationship between money and prices, especially in low inflation economies [57]. Until now, each event has been considered individually. Calculating the conditional probability is one way of incorporating the two types of information: interest rate hikes and cuts. Fig. 7 represents the conditional probabilities of interest rate hikes and cuts for small and large variations. The conditional probability of a rate hike before t days is the probability of a rate hike before time t knowing that a rate cut has not occurred for that period of time. Significant differences across the magnitude of the interest rate change appear. When considering the conditional probability of an IFFR hike, we note that for small changes the trend is similar to the conditional probability of an IFRR rate cut for large changes. There is approximately a 60% chance of a small interest rate hike (25 b.p.) before 50 days. The conditional probability for small interest rate

Probability of rate cut conditioned on small and large rate cut

01/01/2000 01/01/2002 01/01/2004 01/01/2006 01/01/2008

100

200 Days

300

1.0

400 Large

0.8

Small

0.6 0.4 0.2 0.0 0

100

200 Days

300

400

Fig. 7. Conditional probability for the IFFR by magnitude of change (small, large).

hikes is higher than for large interest rate hikes at any time. On the contrary, the conditional probability for large interest rate cuts is higher than for a small interest rate cuts at any time.

5. Summary and conclusions This paper analyses CR data aimed at understanding how federal funds target rates durations respond to changes in main macroeconomic information. The results imply a clear link between macroeconomic aggregates and interest rates duration. Such a relationship has an important policy dimension for financial analysis. Our results also illustrate the importance of distinguishing the different competing risks associated to interest rate hike and cut. They also suggest the overriding importance of considering factors that can explain the structure in Fed fund rates duration. Thus, it has been found that the current IFFR reduces the risk of rate hikes

E. Gutie´rrez, S. Lozano / Computers & Operations Research 39 (2012) 785–791

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