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The term premium, time varying interest rate volatility and central bank policy reaction
Economics Letters 76 (2002) 311–316 www.elsevier.com / locate / econbase
The term premium, time varying interest rate volatility and central bank policy reaction Peter Kugler* WWZ /University of Basel, Petersgraben 51, CH-4003 Basel, Switzerland Received 19 August 2001; accepted 31 January 2002
Abstract The solution of the McCallum policy reaction model of the term structure of interest rates including a volatility-dependent term premium indicates that the performance of the expectations hypothesis of the term structure (EHTS) in simple regression and ARCH in mean framework is strongly affected by policy reaction. This finding is illustrated with US data for the 1973–1995 period. 2002 Elsevier Science B.V. All rights reserved. Keywords: Term structure; Volatility; Term premium; GARCH-M; Policy reaction JEL classification: E43
1. Introduction In a series of recent papers McCallum (1994), Kugler (1997) and Hsu and Kugler (1997) present theoretical models and empirical results supporting a monetary policy explanation of the empirical rejection of the expectations hypothesis of the term structure of interest rates (EHTS). These papers solve and estimate models combining a time variant term premium formulation of the expectations hypothesis with a reaction of the central bank to the long-short spread. For analytical convenience, the term premium is modeled as an autoregressive process. This assumption is descriptive and is therefore not entirely satisfactory. Standard theoretical finance models of the term structure indicate that the term premium depends on short rate volatility. In fact, there is empirical literature following Engle et al. (1987) which take this dependence into account by adding a ARCH in mean term to EHTS test regression of the short rate change on the spread. This paper addresses two issues. First we show the effects of a volatility-dependent term premium * Tel.: 141-61-267-3344; fax: 141-61-267-1236. E-mail address: [email protected] (P. Kugler). 0165-1765 / 02 / $ – see front matter PII: S0165-1765( 02 )00059-9
2002 Elsevier Science B.V. All rights reserved.
P. Kugler / Economics Letters 76 (2002) 311 – 316
312
on standard tests of the expectations hypothesis in the framework of a policy reaction model. Second, we analyze the effects of policy reaction on the Engle–Lilien–Robins approach to model a time variant risk premium. To these ends we solve a rational expectations model of the term structure incorporating policy reaction as well as a GARCH(1,1) volatility-dependent term premium and illustrate the theoretical findings with 1- and 3-month US interest rate data.
2. The model Consider the following linearized version of the term structure equation: 1 R t 5 ]sr t 1 r et11 1 r et12 1 ? ? ? 1 r et1N 21d 1 gst11 1 j t N
(1)
e where R t is the N period (long) rate, r t is the one period (short) rate and r t1j denotes expected values 2 given information available in period t. s t 11 is the conditional variance (given information dated time t and earlier) of the short rate shock defined later and j t is an error term. The introduction of the conditional variance term can be motivated by the simple asset pricing model outlined by Engle et al. (1987). Moreover, general equilibrium stochastic models of the term structure as introduced by Cox et al. (1985) and Longstaff and Schwartz (1992) point to the relevance of the short-rate volatility for the term premium. The term premium error term is assumed to follow a stable AR(1) scheme:
j t 5 rj t21 1 u t u t | IID(0,s 2u )
ur u , 1
(2)
For expository purposes a constant term is omitted from Eq. (2). To close the model we postulate a McCallum (1994) policy reaction function: Dr t 5 lsR t 2 r td 1 zt
(3)
zt is the exogenous short rate shock which is assumed to follow a GARCH(1,1) process: 2 s t2 5 v 1 az t2 1 bs t21 zt | IINs0,s 2z d c5a1b,1
(4)
The first term on the RHS of Eq. (3) with coefficient l . 0 accounts for the endogeneity of monetary policy: the central bank increases the short rate when a widening spread signals higher expected future inflation and correspondingly higher future short rates. That is, Eq. (3) mimics a forward-looking counter cyclical policy by the central bank in a highly stylized way. A time varying short rate volatility is accounted for by a GARCH process for the exogenous short rate shock. The solution of this model using the method of undetermined coefficients is a simple extension of that given in detail by Kugler (1997) for the model without GARCH term. Therefore, the details, which are available from the author on request, are not given here and we proceed directly to the reduced form expression obtained:
P. Kugler / Economics Letters 76 (2002) 311 – 316
Ng R t 2 r t 5 rsR t21 2 r t 21d 1 ]]]]]] sv 1 (b 2 r ) s 2t d N21 j N2l sN 2 jd c j51 . N N 2 1 ]]]]]] az t 1 ]]]]]] ut N21 N 21 j j N2l (N 2 j) c N2l (N 2 j) r
313
O
O
(5)
O
j51
j51
Ng Dr t 5 lrsR t 21 2 r t21d 1 l ]]]]]] sv 1 (b 2 r ) s 2t d N 21 j N2l (N 2 j) c
O
j51
N N 1 l ]]]]]] az t2 1 zt 1 l ]]]]]] ut N 21 N 21 j j N2l (N 2 j) c N2l (N 2 j) r
O
j51
(6)
O
j 51
Now let us consider the implications of this model solution for testing the EHTS. We will discuss this issue under the simplifying assumption that N is equal to 2, but our findings are essentially relevant for the general N-period case. In the two period case, the reduced form Eq. (6) corresponds nicely to the usual framework for testing the expectations hypothesis: under the constant term premium assumption a simple regression of the short rate change on the lagged spread is run and the significance of the slope coefficient, which should be 2 under the EHTS, is tested. Assuming a GARCH time varying term premium a` la Engle–Lilien–Robbins, this regression equation is augmented by a GARCH in mean term. First of all we note that the coefficient of the lagged spread and the GARCH-M term is zero if l is equal to zero. This result is easily understood, as (3) indicates a random walk behavior for the short rate for l 5 0 and we have therefore no predictable variation of the short rate and we get results at odds with the EHTS as in the model without a GARCH term premium. Second we note that the standard simple regression test for the EHTS produces a regression coefficient which plim is different from lr in general: when the persistence coefficient ( r ) of the AR(1) process is different from that of the GARCH(1,1) process (b), this simple regression equation suffers from an omitted variable bias. As Eq. (5) indicates positive correlation between the spread and the conditional variance we have a positive (negative) bias of the two variable least-squares regression estimate of lr in Eq. (6) when b 2 r is larger (lower) than zero. Thus, the presence of a volatility dependent term premium may increase or decrease the slope coefficient of the simple EHTS regression depending on the values of b and r.1 . Third let us turn to the consequences of this model for the GARCH-M in mean coefficient estimate. Eq. (6) shows that the sign of the GARCH-M coefficients depends on the sign of b 2 r. Thus we may obtain a positive sign of this coefficient although we expect a negative sign when we 1
Moreover, we see from Eqs. (5) and (6) that the plim of the least-squares coefficient of a regression of Dr t on (R t 21 2 r t 21 ) is l times the plim of the least-squares coefficient of a regression of (R t 2 r t ) on (R t 21 2 r t 21 ). Thus, the indirect least-squares estimate of l is consistent and only the least-squares estimate of r is inconsistent. Alternatively this result can be obtained by the fact that the indirect least squares estimate of l is an instrumental variable estimate of (3) with the lagged spread as instrument. This is obviously a consistent but inefficient estimate.
314
P. Kugler / Economics Letters 76 (2002) 311 – 316
derive this test equation in the standard way by replacing expected values by observed ones and assuming j 5 0 in Eq. (1). Thus policy reaction may easily be conceived to result in unsuccessful attempts to model a time variant term premium in a standard GARCH in mean framework. Moreover, the GARCH process in the framework of the standard test equation is applied to an error term which is not only an expectations error but contains also the disturbance term of the premium. The coefficients of our model with the bivariate normal distribution for the error term u and z can be estimated by the application of maximum likelihood to the reduced form Eqs. (5) and (6) including the GARCH Eq. (4) and assuming a constant correlation between the two structural errors. In the next section we turn to the results of such an exercise with recent US data.
3. Empirical results The reduced form Eqs. (5) and (6) were estimated using weekly end of period US data for 1- and 3-months interest rates covering the period from 1973 to 1995. This data is of some interest in the present context as there is a striking break in the performance of the expectations hypothesis in 1987 (Hsu and Kugler, 1997) that can be rationalized by the policy reaction model. Moreover, interest rate volatility was subjected to considerable changes in this period: in particular we have the very high short rate volatility during the non-borrowed reserve targeting period from fall 1979 to fall 1982. We used four subsamples of our data set: 1 / 1 / 73–9 / 30 / 79, 10 / 1 / 79–9 / 30 / 82, 10 / 1 / 82–9 / 30 / 87 and 10 / 1 / 87–11 / 30 / 95. Of course, the first two subsamples reflect the introduction and abolishment of the unborrowed reserves targeting procedure. The split of the post-1982 sample is motivated by the fact that in the late 1980s, the use of the spread as a monetary policy indicator was advocated by FED officials. The empirical results are presented in Table 1. The model estimated differs slightly from our theoretical model as we used weekly data and the 1-month rate is not the one period rate. However, it can be easily shown that we only have to change the autoregressive coefficient r and c by the respective 4 weeks sum in the denominators in the two equations (Kugler, 1997). With respect to the policy reaction coefficient l and the AR coefficient r we note hardly any difference to the least-squares estimate of the model without GARCH in mean term reported in Hsu and Kugler (1997, Table 2). The result with respect to l is not surprising given our finding of the consistency of the indirect least-squares estimate of l. The result for r is also not surprising for the first two subsamples given the statistically insignificant parameter estimate for the volatility term g. However, for the third and, in particular the fourth subsample, the g estimate has the expected positive sign and is statistically significantly different from zero. The relatively low difference between the b and r estimates in Table 1 points to a relatively small bias of the simple regression estimate of r. The main difference between the third and the fourth subsample is the value of the l estimate. It is insignificant in the former case but highly significant in the latter case. Thus we would expect a good (bad) result for the EHTS in the fourth (third) subsample as obtained by Hsu and Kugler (1997). Moreover, we note that r is larger than b in the last subsample which means that we can expect the GARCH-M term in Engle–Lilien–Robins test equation to have the ‘right’ negative sign. In other words, US data since 1987 are very favorable to the EHTS. Finally note that with the exception of the first subsample parameter estimates for the GARCH(1,1) model do not indicate a unit root problem in the conditional variance process which is typical of many interest rate time series.
P. Kugler / Economics Letters 76 (2002) 311 – 316
315
Table 1 Maximum likelihood estimation result for the policy reaction model for 1- and 3-month Euro US dollar rates, weekly data 1973–1995 Sample
1 / 1 / 73–9 / 30 / 79
10 / 1 / 79–9 / 30 / 82
10 / 1 / 82–9 / 30 / 87
10 / 1 / 87–11 / 30 / 95
l
20.02859 (0.03783) 0.8458 (0.0241) 0.1624 (0.1141) 0.003820 (0.000873) 0.3593 (0.0613) 0.6769 (0.0361) 0.02087 (0.00141) 20.1079 (0.0679)
0.2627 (0.2314) 0.8457 (0.0387) 20.01024 (0.02538) 0.3260 (0.1026) 0.2673 (0.1268) 0.3733 (0.1920) 0.06456 (0.00755) 20.2936 (0.0978)
20.02348 (0.07631) 0.8398 (0.0224) 1.5812 (0.7610) 0.01141 (0.00278) 0.1512 (0.0523) 0.6429 (0.0544) 0.01399 (0.00080) 20.2585 (0.05313)
0.5647 (0.0225) 0.7455 (0.0145) 0.3094 (0.1196) 0.00670 (0.00101) 0.5834 (0.0738) 0.3950 (0.0388) 0.01215 (0.00049) 20.6272 (0.0334)
r g v a b
s 2u k
Estimated standard errors are given in parentheses. 3g 3g az t2 1 u tm R t 2 r t 21 5 c 1 1 rsR t 21 2 r t 21d 1 ]]]]]] sv 1 (b 2 r )s t2d 1 ]]]]]] 4 4 32l
Os2c 1 c j
j 14
d
32l
j 51
Os2c 1 c j
j 14
d
j 51
3g 3g az t2 1 zt 1 lu tm Dr t 5 lc 1 1 lrsR t 21 2 r t 21d 1 l ]]]]]] sv 1 (b 2 r )s t2d 1 l ]]]]]] 4 4 32l
Os2c 1 c j
j 14
d
32l
j 51
3 m 2 2 2 u t 5 ]]]]]] u t ,s t 5 v 1 az t 1 bs t 21 , 4 32l
Os2c 1 c j
j 14
d
Os2c 1 c j
j 14
d
j 51
f zt ,u tmg9 | N
SF GF 0 0
,
s 2t
kst su
kst su s 2u
GD
j 51
4. Conclusion This paper extends the policy reaction model of the term structure of interest rates of McCallum (1994) and Kugler (1997) by the inclusion of a short rate volatility dependent term premium. The solution of this model, assuming a GARCH(1,1) model for the short rate volatility, provides three main results. First the performance of the EHTS in a simple regression framework as well as the ARCH in mean framework introduced by Engle et al. (1987) depend crucially on the policy reaction to the spread. Second the slope coefficient of the regression of the short rate change on the spread, and
316
P. Kugler / Economics Letters 76 (2002) 311 – 316
therefore the empirical performance of the EHTS, may be increased or decreased by a volatility dependent term premium, depending on the parameter values. Third the mean effects of the volatility term in the standard GARCH-M test equation for the EHTS may have a ‘right’ negative or ‘wrong’ positive sign depending on the parameter values. Maximum likelihood estimation of our model with US data for four subperiods of 1973–1995 illustrate the theoretical findings outlined above.
Acknowledgements Earlier versions of this paper were presented at research seminars at Aarhus Business School and the University of Maastricht. Helpful comments from participants of these seminars as well as of an anonymous referee are gratefully acknowledged.
References Cox, J., Ingersoll, J.E., Ross, S.A., 1985. A theory of the term structure of interest rates. Econometrica 53, 385–407. Engle, R.F., Lilien, D.M., Robins, R.P., 1987. Estimating time varying risk premia in the term structure: The Arch-M model. Econometrica 55 (2), 391–407. Hsu, C.T., Kugler, P., 1997. The revival of the expectations hypothesis for the US term structure. Economics Letters 55, 115–120. Kugler, P., 1997. Central bank policy reaction and the expectations hypothesis of the term structure. International Journal of Finance and Economics 2 (3), 217–224. Longstaff, F.A., Schwartz, E.S., 1992. Interest rate volatility and the term structure: A two-factor model. Journal of Finance 47, 1259–1282. McCallum, B., 1994. Monetary Policy and the Term Structure of Interest Rates. NBER Working Paper No. 4938.
The term premium, time varying interest rate volatility and central bank policy reaction Peter Kugler* WWZ /University of Basel, Petersgraben 51, CH-4003 Basel, Switzerland Received 19 August 2001; accepted 31 January 2002
Abstract The solution of the McCallum policy reaction model of the term structure of interest rates including a volatility-dependent term premium indicates that the performance of the expectations hypothesis of the term structure (EHTS) in simple regression and ARCH in mean framework is strongly affected by policy reaction. This finding is illustrated with US data for the 1973–1995 period. 2002 Elsevier Science B.V. All rights reserved. Keywords: Term structure; Volatility; Term premium; GARCH-M; Policy reaction JEL classification: E43
1. Introduction In a series of recent papers McCallum (1994), Kugler (1997) and Hsu and Kugler (1997) present theoretical models and empirical results supporting a monetary policy explanation of the empirical rejection of the expectations hypothesis of the term structure of interest rates (EHTS). These papers solve and estimate models combining a time variant term premium formulation of the expectations hypothesis with a reaction of the central bank to the long-short spread. For analytical convenience, the term premium is modeled as an autoregressive process. This assumption is descriptive and is therefore not entirely satisfactory. Standard theoretical finance models of the term structure indicate that the term premium depends on short rate volatility. In fact, there is empirical literature following Engle et al. (1987) which take this dependence into account by adding a ARCH in mean term to EHTS test regression of the short rate change on the spread. This paper addresses two issues. First we show the effects of a volatility-dependent term premium * Tel.: 141-61-267-3344; fax: 141-61-267-1236. E-mail address: [email protected] (P. Kugler). 0165-1765 / 02 / $ – see front matter PII: S0165-1765( 02 )00059-9
2002 Elsevier Science B.V. All rights reserved.
P. Kugler / Economics Letters 76 (2002) 311 – 316
312
on standard tests of the expectations hypothesis in the framework of a policy reaction model. Second, we analyze the effects of policy reaction on the Engle–Lilien–Robins approach to model a time variant risk premium. To these ends we solve a rational expectations model of the term structure incorporating policy reaction as well as a GARCH(1,1) volatility-dependent term premium and illustrate the theoretical findings with 1- and 3-month US interest rate data.
2. The model Consider the following linearized version of the term structure equation: 1 R t 5 ]sr t 1 r et11 1 r et12 1 ? ? ? 1 r et1N 21d 1 gst11 1 j t N
(1)
e where R t is the N period (long) rate, r t is the one period (short) rate and r t1j denotes expected values 2 given information available in period t. s t 11 is the conditional variance (given information dated time t and earlier) of the short rate shock defined later and j t is an error term. The introduction of the conditional variance term can be motivated by the simple asset pricing model outlined by Engle et al. (1987). Moreover, general equilibrium stochastic models of the term structure as introduced by Cox et al. (1985) and Longstaff and Schwartz (1992) point to the relevance of the short-rate volatility for the term premium. The term premium error term is assumed to follow a stable AR(1) scheme:
j t 5 rj t21 1 u t u t | IID(0,s 2u )
ur u , 1
(2)
For expository purposes a constant term is omitted from Eq. (2). To close the model we postulate a McCallum (1994) policy reaction function: Dr t 5 lsR t 2 r td 1 zt
(3)
zt is the exogenous short rate shock which is assumed to follow a GARCH(1,1) process: 2 s t2 5 v 1 az t2 1 bs t21 zt | IINs0,s 2z d c5a1b,1
(4)
The first term on the RHS of Eq. (3) with coefficient l . 0 accounts for the endogeneity of monetary policy: the central bank increases the short rate when a widening spread signals higher expected future inflation and correspondingly higher future short rates. That is, Eq. (3) mimics a forward-looking counter cyclical policy by the central bank in a highly stylized way. A time varying short rate volatility is accounted for by a GARCH process for the exogenous short rate shock. The solution of this model using the method of undetermined coefficients is a simple extension of that given in detail by Kugler (1997) for the model without GARCH term. Therefore, the details, which are available from the author on request, are not given here and we proceed directly to the reduced form expression obtained:
P. Kugler / Economics Letters 76 (2002) 311 – 316
Ng R t 2 r t 5 rsR t21 2 r t 21d 1 ]]]]]] sv 1 (b 2 r ) s 2t d N21 j N2l sN 2 jd c j51 . N N 2 1 ]]]]]] az t 1 ]]]]]] ut N21 N 21 j j N2l (N 2 j) c N2l (N 2 j) r
313
O
O
(5)
O
j51
j51
Ng Dr t 5 lrsR t 21 2 r t21d 1 l ]]]]]] sv 1 (b 2 r ) s 2t d N 21 j N2l (N 2 j) c
O
j51
N N 1 l ]]]]]] az t2 1 zt 1 l ]]]]]] ut N 21 N 21 j j N2l (N 2 j) c N2l (N 2 j) r
O
j51
(6)
O
j 51
Now let us consider the implications of this model solution for testing the EHTS. We will discuss this issue under the simplifying assumption that N is equal to 2, but our findings are essentially relevant for the general N-period case. In the two period case, the reduced form Eq. (6) corresponds nicely to the usual framework for testing the expectations hypothesis: under the constant term premium assumption a simple regression of the short rate change on the lagged spread is run and the significance of the slope coefficient, which should be 2 under the EHTS, is tested. Assuming a GARCH time varying term premium a` la Engle–Lilien–Robbins, this regression equation is augmented by a GARCH in mean term. First of all we note that the coefficient of the lagged spread and the GARCH-M term is zero if l is equal to zero. This result is easily understood, as (3) indicates a random walk behavior for the short rate for l 5 0 and we have therefore no predictable variation of the short rate and we get results at odds with the EHTS as in the model without a GARCH term premium. Second we note that the standard simple regression test for the EHTS produces a regression coefficient which plim is different from lr in general: when the persistence coefficient ( r ) of the AR(1) process is different from that of the GARCH(1,1) process (b), this simple regression equation suffers from an omitted variable bias. As Eq. (5) indicates positive correlation between the spread and the conditional variance we have a positive (negative) bias of the two variable least-squares regression estimate of lr in Eq. (6) when b 2 r is larger (lower) than zero. Thus, the presence of a volatility dependent term premium may increase or decrease the slope coefficient of the simple EHTS regression depending on the values of b and r.1 . Third let us turn to the consequences of this model for the GARCH-M in mean coefficient estimate. Eq. (6) shows that the sign of the GARCH-M coefficients depends on the sign of b 2 r. Thus we may obtain a positive sign of this coefficient although we expect a negative sign when we 1
Moreover, we see from Eqs. (5) and (6) that the plim of the least-squares coefficient of a regression of Dr t on (R t 21 2 r t 21 ) is l times the plim of the least-squares coefficient of a regression of (R t 2 r t ) on (R t 21 2 r t 21 ). Thus, the indirect least-squares estimate of l is consistent and only the least-squares estimate of r is inconsistent. Alternatively this result can be obtained by the fact that the indirect least squares estimate of l is an instrumental variable estimate of (3) with the lagged spread as instrument. This is obviously a consistent but inefficient estimate.
314
P. Kugler / Economics Letters 76 (2002) 311 – 316
derive this test equation in the standard way by replacing expected values by observed ones and assuming j 5 0 in Eq. (1). Thus policy reaction may easily be conceived to result in unsuccessful attempts to model a time variant term premium in a standard GARCH in mean framework. Moreover, the GARCH process in the framework of the standard test equation is applied to an error term which is not only an expectations error but contains also the disturbance term of the premium. The coefficients of our model with the bivariate normal distribution for the error term u and z can be estimated by the application of maximum likelihood to the reduced form Eqs. (5) and (6) including the GARCH Eq. (4) and assuming a constant correlation between the two structural errors. In the next section we turn to the results of such an exercise with recent US data.
3. Empirical results The reduced form Eqs. (5) and (6) were estimated using weekly end of period US data for 1- and 3-months interest rates covering the period from 1973 to 1995. This data is of some interest in the present context as there is a striking break in the performance of the expectations hypothesis in 1987 (Hsu and Kugler, 1997) that can be rationalized by the policy reaction model. Moreover, interest rate volatility was subjected to considerable changes in this period: in particular we have the very high short rate volatility during the non-borrowed reserve targeting period from fall 1979 to fall 1982. We used four subsamples of our data set: 1 / 1 / 73–9 / 30 / 79, 10 / 1 / 79–9 / 30 / 82, 10 / 1 / 82–9 / 30 / 87 and 10 / 1 / 87–11 / 30 / 95. Of course, the first two subsamples reflect the introduction and abolishment of the unborrowed reserves targeting procedure. The split of the post-1982 sample is motivated by the fact that in the late 1980s, the use of the spread as a monetary policy indicator was advocated by FED officials. The empirical results are presented in Table 1. The model estimated differs slightly from our theoretical model as we used weekly data and the 1-month rate is not the one period rate. However, it can be easily shown that we only have to change the autoregressive coefficient r and c by the respective 4 weeks sum in the denominators in the two equations (Kugler, 1997). With respect to the policy reaction coefficient l and the AR coefficient r we note hardly any difference to the least-squares estimate of the model without GARCH in mean term reported in Hsu and Kugler (1997, Table 2). The result with respect to l is not surprising given our finding of the consistency of the indirect least-squares estimate of l. The result for r is also not surprising for the first two subsamples given the statistically insignificant parameter estimate for the volatility term g. However, for the third and, in particular the fourth subsample, the g estimate has the expected positive sign and is statistically significantly different from zero. The relatively low difference between the b and r estimates in Table 1 points to a relatively small bias of the simple regression estimate of r. The main difference between the third and the fourth subsample is the value of the l estimate. It is insignificant in the former case but highly significant in the latter case. Thus we would expect a good (bad) result for the EHTS in the fourth (third) subsample as obtained by Hsu and Kugler (1997). Moreover, we note that r is larger than b in the last subsample which means that we can expect the GARCH-M term in Engle–Lilien–Robins test equation to have the ‘right’ negative sign. In other words, US data since 1987 are very favorable to the EHTS. Finally note that with the exception of the first subsample parameter estimates for the GARCH(1,1) model do not indicate a unit root problem in the conditional variance process which is typical of many interest rate time series.
P. Kugler / Economics Letters 76 (2002) 311 – 316
315
Table 1 Maximum likelihood estimation result for the policy reaction model for 1- and 3-month Euro US dollar rates, weekly data 1973–1995 Sample
1 / 1 / 73–9 / 30 / 79
10 / 1 / 79–9 / 30 / 82
10 / 1 / 82–9 / 30 / 87
10 / 1 / 87–11 / 30 / 95
l
20.02859 (0.03783) 0.8458 (0.0241) 0.1624 (0.1141) 0.003820 (0.000873) 0.3593 (0.0613) 0.6769 (0.0361) 0.02087 (0.00141) 20.1079 (0.0679)
0.2627 (0.2314) 0.8457 (0.0387) 20.01024 (0.02538) 0.3260 (0.1026) 0.2673 (0.1268) 0.3733 (0.1920) 0.06456 (0.00755) 20.2936 (0.0978)
20.02348 (0.07631) 0.8398 (0.0224) 1.5812 (0.7610) 0.01141 (0.00278) 0.1512 (0.0523) 0.6429 (0.0544) 0.01399 (0.00080) 20.2585 (0.05313)
0.5647 (0.0225) 0.7455 (0.0145) 0.3094 (0.1196) 0.00670 (0.00101) 0.5834 (0.0738) 0.3950 (0.0388) 0.01215 (0.00049) 20.6272 (0.0334)
r g v a b
s 2u k
Estimated standard errors are given in parentheses. 3g 3g az t2 1 u tm R t 2 r t 21 5 c 1 1 rsR t 21 2 r t 21d 1 ]]]]]] sv 1 (b 2 r )s t2d 1 ]]]]]] 4 4 32l
Os2c 1 c j
j 14
d
32l
j 51
Os2c 1 c j
j 14
d
j 51
3g 3g az t2 1 zt 1 lu tm Dr t 5 lc 1 1 lrsR t 21 2 r t 21d 1 l ]]]]]] sv 1 (b 2 r )s t2d 1 l ]]]]]] 4 4 32l
Os2c 1 c j
j 14
d
32l
j 51
3 m 2 2 2 u t 5 ]]]]]] u t ,s t 5 v 1 az t 1 bs t 21 , 4 32l
Os2c 1 c j
j 14
d
Os2c 1 c j
j 14
d
j 51
f zt ,u tmg9 | N
SF GF 0 0
,
s 2t
kst su
kst su s 2u
GD
j 51
4. Conclusion This paper extends the policy reaction model of the term structure of interest rates of McCallum (1994) and Kugler (1997) by the inclusion of a short rate volatility dependent term premium. The solution of this model, assuming a GARCH(1,1) model for the short rate volatility, provides three main results. First the performance of the EHTS in a simple regression framework as well as the ARCH in mean framework introduced by Engle et al. (1987) depend crucially on the policy reaction to the spread. Second the slope coefficient of the regression of the short rate change on the spread, and
316
P. Kugler / Economics Letters 76 (2002) 311 – 316
therefore the empirical performance of the EHTS, may be increased or decreased by a volatility dependent term premium, depending on the parameter values. Third the mean effects of the volatility term in the standard GARCH-M test equation for the EHTS may have a ‘right’ negative or ‘wrong’ positive sign depending on the parameter values. Maximum likelihood estimation of our model with US data for four subperiods of 1973–1995 illustrate the theoretical findings outlined above.
Acknowledgements Earlier versions of this paper were presented at research seminars at Aarhus Business School and the University of Maastricht. Helpful comments from participants of these seminars as well as of an anonymous referee are gratefully acknowledged.
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