Frequency domain inference for univariate impulse responses

Economics Letters 63 (1999) 269–277

Frequency domain inference for univariate impulse responses Jonathan H. Wright* a

University of Virginia, Charlottesville, VA 22903, USA Received 2 February 1998; accepted 12 April 1998

Abstract Impulse response analysis is a major topic in applied econometrics where it is invariably conducted by fitting an autoregression to the time series being considered. This is not however the only approach that is available: it is also possible to estimate univariate impulse response coefficients in the frequency domain by smoothing the periodogram and then calculating the corresponding impulse response coefficients. In this paper, I compare the properties of these estimates with those obtained from estimating an autoregression. I also suggest forming confidence intervals for the impulse response coefficients using a frequency domain bootstrap and I evaluate the properties of these confidence intervals.  1999 Elsevier Science S.A. All rights reserved. Keywords: Impulse responses; Confidence intervals; Spectrum; Bootstrap JEL classification: C12

1. Introduction Impulse response analysis is a major topic in empirical macroeconomics and finance. The impulse response function gives the effect of a shock on future values of the time series being considered. It is invariably estimated by fitting an autoregression to the time series and solving for the impulse response’s coefficients corresponding to the estimated parameters. Confidence intervals are generally ¨ constructed either by the bootstrap (Runkle, 1987) or by the delta method (Lutkepohl, 1990). This is however not the only approach to estimating impulse response coefficients that is available. In the case of a univariate time series, it is also possible to estimate the spectrum by smoothing the periodogram and then calculating the corresponding impulse response coefficients. In the univariate case, there is a closed form expression for the impulse response coefficients in terms of the spectrum, which makes this calculation easy (there is no such expression in the multivariate case). This approach *Tel.: 11-804-924-3177; fax: 11-804-982-2904. E-mail address: [email protected] (J.H. Wright) 0165-1765 / 99 / $ – see front matter PII: S0165-1765( 99 )00043-9

 1999 Elsevier Science S.A. All rights reserved.

J.H. Wright / Economics Letters 63 (1999) 269 – 277

270

to estimation was proposed by Bhansali (1976). It is very closely related to the literature on predicting time series in the frequency domain which calculates the m-step ahead conditional expectation of a time series from the smoothed periodogram estimate of its spectrum. In this paper I compare the properties of the frequency domain estimates of the impulse response coefficients to those of the usual autoregressive estimates in Monte Carlo simulations. I also show ¨ how the frequency domain bootstrap algorithm of Franke and Hardle (1992) may be used to construct confidence intervals. I compare the properties of these confidence intervals to the properties of the usual bootstrap confidence intervals and find that they can have better coverage properties for the designs considered here. The plan of the remainder of this paper is as follows. In Section 2, I briefly review the standard methods of inference for impulse responses. Section 3 describes the frequency domain approach to inference and Section 4 contains the Monte Carlo evidence. Section 5 concludes and contains discussion of the issues associated with the multivariate extension of the methods considered in this paper.

2. Impulse response analysis Most of the literature on impulse response analysis is concerned with the case of a vector autoregression, but the impulse response function of univariate series such as GNP and real exchange rates has also attracted attention (e.g. Diebold and Rudebusch, 1989 and Koop, 1996). This paper is concerned with univariate impulse response analysis and the discussion of the conventional approach to inference in this section is specialized to the univariate case. Unit roots and near unit roots are known to cause difficulties (e.g. Sims and Zha, 1996 or Kilian, 1998). Though these issues are important, they are not the focus of the current paper which assumes that the researcher is considering the impulse response function of a univariate stationary time series y t (perhaps the first differences of some nonstationary variable). Let the Wold representation of this time series be

Ouu `

yt 5 m 1

j t2j

1 ut

(1)

j 51

2

where u t is i.i.d. with mean zero and variance s . The impulse response function gives the effect of a shock to u t on y t 1 , . This shock to u t could be of one unit, or it could be a shock of one standard deviation. The latter definition of the impulse response function does not depend on the units in which y t is measured. I adopt this definition henceforth throughout this paper. So the impulse response at lead time , is simply su, . The usual approach to estimation of the impulse response function is to fit a pth order autoregression to y t , by least squares, and then to solve for the corresponding impulse ¨ response coefficients. Confidence intervals may be constructed by the delta method (Lutkepohl, 1990) or by the bootstrap (Runkle, 1987). The asymptotic justification for these methods does not necessarily require the time series to be generated by a finite order autoregression, provided that p (the order of the autoregression fitted to the data) is an increasing function of the sample size. But the selection of p is a substantial practical issue. Kilian (1996) has discussed the use of information criteria to select p. He has found in Monte Carlo simulations that, although the Akaike information

J.H. Wright / Economics Letters 63 (1999) 269 – 277

271

criterion (AIC) is known to be inconsistent (overfitting the order of the autoregression with positive probability, even asymptotically), it gives the best finite sample properties when p is unknown.

3. The frequency domain approach to inference In this paper I consider instead a nonparametric approach to inference for univariate impulse responses that avoids fitting any autoregression to the data but rather calculates a smoothed periodogram estimate of the spectrum and backs out the corresponding impulse response coefficients. I suggest using a frequency domain bootstrap, that exploits the asymptotic independence of the periodogram ordinates, to form confidence intervals for the impulse responses. The series is assumed to have a spectrum f( l) and a periodogram 1 I( l) 5 ]] 2p T

UUO T

( y t 2 y¯ ) e

t51

UU

2

it l

such that the usual smoothed estimate of the spectrum

O S

D

l 2 lk 1 n ˆf( l) 5 ] K ]] I( lk ) Th k 52n h 21 T is consistent for f( l), uniformly over [2p,p ], where y¯ 5 T o t51 y t , n 5 T / 2, lk 5 (2p k)T, K is a kernel function, h is a bandwidth parameter and the sample size is T, which is henceforth assumed to be an even number, without loss of generality. One example of a kernel function is the Epanechnikov kernel which sets

K(v ) 5 0.75(1 2 w 2 )1(uv u # 1), which I use in the Monte Carlo work below. Primitive conditions for the uniform consistency of this estimate are well known (Brillinger, 1981). The time series has an infinite order moving average representation of the form y t 5 m 1 o `j 50 b j vt2j where hvt j is a sequence of uncorrelated random variables with mean zero and unit variance. Note that b 0 is not necessarily equal to one and in fact s 2 5 b 02 and uj 5 b j /b 0 in the usual Wold representation (1). The coefficients b 1 ,b 2 ,... are the impulse response coefficients. If the spectrum f( l) is known, it is possible to solve for these coefficients by a standard result (see, for example, Brillinger, 1981) which specifies that p

1 bj 5 ] 2p

E expS O c(v) exp(2ivl)Dexp(ijl)dl `

(2)

v51

2p

where p

1 c(v) 5 ] 2p

E log( f( l)) exp(ivl)dl . 2p

Of course the spectrum f( l) is not known, but is consistently estimable, uniformly over [2p,p ]. Since

J.H. Wright / Economics Letters 63 (1999) 269 – 277

272

ˆ l) each b j is a continuous functional of f( l), solving by this algorithm for hb j j corresponding to f( hence gives a consistent estimate of each impulse response. This approach to estimating the b j s was proposed by Bhansali (1976) and is closely related to the literature on forecasting time series in the frequency domain. Point estimates of the impulse responses are however of little value without associated confidence intervals. One way in which these may be provided is by a frequency domain bootstrap, which consists of the following steps:

21 1. Draw a series h j k j kn51 of i.i.d. random variables each of which is x 2 distributed on 2 degrees of freedom. ˆ l )j where l 5 (2p k /T ), for k 5 1,2, . . . , n 2 1. Also let I(0)* 5 I(p )* 5 0. 2. Let I( lk )* 5 ]12 f( k k k ˆ * the smoothed periodogram corresponding to I(.)* , i.e. 3. Let f(.)

O S

D

l 2 lk 1 n ˆ l)* 5 ] f( K ]] I( lk )* . Th k52n h ˆ *. 4. Let bˆ *1 ,bˆ 2* ,...denote the impulse response coefficients corresponding to f(.) 5. Repeat steps (i)–(iv) to generate a bootstrap sample of the estimated impulse responses. The confidence intervals for the impulse responses may be constructed from the percentiles of this bootstrap sample. ¨ This algorithm was proposed by Franke and Hardle (1992) except that they formed confidence intervals for the spectrum, not for the impulse responses, so they omitted step (iv). The motivation for this algorithm is that the periodogram ordinates are asymptotically independent and 2I( lk ) / ¨ f( lk ) → d x 2 (2), k 5 1, . . . , n 2 1. Franke and Hardle showed that their bootstrap distribution was asymptotically equal to the true distribution of the smoothed periodogram, even to second order.

4. Monte Carlo evidence In this section, I report Monte Carlo evidence on the finite sample behavior of the frequency domain estimator and bootstrap confidence intervals for the impulse response coefficients, in comparison with their usual autoregressive counterparts. The models I consider are an autoregressive (AR) model y t 5 m 1 f y t 21 1 u t and a moving average (MA) model y t 5 m 1 u t 1 u u t21 where u t is i.i.d. N(0,1). The true value of m is zero in all cases, but this is not imposed. The sample size is 100 and the parameters f and u take on the values 0.5 and 0.8. The impulse responses at lead times , 51–10 are estimated by fitting an autoregression with order determined by minimizing the AIC (as discussed in Section 2) and by the frequency domain estimate. The smoothed estimate of the spectrum was obtained using the Epanechnikov kernel with h 5 0.15, 0.3, h * where h * is the data-dependent bandwidth proposed by Beltrao and Bloomfield (1987). In each experiment, 1000 replications were conducted. The simulated bias and mean square error of the estimates are shown in Figs. 1 and 2, respectively. I also considered the coverage and average lengths of confidence intervals (90% nominal coverage), which are shown in Figs. 3 and 4, using the

J.H. Wright / Economics Letters 63 (1999) 269 – 277

273

Fig. 1. Bias of estimators. The dotted line refers to the usual autoregressive estimator. The solid, dashed and solid line with pluses refer to the frequency domain estimators with bandwidth h50.15, 0.3 and h * , respectively. In this and all subsequent figures, the lead time of the impulse response, ,, is plotted on the horizontal axis, while h * refers to the data-dependent bandwidth of Beltrao and Bloomfield (1987).

standard 1 and frequency domain bootstraps, using 500 bootstrap replications in each application of either bootstrap. The frequency domain estimates have generally less bias than the usual autoregressive estimates in the MA model. This is also true in the AR model, for bandwidth h50.15. In the AR model, the autoregressive estimates have smaller mean square error. The reverse can be true in the MA model. It makes sense that the autoregressive approach should perform best when the true data generating process is a finite order autoregression. The ordinary bootstrap confidence intervals have coverage that can be well below the nominal level. The frequency domain bootstrap with h50.15 gives higher coverage at all lead times and in all

1

In each replication, the order of the autoregression was determined by minimizing the AIC in the original sample and this order was then treated as fixed as the bootstrapping was conducted. Likewise, in the frequency domain bootstrap using the data-dependent bandwidth, h * was calculated for each replication and then treated as fixed in the bootstrap.

274

J.H. Wright / Economics Letters 63 (1999) 269 – 277

Fig. 2. Mean square error of estimators. The dotted line refers to the usual autoregressive estimator. The solid, dashed and solid line with pluses refer to the frequency domain estimators with bandwidth h50.15, 0.3 and h * , respectively.

models considered. The coverage of these confidence intervals is generally quite close to the nominal level.2 This is the key advantage of the frequency domain approach that emerges from these simulations. The price that is paid for controlling coverage is that the average width of these confidence intervals is high. At long lead times, they can be considerably wider than the usual bootstrap confidence intervals. In most cases, the properties of the frequency domain estimators and confidence intervals are not too sensitive to the choice of the bandwidth parameter h. The exception is the AR model with f 5 0.8. In this case, the bandwidth h50.15 gives the best results, while the data-dependent rule of Beltrao and Bloomfield (1987) consistently yields a value of h which is too large for reliable inference on impulse responses in this model.

2

I have also simulated the coverage of the confidence intervals in a sample of size 1000, which shows the coverage of either bootstrap method converging to the 90% nominal level, as the sample size gets large. These are not shown, to conserve space, but are available from the author on request.

J.H. Wright / Economics Letters 63 (1999) 269 – 277

275

Fig. 3. Coverage of confidence intervals. The dotted line refers to the usual bootstrap confidence intervals. The solid, dashed and solid line with pluses refer to the frequency domain bootstrap confidence intervals with bandwidth h50.15, 0.3 and h * , respectively. A horizontal solid line has been drawn at the 90% nominal coverage rate, for reference.

5. Conclusion and future research Frequency domain approaches to econometric inference have received considerable attention in recent literature (e.g. King and Watson, 1996 or Diebold and Kilian, 1997). This paper has compared the usual autoregressive approach to impulse response analysis to a frequency domain approach, in the case of a univariate time series. I have found the frequency domain approach to provide reliable inference in the design considered here. In future research, frequency domain inference could be applied to multivariate impulse responses, where the focus could be on either orthogonalized or nonorthogonalized impulse responses ¨ ¨ (Lutkepohl, 1990). The bootstrap algorithm of Franke and Hardle (1992) has recently been extended to the multivariate case by Berkowitz and Diebold (1998). The key difficulty in a frequency domain approach to multivariate impulse responses lies in solving for the impulse response coefficients from the spectrum. There is no multivariate counterpart of Eq. (2): there is no closed form expression for the moving average matrices in terms of the multivariate spectrum. There are some numerical algorithms which allow a researcher to solve for a finite order vector autoregression or a finite order

276

J.H. Wright / Economics Letters 63 (1999) 269 – 277

Fig. 4. Average width of confidence intervals. The dotted line refers to the usual bootstrap confidence intervals. The solid, dashed and solid line with pluses refer to the frequency domain bootstrap confidence intervals with bandwidth h50.15, 0.3 and h * , respectively.

vector moving average process that approximates a given spectrum (see Whittle, 1963, Wilson, 1972 and Wilson, 1993 among others). These may be helpful in providing a multivariate extension of the work in this paper, though it does not seem appropriate to use a low order parametric model to approximate the multivariate spectrum when the motivation for the frequency domain approach to inference is to avoid having to make strong parametric assumptions.

Acknowledgements I am grateful to Tim Bollerslev, Lutz Kilian, Granville Wilson and an anonymous referee for helpful comments. All errors are the sole responsibility of the author.

References Beltrao, K.I., Bloomfield, P., 1987. Determining the bandwidth of a kernel spectrum estimate. Journal of Time Series Analysis 8, 21–38.

J.H. Wright / Economics Letters 63 (1999) 269 – 277

277

Berkowitz, J., Diebold, F.X. Bootstrapping multivariate spectra, Review of Economics and Statistics (1998) forthcoming. Bhansali, R.J., 1976. Estimation of the moving average representation of a stationary nondeterministic process. Biometrika 63, 408–410. Brillinger, D.R., 1981. Time Series: Data Analysis and Theory, Holden–Day, San Francisco. Diebold, F.X., Rudebusch, G.D., 1989. Long Memory and Persistence in Aggregate Output. Journal of Monetary Economics 24, 189–209. Diebold, F.X., Kilian, L., 1997. Measuring Predictability: Theory and Macroeconomic Applications, NBER Technical Working Paper. ¨ Franke, J., Hardle, W., 1992. On bootstrapping kernel spectral estimates. Annals of Statistics 20, 121–145. Kilian, L., 1996. Impulse Response Analysis in Vector Autoregressions with Unknown Lag Order. Mimeo. Kilian, L., 1998. Small-Sample confidence intervals for impulse response functions. Review of Economics and Statistics 80, 218–230. King, R.G., Watson, M.W., 1996. Money, prices, interest rates and the business cycle. Review of Economics and Statistics 78, 35–53. Koop, G., 1996. Parameter uncertainty and impulse response analysis. Journal of Econometrics 72, 135–149. ¨ Lutkepohl, H., 1990. Asymptotic distributions of impulse response functions and forecast error variance decompositions of vector autoregressive models. Review of Economics and Statistics 72, 116–125. Runkle, D.E., 1987. Vector autoregressions and reality. Journal of Business and Economic Statistics 5, 437–442. Sims, C.A., Zha, T. (1996). Error Bands for Impulse Responses. Mimeo. Whittle, P., 1963. On the fitting of multivariate autoregression and the approximate canonical factorisation of a spectral density matrix. Biometrika 50, 129–134. Wilson, G.T., 1972. The factorisation of matricial spectral densities. SIAM Journal of Applied Mathematics 23, 420–426. Wilson, G.T., 1993. Developments in multivariate covariance generation and factorization. In: Subba Rao, T. (Ed.), Developments in Time Series Analysis, Chapman and Hall, London.