×

Long memory processes and fractional integration in econometrics. (English) Zbl 0854.62099

Summary: This paper provides a survey and review of the major econometric work on long memory processes, fractional integration, and their applications in economics and finance. Some of the definitions of long memory are reviewed, together with previous work in other disciplines. Section 3 describes the population characteristics of various long memory processes in the mean, including ARFIMA. Section 4 is concerned with estimation and examines semiparametric procedures in both the frequency and time domain, and also the properties of various regression based and maximum likelihood techniques. Long memory volatility processes are discussed in Section 5, while Section 6 discusses applications in economics and finance. The paper also has a concluding section.

MSC:

62P20 Applications of statistics to economics
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
91B84 Economic time series analysis
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adelman, I.: Long cycles: fact or artefact?. American economic review 55, 444-463 (1965)
[2] Adenstedt, R. K.: On large sample estimation for the mean of a stationary random sequence. Annals of mathematical statistics 2, 1095-1107 (1974) · Zbl 0296.62081
[3] Agiakloglou, C.; Newbold, P.: Lagrange multiplier tests for fractional difference. Journal of time series analysis 15, 253-262 (1994) · Zbl 0800.62551
[4] Agiakloglou, C.; Newbold, P.; Wohar, M.: Bias in an estimator of the fractional difference parameter. Journal of time series analysis 14, 235-246 (1992)
[5] Anis, A. A.; Lloyd, E. H.: The expected value of the adjusted rescaled Hurst range of independent normal summands. Biometrika 63, 111-116 (1976) · Zbl 0347.60061
[6] Avram, F.; Taqqu, M. S.: Noncentral limit theorems and Appell polynomials. Annals of probability 15, 767-775 (1987) · Zbl 0624.60049
[7] Aydogan, K.; Booth, G. G.: Are there long cycles in common stock returns?. Southern economic journal 55, 141-149 (1988)
[8] Backus, D. K.; Zin, S. E.: Long-memory inflation uncertainty: evidence from the term structure of interest rates. Journal of money, credit and banking, 681-700 (1993)
[9] Baillie, R. T.: Tests of rational expectations and market efficiency. Econometric reviews 8, 151-186 (1989) · Zbl 0718.62284
[10] Baillie, R. T.; Bollerslev, T.: Common stochastic trends in a system of exchange rates. Journal of finance 44, 167-181 (1989)
[11] Baillie, R. T.; Bollerslev, T.: Prediction in dynamic models with time-dependent conditional variances. Journal of econometrics 52, 91-113 (1992) · Zbl 0850.62902
[12] Baillie, R. T.; Bollerslev, T.: Cointegration, fractional cointegration and exchange rate dynamics. Journal of finance 49, 737-745 (1994)
[13] Baillie, R. T.; Bollerslev, T.: Long memory in the forward premium. Journal of international money and finance 13, 565-571 (1994)
[14] Baillie, R. T.; Pecchenino, R. A.: The search for equilibrium relationships in international finance. Journal of international money and finance 10, 582-593 (1991)
[15] Baillie, R. T.; Bollerslev, T.; Mikkelsen, H. -O.: Fractionally integrated generalized autoregressive conditional heteroskedasticity. Journal of econometrics (1996) · Zbl 0865.62085
[16] Baillie, R. T.; Chung, C. -F.; Tieslau, M. A.: Analyzing inflation by the fractionally integrated ARFIMA-GARCH model. Journal of applied econometrics (1995)
[17] Beran, J. A.: A test of location for data with slowly decaying serial correlations. Biometrika 76, 261-269 (1989) · Zbl 0669.62080
[18] Beran, J. A.: A goodness of fit test for time series with long range dependence. Journal of the royal statistical society B 54, 749-760 (1992) · Zbl 0775.62226
[19] Beran, J. A.: Statistical methods for data with long-range dependence. Statistical science 7, 404-427 (1992)
[20] Beran, J. A.; Terrin, N.: Estimation of the long-memory parameter, based on a multivariate central limit theorem. Journal of time series analysis 15, 269-278 (1994) · Zbl 0794.62054
[21] Beveridge: Weather and harvest cycles. Economic journal 31, 429-452 (1925)
[22] Blough, S. R.: The relationship between power and level for generic unit root tests in finite samples. Journal of applied econometrics 7, 295-308 (1992)
[23] Boes, D. C.; Davis, R. A.; Gupta, S. N.: Parameter estimation in low order fractionally differenced ARMA models. Stochastic hydrology and hydrolics 3, 97-110 (1989) · Zbl 0687.62079
[24] Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity. Journal of econometrics 31, 307-327 (1986) · Zbl 0616.62119
[25] Bollerslev, T.; Mikkelsen, H. -O.: Modeling and pricing long-memory in stock market volatility. Journal of econometrics (1996) · Zbl 0960.62560
[26] Bollerslev, T.; Chou, R. Y.; Kroner, K. F.: ARCH modeling in finance. Journal of econometrics 52, 5-59 (1992) · Zbl 0825.90057
[27] Booth, G. G.; Kaen, F. R.; Koveos, P. E.: R/S analysis of foreign exchange rates under two international money regimes. Journal of monetary economics 10, 407-415 (1982)
[28] Breidt, F. J.; Crato, N.; De Lima, P. J. F.: Modeling long-memory stochastic volatility. (1993) · Zbl 0905.62116
[29] Brockwell, P. J.; Davis, R.: Time series: theory and methods. (1987) · Zbl 0604.62083
[30] Campbell, J. Y.; Mankiw, N. G.: Are output fluctuations transitory?. Quarterly journal of economics 102, 857-880 (1987) · Zbl 1256.91037
[31] Campbell, J. Y.; Shiller, R.: Cointegration and tests of present value models. Journal of political economy 95, 1062-1088 (1987)
[32] Carlin, J. B.; Dempster, P.: Sensitivity analysis of seasonal adjustments: empirical case studies. Journal of the American statistical association 84, 6-32 (1989)
[33] Carlin, J. B.; Dempster, P.; Jonas, A. B.: On methods and models for Bayesian: time series analysis. Journal of econometrics 30, 67-90 (1985) · Zbl 0584.62191
[34] Chen, G.; Abraham, B.; Peiris, S.: Lag window estimation of the degree of differencing in fractionally integrated time series models. Journal of time series analysis 15, 473-487 (1994) · Zbl 0825.62685
[35] Cheung, Y. -W.: Long memory in foreign-exchange rates. Journal of business and economic statistics 11, 93-101 (1993)
[36] Cheung, Y. -W.: Tests for fractional integration: A Monte Carlo investigation. Journal of time series analysis 14, 331-345 (1993) · Zbl 0800.62546
[37] Cheung, Y. -W.; Diebold, F. X.: On maximum likelihood estimation of the differencing parameter of fractionally integrated noise with unknown mean. Journal of econometrics 62, 301-316 (1994)
[38] Cheung, Y. -W.; Lai, K. S.: A fractional cointegration analysis of purchasing power parity. Journal of business and economic statistics 11, 103-112 (1993)
[39] Choi, S.; Wohar, M. E.: The performance of the GPH estimator of the fractional difference parameter. Review of quantitative finance and accounting 2, 409-417 (1992)
[40] Chung, C. -F.: A note on calculating the autocovariances of fractionally integrated ARMA models. Economics letters 45, 293-297 (1994) · Zbl 0825.62672
[41] Chung, C. -F.: A generalized fractionally integrated ARMA process. (1994)
[42] Chung, C. -F.: On estimating a generalized long memory model. Journal of econometrics (1996)
[43] Chung, C. -F.; Baillie, R. T.: Small sample bias in conditional sum of squares estimators of fractionally integrated ARMA models. Empirical economics 18, 791-806 (1993)
[44] Crato, N.; De Lima, P. J. F.: Long-range dependence in the conditional variance of stock returns. Economics letters 45, 281-285 (1994) · Zbl 0800.62791
[45] Crato, N.; Rothman, P.: Fractional integration analysis of long-run behavior for US macroeconomic time series. Economics letters 45, 287-291 (1994) · Zbl 0800.90232
[46] Crato, N.; Rothman, P.: A reappraisal of parity reversion for UK real exchange rates. Applied economics letters 1, 139-141 (1994)
[47] Dacorogna, M. M.; Muller, U. A.; Nagler, R. J.; Olsen, R. B.; Pictet, O. V.: A geographical model for the daily and weekly seasonal volatility in the foreign exchange market. Journal of international money and finance 12, 413-438 (1993)
[48] Dahlhaus, R.: Small sample effects in time series analysis: A new asymptotic theory and a new estimator. Annals of statistics 16, 808-841 (1988) · Zbl 0662.62100
[49] Dahlhaus, R.: Efficient parameter estimation for self similar processes. Annals of statistics 17, 1749-1766 (1989) · Zbl 0703.62091
[50] Davies, R. B.; Harte, D. S.: Tests for Hurst effect. Biometrika 74, 95-102 (1987) · Zbl 0612.62123
[51] Davydov, Y. A.: The invariance principle for stationary processes. Theory of probability and its applications 15, 487-489 (1970)
[52] De Haan, L.: Fighting the arch-enemy with mathematics. Statistica neerlandica 44, 45-68 (1990) · Zbl 0713.62104
[53] Diba, B. T.; Grossman, H. T.: Explosive rational bubbles in stock prices?. American economic review 78, 520-530 (1988)
[54] Diebold, F. X.: Random walks versus fractional integration: power comparisons of scalar and joint tests of the variance-time function. Advances in econometrics, 29-45 (1989)
[55] Diebold, F. X.; Rudebusch, G. D.: Long memory and persistence in aggregate output. Journal of monetary economics 24, 189-209 (1989)
[56] Diebold, F. X.; Rudebusch, G. D.: Is consumption too smooth? long memory and the deaton paradox. Review of economics and statistics 73, 1-9 (1991)
[57] Diebold, F. X.; Rudebusch, G. D.: On the power of Dickey fuller tests against fractional alternatives. Economics letters 35, 155-160 (1991)
[58] Diebold, F. X.; Gardeazabal, J.; Yilmaz, K.: On cointegration and exchange rate dynamics. Journal of finance 49, 727-735 (1994)
[59] Diebold, F. X.; Husted, S.; Rush, M.: Real exchange rates under the gold standard. Journal of political economy 99, 1252-1271 (1991)
[60] Ding, Z.; Granger, C. W. J.; Engle, R. I.: A long memory property of stock returns and a new model. Journal of empirical finance 1, 83-106 (1993)
[61] Engel, C.: On the foreign exchange risk premium in a general equilibrium model. Journal of international economics 32, 305-319 (1991)
[62] Engle, R. F.: Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation. Econometrica 50, 987-1008 (1982) · Zbl 0491.62099
[63] Evans, M. D. D.; Lewis, K. K.: Do long term swings in the dollar affect estimates of the risk premia?. (1993)
[64] Fama, E. F.: Spot and forward exchange rates. Journal of monetary economics 14, 319-338 (1984)
[65] Faust, J.: Near observational equivalence and unit root processes: formal concepts and implications. (1994)
[66] Feller, W.: The asymptotic distribution of the range of sums of independent random variables. Annals of mathematical statistics 22, 427-432 (1951) · Zbl 0043.34201
[67] Fox, R.; Taqqu, M. S.: Noncentral limit theorems for quadratic forms in random variables having long range dependence. Annals of probability 13, 428-446 (1985) · Zbl 0569.60016
[68] Fox, R.; Taqqu, M. S.: Large sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Annals of statistics 14, 517-532 (1986) · Zbl 0606.62096
[69] Fox, R.; Taqqu, M. S.: Central limit theorems for quadratic forms in random variables having long-range dependence. Probability theory and related fields 74, 213-240 (1987) · Zbl 0586.60019
[70] Galbraith, R. F.; Galbraith, J. F.: On the inverse of some patterned matrices arising in the theory of stationary time series. Journal of applied probability 11, 63-71 (1974) · Zbl 0278.62016
[71] Geweke, J.; Porter-Hudak, S.: The estimation and application of long memory time series models. Journal of time series analysis 4, 221-238 (1983) · Zbl 0534.62062
[72] Giraitis, L.; Surgailis, D.: A central limit theorem for quadratic form in strongly dependent linear variables and its application to whittle’s estimate. Probability theory and related fields 86, 87-104 (1990) · Zbl 0717.62015
[73] Gradszteyn, I. S.; Ryzhnik, I. M.: Tables of integrals, series and products. (1980)
[74] Granger, C. W. J.: The typical spectral shape of an economic variable. Econometrica 34, 150-161 (1966)
[75] Granger, C. W. J.: Long memory relationships and the aggregation of dynamic models. Journal of econometrics 14, 227-238 (1980) · Zbl 0466.62108
[76] Granger, C. W. J.: Some properties of time series data and their use in econometric model specification. Journal of econometrics 16, 121-130 (1981)
[77] Granger, C. W. J.: Cointegrated variables and error correction models. (1983) · Zbl 0547.62060
[78] Granger, C. W. J.; Anderson, A.: On the invertibility of time series models. Stochastic process applications 8, 87-92 (1978) · Zbl 0387.62076
[79] Granger, C. W. J.; Joyeux, R.: An introduction to long memory time series models and fractional differencing. Journal of time series analysis 1, 15-39 (1980) · Zbl 0503.62079
[80] Gray, H. L.; Zhang, N. -F.; Woodward, W. A.: On generalized fractional processes. Journal of time series analysis 10, 233-257 (1989) · Zbl 0685.62075
[81] Greene, M.; Fielitz, B.: Long-term dependence in common stock returns. Journal of financial economics 4, 339-349 (1977)
[82] Hakkio, C. S.; Rush, M.: Cointegration: how short is the long run?. Journal of international money and finance 10, 571-581 (1991)
[83] Harvey, A. C.: Long memory in stochastic volatility. (1993)
[84] Haslett, J.; Raftery, A. E.: Space-time modelling with long-memory dependence: assessing Ireland’s wind power resource. Journal of the royal statistical society C 38, 1-50 (1989)
[85] Hassler, U.: Regression of spectral estimators with fractionally integrated time series. Journal of time series analysis 14, 369-380 (1993) · Zbl 0782.62085
[86] Hassler, U.: (Mis)specification of long memory in seasonal time series. Journal of time series analysis 15, 19-30 (1994) · Zbl 0794.62059
[87] Hassler, U.; Wolters, J.: On the power of unit roots against fractionally integrated alternatives. Economics letters 45, 1-5 (1994) · Zbl 0800.62532
[88] Hassler, U.; Wolters, J.: Long memory in inflation rates: international evidence. Journal of business and economic statistics 13, 37-45 (1995)
[89] Haubrich, J. G.: Consumption and fractional differencing: old and new anomalies. (1992)
[90] Haubrich, J. G.; Lo, A. W.: The sources and nature of long-term memory in the business cycle. (1993)
[91] Hauser, M. A.: Semiparametric and nonparametric testing for long memory. (1994)
[92] Helson, J.; Sarason, Y.: Past and future. Mathematica Scandinavia 21, 5-16 (1967) · Zbl 0241.60029
[93] Hipel, K. W.; Mcleod, A. I.: Preservation of the rescaled adjusted range, 3: fractional Gaussian noise algorithms. Water resources research 14, 517-518 (1978)
[94] Hipel, K. W.; Mcleod, A. I.: Preservation of the rescaled adjusted range, 2: simulation studies using box-Jenkins models. Water resources research 14, 509-516 (1978)
[95] Hols, M. C. A.B.; De Vries, C. G.: The limiting distribution of extremal exchange rate returns. Journal of applied econometrics 6, 287-302 (1991) · Zbl 0735.62109
[96] Hosking, J. R. M.: Fractional differencing. Biometrika 68, 165-176 (1981) · Zbl 0464.62088
[97] Hosking, J. R. M.: Modeling persistence in hydrological time series using fractional differencing. Water resources research 20, 1898-1908 (1984)
[98] Hurst, H. E.: Long-term storage capacity of reservoirs. Transactions of the American society of civil engineers 116, 770-799 (1951)
[99] Hurst, H. E.: Methods of using long term storage in reservoirs. Proceedings of the institute of civil engineers 1, 519-543 (1956)
[100] Hurst, H. E.: A suggested statistical model of some time series that occur in nature. Nature 180, 494 (1957) · Zbl 0077.33601
[101] Hurvich, C. M.; Beltrao, K. I.: Automatic semiparametric estimation of the long memory parameter of a long memory time series. Journal of time series analysis 15, 285-302 (1994) · Zbl 0807.62076
[102] Hurvich, C. M.; Ray, B. K.: Estimation of the memory parameter for non-stationary or non-invertible fractionally integrated processes. Journal of time series analysis 16, 17-41 (1995) · Zbl 0813.62081
[103] Janacek, G. J.: Determining the degree of differencing for time series via the log spectrum. Journal of time series analysis 3, 177-183 (1982)
[104] Kaen, F. R.; Rosenman, R. E.: Predictable behavior in financial markets: some evidence in support of heinber’s hypothesis. American economic review 76, 212-220 (1986)
[105] Kennedy, D.: The distribution of the maximum Brownian excursion. Journal of applied probability 13, 371-376 (1976) · Zbl 0338.60048
[106] Kim, Y.: Purchasing power parity in the long run: A cointegration approach. Journal of money, credit and banking 22, 491-503 (1990)
[107] King, M. L.; Hillier, G.: Locally best invariant tests of the error covariance matrix of the linear regression model. Journal of the royal statistical society B 47, 98-102 (1985) · Zbl 0588.62087
[108] Koedijk, K. G.; Schafgans, M. M. A.; De Vries, C. G.: The tail index of exchange rate returns. Journal of international economics 29, 93-108 (1990)
[109] Kolmogorov, A. N.: Wienersche spiralen und einige andere interessante kurven im hilbertschen raum, comptes rendues (Doklady). Urss ns 26, 115-118 (1940) · JFM 66.0552.03
[110] Kunsch, H.: Discrimination between monotonic trends and long-range dependence. Journal of applied probability 23, 1025-1030 (1986) · Zbl 0623.62085
[111] Kwiatkowski, D.; Phillips, P. C. B.; Schmidt, P.; Shin, Y.: Testing the null hypothesis of stationarity against the alternative of a unit root: how sure are we that economic time series have a unit root?. Journal of econometrics 54, 159-178 (1992) · Zbl 0871.62100
[112] Lawrance, A. J.; Kottegoda, N. T.: Stochastic modelling of riverflow time series. Journal of the royal statistical society A 140, 1-47 (1977)
[113] Leadbetter, M. R.; Lindgren, G.; Rootzen, H.: Extremes and related properties of random sequences and processes. (1984) · Zbl 0518.60021
[114] Lee, D.; Schmidt, P.: On the power of the KPSS test of stationarity against fractionally integrated alternatives. Journal of econometrics (1996) · Zbl 0856.62075
[115] Li, W. K.; Mcleod, A. I.: Fractional time series modeling. Biometrika 73, 217-221 (1986)
[116] Lin, J. -Y.: Generalized integrated process and the aggregation of dynamic time series. Academia economic papers 19, 207-226 (1991)
[117] Lo, A. W.: Long term memory in stock market prices. Econometrica 59, 1279-1313 (1991) · Zbl 0781.90023
[118] Lobato, I.; Robinson, P. M.: Averaged periodogram estimation of long memory. Journal of econometrics (1996) · Zbl 0854.62088
[119] Macneill, I.: Properties of sequences of partial sums of polynomial regression residuals with application to tests of change of regression at unknown times. Annals of statistics 6, 422-433 (1978) · Zbl 0375.62064
[120] Mcleod, A. I.; Hipel, K. W.: Preservation of the rescaled adjusted range, I: A reassessment of the Hurst phenomenon. Water resources research 14, 491-508 (1978)
[121] Mandelbrot, B. B.: When can price be arbitraged efficiently? A limit to the validity of the random walk and martingale models. Review of economics and statistics 53, 225-236 (1971)
[122] Mandelbrot, B. B.: Statistical methodology for non periodic cycles: from the covariance to R/S analysis. Annals of economic and social measurement 1, 259-290 (1972)
[123] Mandelbrot, B. B.: A fast fractional Gaussian noise generator. Water resources research 7, 543-553 (1975)
[124] Mandelbrot, B. B.; Taqqu, M.: Robust R/S analysis of long run serial correlation. Bulletin of international statistical institute 48, 59-104 (1979) · Zbl 0518.62036
[125] Mandelbrot, B. B.; Van Ness, J. W.: Fractional Brownian motions, fractional Brownian noises and applications. SIAM review 10, 422-437 (1968) · Zbl 0179.47801
[126] Mandelbrot, B. B.; Wallis, J.: Noah, Joseph and operational hydrology. Water resources research 4, 909-918 (1968)
[127] Mandelbrot, B. B.; Wallis, J.: Computer experiments with fractional Gaussian noises, parts 1,2,3. Water resources research 5, 228-267 (1969)
[128] Mandelbrot, B. B.; Wallis, J.: Robustness of the rescaled range R/S in the measurement of noncyclic long-run statistical dependence. Water resources research 5, 967-988 (1969)
[129] Meese, R. A.; Singleton, K. L.: On unit roots and the empirical modeling of exchange rates. Journal of finance 37, 1029-1035 (1981)
[130] Moehring, R.: Parameter estimation in Gaussian intermediate memory time series. (1990)
[131] Nelson, D. B.: Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59, 347-370 (1991) · Zbl 0722.62069
[132] Nelson, C. R.; Plosser, C. I.: Trends and random walks in macroeconomic time series: some evidence and implications. Journal of monetary economics 10, 139-162 (1982)
[133] Newbold, P.: The exact likelihood function for a mixed autoregressive-moving average process. Biometrika 61, 423-426 (1974) · Zbl 0292.62061
[134] Newbold, P.; Agiakloglou, C.: Bias in the sample autocorrelations of fractional white noise. Biometrika 80, 698-702 (1993) · Zbl 0800.62519
[135] Noakes, D. J.; Hipel, K. W.; Mcleod, A. I.; Jimenez, C.; Yakowitz, S.: Forecasting annual geophysical time series. International journal of forecasting 4, 103-115 (1988)
[136] Odaki, M.: On the invertibility of fractionally differenced ARIMA processes. Biometrika 80, 703-709 (1993) · Zbl 0785.62089
[137] Peiris, M. S.: A note on the predictors of differenced sequences. Australian journal of statistics 29, 42-48 (1987) · Zbl 0631.62101
[138] Peiris, M. S.; Perera, B. J. C.: On prediction with fractionally differenced ARMA models. Journal of time series analysis 9, 215-220 (1988) · Zbl 0668.62069
[139] Phillips, P. C. B.: Time series regression with a unit root. Econometrica 55, 277-301 (1987) · Zbl 0613.62109
[140] Porter-Hudak, S.: An application of the seasonally fractionally differenced model to the monetary aggregates. Journal of the American statistical association 85, 338-344 (1990)
[141] Ray, B. K.: Modeling long-memory processes for optimal long-range prediction. Journal of time series analysis 14, 511-525 (1993) · Zbl 0779.62088
[142] Ray, B. K.: Long-range forecasting of IBM product revenues using a seasonal fractionally differenced ARMA model. International journal of forecasting 9, 255-269 (1993)
[143] Reisen, V. A.: Estimation of the fractional difference parameter in the ARFIMA (p, d, q) model using the smoothed periodogram. Journal of time series analysis 15, 335-351 (1994) · Zbl 0803.62084
[144] Resnick: Extreme values, regular variation and point processes. (1987) · Zbl 0633.60001
[145] Robinson, P. M.: Statistical inference for a random coefficient autoregressive model. Scandinavian journal of statistics 5, 163-168 (1978) · Zbl 0392.62072
[146] Robinson, P. M.: Time series with strong dependence. Advances in econometrics, 6th world congress (1990)
[147] Robinson, P. M.: Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. Journal of econometrics 47, 67-84 (1991) · Zbl 0734.62070
[148] Robinson, P. M.: Semiparametric analysis of long-memory time series. Annals of statistics 22, 515-539 (1992) · Zbl 0795.62082
[149] Rosenblatt, M.: A central limit theorem and a strong mixing condition. Proceedings of the national Academy of sciences 42, 43-47 (1956) · Zbl 0070.13804
[150] Samarov, A.; Taqqu, M. S.: On the efficiency of the sample mean in long memory noise. Journal of time series analysis 9, 191-200 (1988) · Zbl 0637.62085
[151] Schmidt, P.; Phillips, P. C. B.: LM tests for a unit root in the presence of deterministic trend. Oxford bulletin of economics and statistics 54, 257-287 (1992)
[152] Seater, J. J.: World temperature-trend uncertainties and their implications for economic policy. Journal of business and economic statistics 11, 265-277 (1993)
[153] Sephton, P. S.; Larsen, H. K.: Tests of exchange market efficiency: fragile evidence from cointegration tests. Journal of international money and finance 10, 561-570 (1991)
[154] Shea, G. S.: Uncertainty and implied variance bounds in long memory models of the interest rate term structure. Empirical economics 16, 287-312 (1991)
[155] Shiller, R. J.; Perron, P.: Testing the random walk hypothesis: power versus frequency of observations. Economics letters 18, 381-386 (1985) · Zbl 1273.91383
[156] Siddiqui, M.: The asymptotic distribution of the range and other functions of partial sums of stationary processes. Water resources research 12, 1271-1276 (1976)
[157] Sinai, Y. G.: Self-similar probability distributions. Theory of probability and its applications 21, 64-80 (1976) · Zbl 0358.60031
[158] Smith; Sowell, F. B.; Zin, S. E.: Fractional integration with drift: estimation in small samples. (1993)
[159] Sowell, F. B.: Fractionally integrated vector time series. Ph.d. dissertation (1986)
[160] Sowell, F. B.: The fractional unit root distribution. Econometrica 58, 495-505 (1990) · Zbl 0727.62025
[161] Sowell, F. B.: Maximum likelihood estimation of stationary univarite fractionally integrated time series models. Journal of econometrics 53, 165-188 (1992)
[162] Sowell, F. B.: Modeling long run behavior with the fractional ARIMA model. Journal of monetary economics 29, 277-302 (1992)
[163] Steigerwald, D. G.: Purchasing power parity, unit roots and dynamic structure. Journal of empirical finance (1994)
[164] Taqqu, M. S.: Weak convergence to fractional Brownian motion and to the rosenblatt process. Zeitschrift für wahrscheinlichkeitstheorie und verwandte gebiete 31, 287-302 (1975) · Zbl 0303.60033
[165] Taqqu, M. S.: Law of the iterated logarithms for sums of non-linear functions of Gaussian variables that exhibit a long range dependence. Zeitschrift für wahrscheinlichkeitstheorie und verwandte gebiete 40, 203-238 (1977) · Zbl 0358.60048
[166] Taylor, S.: Modeling financial time series. (1986) · Zbl 1130.91345
[167] Tieslau, M. A.: Strongly dependent economic time series: theory and applications. Ph.d. dissertation (1992)
[168] Tieslau, M. A.; Schmidt, P.; Baillie, R. T.: A minimum-distance estimator for long memory processes. Journal of econometrics (1995) · Zbl 0850.62663
[169] Tschernig, R.: Wechselkurse, unsicherheit und long memory. Ph.d. dissertation (1992)
[170] Whittle, P.: Hypothesis testing in time series analysis. (1951) · Zbl 0045.41301
[171] Whittle, P.: Variation of yield variance with plot size. Biometrika 43, 337-343 (1956) · Zbl 0074.14302
[172] Wu, P.: Testing fractionally integrated time series. (1992)
[173] Yajima, Y.: On estimation of long memory time series models. Australian journal of statistics 27, 303-320 (1985) · Zbl 0584.62142
[174] Yajima, Y.: On estimation of a regression model with long memory stationary errors. Annals of statistics 16, 791-807 (1988) · Zbl 0661.62090
[175] Yajima, Y.: A central limit theorem of Fourier transforms of strongly dependent stationary processes. Journal of time series analysis 10, 373-383 (1989) · Zbl 0692.60024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.