Abstract
The terminology multiplicative error model (MEM) has been introduced by Engle (2002b) for a general class of time series models for positive-valued random variables which are decomposed into the product of their conditional mean and a positive-valued error term. Such models might be alternatively classified as autoregressive conditional mean models where the conditional mean of a distribution is assumed to follow a stochastic process. The idea of a MEM is well known in financial econometrics and originates from the structure of the autoregressive conditional heteroscedasticity (ARCH) model introduced by Engle (1982) or the stochastic volatility (SV) model proposed by Taylor (1982) where the conditional variance is dynamically parameterized and multiplicatively interacts with an innovation term. In high-frequency econometrics, a MEM has been firstly introduced by Engle and Russell (1997, 1998) to model the dynamic behavior of the time between trades and was referred to as autoregressive conditional duration (ACD) model.
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Notes
- 1.
For more details, see Sect. 5.3.1, where the asymptotic properties of the GARCH QML estimator are carried over to ACD models.
- 2.
In the case \(P = Q = 1\), we set α : = α1 and β : = β1.
- 3.
See also the descriptive statistics in Chap. 3.
- 4.
UWLLN: Uniform Weak Law of Large Numbers.
- 5.
For an overview of mixture distributions, see, e.g., Lancaster (1997).
- 6.
In some studies, this model is also called “Nelson type” ACD model since it resembles the EGARCH specification proposed by Nelson (1991).
- 7.
References
Aït-Sahalia Y (1996) Testing continuous-time models of the spot interest rate. Rev Financ Stud 9:385–426
Allen D, Chan F, McAleer M, Peiris S (2008) Finite sample properties of the QMLE for the Log-ACD model: application to Australian stocks. J Econom 147:163–185
Andersen TG, Bollerslev T (1998b) Deutsche mark-dollar volatility: intraday activity patterns, macroeconomic announcements, and longer run dependencies. J Finance 53:219–265
Andrews D (1991) Heteroscedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59:817–858
Bauwens L, Giot P, Grammig J, Veredas D (2004) A comparison of financial duration models via density forecasts. Int J Forecast 20:589–609
Bauwens L, Galli F, Giot P (2008) The moments of Log-ACD models. Quant Qual Anal Soc Sci 2:1–28
Bauwens L, Giot P (2000) The logarithmic ACD model: an application to the bid/ask quote process of two NYSE stocks. Annales d’Economie et de Statistique 60:117–149
Bierens HJ (1982) Consistent model specification tests. J Econom 20:105–134
Bierens HJ (1990) A consistent conditional moment test of functional form. Econometrica 58:1443–1458
Bollerslev T (1986) Generalized autoregressive conditional heteroskedasticity. J Econom 31:307–327
Bollerslev T, Wooldridge J (1992) Quasi-maximum likelihood estimation and inference in dynamic models with time varying covariances. Econom Rev 11:143–172
Box GEP, Pierce DA (1970) Distribution of residual autocorrelations in the autoregressive-integrated moving average time series models. J Am Stat Assoc 65:1509–1526
Breusch TS (1978) Testing for autocorrelation in dynamic linear models. Aust Econ Pap 17:334–355
Breusch TS, Pagan RA (1979) A simple test for heteroskedasticity and random coefficient variation. Econometrica 47:203–207
Brock W, Dechert WD, Scheinkman J, LeBaron B (1996) A test for independence based on the correlation dimension. Econom Rev 15:197–235
Brownlees C, Cipollini F, Gallo GM (2011) Journal of Financial Econometrics 9:489–518
Brownlees C, Gallo GM (2011) International Journal of Forecasting 27:365–378
Chen X (2000) A beta kernel estimation for the density functions. Comput Stat Data Anal 31:131–145
Chen Y-T, Hsieh C-S (2010) Generalized moment tests for autoregressive conditional duration models. J Financ Econom 8:345–391
de Boor C (1978) A practical guide to splines. Springer Verlag, Berlin, Heidelberg
de Jong RM (1996) The Bierens test under data dependence. J Econom 72:1–32
De Luca G, Gallo G (2009) Time-varying mixing weights in mixture autoregressive conditional duration models. Econom Rev 28:101–120
Diebold FX, Gunther TA, Tay AS (1998) Evaluating density forecasts, with applications to financial risk management. Int Econ Rev 39:863–883
Drost FC, Werker BJM (2004) Semiparametric duration models. J Bus Econ Stat 22:40–50
Duchesne P, Pacurar M (2008) Evaluating financial time series models for irregularly spaced data: a spectral density approach. Comput Oper Res 35:130–155
Dufour A, Engle RF (2000) The ACD model: predictability of the time between consecutive trades. Working Paper, ISMA Centre, University of Reading
Dufour JM, Roy R (1985) Some robust exact results on sample autocorrelations and tests of randomnes. J Econom 29:257–273
Dufour JM, Roy R (1986) Generalized Portmanteau statistics and tests of randomness. Commun Stat Theory Methods 15:2953–2972
Engle RF (2000) The econometrics of ultra-high-frequency data. Econometrica 68(1):1–22
Engle RF (2002b) New frontiers for ARCH models. J Appl Econom 17:425–446
Engle RF (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50:987–1006
Engle RF (1984) Wald, likelihood ratio and lagrange multiplier tests in econometrics. In: Griliches Z, Intriligator MD (eds) Handbook of econometrics, vol. II, chap. 13. Elsevier Science, pp. 775–826
Engle RF (1996) The Econometrics of ultra-high frequency data. Discussion Paper 96-15, University of California San Diego
Engle RF, Gallo GM (2006) A multiple indicators model for volatility using intra-daily data. J Econom 131:3–27
Engle RF, Ng VK (1993) Measuring and testing the impact of news on volatility. J Finance 48:1749–1778
Engle RF, Russell JR (1997) Forecasting the frequency of changes in quoted foreign exchange prices with the autoregressive conditional duration model. J Empir Financ 4:187–212
Engle RF, Russell JR (1998) Autoregressive conditional duration: a new model for irregularly spaced transaction data. Econometrica 66:1127–1162
Fernandes M (2004) Bounds for the probability distribution function of the linear ACD process. Stat Probab Lett 68:169–176
Fernandes M, Grammig J (2005) Non-parametric specification tests for conditional duration models. J Econom 127:35–68
Gallant RA (1981) On the bias in flexible functional forms and an essential unbiased form: The Fourier flexible form. J Econom 15:211–245
Gosh BK, Huang W-M (1991) The power and optimal kernel of the Bickel–Rosenblatt test for goodness-of-fit. Ann Stat 19:999–1009
Ghysels E, Gouriéroux C, Jasiak J (1998) Stochastic volatility duration models. Discussion paper, CIRANO
Godfrey LG (1978) Testing against general autoregressive and moving average error models when the regressors include lagged dependent variables. Econometrica 46:1293–1302
Godfrey LG (1996) Misspecification tests and their use in econometrics. J Stat Plan Inference 49:241–260
Gouriéroux C, Monfort A, Trognon A (1984) Pseudo maximum likehood methods: theory. Econometrica 52:681–700
Grammig J, Maurer K-O (2000) Non-monotonic hazard functions and the autoregressive conditional duration model. Econom J 3:16–38
Hagmann M, Scaillet O (2007) Local multiplicative bias correction for asymmetric kernel density estimators. J Econom 141:213–249
Hamilton JD, Jorda O (2002) A model of the federal funds rate target. J Polit Econ 110:1135
Hautsch N (2003) Assessing the risk of liquidity suppliers on the basis of excess demand intensities. J Financ Econom 1:189–215
Hautsch N, Malec P, Schienle M (2010) Capturing the zero: a new class of zero-augmented distributions and multiplicative error processes. Discussion Paper 2010-055, Humboldt-Universität zu Berlin
Hendry DF (1995) Dynamic econometrics. Oxford University Press, Oxford
Hjort NL, Glad IK (1995) Nonparametric density estimation with a parametric start. Ann Stat 23:882–904
Hong Y (1996) Consistent testing for serial correlation of unknown form. Econometrica 64:837–864
Hong Y (1999) Hypothesis testing in time series via the empirical characteristic function: a generalized spectral density approach. J Am Stat Assoc 84:1201–1220
Hong Y, Lee T-H (2003) Diagnostic checking for the adequacy of nonlinear time series models. Econom Theory 19:1065–1121
Hong Y, Lee Y-J (2011) Journal of Time Series Analysis 32:1–32
Kalbfleisch JD, Prentice RL (1980) The statistical analysis of failure time data. Wiley, New York
Karanasos M (2004) The statistical properties of long-memory ACD models. WSEAS Trans Bus Econ 2:169–175
Karanasos M (2008) The statistical properties of exponential ACD models. Quant Qual Anal Soc Sci 2:29–49
Kwan ACC, Sim A-B (1996a) On the finite-sample distribution of modified Portmanteau tests for randomness of a Gaussian time series. Biometrika 83:938–943
Kwan ACC, Sim A-B (1996b) Portmanteau tests of randomness and Jenkins’ variance-stabilizing transformation. Econ Lett 50:41–49
Kwan ACC, Sim A-B, Wu Y (2005) A comparative study of the finite-sample performance of some Portmanteau tests for randomness of a time series. Comput Stat Data Anal 48:391–413
Lancaster T (1997) The econometric analysis of transition data. Cambridge University Press
Lawrence AL, Lewis PA (1980) The exponential autoregressive-moving average EARMA(P,Q) model. J R Stat Soc Series B 42:150–161
Lee S, Hansen B (1994) Asymptotic theory for the GARCH(1 1) quasi-maximum likelihood estimator. Econom Theory 10:29–52
Li WK, Mak TK (1994) On the squared residual autorrelations in non-linear time series with conditional heteroscedasticity. J Time Series Anal 15:627–636
Li WK, Yu LH (2003) On the residual autocorrelation of the autoregressive conditional duration model. Econ Lett 79:169–175
Ljung GM, Box GEP (1978) On a measure of lack of fit in time series models. Biometrika 65:297–303
Luca GD, Zuccolotto P (2006) Regime-switching Pareto distributions for ACD models. Comput Stat Data Anal 51:2179–2191
Lunde A (2000) A generalized gamma autoregressive conditional duration model. Discussion paper, University of Aarhus
Manganelli S (2005) Duration, volume and volatility impact of trades. J Finan Markets 8:377–399
McLeod AI, Li WK (1983) Diagnostic checking arma time series models using squared residual autocorrelations. J Time Series Anal 4:269–273
Meitz M, Teräsvirta T (2006) Evaluating models of autoregressive conditional duration. J Bus Econ Stat 24:104–124
Nelson D (1991) Conditional heteroskedasticity in asset returns: a new approach. J Econom 43:227–251
Newey WK (1985) Maximum likelihood specification testing and conditional moment tests. Econometrica 5:1047–1070
Newey WK, West KD (1987) A simple, positive semidefinite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55:703–708
Pacurar M (2008) Autoregressive conditional duration models in finance: a survey of the theoretical and empirical literature. J Econ Surveys 22:711–751
Pagan A, Vella F (1989) Diagnostic tests for models based on individual data: a survey. J Appl Econom 4:29–59
Rosenblatt M (1952) Remarks on a multivariate transformation. Ann Math Stat 23:470–472
Silverman BW (1986) Density estimation for statistics and data analysis. Chapman and Hall, London
Tauchen G (1985) Diagnostic testing and evaluation of maximum likelihood models. J Econom 30:415–443
Taylor SJ (1982) Financial returns modelled by the product of two stochastic processes – a study of daily sugar prices. In: Anderson OD (ed) Time series analysis: theory and practice, North-Holland, Amsterdam
Veredas D, Rodriguez-Poo J, Espasa A (2008) Semiparametric estimation for financial durations. In: Bauwens WPL, Veredas D (eds) High frequency financial econometrics. Physica-Verlag, Heidelberg, pp 225–251
White H (1982) Maximum likelihood estimation of misspecified models. Econometrica 50(1):1–25
Wooldridge JM (1990) A unified approach to robust, regression-based specification tests. Econom Theory 6:17–43
Wooldridge JM (1991) Specification testing and quasi-maximum-likelihood estimation. J Econom 48:29–55
Zhang MY, Russell JR, Tsay RS (2001) A nonlinear autoregressive conditional duration model with applications to financial transaction data. J Econom 104:179–207
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Hautsch, N. (2012). Univariate Multiplicative Error Models. In: Econometrics of Financial High-Frequency Data. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21925-2_5
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