## 1. Introduction

## 2. Reconstrusting and Resampling the Signal

## 3. Models in Discrete and Continuous Time

## 4. Autocovariance Functions and Spectra

**Example**

**1.**

## 5. The Continuous-Time Frequency-Limited ARMA Process

## 6. Estimates of the Linear Stochastic Models

#### 6.1. From ARMA to CARMA

**Example**

**2.**

#### 6.2. From ARMA to LSDE

**Example**

**3.**

#### 6.3. From LSDE to ARMA

## 7. Summary and Conclusions

## Supplementary Materials

## Funding

## Conflicts of Interest

## References

- Bartlett, Maurice S. 1946. On the Theoretical Specification and Sampling Properties of Autocorrelated Time Series. Supplement to the Journal of the Royal Statistical Society 8: 27–41. [Google Scholar] [CrossRef]
- Bergstrom, Albert R. 1966. Nonrecursive Models as Discrete Approximations to Systems of Stochastic Differential Equations. Econometrica 34: 173–82. [Google Scholar] [CrossRef]
- Bergstrom, Albert R. 1984. Continuous time stochastic models and issues of aggregation over time. In Handbook of Econometrics. Edited by Zvi Griliches and Michael D. Intriligator. Amsterdam: North-Holland, Volume 2, pp. 1145–212. [Google Scholar]
- Brockwell, Peter J. 1995. A Note on the Embedding Discrete-Time ARMA Processes. Journal of Time Series Analysis 16: 451–60. [Google Scholar] [CrossRef]
- Bundy, Brian D., and Gerald R. Garside. 1978. Optimisation Methods in Pascal. Baltimore: Edward Arnold. [Google Scholar]
- Chambers, Marcus J. 1999. Discrete time representation of stationary and non-stationary con-tinuous time systems. Journal of Economic Dynamics and Control 23: 619–39. [Google Scholar] [CrossRef]
- Chambers, Marcus J., and Michael A. Thornton. 2012. Discrete Time Representation of Continuous Time ARMA Processes. Econometric Theory 28: 219–38. [Google Scholar] [CrossRef]
- Fishman, George S. 1969. Spectral Methods in Econometrics. Cambridge: Harvard University Press. [Google Scholar]
- Granger, Clive William John, and Michio Hatanaka. 1964. Spectral Analysis of Economic Time Series. Princeton: Princeton University Press. [Google Scholar]
- Granger, Clive William John, and Paul Newbold. 1977. Forecasting Economic Time Series. New York: Academic Press. [Google Scholar]
- Harvey, Andrew C., and James Stock. 1985. The Estimation of Higher Order Continuous Time Autoregressive Models. Econometric Theory 1: 97–112. [Google Scholar] [CrossRef]
- Harvey, Andrew C., and James Stock. 1988. Continuous-time Autoregressive Models with Common Stochastic Trends. Journal of Economic Dynamics and Control 12: 365–84. [Google Scholar] [CrossRef]
- Hyndman, Rob J. 1993. Yule–Walker Estimates for Continuous-Time Autoregressive Models. Journal of Time Series Analysis 14: 281–96. [Google Scholar] [CrossRef]
- James, Glyn. 2004. Advanced Modern Engineering Mathematics, 3rd ed. London: Pearson Prentice Hall. [Google Scholar]
- Jones, Richard H. 1981. Fitting a Continuous Time Autoregression to Discrete Data. In Applied Time Series Analysis II. Edited by David F. Findley. New York: Academic Press, pp. 651–82. [Google Scholar]
- Larsson, Erik K., Magnus Mossberg, and Torsten Soderstrom. 2006. An Overview of Important Practical Aspects of Continuous-Time ARMA System Identification. Circuits, Systems and Signal Processing 25: 17–46. [Google Scholar] [CrossRef]
- McCrorie, J. Roderick. 2000. Deriving the Exact Discrete Analog of a Continuous Time System. Econometric Theory 16: 998–1015. [Google Scholar] [CrossRef]
- Müller, David E. 1956. A Method of Solving Algebraic Equations Using an Automatic Computer. Mathematical Tables and Other Aids to Computation (MTAC) 10: 208–15. [Google Scholar] [CrossRef]
- Nelder, John A., and Roger Mead. 1965. A Simplex Method for Function Minimization. Computer Journal 7: 308–13. [Google Scholar] [CrossRef]
- Nyquist, Harry. 1924. Certain Factors Affecting Telegraph Speed. Bell System Technical Journal 3: 324–46. [Google Scholar] [CrossRef]
- Nyquist, Harry. 1928. Certain Topics in Telegraph Transmission Theory. Transactions of the AIEE 47: 617–44, Reprinted in 2002, Proceedings of the IEEE 90: 280–305. [Google Scholar] [CrossRef]
- Oppenheim, Alan V., and R. W. Schafer. 2013. Discrete-Time Signal Processing. Englewood Cliffs: Prentic-Hall. [Google Scholar]
- Pollock, David Stephen Geoffrey. 1999. A Handbook of Time-Series Analysis, Signal Processing and Dynamics. London: Academic Press. [Google Scholar]
- Pollock, David Stephen Geoffrey. 2009. Realisations of Finite-sample Frequency-selective Filters. Journal of Statistical Planning and Inference 139: 1541–58. [Google Scholar] [CrossRef]
- Pollock, David Stephen Geoffrey. 2012. Band-Limited Stochastic Processes in Discrete and Continuous Time. Studies in Nonlinear Dynamics and Econometrics 16: 1–37. [Google Scholar] [CrossRef]
- Pollock, David Stephen Geoffrey. 2015. Econometric Filters. Computational Economics 48: 669–91. [Google Scholar] [CrossRef]
- Pollock, David Stephen Geoffrey. 2018. Filters, Waves and Spectra. Econometrics 6: 35. [Google Scholar] [CrossRef]
- Priestley, M.B. 1981. Spectral Analysis and Time Series. London: Academic Press. [Google Scholar]
- Shannon, Claude Elwood. 1949. Communication in the Presence of Noise. Proceedings of the Institute of Radio Engineers 37: 10–21, Reprinted in 1998, Proceedings of the IEEE 86: 447–57. [Google Scholar]
- Söderström, T. 1990. On Zero Locations for Sampled Stochastic Systems. IEEE Transactions on Automatic Control 35: 1249–53. [Google Scholar] [CrossRef]
- Söderström, T. 1991. Computing Stochastic Continuous-Time Models from ARMA Models. International Journal of Control 53: 1311–26. [Google Scholar] [CrossRef]
- Wilson, G. T. 1969. Factorisation of the Covariance Generating Function of a Pure Moving Average Process. SIAM Journal of Numerical Analysis 6: 1–7. [Google Scholar] [CrossRef]

**Figure 1.**The spectrum of the ARMA(2, 1) process $(1.0-1.273L+0.81{L}^{2})y\left(t\right)=(1-0.5L)\epsilon \left(t\right)$.

**Figure 2.**The discrete autocovariance sequence of the ARMA(2, 1) process and the continuous autocovariance function of the corresponding frequency-limited CARMA(2, 1) process.

**Figure 3.**The spectrum of the LSDE(2, 1) corresponding to the ARMA(2, 1) model of Example 1 plotted on top of the spectrum of that model, represented by the thick grey line. The two spectra virtually coincide over the interval $[0,\pi ]$.

**Figure 4.**(

**Left**) The contours of the criterion function $z=z(a,b)$ together with the minimising values, marked by black dots. (

**Right**) The contours of the function $q=1/(z+d)$.

**Figure 5.**The spectrum of the revised ARMA model, confined to the interval $[0,\pi ]$, superimposed on the spectrum of the derived LSDE—the bold line.

**Table 1.**The various values to the moving-average parameters of the LSDE(2, 1) model inferred from the discrete-time autocovariance function according to the principle of autocovariance equivalence. The parameters a and b are arguments of the continuous autocovariance function of (48).

a | b | ${\mathit{\theta}}_{0}$ | ${\mathit{\theta}}_{1}$ | |
---|---|---|---|---|

(i) | −0.4544 | 0.2956 | −0.9088 | 0.5601 |

(ii) | 0.4544 | 0.4175 | 0.9088 | 0.5601 |

(iii) | −0.4544 | −0.4174 | −0.9088 | −0.5601 |

(iv) | 0.4544 | −0.2956 | 0.9088 | −0.5601 |

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).