Lesson 2: What is a time series Model

Transcription

1 Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@ec.univaq.it

2 Time series forecasts Having observed a time series {x 1, x 2,..., x T } over the period t = 1 up to t = T,we are interested in forecasting a future value x T +h, h > 0 The integer h is called the forecasting horizon and the forecast of x T +h made at time T for h steps ahead will be denoted by ˆx T +h

3 Time series forecasts How can we do it?

4 Time series forecasts

5 We know that that our time series {x 1, x 2,..., x T } has been generated by some mechanism : Data Generating Process (DGP).

6 DGP x 1,..., x T

7 Of course, if we had known the DGP we could have forecast the future of our time series perfectly. The problem is that it is impossible to know the DGP perfectly. The DGP is the reality and the reality is not completely knowable. There are inevitable sources of uncertainty that make perfect knowledge impossible.

8 We can t obtain a complete and perfect knowledge of reality.

9 And then? What can we do to understand the dynamic structure of the time series and to predict its future values?

10 To achieve these goals we have to construct a mathematical model of the DGP using the observations.

11 Schematic showing the relationship among the DGP, the model used to represent the DGP and the time series DGP Model x 1,..., x T

12 Thus the major task of time series analysis is to construct a model of the the DGP A model that provides a useful approximation to the DGP. A model that it is hoped has similar properties to those of the DGP

13 What is a model? A representation of the essential aspects of a system which presents knowledge of that system in usable form.

14 Of course, it is important to keep in mind that the model is not the reality, it is not the DGP. A model is different from reality by definition. A model is a representation of some essential aspects of reality.

15 To exemplify all this, let s consider Berlin

16 and the Berlin s map

17 The map is an example of model which represents some features of Berlin.

18 But of course the map isn t Berlin!

19 So, when we have a model it is incorrect to ask Is this model a realistic model?

20 Every model is unrealistic. The correct question is Is this model useful or isn t with respect our goals?

21 There is often a trade-off between realism and usefulness of a model.

22 We may move back to the example. The Berlin s map in 1:1 scale is a very realistic model of Berlin but it is also a totally useless model. If you use the map to walk in Berlin, of course it is impossible to put it into your pocket.

23 The quotation from G.E.P. Box All models are wrong but some are useful. has become famous.

24 Further, the usefulness of a model cannot be evaluated without to consider the goal of the analysis. Different kinds of model are required in different situations and for different objectives. The best model for forecasting may not be the same as the best model for describing past data.

25 How do we build a model of the mechanism that generated our time series? Consider, for example, the annual series of divorces in Italy. In this case x t represents the number of italian pairs which have divorced in year t. We observe that: 1 As the reasons for which a pair divorces are very complex x t cannot be considered a deterministic variable. 2 Since the time series {x t ; t = 1, 2,..., T } is a set of measurements of the same variable x it is likely that the value of x at moment t reflects (depends on) the past history of the variable. These are intrinsic features of a time series. The modelling strategies of time series must take into account these properties.

26 Thus most (univariate) models have the general form x t = f (x t 1,...x t 1 ; φ) + g(u t,..., u t q ; θ), where f (.) and g(.) are (non-linear or linear) functions that depend on unknown vectors of parameters φ and θ respectively and u t,...u t q are assumed to be random disturbs with mean 0 and common unknown finite variance.

27 DGP x 1,..., x T x t = f (.) + g(.)

28 Time series analysis concerns the way to specify the functions f (x t 1,...x t 1 ; φ) and g(u t,..., u t q ; θ) and the estimate of unknown vectors of parameters φ and θ.

29 This approach were pioneered by Yule and Slutsky in the 1920 s. Yule s researches led to the notion of the autoregressive scheme. Slutsky s researches led to the notion of a moving average scheme.

30 The Autoregressive scheme A time series x t is assumed to be generated as a linear function of its past values, plus a random shock x t = φ 1 x t φ p x t p + u t Conceptualy, an autoregressive scheme is one with a memory in the sense thar each values is correlated with p preceding values. The constants φ 1,...,φ p are weights measuring the influence of preceding values x t 1,...x t p on the value x t.

31 The Moving Average scheme A time series x t is assumed to be generated as a weighted linear sum of the last q + 1 random shocks x t = u t + θ 1 u t φ p u t q

32 The Autoregressive Moving Average scheme If both schemes, the autoregressive and the moving average one, are used, we obtain the so-called Autoregressive Moving Average scheme. x t = φ 1 x t φ p x t p + u t + θ 1 u t φ p u t q

33 The first step in order to obtain a DGP s model of this kind is to consider our time series x 1, x 2..., x T as a part of a realization of a stochastic process..., x 2, x 1, x 0, x 1, x 2..., x T, x T +1, x T +2,...