CURSO DE ECONOMETRÍA II MODELOS ESTACIONARIOS Y METODOLOGÍA BOX-JENKINS. 1

Transcription

1 CURSO DE ECONOMETRÍA II MODELOS ESTACIONARIOS Y METODOLOGÍA BOX-JENKINS. espasa@est-econ.uc3m.es 1

2 LAG AND DIFFERENCE OPERATORS L is an operator such that: LX t = X t-1 (16) L 2 X t = L (LX t ) = X t-2 (17) L j X t = X t-j (18) X t = (1-L) X t = X t -X t-1 (19) X t -X t-s = (1-L s ) X t (2) (1-L s ) = s (21) s = (1-L s ) = (1-L) (1 + L + + L s-1 ) = U s-1 (L) (22) espasa@est-econ.uc3m.es 2

3 CEMENT.2 DLCEMENT D12DLCEMENT DD12DLCEMEN espasa@est-econ.uc3m.es 3

4 CONCLUSION Applying regular and seasonal differences we can remove trend and seasonality. 4

5 A FIRST COMMENT ON STATIONARITY A time series Z t is stationary if E(Z t ) = constant = µ Var(Z t ) = constant =γ o Cov (Z t, Z t+k ) only depends on k = γ k 35 Cement consumption orginal series Cement consumption differenced series ( 12 log X t ) espasa@est-econ.uc3m.es 5

6 Innovations, white noise and random walk Innovations Time series models We denote, wˆ ( t 1) + 1 It is a function of the past. w t = f (past) + a t known at unknown at time t-1 time t-1 : forecast for w t with information up to t-1 Therefore, t = wˆ ( t 1) + 1 w + a t surprise or innovation espasa@est-econ.uc3m.es 6

7 Up to moment t-1 we have the following information Past values of the series: w 1, w 2,..., w t-1 Past innovations: a 1, a 2,..., a t-1 According to the information used we can construct two types of time series models: w t = f (past history in terms of innovations) + a t These are called MOVING AVERAGE MODELS (MA) w t = f (past history in terms of observed values of the series) + a t These are called AUTOREGRESSIVE MODELS (AR) espasa@est-econ.uc3m.es 7

8 White noise time series The simple time series model will generate a purely random series, which is denoted as WHITE NOISE and reperesented by a t A series is white noise if: cor(a t, a t-j ) = for all, j Its mean or expected value is constant and equal to zero. Example: The winning numbers of a state lotery. For many previous records that you know, you dont have any advantage in forecasting the next winning number. There is no dependency between future and past numbers. espasa@est-econ.uc3m.es 8

9 Properties of white noise series The order of the observations does not matter. Being µ =, the series is unforecastable. Because of property (1) a white noise is in a certain sense the denial of a time series. 2 WN espasa@est-econ.uc3m.es 9

10 But the white noise is in the base of any time series model... w t depends on its previous values and is stochastic Stochastic means that can not be predicted without error, therefore, w t f (w t-1, w t-2,...) But, w t = f (w t-1, w t-2,...) + an unpredictable component. w t = f (w t-1, w t-2,...) + white noise. espasa@est-econ.uc3m.es 1

11 TIME SERIES MODEL We consider w t = f (w t-1, w t-2,...) + r t (residuals) (1) If, r t r t is white noise our model cannot be improved is not white noise: is predictable we can construct a model for it and change (1) espasa@est-econ.uc3m.es 11

12 Random walk model Efficient markets: A huge number of agents having perfect information. Therefore they act adapting their behaviours completely to the available information and a price p t results for time t and since there is nothing left to do given the available information this is the price taken for the future. But at time (t+1) some unexpected events will occur and agents adapt immediately and in a full way to the new information and a new price will be formed. p t+1 = p t + a t espasa@est-econ.uc3m.es 12

13 Random walk model The model p t+1 = p t + a t+1 is called a RANDOM WALK MODEL. In this model, the variable p t is not stationary, it is I(1,), therefore it shows local oscillations in level and w t = p t = p t -p t-1 =a t it is unforecastable. espasa@est-econ.uc3m.es 13

14 Changes in prices in efficient markets are unforecastable. Monetary markets, currency markets, etc., are very near to efficiency, therefore the prices in them: interest rates, exchange rates are closed to x t = x t-1 + a t and the forecasting exercise is too simple xˆ ( t) + h = xt h In these markets innovations are completely absorbed when they appear and the price changes only depend on contemporaneous innovations, all previous innovations have been absorbed in x t-1 espasa@est-econ.uc3m.es 14

15 Moving Average Models (MA) w t is explained as a function of past innovations (a 1, a 2,..., a t-1 ) Moving average model of order 1 a t is a white noise time series with: w t = - θ a t-1 + a t MA(1) Var(a t ) = σ 2 a Examples: wt = a t +.5 a t-1 w t = a t -.5 a t-1 MA(1);_-.5 3 MA(1);_ Properties: µ w =, γ o = (1+ θ 2 ) σ 2 a espasa@est-econ.uc3m.es 15

16 The innovation a t has effect on w t an w t+1. Let x t = x t-1 + w t x t = x t-1 - θa t-1 + a t x t-1 = x t-2 - θa t-2 + a t-1 x t = x t-2 - θa t-1 + a t - θa t-2 + a t-1 = x t-2 + a t + (1-θ)a t-1 - θa t-2... x t = x o + a t + (1-θ)a t-1 +(1-θ)a t-2 +. An innovation coming into the system has a permanent effect on the model of magnitud (1-θ) times its value. espasa@est-econ.uc3m.es 16

17 Time dependency in a MA(1) time series. It is measured by the correlation between w t and w t-j corr (w t, w t-j ), j= 1,2,3.. These are called AUTOCORRELATIONS. corr ( w,w ) t t j = cov ( w,w ) var t ( w ) t t j Autocorrelations in a MA(1) time series. Examples: 1. w t = a t +.5 a t-1 w t = a t -.5 a t espasa@est-econ.uc3m.es 17

18 In a MA(1) time series only ρ 1 ρ 1 = θ 1+ θ 2 If θ < ρ 1 > An observation at one side of the mean is inclined to be followed by an observation at the same side of the mean. If θ = The resulting series is a white noise. If θ > ρ 1 < The resulting series oscillates more than a white noise espasa@est-econ.uc3m.es 18

19 Example: w t = a t +.5 a t-1 white noise w t = a t -.5 a t-1 MA(1);_-.5 WN 3 MA(1);_ espasa@est-econ.uc3m.es 19

20 A mesure of forecastability: R 2 R 2 Var( e = 1 Var( w T + 1 ) T + 1) R 2 = 2 σ 2 (1 + θ ) σ 2 θ (1 + θ ) 1 = 2 2 we will see that -1< θ <1 Therefore: (1) A white noise series is unforecastable. (2) A MA(1) series can be forecasted one-step ahead whith a maximun value of R 2 <.5. espasa@est-econ.uc3m.es 2

21 Moving average model of order 2: MA(2) w t = a t - θ 1 a t-1 - θ 2 a t-2 Example: W t = a t -.4 a t a t Wt FAC espasa@est-econ.uc3m.es 21

22 Only current and 2 past innovations enter in the model µ w =, γ o = (1+θ θ 2 2 ) σ a 2 There is a cut in the autocorrelation function after lag 2. Just one and two forecasts are different from zero w T+1 = a T+1 - θ 1 a T - θ 2 a T Persistence of innovations only last 2 periods. Variance of the forecast error increases till var of W t. For X t variance of the forecast error increases without limit. espasa@est-econ.uc3m.es 22

23 A general moving average model of order q: MA(q) w t = a t - θ 1 a t-1 - θ 2 a t-2 - -θ q a t-q. Only current and q previous innovations enter the model µ w =, γ o = (1+θ θ q2 ) σ 2. a There is a cut in the autocorrelation function after lag q. Just one to q steps ahead forecasts are different from zero. Innovations persist q periods..4 Example: FAC MA(6) espasa@est-econ.uc3m.es 23

24 Autocorrelation Function (FAC) and correlogram The autocorrelation function represent dynamics in the time series model. The sample counterpart of the autocorrelation function (r k ) is the correlogram. It is calculated as: n k t = 1 w w rk = k = n t t = 1 w 2 t t + k 1,2,... Example : w t = a t -.5a t-1 1. FAC 1 CORRELOGRAM espasa@est-econ.uc3m.es 24

25 AUTOREGRESSIVE MODELS 25

26 AUTOREGRESSIVE MODELS (AR) w t = f (past in terms of past w s) + a t Simplest model: AR(1) w t = φ w t-1 + a t known at unknown at time t-1 time t-1 INNOVATION Dual representation of dynamic models: MA(q) AR AR(p) MA espasa@est-econ.uc3m.es 26

27 MOVING AVERAGE REPRESENTATION OF AN AR(1) PROCESS φ =.5 w t =.5 w t-1 + a t w t-1 =.5 w t-2 + a t-1 w t =.5 2 w t a t-1 + a t w t-2 =.5 w t-3 + a t-2 w t =.5 3 w t a t a t-1 + a t... w t = φ w t-1 + a t w t = φ t w o +(a t + φ a t φ t-1 a 1 ) sequence of INNOVATIONS An AR model may be represented as an MA model. espasa@est-econ.uc3m.es 27

28 INNOVATIONS PERSISTENCE w t = φ t w o +(a t + φ a t φ t-1 a 1 ) -1< φ <1 The effect on w t of a very distant innovation is not exactly, but it is negligible. AR(1) may be approximated by MA(q) given to q a sufficient large value. The autocorrelation will be longer but restricted. In MA(q) the innovation a t-j, j<q has a free effect given by θ j Phi=.3 In AR(1) the innovation a t-j has a restricted effect to be φ j espasa@est-econ.uc3m.es 28

29 INNOVATIONS PERSISTENCE w t = φ t w o +(a t + φ a t φ t-1 a 1 ) φ > 1 From the initial value w o the model becomes explosive. very distant innovations are much more important then recent ones. Similary happens with φ <-1. Reality does not seem to behave in a permanent explosive way and we reject that φ >1. 35 Phi= espasa@est-econ.uc3m.es 29

30 INNOVATIONS PERSISTENCE w t = φ t w o +(a t + φ a t φ t-1 a 1 ) φ = 1 This situation is not explosive. It is the situation we have considered for trends characterized by showing local oscilations in level. 1 Phi= Now we do not want models for trends, but for deviations from trends, so we restrict the AR(1) model to φ <1-1<φ <1. An AR(1) model w t =φw t-1 + a t with -1< φ <1 is stationary. espasa@est-econ.uc3m.es 3

31 ESTIMATION OF INNOVATIONS - Invertibility An MA(1) model may also be represented as an AR model w t = a t - θ a t-1 a t = w t + θ a t-1 a t-1 = w t-1 + θ a t-2 w t = a t + θ (w t-1 + θ a t-2 ) = = a t + θ w t-1 + θ 2 a t-2 a t-2 = w t-2 + θ a t-3 w t = a t + θ w t-1 + θ 2 w t-2 + θ 3 a t-3... w t = a t - θ a w t-1 t = θ t a o + a t + (θ w t-1 + θ 2 w t θ t-1 w 1 ) We can discuss as in the previous section the effects of the different values for θ. espasa@est-econ.uc3m.es 31

32 ESTIMATION OF INNOVATIONS - Invertibility -1 < θ < 1 The effect on w t of a very distant observation in not exactly, but it is negligible. MA(1) model may be approximated by an AR(p) model with p large enough. θ > 1 From the initial value a o the model becomes explosive. Very distant observations are much more important than recent ones We can not estimate innovations. θ = 1 Very distant observations are important. We can not estimate innovations. espasa@est-econ.uc3m.es 32

33 SUMMARY ON ESTATIONARITY AND INVERTIBILITY STATIONARITY: old innovations lose their importance as time goes by. INVERTIBILITY: old observations of the time series lose their importance as time goes by. Model Stationarity Invertibility AR(1) wt= φ wt-1+ at Only if φ <1 Always MA(1) wt= at- θ at-1 Always Only if θ <1 33

34 STATIONARY: AR(1) MODEL w t = φ w t-1 + a t, φ <1 and so w t = a t + φ w t-1 + φ 2 w t-2 + φ 3 a t j= w t = φ a j t j AR(1)- Model has an infinite moving average - with restricted coefficient which decline to zero. If making ψ j = α j. The stationarity condition φ <1 can also be expressed as ψ j2 =. PROPERTIES: E(w t ) = constant = Var(w t ) = γ o = σ 2 / (1- φ 2 ) Cov (w t, w t-1 ) = ρ k =φ k ; ρ 1 = φ espasa@est-econ.uc3m.es 34

35 THE ACF OF AR(1) Only ρ =. There is not cutting point. But it declines to zero in absolute value. Therefore the correlation between w t and w t-h (H large) is not exactly zero but it neglegible: very distant observations have practically no effect at present time. The declining to zero depends on φ : the higer the value of φ the slowest declining. espasa@est-econ.uc3m.es 35

36 THE ACF OF AR(1) Since the correlation between w t and past values (w t-j ) could last for some long time before it becomes negligible. The forecast horizon can be large before the forecasts become negligible (zero). espasa@est-econ.uc3m.es 36

37 AUTORREGRESIVE MODEL OF ORDER 2 AR(2) w t = φ 1 w t-1 + φ 2 w t-2 + a t Example: (1) Stationary w t =.3 w t-1 +.4w t-2 + a t.25 Var espasa@est-econ.uc3m.es 37

38 The formulation of the dual MA representation is now mathematically more complex. We need to introduce some new concepts. Polynomial and lag operators notation the lag operator -denoted as L- delays one period the variable L w t = w t-1 L 2 w t = L w t-1 = w t-2. Another way for writing AR or MA models is in terms of the lag operator w t = φ 1 w t-1 + φ 1 w +a t-2 t w t = ( φ 1 L+ φ 2 L 2 ) w t + a t (1 - φ 1 L-φ 2 L 2 ) w t = a t w t = a t - θ a t-1 = (1 - θ L) a t. Normal equation Z 2 = φ 1 Z 1 + φ 2 Z. Roots of the normal equation largest absolute value. G1, G2 and denote G the root with espasa@est-econ.uc3m.es 38

39 STATIONARITY AND LARGEST ROOT If G <1 the model AR(2) is stationary. If G >1 the model AR(2) is explosive. IfG= 1 the AR(2) model takes the form: (1-L)(1-G 2 L)W t =a t (1-L)W t =G 2 (1-L)W t-1 +a t. and this model is not stationary but one whith local oscilations in level. espasa@est-econ.uc3m.es 39

40 STATIONARITY AND MA REPRESENTATION Iteratively an AR(2) model can be formulated as an infinite moving average model wt = Ψj at j ; Ψo = 1 j= where Ψ j and Ψ j2 < If AR(2) is stationary espasa@est-econ.uc3m.es 4

41 AUTOCORRELATION FUNCTION (ACF) The first values of the ACF depend on the two roots of the normal equation. Cases: G1, G2 real values (as in AR(1) case). G1, G2 complex. In this case the ACF will show damped sinusoidal oscillations with period grater then two time units. An AR(2) with complex roots is a simple model to capture cyclical oscillation in the data. EXAMPLES ACF for AR(2) Real Roots Complex Roots espasa@est-econ.uc3m.es 41

42 GENERAL AUTOREGRESSIVE MODEL: AR(p) w t = φ 1 w t-1 + φ 2 w t φ p w t-p + a t PROPERTIES OF AR(p) MODEL All the properties are derived mathematically from the normal equation associated to the AR(p) model and its p roots (solutions). Normal equation Z p = φ 1 Z p-1 + φ 2 Z p φ p-1 Z + α p Roots G 1, G 2,,G p Largest root in absolute value G espasa@est-econ.uc3m.es 42

43 STATIONARITY AND LARGEST ROOT If G <1 the model AR(p) is stationary. If G >1 the model AR(p) is explosive. If G= 1 the AR(p) has local oscillations in level. Solving the model iteratively in terms of past innovations we obtain an MA( ). where Ψ j wt = Ψj at j ; Ψo = 1 j= and Ψ j 2 < then model AR(p) is stationary espasa@est-econ.uc3m.es 43

44 AUTOCORRELATION FUNCTION (ACF) The first values of the ACF depend on all the p roots of the normal equation. But if G is a dominant root - e.g. no other root has an absolute size near G -the autocorrelation coefficients. ρ k for k not small. can be approximated by. ρ k A G k or G> (1) ρ k ± A G k otherwise (2) where A is a constant. The dominant root could be a pair of comple conjugate roots in which case ρ k in (2) shows damped sinusoidal oscillation. With period greater than two time units. espasa@est-econ.uc3m.es 44

45 PARTIAL AUTOCORRELATION FUNCTION (PACF) The partial autocorrelacion function is a tool to identify the order of on AR model. Example: FAC and PACF for AR models FAC PACF espasa@est-econ.uc3m.es 45

46 The comparison between on AR(1) and AR(2) shows that, though in both cases each observation is related to the previous ones, the kind of relationship between observations related 2 periods is different. AR(1): w t-3 w t-2 w t-1 w t The only relation between w t-2 and w t is through w t-1. Now in the AR(2) the relation between w t-2 and w t is direct and indirect though w t-1 AR(2): w t-3 w t-2 w t-1 w t The PACFshows the number of direct relations with w t. espasa@est-econ.uc3m.es 46

47 DUAL BEHAVIOUR OF ACF AND PACF IN MA AND AR MODELS ACF PACF AR(p) long structure cuts at lag p MA(q) cuts at lag q losg structure FAC and PACF for MA models FAC PACF FAC and PACF for AR models FAC PACF espasa@est-econ.uc3m.es 47

48 MIXED AUTOREGRESSIVE-MOVING AVERAGE MODELS Wt = f(past) + a t IN TERMS OF W t-j AND IN TERMS OF a t-j. GENERAL FORMULATION: ARMA (p, q) W t = φ 1 W t φ p W t-p - θ 1 a t-1 - -θ q a t-q + a t PAST EXAMPLE: ARMA (1,1) Wt = φ 1 W t-1 - θ 1 a t-1 + a t INNOVATION PAST INNOVATION THE EXPLOSIVE NATURE OF MODEL ARMA(1, 1) DEPENDS ON AUTOREGRESSIVE PARAMETER φ 1. IF φ 1 < 1 THE MODEL IS STATIONARY. THE EXPLOSIVE NATURE OF ARMA(p, q) DEPENDS ON THE AUTOREGRESSIVE PART OF THE MODEL AND THE CONDITON FOR STATIONARITY IS THE SAME THAT IN MODEL AR(p). espasa@est-econ.uc3m.es 48

49 MIXED AUTOREGRESSIVE-MOVING AVERAGE MODELS MOVING AVERAGE REPRESENTATION OF ARMA(p, q). PROCEEDING IN A SIMILAR WAY THAN FOR AN AR MODEL A STATIONARY ARMA (p, q) MODEL CAN BE REPRESENTED BY A MOVING AVERAGE OF INFINITE ORDER WITH DECLINING COEFFICIENTS. espasa@est-econ.uc3m.es 49

50 INTEREST OF MODEL ARMA(p, q) PARSIMONY PRINCIPLE A MA(q) model is only valid for variables in which the innovations are fully absorbed after q observations. This could be very restrictive in cases unless q is large. An AR(p) model incorporates all previous innovations with restricted coefficients declining to zero. This restriction affect all innovations. Examples: W t =.5 W t-1 + a t W t = a t +.5 a t a t-2 + W t = = j.5 j a,, j t j =,1,... This could be very restrictive in cases unless p is large. An ARMA(p, q) model incorporates all previous innovations with restricted coefficients declining to zero, but in a more flexible way than an AR process. Examples: W t =.5 W t a t-1 +a t W t = a t +.7 a t a t a t espasa@est-econ.uc3m.es 5

51 INTEREST OF MODEL ARMA(p, q) CONCLUSION The ARMA(p, q) model can capture longer innovation effects in a less restrictive way with less parameters than MA(q) and AR(p) models. espasa@est-econ.uc3m.es 51

52 AGGREGATION AND ARMA MODELS It can be shown that aggregating MA(q) and AR(p) models one gets an ARMA model. Many economic variables: CPI, industrial production, sales in large firms, etc. are aggregates therefore they can follow an ARMA model. Also, very often the variables which we observed are an aggregation of the true variable and a measurement error process. CONCLUSION: parsimony and aggregation justify that ARMA models appear in practice. AUTOCORRELATION FUNCTION Similar properties to AR(p) model, but with a more flexible declining behaviour that for models in which q p does not come from the beginning. espasa@est-econ.uc3m.es 52

53 FAC AND PACF FOR SOME ARMA(1,1) PROCESSES FAC PACF 53

54 STATIONARY AND INVERTIBILITY IN ARMA MODELS First let s recall to the results in AR(p) and MA(q) models. AR(p) MA(q) Stationarity If largest A R root G has G <1 Always Invertibility Always If largest M A root has H <1 These properties remain in ARMA(p,q) MODELS Stationary: if the largest autoregressive root, G, satisfies G <1. Invertible: If the largest moving average rroot, H satisfies H <1. Stationary and invertible MA(q), AR(p) or ARMA(p,q) models always have (1) An AR representation. (2) An MA representation. espasa@est-econ.uc3m.es 54

55 SEASONAL DEPENDENCY Time series with monthly or quaterly observations usually show dependence on previous realizations but also on the ones occurred a year ago. Jan. Feb. Mar. Apr. Jun. Jul. Aug Dependence on previous months * But also in same month previous years Because of this seasonal dependence the ACF of the models may show long temporal dependence that would lead to ARMA models with high values for p and q. This will be further simplified with the use of multiplicative models. espasa@est-econ.uc3m.es 55

56 MAIN POINTS OF THE BOX-JENKINS METHODOLOGY I. Formulate a theory for a general class of models capable of describing the real times series. II. Construct a procedure to find for a given time series best model within the general class. espasa@est-econ.uc3m.es 56

57 GENERAL CLASS OF MODELS IN B-J METHODOLOGY Is the class of ARIMA models with deterministic factor. In the specification of these models enter different types of parameters which capture different features of the data. SETS OF PARAMETERS IN A ARIMA MODEL A parameter referring if the model is formulated on the original data or on the logarithmic transformation if it shows evolutivity in variance. espasa@est-econ.uc3m.es 57

58 A set of parameters designed to capture the evolutivity in mean of the data. EVOLUTIVITY IN TREND SEASONAL EVOLUTIVITY These parameters correspond to the number of regular differences seasonal differences overall constant seasonal dummies other deterministic factors 58

59 Parameters to capture the time dependency Having fixed the sets parameters [1] and [2] the original data can be transformed in stationary data. For the stationary data, parameters specifying the ARMA model capture the time dependency. Looking at the time dependency appearing in the correlogram we can formulate the main aspects of the time dependency. 59

60 Parameter reflecting the one-step ahead uncertainty The only one step ahead factor which is uncertain is a t. a t Normal (,σ) σ = reflects this uncertainty espasa@est-econ.uc3m.es 6

61 EXAMPLE: MODEL 1 Stationary transformation: 2 log x = w ( 2 ) 2 1 φ1l φ 2L log X t = a t Systematic growth with no seasonality t t I(2,) log x t + [ log x log x ]+ = log xt 1 t 1 t 2 φ1w t 1 + φ2w t 2 + at Evolutivity path Oscillations around the evolutivity path Innovations espasa@est-econ.uc3m.es 61

62 EXAMPLE: MODEL 2 ( s 2 s ) ( )( s 1 Φ ) 1L Φ 2L s log X t = 1 θ 1L 1 θ s L a t Stationary transformation: s log x = w t t I(2,) Systematic growth and stochastic seasonality log x t + [ log x log x ]+ = log xt 1 t s t s 1 + Φ1wt s + Φ2wt 2s θ1at 1 θ sat s + θ sθ s+ 1at s 1 + Evolutivity path Oscillations around the evolutivity path + a t Innovations espasa@est-econ.uc3m.es 62

63 Stationary transformation: w = x EXAMPLE: MODEL 3 Calculate b= 1/s b j s ( 1 θ ) x = bst+ L a x t j j t j = 1 t t t x t 1 = xt 1 + b + b s j= 1 j Define b j* = b j -b and write ( 1 θ ) j a t 1 s S t + j= 1 b j S j t t j j t j = 1 Evolutivity path θ + Oscillations around the a t s x = b + b S t + L a If b is significantly different from zero in model 3 we have systematic growth with a deterministic mean growth (b) and deterministic seasonal factor (bj*). Otherwise model 3 only shows local oscillations in level with deterministic seasonality. evolutivity path Innovations espasa@est-econ.uc3m.es 63

64 PROCEDURE TO FIND THE BEST MODEL FOR A GIVEN TIME SERIES THREE STAGES: INITIAL SPECIFICATION: Looking at plots of original and differenced data and, looking at plots of the correlogram and partial correlogram of those time series (original and differences) Make an initial specification of the values d, D, p, q, P and Q. ESTIMATION: Given the previous specification estimate the corresponding model. DIAGNOSTIC CHECKING: Apply a battery of tests to the estimation results. If these tests do not reject the initial model take it as good and use it for forecasting purposes. If these tests reject the initial model you will have an indication specifying a new model. Do it and go to stage (1). Process finishes when at stage (3) you do not reject the model under consideration. espasa@est-econ.uc3m.es 64

65 Determination of evolutivity parameters Looking at plots X t or X t and log X t or log X t determine in which case the magnitud of the oscillations are more homogeneous. If there is not a clear cut consider if the economic phenomenom can evolve according to proportionality law or not. In the positive case take the logarithmic transformation. In doubt take also the logarithmic transformation. From now on X t will represent the data as decided in this stage. espasa@est-econ.uc3m.es 65

66 Determination of parameters for evolutivity in mean(i) MAINLY number of regular differences number of seasonal differences constant term (other deterministic parameters) CONSIDER FROM THEORY OR EXPERIENCE the type of evolutivity in the economic phenomenom under study. 66

67 Determination of parameters for evolutivity in mean(ii) EXAMINE the plots and tables of X t, X t, s X t, ²X t, and s X t and decide which one can be taken as stationary: constant mean constant variance similar dependency along the sample. If doubt take the transformations with a small number of differences. As an auxiliary intrument use correlograms of the corresponding transformations. The correlograms of the stationary transformation must tend soon to non-significant values. espasa@est-econ.uc3m.es 67

68 These are the series: Serie 1 Spanish Cement Consumption Serie 2 Spanish Expenditure in residential building Serie 3 Spanish Non Residential Building Serie 4 Spanish Civil Work Serie 5 Monthly Spanish Consumption of Cement Serie 6 Japanese Cement Consumption Now, we are going to see their graphics and correlograms. espasa@est-econ.uc3m.es 68

69 Spanish Cement Consumption graphics (Serie 1) CEM.1 D1LCEM D4LCEM DD4LCEM espasa@est-econ.uc3m.es 69

70 Spanish Cement Consumption Correlograms (Serie1) 1 Correlogram LCEM 1 Correlogram D1LCEM Correlogram 1 D4LCEM Correlogram 1 DD4LCEM espasa@est-econ.uc3m.es 7

71 Spanish Expenditure in residential building graphics (Serie 2) RES D1LRES D4LRES DD4LRES espasa@est-econ.uc3m.es 71

72 Spanish Expenditure in residential building correlograms (Serie 2) 1 Correlogram LRES 1 Correlogram D1LRES Correlogram 1 D4LRES Correlogram 1 DD4LRES espasa@est-econ.uc3m.es 72

73 Spanish Non Residential Building graphics (Serie 3) 13.2 LNRES.2 D1LNRES D4LNRES.1 DD4LNRES espasa@est-econ.uc3m.es 73

74 Spanish Non Residential Building correlograms (Serie 3) 1 Correlogram LNRES 1 Correlogram D1LNRES Correlogram 1 D4LNRES Correlogram 1 DD4LNRES espasa@est-econ.uc3m.es 74

75 Spanish Civil Work graphics (Serie 4) e+6 OC.2 D1LOC e+6.1 e+6 e D4LOC DD4LOC espasa@est-econ.uc3m.es 75

76 Spanish Civil Work correlograms (Serie 4) 1 Correlogram LOC 1 Correlogram D1LOC Correlogram 1 D4LOC Correlogram 1 DD4LOC espasa@est-econ.uc3m.es 76

77 Monthly Spanish Consumption of Cement graphics (Serie 5) CEMENT.2 DLCEMENT D12DLCEMENT DD12DLCEMEN espasa@est-econ.uc3m.es 77

78 Monthly Spanish Consumption of Cement correlograms (Serie 5) 1 Correlogram LCEMENT 1 Correlogram DLCEMENT Correlogram 1 D12DLCEMENT Correlogram 1 DD12DLCEMEN espasa@est-econ.uc3m.es 78

79 Japanese Cement Consumption graphics (Serie 6) 2 CEM PIB CT espasa@est-econ.uc3m.es 79

80 Japanese Cement Consumption correlograms (Serie 6) 1 Correlogram CEM.5 1 Correlogram PIB Correlogram CT espasa@est-econ.uc3m.es 8

81 The Stationary Parameters (I) Time dependency in stationary data is reflected in: the correlation parameters (ρ 1, ρ 2,...) of the joint pdf f(w 1,w 2,...,w t ) or the parameters of the conditional mean in the conditional pdf f(w t /w t-1,w t-2,...) which is the same for every w t by the stationarity property. espasa@est-econ.uc3m.es 81

82 The Stationary Parameters (II) These parameters are φ φ,..., φν, ϑ, ϑ,..., 1, ϑ q EXAMPLE (1 w t = φ s 1 L ) w t = (1 ϑ s L ) φ 1 w t 1 ϑ s a t s + a t a t The conditional mean is φ 1 w t 1 ϑ s a t s espasa@est-econ.uc3m.es 82

83 Determination of the stationary parameters For forecasting purposes what matters is the conditional mean. Procedure: look at the time dependency in the data by estimating the correlogram and the partial correlogram. From these estimations specify an ARMA model. Method for initial especification of an ARMA model. By careful examination of the correlogram and partial correelogram an expert can propose an ARMA model for the time series in question. There are also automatic procedures which are quite reliable like TRAMO or SCA. 83

84 Model Users and the specification of ARIMA models It is important that they understand well the specification of the evolutivity parameters and it is advisable that they get experience in doing these especifications, because those parameters determine the medium-term forecasts. For the stationary parameters it is enough that the user could understand the type of time dependency in the model provided by reliable computer programs. Long or short dependency seasonal dependency 84

85 Practical rules to determine the orders of an ARMA multiplicative model (I) Divide the correlogram and partial correlogram in three parts: REGULAR CORRELOGRAM: formed by the first values of the correlogram. SEASONAL CORRELOGRAM: formed by the values at seasonal lags. INTERMEDIATE CORRELOGRAM: formed by the remaining values 85

86 Practical rules to determine the orders of an ARMA multiplicative model (II) Remember the properties of the autocorrelation function in MA(q), AR(p) and ARMA(p,q): PROCESS ACF PACF NOTE M A(q) Cutting point at k=q No cutting point AR(p) No cutting point Cutting point at k=p A R M A (p,q ) N o cu ttin g p oint No cutting point Useful to capture short dependency Useful to capture regulary decrasing long dependency Useful to capture decreasing long dependency. To avoid com m on factor problem s do not use m ore espasa@est-econ.uc3m.es 86

87 Practical rules to determine the orders of an ARMA multiplicative model (III) Take the regular correlogram Is there a cutting point? YES: Look at which lag is the cutting point. Say it occurs at lag q. Then specify an MA(q). NOT: Look at the partial correlogram Is there a cutting point?» YES: Look at which lag is the cutting point. Say it occur at point p. Then specify an AR(p).» NOT: Specify an ARMA (1,1) espasa@est-econ.uc3m.es 87

88 Practical rules to determine the orders of an ARMA multiplicative model (IV) Take the seasonal correlogram Apply same rules as point 3 and specify on: MA(Q) if there is a cutting point in lag Q S of the correlogram. AR(P) if there is not a cutting point in the correlogram and there is one in the partial correlogram at lag P S. For series with long cycles you could specify on AR(2, 1) espasa@est-econ.uc3m.es 88

89 Practical rules to determine the orders of an ARMA multiplicative model (V) Take the Intermediate correlogram to confirm previous decisions. For multiplicative ARMA models the intermediate autocorrelation function take values as: ρ = ( I ) ( I ) ( R) h S+ K = ρh S K ρk where the subscripts mean: (I): intermediate ACF (R): regular ACF (S): seasonal ACF ρ ( S ) hs espasa@est-econ.uc3m.es 89

90 Practical rules to determine the orders of an ARMA multiplicative model (VI) In applying previous rules you could have reasonable doubts: make a decision on which could be the most reasonable values for p, q, P and Q, and make a use of all other possibilities. We will consider them at the diagnostic check stage. espasa@est-econ.uc3m.es 9

91 Examples Stationary transformation for Spanish Non Residential Building correlogram and partial correlogram 91

92 Examples Stationary transformation for Spanish Civil Works correlogram and partial correlogram 92

93 MAIN RESULTS FROM ESTIMATION PROGRAMS Estimation of parameters φ j, Φ h, θ j and Θ h whith their corresponding variances and correlations between then. Estimation of the innovations â espasa@est-econ.uc3m.es 93

94 TESTS ON PARAMETER ESTIMATES AND POSSIBLE ALTERNATIVE MODELS Apply t-tests to each one of the parameter (τ j ) to test H : τ j = If t-test less than 2 do not reject H and simplify the model eliminating this parameter Look at the roots of the MA(q) polynomial θ q q (L) = (1 H jl) j= 1 If one H j is close to one it can be canceled with a difference operator obtaining a simplified model. espasa@est-econ.uc3m.es 94

95 Look at the roots of the AR(p) polynomial φ p p (L) = (1 G jl) j= 1 If one G j is close to unity you could fix it at value one reducing the order of φ(l) to (p-1) and increasing the number of differences to (d+1). Look at the correlations between the parameters φ j, Φ h, θ j and Θ h If you find correlations grater than.9 in absolute value eliminate one of two parameters having large correlation. espasa@est-econ.uc3m.es 95

96 MISSPECIFICATION TESTS BASED ON THE ESTIMATED INNOVATIONS If the model is correct the innovations should be white noise. There we can apply different test to the innovations to ensure that they are white noise. If from these tests results that innovations are not white noise the initial model should be modified and the results of the test will give an indication in order to formulate a new specification. espasa@est-econ.uc3m.es 96

97 TESTING FOR A ZERO MEAN IN THE INNOVATIONS If your model does not include a constant, the mean of the estimared innovations can be significantly different from zero indicating that you need to modify the I(d,m) specification. Test H o : µ = By a t-test as Reject Ho if t ( µ ˆ ) = σ ˆ = σ t( ˆ) µ > mean standard T 2 of estimated deviation of in absolute value. innovations this mean If you reject H o the initial I(d,m) specification is wrong. Consider to: a) apply some segmentation b) increase d by one c) increase m in by one espasa@est-econ.uc3m.es 97

98 TESTING FOR CORRELATION BETWEEN THE ESTIMATED INNOVATIONS If the estimated innovations are white noise then their autocorrelations must not be significatly different from zero. by t test. Test H o : ρ a (k) =, K If you reject Ho for K=1,2,3 s, or 2s your initial model is wrong and you need to modify it. espasa@est-econ.uc3m.es 98

99 (A) The best way to modify the model is by specifying it according with the doubts that we note at the specifications stage. Test H o : { ρ a (1), ρ a (2),.., ρ a (s+2) } = By Ljung-Box test (Q s+2 ) If + 2 Q s + 2 > χs Reject Ho and apply (A). 2 espasa@est-econ.uc3m.es 99

100 FORMULATE OF ALTERNATIVE MODELS Even when the initial model has passed the test on the estimation results and on the innovation, still the model can be insatisfactory because that model was just a reasonable choice done at the specification. Estimate all the reasonable alternative models including the original model choose the one with less AIC value. If you have not douhts about the initial model try alternatives by substituting p by (p+1),or q by (q+1) or P by (P+1), or Q by (Q+1). Estimate these alternative including the original model and choose the one with less AIC value. espasa@est-econ.uc3m.es 1