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ESGF 5IFM Q1 2012
Financial Econometric Models
  Vincent JEANNIN – ESGF 5IFM
            Q1 2012




                                vinzjeannin@hotmail.com
                                      1
ESGF 5IFM Q1 2012
Summary of the session (Est. 3h)

•   Reminder of Last Session
•   Time Series Analysis Principles
•   Auto Regressive Process




                                      vinzjeannin@hotmail.com
•   Moving Average Process
•   ARMA
•   Conclusion




                                            2
Be logic!
                                                          Reminder of Last Session




            vinzjeannin@hotmail.com   ESGF 5IFM Q1 2012
  3
Differentiation possible




                                                  ESGF 5IFM Q1 2012
                                                  vinzjeannin@hotmail.com
������������������������������ = ln⁡ ������)
               (
                                                        4
Time can be a factor of a regression




                                       vinzjeannin@hotmail.com   ESGF 5IFM Q1 2012
          5
Differentiation can add value




                                    vinzjeannin@hotmail.com   ESGF 5IFM Q1 2012
                                6
Check ACF/PACF for autocorrelation




                                     vinzjeannin@hotmail.com   ESGF 5IFM Q1 2012
         7
Time Series Analysis Principles




                                  ESGF 4IFM Q1 2012
    Reminders of the 3 steps




                                  vinzjeannin@hotmail.com
                   Identify

                   Fit

                   Forecast




                                        8
ESGF 4IFM Q1 2012
Reminders of the 3 components




                                vinzjeannin@hotmail.com
              Trend

              Seasonality

              Residual




                                      9
Lag



������������������ = ������������−1




                               ESGF 4IFM Q1 2012
      Difference




                               vinzjeannin@hotmail.com
∆������������ = ������������ − ������������−1



      Seasonality Difference



∆30 ������������ = ������������ − ������������−30      10
ESGF 4IFM Q1 2012
Differentiate series to obtain stationary series



Time series analysis and forecast simpler with stationary series




                                                                   vinzjeannin@hotmail.com
Different models involved with stationary or heteroscedasticity




                                                                   11
Properties of stationary series


          Same distribution of the following




                                                              ESGF 4IFM Q1 2012
     (������1 , ������2 , ������3 , … , ������������ )
     (������2 , ������3 , ������4 , … , ������������+1 )

          Distribution not time dependent




                                                              vinzjeannin@hotmail.com
          Rare occurrence


                  Stationarity accepted if

          ������(������������ ) = ������               Constant in the time
                                                              12
     ������������������(������������ , ������������−������ )           Depends only on n
Acceptable Shortcut




                                                                 ESGF 4IFM Q1 2012
A series is stationary if the mean and the variance are stable


Which one is more likely to be stationary?




                                                                 vinzjeannin@hotmail.com
                                                                 13
About the residuals…


        White noise!




                                                       ESGF 4IFM Q1 2012
Normality test




                                                       vinzjeannin@hotmail.com
     Have an idea with

           Skewness

           Kurtosis


      Proper tests: KS, Durbin Watson, Portmanteau,…
                                                       14
Auto Regressive Process
   Main principle




                                                                         ESGF 4IFM Q1 2012
    There is a correlation between current data and previous data



         ������������ = ������ + ������1 ������������−1 + ������2 ������������−2 + ⋯ + ������������ ������������−������ + ������������




                                                                         vinzjeannin@hotmail.com
    ������������ Parameters of the model
    ������������ White noise


    AR(n)


    If the parameters are identified, the prediction will be easy        15
Let’s upload some data

DATA<-read.csv(file="C:/Users/vin/Desktop/Series1.csv",header=T)
plot(DATA$Val, type="l")




                                                                   ESGF 4IFM Q1 2012
                                                                   vinzjeannin@hotmail.com
                                                                   16
Is this a white noise?


              hist(DATA$Val, breaks=20)




                                          ESGF 4IFM Q1 2012
                                          vinzjeannin@hotmail.com
                                          17
Probably not…


                 Portmanteau test




                                                                                                 ESGF 4IFM Q1 2012
                           Test the autocorrelation of a series

                           If there is autocorrelation, data aren’t independently distributed




                                                                                                 vinzjeannin@hotmail.com
                                      Let’s use Ljung–Box statistics

                             H0: Data are independently distributed
                            H1: Data aren’t independently distributed

                      ������
                             ������2 ������                       With α confidence interval rejection
   ������ = ������(������ + 2)                                        following a Chi Square distribution
                            ������ − ������
                     ������=1
                                                                                                 18
������������    Autocorrelation at the lag k                                   ������ > Χ 21−������,ℎ
> Box.test(DATA$Val)




                                                                 ESGF 4IFM Q1 2012
 Box-Pierce test

data: DATA$Val
X-squared = 188.3263, df = 1, p-value < 2.2e-16




                                                                 vinzjeannin@hotmail.com
     H0 is rejected, the data aren’t independently distributed




                                                                 19
Let’s try a regression and analyse residuals

      TReg<-lm(DATA$Val~DATA$t)
      plot(DATA$Val, type="l")




                                               ESGF 4IFM Q1 2012
      abline(TReg, col="blue")




                                               vinzjeannin@hotmail.com
                                               20
eps<-resid(TReg)
ks.test(eps, "pnorm")
layout(matrix(1:4,2,2))
plot(TReg)




                          ESGF 4IFM Q1 2012
                          vinzjeannin@hotmail.com
                          21
Box-Pierce test




                                                       ESGF 4IFM Q1 2012
data: eps
X-squared = 187.6299, df = 1, p-value < 2.2e-16




Residuals aren’t a white noise




                                                       vinzjeannin@hotmail.com
Regression rejected




     Not a surprise, did the series look stationary?

                                                       22
     What next then?
lag.plot(DATA$Val, 9, do.lines=FALSE)




                                            ESGF 4IFM Q1 2012
                                            vinzjeannin@hotmail.com
                                            23
Differentiation seems to be interesting
Does the differentiation create a stationary series?

           plot(diff(DATA$Val), type="l")




                                                       ESGF 4IFM Q1 2012
                                                       vinzjeannin@hotmail.com
                                                       24
ACF & PACF

    par(mfrow=c(2,1))
    acf(diff(DATA$Val),20)




                                                              ESGF 4IFM Q1 2012
    pacf(diff(DATA$Val),20)




                                                              vinzjeannin@hotmail.com
                              ACF decreasing




                              PACF cancelling after order 1
                                                              25
Decreasing ACF




                                            ESGF 4IFM Q1 2012
PACF cancel after order 1




                                            vinzjeannin@hotmail.com
      Typically an Autoregressive Process



      AR(1) ?


                                            26
Let’s try to fit an AR(1) model
     Modl<-ar(diff(DATA$Val),order.max=20)
     plot(Modl$aic)




                                                ESGF 4IFM Q1 2012
                                                vinzjeannin@hotmail.com
                                                27
The likelihood for the order 1 is significant
> ar(diff(DATA$Val),order.max=20)

       Call:
       ar(x = diff(DATA$Val), order.max = 20)

       Coefficients:
             1       2            3




                                                                 ESGF 4IFM Q1 2012
        0.5925 -0.1669       0.1385

       Order selected 3     sigma^2 estimated as        0.8514




                                                                 vinzjeannin@hotmail.com
We have our coefficient and standard deviation

          > ARDif<-diff(DATA$Val)
          > ARDif[1]
          [1] 0.3757723


We know the first term of our series

             ������������ = 0.3757723 + 0.5925. ������������−1 + ������������
                                                                 28
Here is our model
Need to test the residuals




                                                                        ESGF 4IFM Q1 2012
Box.test(Modl$resid)

        Box-Pierce test




                                                                        vinzjeannin@hotmail.com
data: Modl$resid
X-squared = 7e-04, df = 1, p-value = 0.9789




   H0 accepted, residuals are independently distributed (white noise)


      The differentiated series is a AR(1)
                                                                        29
> predict(arima(diff(DATA$Val), order = c(1,0,0)), n.ahead = 7)
          $pred
          Time Series:
          Start = 193
          End = 199
          Frequency = 1
          [1] -0.81359048 -0.43300609 -0.22850452 -0.11861853 -0.05957287 -




                                                                                                                                  ESGF 4IFM Q1 2012
          0.02784553 -0.01079729

          $se
          Time Series:
          Start = 193
          End = 199
          Frequency = 1




                                                                                                                                  vinzjeannin@hotmail.com
          [1] 0.923352 1.048210 1.081582 1.091027 1.093739 1.094521 1.094747
120

115

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95

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85

80
      1    6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101106111116121126131136141146151156161166171176181186191196
ESGF 4IFM Q1 2012
Another typical example?




                                vinzjeannin@hotmail.com
       You make the comments!




                                31
DATA<-read.csv(file="C:/Users/vin/Desktop/Series2.csv",header=T)
plot(DATA$Ser2, type="l")
hist(DATA$Ser2, breaks=20)




                                                                   ESGF 4IFM Q1 2012
                                                                   vinzjeannin@hotmail.com
                                                                   32
> Box.test(DATA$Ser2)


        Box-Pierce test

data: DATA$Ser2
X-squared = 149.9227, df = 1, p-value < 2.2e-16




                                                                  ESGF 4IFM Q1 2012
                                                                  vinzjeannin@hotmail.com
                                     TReg<-lm(DATA$Ser2~DATA$t)
                                     plot(DATA$Ser2, type="l")
                                     abline(TReg, col="blue")




                                                                  33
> eps<-resid(TReg)
> Box.test(eps)

 Box-Pierce test

data: eps
X-squared = 148.5669, df = 1, p-value < 2.2e-16




                                                  ESGF 4IFM Q1 2012
  > layout(matrix(1:4,2,2))
  > plot(TReg)




                                                  vinzjeannin@hotmail.com
                                                  34
> lag.plot(DATA$Ser2, 9, do.lines=FALSE)




                                                ESGF 4IFM Q1 2012
                                                vinzjeannin@hotmail.com
                                                35

Much less obvious but clues of autoregression
par(mfrow=c(2,1))
plot(diff(DATA$Ser2), type="l")
plot(diff(DATA$Ser2, lag=2), type="l")




                                         ESGF 4IFM Q1 2012
                                         vinzjeannin@hotmail.com
                                         36
par(mfrow=c(2,1))
plot(diff(DATA$Ser2), type="l")
plot(diff(DATA$Ser2, lag=2), type="l")




                                                             ESGF 4IFM Q1 2012
                                     ACF decreases 2 by 2




                                                             vinzjeannin@hotmail.com
                                     PACF cancelling after
                                     order 2

                                                             37
First order differentiation, strong AR(2) clues

      par(mfrow=c(1,1))




                                                  ESGF 4IFM Q1 2012
      Modl<-ar(diff(DATA$Ser2),order.max=20)
      plot(Modl$aic)




                                                  vinzjeannin@hotmail.com
                                                  38
Parameters estimation

> ar(diff(DATA$Ser2),order.max=20)




                                                  ESGF 4IFM Q1 2012
Call:
ar(x = diff(DATA$Ser2), order.max = 20)

Coefficients:
      1       2          3




                                                  vinzjeannin@hotmail.com
 0.5919 -0.8326     0.1086

Order selected 3   sigma^2 estimated as   0.877



> ARDif<-diff(DATA$Ser2)
> ARDif[1]
[1] 0.3757723




                                                  39
> predict(arima(diff(DATA$Ser2), order = c(2,0,0)), n.ahead = 7)
$pred
Time Series:
Start = 193
End = 199
Frequency = 1
[1] 0.4505213 2.0075741 0.6639701 -1.2321156 -1.1409989 0.3866745
1.0879588




                                                                            ESGF 4IFM Q1 2012
$se
Time Series:
Start = 193
End = 199
Frequency = 1
[1] 0.9220713 1.0332515 1.1413067 1.2938326 1.2957576 1.3932158 1.4080266




                                                                            vinzjeannin@hotmail.com
115


110


105


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85


80
      101
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        1
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> Box.test(Modl$resid)

        Box-Pierce test




                                                              ESGF 4IFM Q1 2012
data: Modl$resid
X-squared = 0.0023, df = 1, p-value = 0.9619




                                                              vinzjeannin@hotmail.com
  Model accepted



  The more factors the harder the prediction is


  The more factors there are the more stationary need to be
  the series for a good prediction
                                                              41
Moving Average Process
   Main principle




                                                                        ESGF 4IFM Q1 2012
   Stationary series with auto correlation of errors



        ������������ = ������ + ������������ + ������1 ������������−1 + ������2 ������������−2 + ⋯ + ������������ ������������−������




                                                                        vinzjeannin@hotmail.com
   ������������ Parameters of the model
    ������������ White noise


   MA(n)


   More difficult to estimate than a AR(n)                              42
plot(Data, type="l")
hist(Data, breaks=20)




                        ESGF 4IFM Q1 2012
                        vinzjeannin@hotmail.com
                        43
acf(Data,20)
pacf(Data,20)




                                                 ESGF 4IFM Q1 2012
                              ACF cancels
                              after order 1




                                                 vinzjeannin@hotmail.com
                              PACF decays to 0




                                                 44
   ACF & PACF suggest MA(1)
> arima(Data, order = c(0, 0, 1),include.mean = FALSE)

Call:
arima(x = Data, order = c(0, 0, 1), include.mean = FALSE)

Coefficients:
          ma1




                                                                            ESGF 4IFM Q1 2012
      -0.4621
s.e.   0.0903

sigma^2 estimated as 0.937:    log likelihood = -138.76,     aic = 281.52




                                                                            vinzjeannin@hotmail.com
     > Box.test(Rslt$residuals)

      Box-Pierce test

     data: Rslt$residuals
     X-squared = 0, df = 1, p-value = 0.9967




                                                                            45
                It works, MA(1), 0 mean, parameter -0.4621
Fore<-predict(Rslt, n.ahead=5)
U = Fore$pred + 2*Fore$se
L = Fore$pred - 2*Fore$se
minx=min(Data,L)
maxx=max(Data,U)
ts.plot(Data,Fore$pred,col=1:2,
ylim=c(minx,maxx))
lines(U, col="blue", lty="dashed")




                                     ESGF 4IFM Q1 2012
lines(L, col="blue", lty="dashed")




                                     vinzjeannin@hotmail.com
                                     46
ESGF 4IFM Q1 2012
Another typical example?




                                vinzjeannin@hotmail.com
       You make the comments!




                                47
plot(Data, type="l")
hist(Data, breaks=20)




                        ESGF 4IFM Q1 2012
                        vinzjeannin@hotmail.com
                        48
vinzjeannin@hotmail.com   ESGF 4IFM Q1 2012
49
> arima(Data, order = c(0, 0, 2),include.mean = FALSE)

Call:




                                                                         ESGF 4IFM Q1 2012
arima(x = Data, order = c(0, 0, 2), include.mean = FALSE)

Coefficients:
          ma1      ma2
      -0.5365   0.6489
s.e.   0.0701   0.1044




                                                                         vinzjeannin@hotmail.com
sigma^2 estimated as 1.005:   log likelihood = -142.74,   aic = 291.48

> Box.test(Rslt$residuals)

        Box-Pierce test

data: Rslt$residuals
X-squared = 0.0283, df = 1, p-value = 0.8664



                                                                         50
                  MA(2)
Fore<-predict(Rslt, n.ahead=5)
U = Fore$pred + 2*Fore$se
L = Fore$pred - 2*Fore$se
minx=min(Data,L)
maxx=max(Data,U)
ts.plot(Data,Fore$pred,col=1:2,
ylim=c(minx,maxx))




                                     ESGF 4IFM Q1 2012
lines(U, col="blue", lty="dashed")
lines(L, col="blue", lty="dashed")




                                     vinzjeannin@hotmail.com
                                     51
ARMA
      Main principle




                                                                                     ESGF 4IFM Q1 2012
 The series is a function of past values plus current and past values of the noise




                                                                                     vinzjeannin@hotmail.com
             ARMA(p,q)



                    Combines AR(p) & MA(q)




                                                                                     52
plot(Data, type="l")
hist(Data, breaks=20)




                        ESGF 4IFM Q1 2012
                        vinzjeannin@hotmail.com
                        53
ESGF 4IFM Q1 2012
                                                          vinzjeannin@hotmail.com
                                                          54
Both ACF and PACF decreases exponentially after order 1
> Rslt<-arima(Data, order = c(1, 0, 1),include.mean = FALSE)
> Rslt

Call:
arima(x = Data, order = c(1, 0, 1), include.mean = FALSE)




                                                                         ESGF 4IFM Q1 2012
Coefficients:
         ar1    ma1
      0.7214 0.7563
s.e. 0.0716 0.0721

sigma^2 estimated as 0.961:   log likelihood = -141.13,   aic = 288.27




                                                                         vinzjeannin@hotmail.com
> Box.test(Rslt$residuals)

        Box-Pierce test

data: Rslt$residuals
X-squared = 0.0098, df = 1, p-value = 0.9213




            ARMA(1,1) fits
                                                                         55
>   par(mfrow=c(1,1))
>   Fore<-predict(Rslt, n.ahead=5)
>   U = Fore$pred + 2*Fore$se
>   L = Fore$pred - 2*Fore$se
>   minx=min(Data,L)
>   maxx=max(Data,U)




                                     ESGF 4IFM Q1 2012
                                     vinzjeannin@hotmail.com
                                     56
Identification can get tricky at this stage




                                              ESGF 4IFM Q1 2012
                                              vinzjeannin@hotmail.com
                                              57
What do you think?




     vinzjeannin@hotmail.com   ESGF 4IFM Q1 2012
58
> Rslt<-arima(Data, order = c(4, 0, 3),include.mean = FALSE)
> Rslt

Call:
arima(x = Data, order = c(4, 0, 3), include.mean = FALSE)

Coefficients:




                                                                             ESGF 4IFM Q1 2012
         ar1     ar2       ar3       ar4      ma1       ma2        ma3
      0.2722 -0.5276    0.0202   -0.2663   0.8765   -0.4672    -0.5248
s.e. 0.2018   0.2308    0.1968    0.1546   0.1992    0.1690     0.1882

sigma^2 estimated as 1.140:    log likelihood = -151.19,      aic = 318.38
> Box.test(Rslt$residuals)




                                                                             vinzjeannin@hotmail.com
        Box-Pierce test

data: Rslt$residuals
X-squared = 0.2953, df = 1, p-value = 0.5869




        Was supposed to fit pretty wel….



         Data<-arima.sim(model=list(ar=c(0.5,-0.5,0.3,-
                                                                             59
         0.3),ma=c(0.75,-0.5,-0.5)),n=100)
Identification can be difficult




                                                ESGF 4IFM Q1 2012
Easiest model is AR



Imagine when the series is not stationary…




                                                vinzjeannin@hotmail.com
Step by step approach, exploration, tries,…



        Sometimes you find a satisfying model

                                                60
        Sometimes you don’t!
Conclusion



                               AR

                       MA

                ARMA
                                      Times series




     vinzjeannin@hotmail.com   ESGF 5IFM Q1 2012
61

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Financial Econometric Models III

  • 1. ESGF 5IFM Q1 2012 Financial Econometric Models Vincent JEANNIN – ESGF 5IFM Q1 2012 vinzjeannin@hotmail.com 1
  • 2. ESGF 5IFM Q1 2012 Summary of the session (Est. 3h) • Reminder of Last Session • Time Series Analysis Principles • Auto Regressive Process vinzjeannin@hotmail.com • Moving Average Process • ARMA • Conclusion 2
  • 3. Be logic! Reminder of Last Session vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 3
  • 4. Differentiation possible ESGF 5IFM Q1 2012 vinzjeannin@hotmail.com ������������������������������ = ln⁡ ������) ( 4
  • 5. Time can be a factor of a regression vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 5
  • 6. Differentiation can add value vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 6
  • 7. Check ACF/PACF for autocorrelation vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 7
  • 8. Time Series Analysis Principles ESGF 4IFM Q1 2012 Reminders of the 3 steps vinzjeannin@hotmail.com Identify Fit Forecast 8
  • 9. ESGF 4IFM Q1 2012 Reminders of the 3 components vinzjeannin@hotmail.com Trend Seasonality Residual 9
  • 10. Lag ������������������ = ������������−1 ESGF 4IFM Q1 2012 Difference vinzjeannin@hotmail.com ∆������������ = ������������ − ������������−1 Seasonality Difference ∆30 ������������ = ������������ − ������������−30 10
  • 11. ESGF 4IFM Q1 2012 Differentiate series to obtain stationary series Time series analysis and forecast simpler with stationary series vinzjeannin@hotmail.com Different models involved with stationary or heteroscedasticity 11
  • 12. Properties of stationary series Same distribution of the following ESGF 4IFM Q1 2012 (������1 , ������2 , ������3 , … , ������������ ) (������2 , ������3 , ������4 , … , ������������+1 ) Distribution not time dependent vinzjeannin@hotmail.com Rare occurrence Stationarity accepted if ������(������������ ) = ������ Constant in the time 12 ������������������(������������ , ������������−������ ) Depends only on n
  • 13. Acceptable Shortcut ESGF 4IFM Q1 2012 A series is stationary if the mean and the variance are stable Which one is more likely to be stationary? vinzjeannin@hotmail.com 13
  • 14. About the residuals… White noise! ESGF 4IFM Q1 2012 Normality test vinzjeannin@hotmail.com Have an idea with Skewness Kurtosis Proper tests: KS, Durbin Watson, Portmanteau,… 14
  • 15. Auto Regressive Process Main principle ESGF 4IFM Q1 2012 There is a correlation between current data and previous data ������������ = ������ + ������1 ������������−1 + ������2 ������������−2 + ⋯ + ������������ ������������−������ + ������������ vinzjeannin@hotmail.com ������������ Parameters of the model ������������ White noise AR(n) If the parameters are identified, the prediction will be easy 15
  • 16. Let’s upload some data DATA<-read.csv(file="C:/Users/vin/Desktop/Series1.csv",header=T) plot(DATA$Val, type="l") ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com 16
  • 17. Is this a white noise? hist(DATA$Val, breaks=20) ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com 17
  • 18. Probably not… Portmanteau test ESGF 4IFM Q1 2012 Test the autocorrelation of a series If there is autocorrelation, data aren’t independently distributed vinzjeannin@hotmail.com Let’s use Ljung–Box statistics H0: Data are independently distributed H1: Data aren’t independently distributed ������ ������2 ������ With α confidence interval rejection ������ = ������(������ + 2) following a Chi Square distribution ������ − ������ ������=1 18 ������������ Autocorrelation at the lag k ������ > Χ 21−������,ℎ
  • 19. > Box.test(DATA$Val) ESGF 4IFM Q1 2012 Box-Pierce test data: DATA$Val X-squared = 188.3263, df = 1, p-value < 2.2e-16 vinzjeannin@hotmail.com H0 is rejected, the data aren’t independently distributed 19
  • 20. Let’s try a regression and analyse residuals TReg<-lm(DATA$Val~DATA$t) plot(DATA$Val, type="l") ESGF 4IFM Q1 2012 abline(TReg, col="blue") vinzjeannin@hotmail.com 20
  • 22. Box-Pierce test ESGF 4IFM Q1 2012 data: eps X-squared = 187.6299, df = 1, p-value < 2.2e-16 Residuals aren’t a white noise vinzjeannin@hotmail.com Regression rejected Not a surprise, did the series look stationary? 22 What next then?
  • 23. lag.plot(DATA$Val, 9, do.lines=FALSE) ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com 23 Differentiation seems to be interesting
  • 24. Does the differentiation create a stationary series? plot(diff(DATA$Val), type="l") ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com 24
  • 25. ACF & PACF par(mfrow=c(2,1)) acf(diff(DATA$Val),20) ESGF 4IFM Q1 2012 pacf(diff(DATA$Val),20) vinzjeannin@hotmail.com ACF decreasing PACF cancelling after order 1 25
  • 26. Decreasing ACF ESGF 4IFM Q1 2012 PACF cancel after order 1 vinzjeannin@hotmail.com Typically an Autoregressive Process AR(1) ? 26
  • 27. Let’s try to fit an AR(1) model Modl<-ar(diff(DATA$Val),order.max=20) plot(Modl$aic) ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com 27 The likelihood for the order 1 is significant
  • 28. > ar(diff(DATA$Val),order.max=20) Call: ar(x = diff(DATA$Val), order.max = 20) Coefficients: 1 2 3 ESGF 4IFM Q1 2012 0.5925 -0.1669 0.1385 Order selected 3 sigma^2 estimated as 0.8514 vinzjeannin@hotmail.com We have our coefficient and standard deviation > ARDif<-diff(DATA$Val) > ARDif[1] [1] 0.3757723 We know the first term of our series ������������ = 0.3757723 + 0.5925. ������������−1 + ������������ 28 Here is our model
  • 29. Need to test the residuals ESGF 4IFM Q1 2012 Box.test(Modl$resid) Box-Pierce test vinzjeannin@hotmail.com data: Modl$resid X-squared = 7e-04, df = 1, p-value = 0.9789 H0 accepted, residuals are independently distributed (white noise) The differentiated series is a AR(1) 29
  • 30. > predict(arima(diff(DATA$Val), order = c(1,0,0)), n.ahead = 7) $pred Time Series: Start = 193 End = 199 Frequency = 1 [1] -0.81359048 -0.43300609 -0.22850452 -0.11861853 -0.05957287 - ESGF 4IFM Q1 2012 0.02784553 -0.01079729 $se Time Series: Start = 193 End = 199 Frequency = 1 vinzjeannin@hotmail.com [1] 0.923352 1.048210 1.081582 1.091027 1.093739 1.094521 1.094747 120 115 110 105 100 95 90 30 85 80 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101106111116121126131136141146151156161166171176181186191196
  • 31. ESGF 4IFM Q1 2012 Another typical example? vinzjeannin@hotmail.com You make the comments! 31
  • 33. > Box.test(DATA$Ser2) Box-Pierce test data: DATA$Ser2 X-squared = 149.9227, df = 1, p-value < 2.2e-16 ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com TReg<-lm(DATA$Ser2~DATA$t) plot(DATA$Ser2, type="l") abline(TReg, col="blue") 33
  • 34. > eps<-resid(TReg) > Box.test(eps) Box-Pierce test data: eps X-squared = 148.5669, df = 1, p-value < 2.2e-16 ESGF 4IFM Q1 2012 > layout(matrix(1:4,2,2)) > plot(TReg) vinzjeannin@hotmail.com 34
  • 35. > lag.plot(DATA$Ser2, 9, do.lines=FALSE) ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com 35 Much less obvious but clues of autoregression
  • 36. par(mfrow=c(2,1)) plot(diff(DATA$Ser2), type="l") plot(diff(DATA$Ser2, lag=2), type="l") ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com 36
  • 37. par(mfrow=c(2,1)) plot(diff(DATA$Ser2), type="l") plot(diff(DATA$Ser2, lag=2), type="l") ESGF 4IFM Q1 2012 ACF decreases 2 by 2 vinzjeannin@hotmail.com PACF cancelling after order 2 37
  • 38. First order differentiation, strong AR(2) clues par(mfrow=c(1,1)) ESGF 4IFM Q1 2012 Modl<-ar(diff(DATA$Ser2),order.max=20) plot(Modl$aic) vinzjeannin@hotmail.com 38
  • 39. Parameters estimation > ar(diff(DATA$Ser2),order.max=20) ESGF 4IFM Q1 2012 Call: ar(x = diff(DATA$Ser2), order.max = 20) Coefficients: 1 2 3 vinzjeannin@hotmail.com 0.5919 -0.8326 0.1086 Order selected 3 sigma^2 estimated as 0.877 > ARDif<-diff(DATA$Ser2) > ARDif[1] [1] 0.3757723 39
  • 40. > predict(arima(diff(DATA$Ser2), order = c(2,0,0)), n.ahead = 7) $pred Time Series: Start = 193 End = 199 Frequency = 1 [1] 0.4505213 2.0075741 0.6639701 -1.2321156 -1.1409989 0.3866745 1.0879588 ESGF 4IFM Q1 2012 $se Time Series: Start = 193 End = 199 Frequency = 1 [1] 0.9220713 1.0332515 1.1413067 1.2938326 1.2957576 1.3932158 1.4080266 vinzjeannin@hotmail.com 115 110 105 100 95 90 40 85 80 101 106 111 116 121 126 131 136 141 146 151 156 161 166 171 176 181 186 191 196 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96
  • 41. > Box.test(Modl$resid) Box-Pierce test ESGF 4IFM Q1 2012 data: Modl$resid X-squared = 0.0023, df = 1, p-value = 0.9619 vinzjeannin@hotmail.com Model accepted The more factors the harder the prediction is The more factors there are the more stationary need to be the series for a good prediction 41
  • 42. Moving Average Process Main principle ESGF 4IFM Q1 2012 Stationary series with auto correlation of errors ������������ = ������ + ������������ + ������1 ������������−1 + ������2 ������������−2 + ⋯ + ������������ ������������−������ vinzjeannin@hotmail.com ������������ Parameters of the model ������������ White noise MA(n) More difficult to estimate than a AR(n) 42
  • 43. plot(Data, type="l") hist(Data, breaks=20) ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com 43
  • 44. acf(Data,20) pacf(Data,20) ESGF 4IFM Q1 2012 ACF cancels after order 1 vinzjeannin@hotmail.com PACF decays to 0 44 ACF & PACF suggest MA(1)
  • 45. > arima(Data, order = c(0, 0, 1),include.mean = FALSE) Call: arima(x = Data, order = c(0, 0, 1), include.mean = FALSE) Coefficients: ma1 ESGF 4IFM Q1 2012 -0.4621 s.e. 0.0903 sigma^2 estimated as 0.937: log likelihood = -138.76, aic = 281.52 vinzjeannin@hotmail.com > Box.test(Rslt$residuals) Box-Pierce test data: Rslt$residuals X-squared = 0, df = 1, p-value = 0.9967 45 It works, MA(1), 0 mean, parameter -0.4621
  • 46. Fore<-predict(Rslt, n.ahead=5) U = Fore$pred + 2*Fore$se L = Fore$pred - 2*Fore$se minx=min(Data,L) maxx=max(Data,U) ts.plot(Data,Fore$pred,col=1:2, ylim=c(minx,maxx)) lines(U, col="blue", lty="dashed") ESGF 4IFM Q1 2012 lines(L, col="blue", lty="dashed") vinzjeannin@hotmail.com 46
  • 47. ESGF 4IFM Q1 2012 Another typical example? vinzjeannin@hotmail.com You make the comments! 47
  • 48. plot(Data, type="l") hist(Data, breaks=20) ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com 48
  • 49. vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 49
  • 50. > arima(Data, order = c(0, 0, 2),include.mean = FALSE) Call: ESGF 4IFM Q1 2012 arima(x = Data, order = c(0, 0, 2), include.mean = FALSE) Coefficients: ma1 ma2 -0.5365 0.6489 s.e. 0.0701 0.1044 vinzjeannin@hotmail.com sigma^2 estimated as 1.005: log likelihood = -142.74, aic = 291.48 > Box.test(Rslt$residuals) Box-Pierce test data: Rslt$residuals X-squared = 0.0283, df = 1, p-value = 0.8664 50 MA(2)
  • 51. Fore<-predict(Rslt, n.ahead=5) U = Fore$pred + 2*Fore$se L = Fore$pred - 2*Fore$se minx=min(Data,L) maxx=max(Data,U) ts.plot(Data,Fore$pred,col=1:2, ylim=c(minx,maxx)) ESGF 4IFM Q1 2012 lines(U, col="blue", lty="dashed") lines(L, col="blue", lty="dashed") vinzjeannin@hotmail.com 51
  • 52. ARMA Main principle ESGF 4IFM Q1 2012 The series is a function of past values plus current and past values of the noise vinzjeannin@hotmail.com ARMA(p,q) Combines AR(p) & MA(q) 52
  • 53. plot(Data, type="l") hist(Data, breaks=20) ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com 53
  • 54. ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com 54 Both ACF and PACF decreases exponentially after order 1
  • 55. > Rslt<-arima(Data, order = c(1, 0, 1),include.mean = FALSE) > Rslt Call: arima(x = Data, order = c(1, 0, 1), include.mean = FALSE) ESGF 4IFM Q1 2012 Coefficients: ar1 ma1 0.7214 0.7563 s.e. 0.0716 0.0721 sigma^2 estimated as 0.961: log likelihood = -141.13, aic = 288.27 vinzjeannin@hotmail.com > Box.test(Rslt$residuals) Box-Pierce test data: Rslt$residuals X-squared = 0.0098, df = 1, p-value = 0.9213 ARMA(1,1) fits 55
  • 56. > par(mfrow=c(1,1)) > Fore<-predict(Rslt, n.ahead=5) > U = Fore$pred + 2*Fore$se > L = Fore$pred - 2*Fore$se > minx=min(Data,L) > maxx=max(Data,U) ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com 56
  • 57. Identification can get tricky at this stage ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com 57
  • 58. What do you think? vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 58
  • 59. > Rslt<-arima(Data, order = c(4, 0, 3),include.mean = FALSE) > Rslt Call: arima(x = Data, order = c(4, 0, 3), include.mean = FALSE) Coefficients: ESGF 4IFM Q1 2012 ar1 ar2 ar3 ar4 ma1 ma2 ma3 0.2722 -0.5276 0.0202 -0.2663 0.8765 -0.4672 -0.5248 s.e. 0.2018 0.2308 0.1968 0.1546 0.1992 0.1690 0.1882 sigma^2 estimated as 1.140: log likelihood = -151.19, aic = 318.38 > Box.test(Rslt$residuals) vinzjeannin@hotmail.com Box-Pierce test data: Rslt$residuals X-squared = 0.2953, df = 1, p-value = 0.5869 Was supposed to fit pretty wel…. Data<-arima.sim(model=list(ar=c(0.5,-0.5,0.3,- 59 0.3),ma=c(0.75,-0.5,-0.5)),n=100)
  • 60. Identification can be difficult ESGF 4IFM Q1 2012 Easiest model is AR Imagine when the series is not stationary… vinzjeannin@hotmail.com Step by step approach, exploration, tries,… Sometimes you find a satisfying model 60 Sometimes you don’t!
  • 61. Conclusion AR MA ARMA Times series vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 61