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STAT 497 LECTURE NOTES 9

STAT 497 LECTURE NOTES 9. ESTIMATION. ESTIMATION. After specifying the order of a stationary ARMA process, we need to estimate the parameters. We will assume (for now) that: 1. The model order ( p and q ) is known, and 2. The data has zero mean.

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STAT 497 LECTURE NOTES 9

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  1. STAT 497LECTURE NOTES 9 ESTIMATION

  2. ESTIMATION • After specifying the order of a stationary ARMA process, we need to estimate the parameters. • We will assume (for now) that: 1. The model order (p and q) is known, and 2. The data has zero mean. • If (2) is not a reasonable assumption, we can subtract the sample mean , fit a zero-mean ARMA model: Then use as the model for Yt.

  3. ESTIMATION • Method of Moment Estimation (MME) • Ordinary Least Squares (OLS) Estimation • Maximum Likelihood Estimation (MLE) • Least Squares Estimation • Conditional • Unconditional

  4. THE METHOD OF MOMENT ESTIMATION • It is also known as Yule-Walker estimation. Easy but not efficient estimation method. Works for only AR models for large n. • BASIC IDEA: Equating sample moment(s) to population moment(s), and solve these equation(s) to obtain the estimator(s) of unknown parameter(s).

  5. THE METHOD OF MOMENT ESTIMATION • Let nis the variance/covariance matrix of X with the given parameter values. • Yule-Walker for AR(p): Regress Xtonto Xt−1, . . ., Xt−p. • Durbin-Levinson algorithm with replaced by . • Yule-Walker for ARMA(p,q): Method of moments. Not efficient.

  6. THE YULE-WALKER ESTIMATION • For a stationary (causal) AR(p)

  7. THE YULE-WALKER ESTIMATION • To find the Yule-Walker estimators, we are using, • These are forecasting equations. • We can use Durbin-Levinson algorithm.

  8. THE YULE-WALKER ESTIMATION • If • If {Xt} is an AR(p) process, Hence, we can use the sample PACF to test for AR order, and we can calculate approximate confidence intervals for the parameters.

  9. THE YULE-WALKER ESTIMATION • If Xt is an AR(p) process, and n is large, • 100(1)% approximate confidence interval for j is

  10. THE YULE-WALKER ESTIMATION • AR(1) Find the MME of . It is known that 1 = .

  11. THE YULE-WALKER ESTIMATION • So, the MME of  is • Also, is unknown. • Therefore, using the variance of the process, we can obtain MME of .

  12. THE YULE-WALKER ESTIMATION

  13. THE YULE-WALKER ESTIMATION • AR(2) Find the MME of all unknown parameters. • Using the Yule-Walker Equations

  14. THE YULE-WALKER ESTIMATION • So, equate population autocorrelations to sample autocorrelations, solve for 1 and 2.

  15. THE YULE-WALKER ESTIMATION Using these we can obtain the MME of To obtain MME of , use the process variance formula.

  16. THE YULE-WALKER ESTIMATION • AR(1) • AR(2)

  17. THE YULE-WALKER ESTIMATION • MA(1) • Again using the autocorrelation of the series at lag 1, Choose the root so that the root satisfying the invertibility condition

  18. THE YULE-WALKER ESTIMATION • For real roots, If , unique real roots but non-invertible. If , no real roots exists and MME fails. If , unique real roots and invertible.

  19. THE YULE-WALKER ESTIMATION • This example shows that the MMEs for MA and ARMA models are complicated. • More generally, regardless of AR, MA or ARMA models, the MMEs are sensitive to rounding errors. They are usually used to provide initial estimates needed for a more efficient nonlinear estimation method. • The moment estimators are not recommended for final estimation results and should not be used if the process is close to being nonstationary or noninvertible.

  20. THE MAXIMUM LIKELIHOOD ESTIMATION • Assume that • By this assumption we can use the joint pdf instead of which cannot be written as multiplication of marginal pdfs because of the dependency between time series observations.

  21. MLE METHOD • For the general stationary ARMA(p,q) model or

  22. MLE • The joint pdf of (a1,a2,…, an) is given by • Let Y=(Y1,…,Yn) and assume that initial conditions Y*=(Y1-p,…,Y0)’and a*=(a1-q,…,a0)’ are known.

  23. MLE • The conditional log-likelihood function is given by Initial Conditions:

  24. MLE • Then, we can find the estimators of =(1,…,p), =(1,…, q) and  such that the conditional likelihood function is maximized. Usually, numerical nonlinear optimization techniques are required. After obtaining all the estimators, where d.f.=  of terms used in SS   of parameters = (np)  (p+q+1) = n  (2p+q+1).

  25. MLE • AR(1)

  26. MLE The Jacobian will be

  27. MLE • Then, the likelihood function can be written as

  28. MLE • Hence, • The log-likelihood function:

  29. MLE • Here, S*() is the conditional sum of squares and S() is the unconditional sum of squares. • To find the value of  where the likelihood function is maximized, • Then,

  30. MLE • If we neglect ln(12), then MLE=conditional LSE. • If we neglect both ln(12) and , then

  31. MLE • Asymptotically unbiased, efficient, consistent, sufficient for large sample sizes but hard to deal with joint pdf.

  32. CONDITIONAL LEST SQUARES ESTIMATION • AR(1)

  33. CONDITIONAL LSE • If the process mean is different than zero

  34. CONDITIONAL LSE • MA(1) • Non-linear in terms of parameters • LS problem • S*() cannot be minimized analytically • Numerical nonlinear optimization methods like Newton-Raphson or Gauss-Newton,... *There are similar problem is ARMA case.

  35. UNCONDITIONAL LSE • This nonlinear in . • We need nonlinear optimization techniques.

  36. BACKCASTING METHOD • Obtain the backward form of ARMA(p,q) • Instead of forecasting, backcast the past values of Ytand at, t  0. Obtain the unconditional log-likelihood function, then obtain the estimators.

  37. EXAMPLE • If there are only 2 observations in time series (not realistic) Find the MLE of  and .

  38. EXAMPLE • US Quarterly Beer Production from 1975 to 1990 library(TSA) library(uroot) library(aTSA) data(beersales) beer=ts(beersales, start=1975, frequency=12) par(mfrow=c(1,3)) plot(beer, main='Monthly Beer Sales 1975 to 1990', xlab='Time',ylab='') acf(beer, lag.max=36) pacf(beer, lag.max=36)

  39. EXAMPLE (contd.) • library(uroot) • hegy.test(beer, deterministic = c(1,0,0), lag.method = "BIC", maxlag = 12) HEGY test for unit roots statistic p-value t_1 -2.2779 0.1281 t_2 -2.2102 0.0185 * F_3:4 0.2129 0.8227 F_5:6 14.1673 0 *** F_7:8 7.3787 9e-04 *** F_9:10 6.2202 0.0027 ** F_11:12 0.4434 0.6645 F_2:12 6.416 0 *** F_1:12 6.3383 0 *** Deterministic terms: constant Lag selection criterion and order: BIC, 2 P-values: based on response surface regressions > CH.test(beer) ------ - ------ ---- Canova & Hansen test ------ - ------ ---- Null hypothesis: Stationarity. Alternative hypothesis: Unit root. L-statistic: 0.817 Critical values: 0.10 0.05 0.025 0.01 0.846 1.01 1.16 1.35

  40. hegy.test(diff(beer), deterministic = c(1,0,0), lag.method = "BIC", maxlag = 12) HEGY test for unit roots statistic p-value t_1 -8.6882 0 *** t_2 -2.2368 0.0159 * F_3:4 0.2065 0.8316 F_5:6 13.73 0 *** F_7:8 7.2511 0.0011 ** F_9:10 6.0517 0.0034 ** F_11:12 0.4189 0.6864 F_2:12 6.2823 0 *** F_1:12 12.5214 0 *** --- Deterministic terms: constant Lag selection criterion and order: BIC, 1 P-values: based on response surface regressions

  41. hegy.test(diff(beer,12), deterministic = c(1,0,0), lag.method = "BIC", maxlag = 12) statistic p-value t_1 -2.6702 0.0518 . t_2 -3.775 1e-04 *** F_3:4 16.8115 0 *** F_5:6 22.1817 0 *** F_7:8 17.7135 0 *** F_9:10 15.5982 0 *** F_11:12 11.1838 0 *** F_2:12 24.7171 0 *** F_1:12 24.5112 0 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Deterministic terms: constant Lag selection criterion and order: BIC, 2 P-values: based on response surface regressions

  42. EXAMPLE (contd.) > plot(diff(beer),ylab='First Difference of Beer Production',xlab='Time') > acf(as.vector(diff(beer)),lag.max=36) > pacf(as.vector(diff(beer)),lag.max=36)

  43. EXAMPLE (contd.) > HEGY.test(wts =diff(beer), itsd = c(1, 1, c(1:3)), regvar = 0,selectlags = list(mode = "bic", Pmax = 12)) ---- ---- HEGY test ---- ---- Null hypothesis: Unit root. Alternative hypothesis: Stationarity. ---- HEGY statistics: Stat. p-value tpi_1 -6.067 0.01 tpi_2 -1.503 0.10 Fpi_3:4 9.091 0.01 Fpi_2:4 7.136 NA Fpi_1:4 26.145 NA

  44. > fit1=arima(beer,order=c(0,1,1),seasonal=list(order=c(0,1,1), period=12)) > fit1 Coefficients: ma1 sma1 -0.8883 -0.6154 s.e. 0.0361 0.0981 sigma^2 estimated as 0.357: log likelihood = -165.6, aic = 335.21

  45. To check first order overdifferecing To check seasonal overdifferecing fit1=arima(x,order=c(0,1,1),seasonal=list(order=c(0,1,1), period=12)) fit1 Coefficients: ma1 sma1 0.0034 -1.000 s.e. 0.0578 0.038 sigma^2 estimated as 1.607: log likelihood = -317.4, aic = 638.8 Sign of seasonal overdifferencing

  46. EXAMPLE (contd.) fit1=arima(beer,order=c(3,1,0),seasonal=list(order=c(2,0,0), period=12) fit1 Coefficients: ar1 ar2 ar3 sar1 sar2 -0.5079 -0.2535 0.0548 0.6657 0.2469 s.e. 0.0796 0.0846 0.0746 0.0738 0.0786 sigma^2 estimated as 0.4441: log likelihood = -203.56, aic = 417.11 fit2=arima(beer,order=c(3,1,2),seasonal=list(order=c(1,0,1), period=12)) fit2 Coefficients: ar1 ar2 ar3 ma1 ma2sar1 sma1 1.0363 -0.0977 -0.2302 -1.7846 0.8230 0.9754 -0.6183 s.e. 0.1028 0.1055 0.0905 0.0785 0.0732 0.0183 0.1164 sigma^2 estimated as 0.3411: log likelihood = -179.52, aic = 373.03

  47. > fit2=arima(beer,order=c(3,1,4),seasonal=list(order=c(2,1,1), period=12)) > fit2 Coefficients: ar1 ar2 ar3 ma1 ma2 ma3 ma4 sar1 0.4832 0.1634 -0.8272 -1.2107 0.0877 1.1191 -0.8587 0.3855 s.e. 0.0939 0.1258 0.0934 0.0783 0.1491 0.1452 0.0739 0.1263 sar2sma1 -0.3199 -0.6998 s.e. 0.0972 0.1307 sigma^2 estimated as 0.2588: log likelihood = -141.6, aic = 303.19

  48. Forecasting forecast=predict(fit3, n.ahead = 12) LCI=ts(forecast$pred-1.96*forecast$se,start=1991,frequency=12) UCI=ts(forecast$pred+1.96*forecast$se,start=1991,frequency=12) plot(fitted(fit3),lty=3,n1=1991,n.ahead=12,main='',xlim=c(1975,1992),ylim=c(10,20)) lines(forecast$pred, col="blue",lty=5,lwd=2) lines(beer) lines(LCI,col="3",lty=3) lines(UCI,col="3",lty=3)

  49. Forecasting library(forecast) fit_ets <- ets(beer) fit_ets Smoothing parameters: alpha = 0.1138 beta = 2e-04 gamma = 1e-04 phi = 0.98 sigma: 0.5807 AIC AICc BIC 818.9508 822.9046 877.5857 fc <- forecast(fit_ets) plot(fc)

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