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Lecture 4: Estimation of ARIMA Models

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Lecture 4: Estimation of ARIMA Models Lecture 4: Estimation of ARIMA models Florian Pelgrin University of Lausanne, Ecole´ des HEC Department of mathematics (IMEA-Nice) Sept. 2011 - Dec. 2011 Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Dec. 2011 1 / 50 Road map 1 Introduction Overview Framework 2 Sample moments 3 (Nonlinear) Least squares method Least squares estimation Nonlinear least squares estimation Discussion 4 (Generalized) Method of moments Methods of moments and Yule-Walker estimation Generalized method of moments 5 Maximum likelihood estimation Overview Estimation Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Dec. 2011 2 / 50 Introduction Overview 1. Introduction 1.1. Overview Provide an overview of some estimation methods for linear time series models : Sample moments estimation ; (Nonlinear) Linear least squares method ; Maximum likelihood estimation (Generalized) Method of moments Other methods are available (e.g., Bayesian estimation or Kalman filtering through state-space models) ! Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Dec. 2011 3 / 50 Introduction Overview Broadly speaking, these methods consist in estimating the parameters of interest (autoregressive coefficients, moving average coefficients, and variance of the innovation process) as follows : Optimize QT (δ) δ s.t. (nonlinear) linear equality and inequality constraints where QT is the objective function and δ is the vector of parameters of interest. Remark : (Nonlinear) Linear equality and inequality constraints may represent the fundamentalness conditions. Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Dec. 2011 4 / 50 Introduction Overview Examples (Nonlinear) Linear least squares methods aims at minimizing the sum of squared residuals ; Maximum likelihood estimation aims at maximizing the (log-) likelihood function ; Generalized method of moments aims at minimizing the distance between the theoretical moments and zero (using a weighting matrix). Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Dec. 2011 5 / 50 Introduction Overview These methods differ in terms of : Assumptions ; ”Nature” of the model (e.g., MA versus AR) ; Finite and large sample properties ; Etc. High quality software programs (Eviews, SAS, Splus, Stata, etc) are available ! However, it is important to know the estimation options (default procedure, optimization algorithm, choice of initial conditions) and to keep in mind that all these estimation techniques do not perform equally and do depend on the nature of the model... Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Dec. 2011 6 / 50 Introduction Framework 1.2. Framework Suppose that (Xt ) is correctly specified as an ARIMA(p,d,q) model d Φ(L)∆ Xt = Θ(L)t 2 where t is a weak white noise (0, σ ) and p Φ(L) = 1 − φ1L − · · · − φpL with φp 6= 0 q Θ(L) = 1 + θ1L + ··· + θqL with θq 6= 0. Assume that 1 The model order (p,d, and q) is known ; 2 The data has zero mean ; 3 The series is weakly stationary (e.g., after d-difference transformation, etc). Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Dec. 2011 7 / 50 Sample moments 2. Sample moments Let (X1, ··· , XT ) represent a sample of size T from a covariance-stationary process with E [Xt ] = mX for all t Cov [Xt , Xt−h] = γX (h) for all t Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Dec. 2011 8 / 50 Sample moments If nothing is known about the distribution of the time series, (non-parametric) estimation can be provided by : Mean T 1 X mˆ ≡ X¯ = X X T T t t=1 Autocovariance T −|h| 1 X γˆ (h) = (X − X¯ )(X − X¯ ) X T t T t+|h| T t=1 or T −|h| 1 X γˆ (h) = (X − X¯ )(X − X¯ ). X T − |h| t T t+|h| T t=1 Autocorrelation γˆX (h) ρˆX (h) = . γˆX (0) Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Dec. 2011 9 / 50 (Nonlinear) Least squares method Least squares estimation 3. (Nonlinear) Least squares method 3.1. Least squares estimation Consider the linear regression model (t = 1, ··· , T ): 0 yt = xt b0 + t 0 0 where (yt , xt ) are i.i.d. (H1), the regressors are stochastic (H2), xt is the tth-row of the X matrix, and the following assumptions hold : 1 Exogeneity (H3) E [t |xt ] = 0 for all t; 2 Spherical error terms (H4) 2 V [t |xt ] = σ0 for all t; 0 Cov [t , t0 | xt , xt0 ] = 0 for t 6= t 3 Rank condition (H5) 0 rank [E (xt xt )] = k. Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Dec. 2011 10 / 50 (Nonlinear) Least squares method Least squares estimation Definition The ordinary least squares estimation of b0, bols , defined from the following minimization program T ˆ X 2 bols = argmin t b0 t=1 T X 0 2 = argmin yt − xt b0 b0 t=1 is given by T !−1 T ! ˆ X 0 X bols = xt xt xt yt . t=1 t=1 Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Dec. 2011 11 / 50 (Nonlinear) Least squares method Least squares estimation Proposition Under H1-H5, the ordinary least squares estimator of b is weakly consistent p bˆols −→ b0 T →∞ and is asymptotically normally distributed √ ˆ ` 2 0−1 T bols − b0 → N 0k×1, σ0E xt xt . Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Dec. 2011 12 / 50 (Nonlinear) Least squares method Least squares estimation Example : AR(1) estimation Let (Xt ) be a covariance-stationary process defined by the fundamental representation (|φ| < 1) : Xt = φXt−1 + t where (t ) is the innovation process of (Xt ). The ordinary least squares estimation of φ is defined to be : T !−1 T ! ˆ X 2 X φols = xt−1 xt−1xt t=2 t=2 Moreover √ ` 2 T φˆols − φ → N 0, 1 − φ . Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Dec. 2011 13 / 50 (Nonlinear) Least squares method Nonlinear least squares estimation 3.2. Nonlinear least squares estimation Definition Consider the nonlinear regression model defined by (t = 1, ··· , T ): 0 yt = h(xt ; b0) + t where h is a nonlinear function (w.r.t. b0). The nonlinear least squares estimator of b0, bˆnls , satisfies the following minimization program : T ˆ X 2 bnls = argmin t b0 t=1 T X 0 2 = argmin yt − h(xt ; b0) . b0 t=1 Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Dec. 2011 14 / 50 (Nonlinear) Least squares method Nonlinear least squares estimation Example : MA(1) estimation Let (Xt ) be a covariance-stationary process defined by the fundamental representation (|θ| < 1) : Xt = t + θt−1 where (t ) is the innovation process of (Xt ). The ordinary least squares estimation of θ, which is defined by T X 2 θˆols = argmin (xt − θt−1) , θ t=2 is not feasible (since t−1 is not observable !). Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Dec. 2011 15 / 50 (Nonlinear) Least squares method Nonlinear least squares estimation Example : MA(1) estimation (cont’d) Conditionally on 0, x1 = 1 + θ0 ⇔ 1 = x1 − θ0 x2 = 2 + θ1 ⇔ 2 = x2 − θ1 ⇔ 2 = x2 − θ(x1 − θ0) . t−2 X k t−1 xt−1 = t−1 − θt−2 ⇔ t−1 = (−θ) xt−1−k + (−θ) 0. k=0 The (conditional) nonlinear least squares estimation of θ is defined by (assuming that 0 is negligible) T t−2 !2 X X k θˆnls = argmin xt − θ (−θ) xt−1−k θ t=2 k=0 √ ` 2 and T θˆnls − θ → N 0, 1 − θ . Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Dec. 2011 16 / 50 (Nonlinear) Least squares method Nonlinear least squares estimation Example : Least squares estimation using backcasting procedure Consider the fundamental representation of a MA(q) : Xt = ut ut = t + θ1t−1 + ··· + θqt−q 0 (0) (0) (0) Given some initial values δ = θ1 , ··· , θq , Compute the unconditional residuals uˆt uˆt = Xt ! Backcast values of t for t = −(q − 1), ··· , T using the backward recursion (0) (0) ˜t =u ˆt − θ1 ˜t+1 − · · · − θq ˜t+q where the q values for t beyond the estimation sample are set to zero : ˜T +1 = ··· = ˜T +q = 0, andu ˆt = 0 for t < 1. Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Dec. 2011 17 / 50 (Nonlinear) Least squares method Nonlinear least squares estimation Example : Least squares estimation using backcasting procedure (cont’d) Estimate the values of t using a forward recursion (0) (0) ˆt =u ˆt − θ1 ˜t−1 − · · · − θq ˜t−q Minimize the sum of squared residuals using the fitted values ˆt : T X 2 (Xt − ˆt − θ1ˆt−1 − · · · − θqˆt−q) q+1 0 ˆ(1) (1) (1) and find δ = θ1 , ··· , θq . Repeat the backcast step, forward recursion, and minimization procedures until the estimates of the moving average part converge. Remark : The backcast step can be turned off (˜−(q−1) = ··· = ˜0 = 0) and the forward recursion is only used. Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Dec. 2011 18 / 50 (Nonlinear) Least squares method Discussion 3.3. Discussion (Linear and Nonlinear) Least squares estimation are often used in practise. These methods lead to simple algorithms (e.g., backcasting procedure for ARMA models) and can often be used to initialize more ”sophisticated” estimation methods. These methods are opposed to exact methods (see likelihood estimation). Nonlinear estimation requires numerical optimization algorithms : Initial conditions ; Number of iterations and stopping criteria ; Local and global convergence ; Fundamental representation and constraints. Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Dec. 2011 19 / 50 (Generalized) Method of moments Methods of moments and Yule-Walker estimation 4. (Generalized) Method of moments 4.1. Methods of moments and Yule-Walker estimation Definition Suppose there is a set of k conditions ST − g (δ) = 0k×1 k k where ST ∈ R denotes a vector of theoretical moments , δ ∈ R is a k k vector of parameters, and g : R → R defines a (bijective) mapping between ST and δ.
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