4. MA(2) +drift: y t = µ + ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t 2. Mean: where θ(l) = 1 + θ 1 L + θ 2 L 2. Therefore,

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1 61 4. MA(2) +drift: y t = µ + ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t 2 Mean: y t = µ + θ(l)ɛ t, where θ(l) = 1 + θ 1 L + θ 2 L 2. Therefore, E(y t ) = µ + θ(l)e(ɛ t ) = µ

2 62 Example: MA(q) Model: y t = ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t θ q ɛ t q 1. Mean of MA(q) Process: E(y t ) = E(ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t θ q ɛ t q ) = 0 2. Autocovariance Function of MA(q) Process: q τ σ 2 ɛ(θ 0 θ τ + θ 1 θ τ θ q τ θ q ) = σ 2 ɛ θ i θ τ+i, τ = 1, 2,, q, γ(τ) = i=0 0, τ = q + 1, q + 2,, where θ 0 = 1.

3 63 3. MA( q) process is stationary. 4. MA(q) +drift: y t = µ + ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t θ q ɛ t q Mean: y t = µ + θ(l)ɛ t, where θ(l) = 1 + θ 1 L + θ 2 L θ q L q. Therefore, we have: E(y t ) = µ + θ(l)e(ɛ t ) = µ.

4 ARMA Model ARMA (Autoregressive Moving Average ) Process 1. ARMA(p, q) y t = φ 1 y t 1 + φ 2 y t φ p y t p + ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t θ q ɛ t q, which is rewritten as: φ(l)y t = θ(l)ɛ t, where φ(l) = 1 φ 1 L φ 2 L 2 φ p L p and θ(l) = 1+θ 1 L+θ 2 L 2 + +θ q L q.

5 65 2. Likelihood Function: The variance-covariance matrix of Y, denoted by V, has to be computed. Example: ARMA(1,1) Process: y t = φ 1 y t 1 + ɛ t + θ 1 ɛ t 1 Obtain the autocorrelation coefficient. The mean of y t is to take the expectation on both sides. E(y t ) = φ 1 E(y t 1 ) + E(ɛ t ) + θ 1 E(ɛ t 1 ), where the second and third terms are zeros.

6 66 Therefore, we obtain: E(y t ) = 0. The autocovariance of y t is to take the expectation, multiplying y t τ on both sides. E(y t y t τ ) = φ 1 E(y t 1 y t τ ) + E(ɛ t y t τ ) + θ 1 E(ɛ t 1 y t τ ). Each term is given by: E(y t y t τ ) = γ(τ), E(y t 1 y t τ ) = γ(τ 1),

7 67 σ 2 ɛ, τ = 0, E(ɛ t y t τ ) = 0, τ = 1, 2,, (φ 1 + θ 1 )σ 2 ɛ, τ = 0, E(ɛ t 1 y t τ ) = σ 2 ɛ, τ = 1, 0, τ = 2, 3,. Therefore, we obtain; γ(0) = φ 1 γ(1) + (1 + φ 1 θ 1 + θ 2 1 )σ2 ɛ, γ(1) = φ 1 γ(0) + θ 1 σ 2 ɛ, γ(τ) = φ 1 γ(τ 1), τ = 2, 3,.

8 68 From the first two equations, γ(0) and γ(1) are computed by: ( 1 φ1 ) ( γ(0) ) ( 1 + φ1 θ 1 + θ 2 ) 1 φ 1 1 γ(1) = σ 2 ɛ θ 1 ( γ(0) ) ( 1 ) 1 φ1 ( 1 + φ1 θ 1 + θ 2 ) 1 γ(1) = σ 2 ɛ = σ2 ɛ 1 φ 2 1 φ 1 1 θ 1 ( 1 φ1 ) ( 1 + φ1 θ 1 + θ 2 ) 1 φ 1 1 θ 1 ( = σ2 ɛ 1 φ φ 1 θ 1 + θ 2 1 (1 + φ 1 θ 1 )(φ 1 + θ 1 ) ).

9 69 Thus, the initial value of the autocorrelation coefficient is given by: ρ(1) = (1 + φ 1θ 1 )(φ 1 + θ 1 ) φ 1 θ 1 + θ 2 1 We have: ρ(τ) = φ 1 ρ(τ 1).

10 70 ARMA(p, q) +drift: y t = µ + φ 1 y t 1 + φ 2 y t 2 + φ p y t p + ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t θ q ɛ t q. Mean of ARMA(p, q) Process: φ(l)y t = µ + θ(l)ɛ t, where φ(l) = 1 φ 1 L φ 2 L 2 φ p L p and θ(l) = 1 + θ 1 L + θ 2 L θ q L q. Therefore, y t = φ(l) 1 µ + φ(l) 1 θ(l)ɛ t. E(y t ) = φ(l) 1 µ + φ(l) 1 θ(l)e(ɛ t ) = φ(1) 1 µ = µ 1 φ 1 φ 2 φ p.

11 ARIMA Model Autoregressive Integrated Moving Average (ARIMA ) Model ARIMA(p, d, q) Process φ(l) d y t = θ(l)ɛ t, where d y t = d 1 (1 L)y t = d 1 y t d 1 y t 1 = (1 L) d y t for d = 1, 2,, and 0 y t = y t.

12 SARIMA Model Seasonal ARIMA (SARIMA) Process: 1. SARIMA(p, d, q) φ(l) d s y t = θ(l)ɛ t, where s y t = (1 L s )y t = y t y t s. s = 4 when y t denotes quarterly date and s = 12 when y t represents monthly data.

13 Optimal Prediction 1. AR(p) Process: y t = φ 1 y t φ p y t p + ɛ t (a) Define: E(y t+k Y t ) = y t+k t, where Y t denotes all the information available at time t. Taking the conditional expectation of y t+k = φ 1 y t+k 1 + +φ p y t+k p +ɛ t+k on both sides, y t+k t = φ 1 y t+k 1 t + + φ p y t+k p t,

14 74 where y s t = y s for s t. (b) Optimal prediction is given by solving the above differential equation. 2. MA(q) Process: y t = ɛ t + θ 1 ɛ t θ q ɛ t q (a) Let ˆɛ T, ˆɛ T 1,, ˆɛ 1 be the estimated errors. (b) y t+k = ɛ t+k + θ 1 ɛ t+k θ q ɛ t+k q (c) Therefore, y t+k t = ɛ t+k t + θ 1 ɛ t+k 1 t + + θ q ɛ t+k q t, where ɛ s t = 0 for s > t and ɛ s t = ˆɛ s for s t.

15 75 3. ARMA(p, q) Process: y t = φ 1 y t φ p y t p + ɛ t + θ 1 ɛ t θ q ɛ t q (a) y t+k = φ 1 y t+k φ p y t+k p + ɛ t+k + θ 1 ɛ t+k θ q ɛ t+k q (b) Optimal prediction is: y t+k t = φ 1 y t+k 1 t + + φ p y t+k p t + ɛ t+k t + θ 1 ɛ t+k 1 t + + θ q ɛ t+k q t, where y s t = y s and ɛ s t = ˆɛ s for s t, and ɛ s t = 0 for s > t.

16 Identification 1. Based on AIC or SBIC given d, s, we obtain p, q. (a) AIC (Akaike s Information Criterion) AIC = 2 log(likelihood) + 2k, where k = p + q, which is the number of parameters estimated. (b) SBIC (Shwarz s Bayesian Information Criterion) SBIC = 2 log(likelihood) + k log T,

17 77 where T denotes the number of observations. 2. From the sample autocorrelation coefficient function ˆρ(k) and the partial autocorrelation coefficient function ˆφ k,k for k = 1, 2,, we obtain p, d, q, s. AR(p) Process MA(q) Process Autocorrelation Function Gradually decreasing ρ(k) = 0, k = q + 1, q + 2, Partial Autocorrelation Function φ(k, k) = 0, Gradually decreasing k = p + 1, p + 2, (a) Compute s y t to remove seasonality.

18 78 Compute the autocovariance functions of s y t. If the autocovariance functions have period s, we take (1 L s ), again. (b) Determine the order of difference. Compute the partial autocovariance functions every time. If the autocovariance functions decrease as τ is large, go to the next step. (c) Determine the order of AR terms (i.e., p). Compute the partial autocovariance functions every time. The partial autocovariance functions are close to zero after some τ, go to the next step.

19 79 (d) Determine the order of MA terms (i.e., q). Compute the autocovariance functions every time. If the autocovariance functions are randomly around zero, end of the procedure.

20 Example of SARIMA using Consumption Data Construct SARIMA model using monthly and seasonally unadjusted consumption expenditure data and STATA12. Estimation Period: Jan., 1970 Dec., 2012 (T = 516). gen time=_n. tsset time time variable: time, 1 to 516 delta: 1 unit. corrgram expend

21 LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor] gen dexp=expend-l.expend (1 missing value generated)

22 82. corrgram dexp LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor] gen sdex=dexp-l12.dexp (13 missing values generated)

23 83. corrgram sdex LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor]

24 84. arima sdex, ar(1,2) ma(1) (setting optimization to BHHH) Iteration 0: log likelihood = Iteration 1: log likelihood = Iteration 2: log likelihood = Iteration 3: log likelihood = Iteration 4: log likelihood = (switching optimization to BFGS) Iteration 5: log likelihood = Iteration 6: log likelihood = Iteration 7: log likelihood = Iteration 8: log likelihood = ARIMA regression Sample: Number of obs = 503 Wald chi2(3) = Log likelihood = Prob > chi2 =

25 OPG sdex Coef. Std. Err. z P> z [95% Conf. Interval] sdex _cons ARMA ar L L ma L /sigma Note: The test of the variance against zero is one sided, and the two-sided confidence interval is truncated at zero. 85

26 86. estat ic Model Obs ll(null) ll(model) df AIC BIC Note: N=Obs used in calculating BIC; see [R] BIC note

27 ARCH and GARCH Models Autoregressive Conditional Heteroskedasticity (ARCH) Generalized Autoregressive Conditional Heteroskedasticity (GARCH) 1. ARCH (p) Model ɛ t ɛ t 1, ɛ t 2,, ɛ 1 N(0, h t ), where h t = α 0 + α 1 ɛ 2 t α pɛ 2 t p.

28 88 The unconditional variance of ɛ t is: σ 2 ɛ = α 0 1 α 1 α 2 α p 2. GARCH (p, q) Model ɛ t ɛ t 1, ɛ t 2,, ɛ 1 N(0, h t ), where h t = α 0 + α 1 ɛ 2 t α pɛ 2 t p + β 1 h t β q h t q.

29 89 3. Application to OLS (Case of ARCH(1) Model): y t = x t β + ɛ t, ɛ t ɛ t 1, ɛ t 2,, ɛ 1 N(0, α 0 + α 1 ɛ 2 t 1 ). The joint density of ɛ 1, ɛ 2,, ɛ T is: T f (ɛ 1,, ɛ T ) = f (ɛ 1 ) f (ɛ t ɛ t 1,, ɛ 1 ) t=2 ( ) 1/2 ( ) = (2π) 1/2 α 0 1 exp 1 α 1 2α 0 /(1 α 1 ) ɛ2 1 T T (2π) (T 1)/2 (α 0 + α 1 ɛ 2 t 1 ) 1/2 exp 1 ɛt 2 2 α 0 + α 1 ɛ. 2 t 1 t=2 t=2

30 90 The log-likelihood function is: log L(β, α 0, α 1 ; y 1,, y T ) = 1 2 log(2π) 1 2 log ( T T t=2 ) α α 1 2α 0 /(1 α 1 ) (y 1 x 1 β) 2 log(2π) 1 T log ( α 0 + α 1 (y t 1 x t 1 β) 2) 2 t=2 (y t x t β) 2 α 0 + α 1 (y t 1 x t 1 β) 2. Obtain α 0, α 1 and β such that the log-likelihood function is maximized. α 0 > 0 and α 1 > 0 have to be satisfied.

31 91 These two conditions are explicitly included, when the model is modified to: E(ɛ 2 t ɛ t 1, ɛ t 2,, ɛ 1 ) = α α2 1 ɛ2 t 1. Testing the ARCH(1) Effect: (a) Estimate y t = x t β + u t by OLS, and compute ˆβ and û t = y t x t ˆβ. (b) Estimate û 2 t = α 0 + α 1 û 2 t 1 by OLS. If ˆα 1 is significant, there is the ARCH(1) effect in the error term. This test corresponds to LM test.