ECON/FIN 250: Forecasting in Finance and Economics: Section 6: Standard Univariate Models
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1 ECON/FIN 250: Forecasting in Finance and Economics: Section 6: Standard Univariate Models Patrick Herb Brandeis University Spring 2016 Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
2 Course Overview 1 Key Objectives 2 Lag Operators 3 Moving Average Models 4 Autoregressive Models 5 Converting Between AR & MA Models 6 Autoregressive Moving Average (ARMA) Models 7 Nonzero Expected Values 8 Linear Time Series: Big Picture Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
3 Key Objectives 1 Key Objectives 2 Lag Operators 3 Moving Average Models 4 Autoregressive Models 5 Converting Between AR & MA Models 6 Autoregressive Moving Average (ARMA) Models 7 Nonzero Expected Values 8 Linear Time Series: Big Picture Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
4 Key Objectives Basic time series tools Moving average models Autoregressive models Mixed ARMA models Some special models Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
5 Lag Operators 1 Key Objectives 2 Lag Operators 3 Moving Average Models 4 Autoregressive Models 5 Converting Between AR & MA Models 6 Autoregressive Moving Average (ARMA) Models 7 Nonzero Expected Values 8 Linear Time Series: Big Picture Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
6 Lag Operators Useful tools for time series Not too complicated Sometimes can get difficult Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
7 Lag Operators Ly t = y t 1 (1) L(Ly t ) = L 2 y t = y t 2 (2) L(L 2 y t ) = L 3 y t = y t 3 (3) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
8 Lag Operators y t = µ + ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t 2 (4) y t = µ + ɛ t + θ 1 Lɛ t + θ 2 L 2 ɛ t (5) y t = µ + (1 + θ 1 L + θ 2 L 2 )ɛ t (6) y t = µ + θ(l)e t (7) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
9 Other Operators Fy t = y t+1 (8) Dy t = y t y t 1 (9) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
10 Operators in Stata This notation moves easily into Stata: reg y L.y L2.y reg L (0/2). y reg F.y D.y Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
11 Moving Average Models 1 Key Objectives 2 Lag Operators 3 Moving Average Models 4 Autoregressive Models 5 Converting Between AR & MA Models 6 Autoregressive Moving Average (ARMA) Models 7 Nonzero Expected Values 8 Linear Time Series: Big Picture Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
12 MA(1): Moving Average of Order 1 y t = ɛ t + θɛ t 1 (10) ɛ t WN(0, σ 2 ɛ ) (11) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
13 Comparing Two MA(1) Models t θ = 0.4 θ = 0.95 see software file compma1.do Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
14 Comparing Two MA(1) Models t θ = 0.3 θ = 0.9 see software file compma1same.do Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
15 MA(1): Unconditional Mean y t = ɛ t + θɛ t 1 E[y t ] = E[ɛ t ] + θe[ɛ t 1 ] = 0 (12) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
16 MA(1): Unconditional Variance y t = ɛ t + θɛ t 1 Why? Var[y t ] = Var[ɛ t ] + θ 2 Var[ɛ t 1 ] (13) = σ 2 ɛ (1 + θ 2 ) (14) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
17 MA(1): Unconditional Variance y t = ɛ t + θɛ t 1 Var[y t ] = Var[ɛ t ] + θ 2 Var[ɛ t 1 ] (15) = σɛ 2 (1 + θ 2 ) (16) Why? E[ɛ t ɛ t 1 ] = 0 (17) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
18 MA(1): Conditional Mean y t = ɛ t + θɛ t 1 E[y t Ω t 1 ] = E[ɛ t Ω t 1 ] + θe[ɛ t 1 Ω t 1 ] (18) = 0 + θɛ t 1 (19) = θɛ t 1 (20) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
19 MA(1): Conditional Variance y t = ɛ t + θɛ t 1 Var[y t Ω t 1 ] = E[(y t E[y t Ω t 1 ]) 2 Ω t 1 ] (21) = E[(ɛ t + θɛ t 1 θɛ t 1 ) 2 Ω t 1 ] (22) = σ 2 ɛ (23) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
20 MA(1): Autocovariance y t = ɛ t + θɛ t 1 γ j = E[y t y t j ] (24) γ 1 = E[y t y t 1 ] (25) γ 1 = E[(ɛ t + θɛ t 1 )(ɛ t 1 + θɛ t 2 )] (26) γ 1 = θe[ɛ t 1 ɛ t 1 ] = θσ 2 ɛ (27) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
21 MA(1): Autocovariance y t = ɛ t + θɛ t 1 γ 2 = E[y t y t 2 ] (28) γ 2 = E[(ɛ t + θɛ t 1 )(ɛ t 2 + θɛ t 3 )] (29) γ 2 = 0 (30) γ j = 0 j 2 (31) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
22 MA(1): Autocovariance y t = ɛ t + θɛ t 1 θσɛ 2 if j = 1 γ j = 0 if j > 1 (32) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
23 MA(1): Autocorrelation y t = ɛ t + θɛ t 1 ρ j = γ j Var[y t ] = γ j γ j = (33) γ 0 (1 + θ 2 )σ 2 θ ρ j =, if j = 1 (1+θ 2 ) (34) 0, if j > 1 Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
24 MA(q): Moving Average of Order q y t = ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t θ q ɛ t q = Θ(L)ɛ t (35) ɛ t WN(0, σ 2 ɛ ) (36) Θ(L) = 1 + θ 1 L + θ 2 L θ q L q (37) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
25 MA(q): Unconditional Mean y t = ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t θ q ɛ t q = Θ(L)ɛ t E[y t ] = E[ɛ t ] + θ 1 E[ɛ t 1 ] + θ 2 E[ɛ t 2 ] + + θ q E[ɛ t q ] = 0 (38) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
26 MA(q): Unconditional Variance y t = ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t θ q ɛ t q = Θ(L)ɛ t Var[y t ] = Var[ɛ t ] + θ 2 1Var[ɛ t 1 ] + + θ 2 qvar[ɛ t q ] (39) Var[y t ] = σ 2 ɛ (1 + θ θ θ 2 q) (40) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
27 MA(q): Conditional Mean y t = ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t θ q ɛ t q = Θ(L)ɛ t E[y t Ω t 1 ] = E[ɛ t Ω t 1 ]+θ 1 E[ɛ t 1 Ω t 1 ]+ +θ q E[ɛ t q Ω t 1 ] (41) E[y t Ω t 1 ] = 0 + θ 1 ɛ t 1 + θ 2 ɛ t θ q ɛ t q (42) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
28 MA(q): Conditional Variance y t = ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t θ q ɛ t q = Θ(L)ɛ t Var[y t Ω t 1 ] = E[(y t E[y t Ω t 1 ]) 2 Ω t 1 ] (43) Var[y t Ω t 1 ] = E[ɛ 2 t Ω t 1 ] (44) Var[y t Ω t 1 ] = σ 2 ɛ (45) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
29 MA(2): Autocovariance y t = ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t 2 γ j = E[y t y t j ] (46) γ 2 = E[y t y t 2 ] (47) γ 2 = E[(ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t 2 )(ɛ t 2 + θ 1 ɛ t 3 + θ 2 ɛ t 4 )] (48) γ 2 = θ 2 σ 2 ɛ (49) γ j = 0 j > 2 (50) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
30 MA(q): Autocovariance & Autocorrelation y t = ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t θ q ɛ t q 0 if j q γ j = 0 if j > q 0 if j q ρ j = 0 if j > q (51) (52) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
31 Autocorrelation Function: MA(1) & MA(3) Autocorrelations of y Lag Bartlett s formula for MA(q) 95% confidence bands Autocorrelations of y Lag Bartlett s formula for MA(q) 95% confidence bands see software file compma1ma3.do Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
32 MA( ): Moving Average of Order Complicated Important tool y t = ɛ t + θ j ɛ t j (53) j=1 ɛ t WN(0, σ 2 ɛ ) (54) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
33 Autoregressive Models 1 Key Objectives 2 Lag Operators 3 Moving Average Models 4 Autoregressive Models 5 Converting Between AR & MA Models 6 Autoregressive Moving Average (ARMA) Models 7 Nonzero Expected Values 8 Linear Time Series: Big Picture Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
34 AR(1): Autoregressive Model of Order 1 y t = φy t 1 + ɛ t (55) ɛ t WN(0, σ 2 ɛ ) (56) (1 φl)y t = ɛ t (57) φ < 1 (58) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
35 Comparing Two AR(1) Models t φ = 0.2 φ = 0.95 see software file compar1.do Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
36 AR(1): Unconditional Mean y t = φy t 1 + ɛ t E[y t ] = φe[y t 1 ] + E[ɛ t ] (59) E[y t ] = φe[y t ] + E[ɛ t ] (60) (1 φ)e[y t ] = E[ɛ t ] = 0 (61) E[y t ] = 0 (62) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
37 AR(1): Unconditional Variance y t = φy t 1 + ɛ t Var[y t ] = φ 2 Var[y t 1 ] + Var[ɛ t ] (63) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
38 AR(1): Unconditional Variance y t = φy t 1 + ɛ t Why? Var[y t ] = φ 2 Var[y t 1 ] + Var[ɛ t ] (63) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
39 AR(1): Unconditional Variance y t = φy t 1 + ɛ t Why? Var[y t ] = φ 2 Var[y t 1 ] + Var[ɛ t ] (63) E[ɛ t y t 1 ] = 0 (64) Var[y t ] = φ 2 Var[y t ] + Var[ɛ t ] (65) Var[y t ] = E[y 2 t ] = σ2 ɛ 1 φ 2 (66) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
40 AR(1): Conditional Mean y t = φy t 1 + ɛ t E[y t Ω t 1 ] = E[φy t 1 Ω t 1 ] + E[ɛ t Ω t 1 ] (67) E[y t Ω t 1 ] = φy t 1 (68) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
41 AR(1): Conditional Variance y t = φy t 1 + ɛ t Var[y t Ω t 1 ] = E[(y t E[y t Ω t 1 ]) 2 Ω t 1 ] (69) Var[y t Ω t 1 ] = E[ɛ 2 t Ω t 1 ] (70) Var[y t Ω t 1 ] = σ 2 ɛ (71) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
42 AR(1): Autocovariance & Yule-Walker Equation y t = φy t 1 + ɛ t y t y t j = φy t 1 y t j + ɛ t y t j (72) E[y t y t j ] = φe[y t 1 y t j ] + E[ɛ t y t j ] (73) γ j = φγ j 1 (74) Since we know γ 0 = Var[y t ], we can use the Yule-Walker equation to get all the autocovariances. Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
43 AR(1): Autocovariance γ j = φγ j 1 γ 0 = σ2 ɛ 1 φ 2 (75) σ2 ɛ γ 1 = φ (76) 1 φ 2 σ 2 ɛ γ 2 = φ 2 (77) 1 φ 2 σ 2 ɛ γ j = φ j (78) 1 φ 2 Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
44 AR(1): Autocorrelation ρ j = γ j γ 0 ρ 1 = φ (79) ρ 2 = φ 2 (80) ρ j = φ j (81) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
45 AR(1): Comparing Autocorrelation Functions: φ = 0.4, φ = 0.95 Autocorrelations of y Lag Bartlett s formula for MA(q) 95% confidence bands Autocorrelations of y Lag Bartlett s formula for MA(q) 95% confidence bands see software file compar1acf.do Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
46 AR(1): Comparing Autocorrelation Functions: φ = 0.9, φ = 0.9 Autocorrelations of y Lag Bartlett s formula for MA(q) 95% confidence bands Autocorrelations of y Lag Bartlett s formula for MA(q) 95% confidence bands see software file compar1acfsign.do Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
47 Partial Autocorrelation Function (PACF) Regress y t on lags: y t = φ 1 y t 1 + φ 2 y t 2 + φ 3 y t Estimates of ˆφ j are the partial autocorrelations for lag j. For an AR(1) we would get: = φ if j = 1 ˆφ j = 0 if j > 1 (82) For an AR(p): 0 if j p ˆφ j = 0 if j > p (83) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
48 AR(1): Comparing PACF with φ = 0.4, φ = 0.95 Partial autocorrelations of y Lag 95% Confidence bands [se = 1/sqrt(n)] Partial autocorrelations of y Lag 95% Confidence bands [se = 1/sqrt(n)] see software file compar1pacf.do Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
49 AR(p): Autoregressive Model of Order p p y t = φ j y t j + ɛ t (84) j=1 ɛ t WN(0, σɛ 2 ) (85) Φ(L) = (1 φ 1 L φ 2 L 2... φ p L p ) (86) Φ(L)y t = ɛ t (87) y t = ɛ t Φ(L) (88) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
50 AR(p): Autoregressive Model of Order p AR(1) only has two kinds of persistence: φ is positive or negative. If y t = 0.9y t 1 + ɛ t, then the ACF is positive with decay If y t = 0.9y t 1 + ɛ t, then the ACF oscillates between positive and negative with decay AR(p) has more complicated patterns and can have large positive and negative swings Also, stationarity is trickier Example: y t = 1.5y t 1 0.9y t 2 + ɛ t (89) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
51 AR(2): Autocorrelation Function Autocorrelations of y Lag Bartlett s formula for MA(q) 95% confidence bands see software file acfar2.do Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
52 AR(2): Autocovariance & Yule-Walker Equation y t = φ 1 y t 1 + φ 2 y t 2 + ɛ t Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
53 AR(2): Autocovariance & Yule-Walker Equation y t = φ 1 y t 1 + φ 2 y t 2 + ɛ t y t y t j = φ 1 y t 1 y t j + φ 2 y t 2 y t j + ɛ t y t j (90) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
54 AR(2): Autocovariance & Yule-Walker Equation y t = φ 1 y t 1 + φ 2 y t 2 + ɛ t y t y t j = φ 1 y t 1 y t j + φ 2 y t 2 y t j + ɛ t y t j (90) E[y t y t j ] = φ 1 E[y t 1 y t j ] + φ 2 E[y t 2 y t j ] + E[ɛ t y t j ] (91) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
55 AR(2): Autocovariance & Yule-Walker Equation y t = φ 1 y t 1 + φ 2 y t 2 + ɛ t y t y t j = φ 1 y t 1 y t j + φ 2 y t 2 y t j + ɛ t y t j (90) E[y t y t j ] = φ 1 E[y t 1 y t j ] + φ 2 E[y t 2 y t j ] + E[ɛ t y t j ] (91) γ j = φ 1 γ j 1 + φ 2 γ j 2 (92) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
56 AR(2): Autocovariance & Yule-Walker Equation y t = φ 1 y t 1 + φ 2 y t 2 + ɛ t y t y t j = φ 1 y t 1 y t j + φ 2 y t 2 y t j + ɛ t y t j (90) E[y t y t j ] = φ 1 E[y t 1 y t j ] + φ 2 E[y t 2 y t j ] + E[ɛ t y t j ] (91) Divide by γ 0 = Var[y t ] γ j = φ 1 γ j 1 + φ 2 γ j 2 (92) ρ j = φ 1 ρ j 1 + φ 2 ρ j 2 (93) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
57 AR(2): Autocorrelation & Yule-Walker Equation ρ j = φ 1 ρ j 1 + φ 2 ρ j 2 ρ 0 = 1 (94) ρ 1 = φ 1 ρ 0 + φ 2 ρ 1 = φ 1 ρ 0 + φ 2 ρ 1 (95) ρ 1 = φ 1 1 φ 2 (96) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
58 Stationarity Question: Is the process blowing up (nonstationary)? What are the features of a stationary process? Constant unconditional mean Constant and finite unconditional variance Autocorrelations depends only on displacement (time) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
59 Stationary vs. Nonstationary AR(1) t φ = 1.1 φ = 0.9 see software file unstablear1.do Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
60 Quick & Dirty AR(p) Stationarity Check A necessary condition for covariance stationarity is p φ i < 1 (97) i=1 If this condition fails, you know the process is not stationary. However, if the condition holds, it may or may not be stationary. Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
61 AR(2): Covariance Stationarity Covariance stationarity depends upon the roots of the lag operator polynomial. y t = φ 1 y t 1 + φ 2 y t 2 + ɛ t y t (1 φ 1 L φ 2 L 2 ) = ɛ t (98) y t (1 φ 1 z φ 2 z 2 ) = ɛ t (99) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
62 AR(2): Covariance Stationarity Can find roots using quadratic formula: (1 φ 1 z φ 2 z 2 ) = 0 (100) z 1 = φ 1 φ φ 2 (101) 2φ 2 z 2 = φ 1 + φ φ 2 (102) 2φ 2 Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
63 AR(2): Finding Roots y t = 1.5y t 1 0.9y t 2 + ɛ t y t (1 1.5L + 0.9L 2 ) = ɛ t (103) y t (1 1.5z + 0.9z 2 ) = ɛ t (104) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
64 AR(2): Finding Roots (1 1.5z + 0.9z 2 ) = 0 (105) z 1 = 1.5 ( 1.5) 2 4(0.9)(1) = i 2(0.9) (106) z 1 = 1.5 ( 1.5) 2 4(0.9)(1) = i 2(0.9) (107) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
65 AR(2): Covariance Stationarity: Modulus Stationarity: Roots must be greater than 1 in length, and can be calculated from the modulus: a 2 + b 2 = c 2 Modulus = R 2 + C 2 (108) R = Real Part and C = Complex Part. If root length > 1, then root lies outside of unit circle and the process is stationary. Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
66 AR(2): Covariance Stationarity: Modulus y t = 1.5y t 1 0.9y t 2 + ɛ t roots = 0.83 ± 0.65i (109) Modulus = = 1.11 > 1 (110) The roots are outside the unit circle, so this process is stationary If root contains only real numbers, then the correlations decay over time If roots are complex, then correlations have cycles Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
67 Inverse Roots Some books and software alternatively test and report the inverse roots. y t = φ 1 y t 1 + φ 2 y t 2 + ɛ t y t (1 φ 1 L φ 2 L 2 ) = ɛ t (111) (z 2 φ 1 z 1 φ 2 ) = 0 (112) (λ 2 φ 1 λ φ 2 ) = 0, λ z 1 (113) Stationarity requires that the roots of λ must lie inside the unit circle such that: Modulus = R 2 + C 2 < 1 (114) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
68 AR(2): Inverse Roots y t = 1.5y t 1 0.9y t 2 + ɛ t y t (1 1.5L + 0.9L 2 ) = ɛ t (115) (λ 2 1.5λ + 0.9) = 0 (116) From the quadratic formula: Modulus = = < 1 (117) Stata tests the inverse roots using the estat aroots command Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
69 Technical Aside Alternatively we could write the second order difference equation in matrix notation [ yt y t 1 ] = [ φ1 φ ] [ ] yt 1 + y t 2 [ ] et 0 (118) In this case, the inverse roots λ 1, λ 2 are the eigenvalues of the matrix Which is the solution to F λi = F = [ ] φ1 φ [ ] φ1 φ (119) [ ] λ 0 0 λ = 0 (120) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
70 AR(p): Autoregressive of Order p p y t = φ j y t j + ɛ t j=1 Stationarity conditions more complicated Involve pth order polynomials Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
71 Converting Between AR & MA Models 1 Key Objectives 2 Lag Operators 3 Moving Average Models 4 Autoregressive Models 5 Converting Between AR & MA Models 6 Autoregressive Moving Average (ARMA) Models 7 Nonzero Expected Values 8 Linear Time Series: Big Picture Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
72 Finite Order Moving Averages All finite order moving averages are covariance stationary Not all finite order moving averages are invertible Finite order moving averages are invertible if the inverse roots of the lag operator polynomial are inside the unit circle If the finite order moving average is invertible, then it can be written as an AR( ) process Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
73 MA(1) to AR( ) MA(1) y t = ɛ t + θɛ t 1, ɛ t WN(0, σ 2 ), θ < 1 We can solve for the innovation ɛ t = y t θɛ t 1 (121) The lagged innovations are ɛ t 1 = y t 1 θɛ t 2 (122) ɛ t 2 = y t 2 θɛ t 3 (123) ɛ t 3 = y t 3 θɛ t 4 (124) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
74 MA(1) to AR( ) Substitute the innovations into the MA(1) process y t = ɛ t + θɛ t 1, ɛ t WN(0, σ 2 ), θ < 1 to yield an AR( ) y t = ɛ t + θy t 1 θ 2 y t 2 + θ 3 y t 3... (125) Or in lag notation θl y t = ɛ t (126) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
75 MA(1): Comparing PACF with θ = 0.9, θ = 0.9 Partial autocorrelations of y Lag 95% Confidence bands [se = 1/sqrt(n)] Partial autocorrelations of y Lag 95% Confidence bands [se = 1/sqrt(n)] see software file compma1pacf.do Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
76 MA(q) & MA( ) The same approach can be used to determine invertibility of finite MA(q) processes q < When q, stability is no longer guaranteed Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
77 MA( ) For an MA( ) process y t = ψ j ɛ t j = ψ 0 ɛ t + ψ 1 ɛ t 1 + ψ 1 ɛ t (127) j=0 The infinite sequence is covariance stationary provided that ψj 2 < (128) j=0 Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
78 AR(1) to MA( ) by Substitution y t = φy t 1 + ɛ t, y t 1 = φy t 2 + ɛ t 1 y t = φ(φy t 2 + ɛ t 1 ) + ɛ t (129) y t = φ 2 y t 2 + φɛ t 1 + ɛ t (130) y t = φ m y t m + m 1 j=1 φ j ɛ t j + ɛ t (131) y t = φ j ɛ t j + ɛ t = ɛ t j (132) j=1 j=0 Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
79 Stationary AR Processes AR processes are always invertible AR processes are not always stationary Stationarity requires the inverse roots of the lag operator polynomial to be inside the unit circle Any stationary AR(p) process can be converted to an MA( ) Conversion can make some calculations easier Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
80 AR(1) to MA( ) with Lag Operator y t = φy t 1 + ɛ t, φ < 1 y t = (1 φl)y t = ɛ t (133) 1 1 φl ɛ t = φ j ɛ t j (134) j=0 1 1 φl = 1 + φl + φ2 L 2 + φ 3 L (135) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
81 Autoregressive Moving Average (ARMA) Models 1 Key Objectives 2 Lag Operators 3 Moving Average Models 4 Autoregressive Models 5 Converting Between AR & MA Models 6 Autoregressive Moving Average (ARMA) Models 7 Nonzero Expected Values 8 Linear Time Series: Big Picture Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
82 ARMA(p,q) Models p q y t = φ j y t j + θ j ɛ t j + ɛ t (136) j=1 j=1 Combines AR and MA components Richer Dynamics Can often reduce parameter estimates (increase parsimony) Can result from aggregation Sums of AR processes, or sums of AR and MA process can be ARMA processes AR processes observed subject to measurement error also turn out to be ARMA processes Connection to state space models Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
83 ARMA(1,1) In lag notation y t = φy t 1 + θɛ t 1 + ɛ t, ɛ t WN(0, σ 2 ) (137) Stationarity requires φ < 1 Invertibility requires θ < 1 (1 φl)y t = (1 + θl)ɛ t (138) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
84 ARMA(1,1) If the covariance stationarity condition is satisfied, then we can convert this to an infinite MA process y t = (1 + θl) (1 φl) ɛ t (139) If the invertibility condition is satisfied, then we can convert this to the infinite order AR process (1 φl) (1 + θl) y t = ɛ t (140) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
85 ARMA(1,1): Extreme Example y t = 0.99y t ɛ t 1 + ɛ t (141) Run arma11acf.do Note the ACF pattern Extreme persistence and low level Sometimes difficult to see Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
86 Possible Model Origins y t = x t + η t (142) x t = ρx t 1 + µ t (143) Observe only y t x t is hidden With ρ large (close to one), x t moves slowly With Var[η t ] large this is difficult to see/measure Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
87 Put These Together y t = x t + η t x t = ρx t 1 + µ t y t = ρx t 1 + µ t + η t = ρ(y t 1 η t 1 ) + µ t + η t = ρy t 1 ρη t 1 + (µ t + η t ) = ρy t 1 θe t 1 + e t Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
88 Solving for ARMA Parameters (Optional Note) Given that you know the parameters for the noisy random walk, how do you map to an ARMA(1,1)? y t = x t + η t, x t = ρx t 1 + µ t y t = ρy t 1 ρη t 1 + (µ t + η t ) = ρy t 1 θe t 1 + e t = ρy t 1 + z t Var[z t ] = (1 + θ 2 )σ 2 e = (1 + ρ 2 )σ 2 η + σ 2 µ Cov[z t, z t 1 ] = θσ 2 e = ρσ 2 η σe 2 = (ρ/θ)ση 2 (1 + θ 2 )(ρ/θ)ση 2 = (1 + ρ 2 )ση 2 + σµ 2 Now solve quadratic for θ. You know all the other values. Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
89 Nonzero Expected Values 1 Key Objectives 2 Lag Operators 3 Moving Average Models 4 Autoregressive Models 5 Converting Between AR & MA Models 6 Autoregressive Moving Average (ARMA) Models 7 Nonzero Expected Values 8 Linear Time Series: Big Picture Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
90 Adding Constants q y t = θ 0 + θ j ɛ t j + ɛ t (144) j=1 p y t = φ 0 + φ j y t j + ɛ t (145) j=1 For the MA model this is a trivial change. For the AR, it can be trickier. Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
91 AR(1): Adding a Constant y t = φ 0 + φ 1 y t 1 + ɛ t (146) E(y t ) = φ 0 + φ 1 E(y t 1 ) + E(ɛ t ) (147) E(y t ) = φ 0 + φ 1 E(y t ) + 0 (148) E(y t ) = φ 0 1 φ 1 (149) Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
92 Linear Time Series: Big Picture 1 Key Objectives 2 Lag Operators 3 Moving Average Models 4 Autoregressive Models 5 Converting Between AR & MA Models 6 Autoregressive Moving Average (ARMA) Models 7 Nonzero Expected Values 8 Linear Time Series: Big Picture Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
93 Frequently Used Models MA(q) AR(1) AR(2) ARMA(1,1) Random Walks Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
94 Thoughts on Linear Models Many flavors of models (MA, AR, ARMA) AR(2) very different from AR(1) ARMA(1,1) is more useful than you might think Most of these do map to a MA( ) Certain approximate models may have fewer parameters (parsimonious), and be more useful for forecasting. Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
95 Thoughts on Linear Models Many flavors of models (MA, AR, ARMA) AR(2) very different from AR(1) ARMA(1,1) is more useful than you might think Most of these do map to a MA( ) Certain approximate models may have fewer parameters (parsimonious), and be more useful for forecasting. In practice, model identification can be tricky Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
96 Thoughts on Linear Models Many flavors of models (MA, AR, ARMA) AR(2) very different from AR(1) ARMA(1,1) is more useful than you might think Most of these do map to a MA( ) Certain approximate models may have fewer parameters (parsimonious), and be more useful for forecasting. In practice, model identification can be tricky Other families of models Nonlinear Long memory Regime shifting Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN 250: Spring / 88
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