Forecasting. Francis X. Diebold University of Pennsylvania. August 11, / 323

Transcription

1 Forecasting Francis X. Diebold University of Pennsylvania August 11, / 323

2 Copyright c 2013 onward, by Francis X. Diebold. These materials are freely available for your use, but be warned: they are highly preliminary, significantly incomplete, and rapidly evolving. All are licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. (Briefly: I retain copyright, but you can use, copy and distribute non-commercially, so long as you give me attribution and do not modify. To view a copy of the license, visit In return I ask that you please cite the books whenever appropriate, as: Diebold, F.X. (year here), Book Title Here, Department of Economics, University of Pennsylvania, fdiebold/textbooks.html. The painting is Enigma, by Glen Josselsohn, from Wikimedia Commons. 2 / 323

3 Elements of Forecasting in Business, Finance, Economics and Government 1. Forecasting in Action 1.1 Operations planning and control 1.2 Marketing 1.3 Economics 1.4 Financial speculation 1.5 Financial risk management 1.6 Capacity planning 1.7 Business and government budgeting 1.8 Demography 1.9 Crisis management 3 / 323

4 Forecasting Methods: An Overview Review of probability, statistics and regression Six Considerations Basic to Successful Forecasting 1. Forecasts and decisions 2. The object to be forecast 3. Forecast types 4. The forecast horizon 5. The information set 6. Methods and complexity 6.1 The parsimony principle 6.2 The shrinkage principle 4 / 323

5 Statistical Graphics for Forecasting Why graphical analysis is important Simple graphical techniques Elements of graphical style Application: graphing four components of real GNP 5 / 323

6 Modeling and Forecasting Trend Modeling trend Estimating trend models Forecasting trend Selecting forecasting models using the Akaike and Schwarz criteria Application: forecasting retail sales 6 / 323

7 Modeling and Forecasting Seasonality The nature and sources of seasonality Modeling seasonality Forecasting seasonal series Application: forecasting housing starts 7 / 323

8 Characterizing Cycles Covariance stationary time series White noise The lag operator Wold s theorem, the general linear process, and rational distributed lags Estimation and inference for the mean, autocorrelation and partial autocorrelation functions Application: characterizing Canadian employment dynamics 8 / 323

9 Modeling Cycles: MA, AR and ARMA Models Moving-average (MA) models Autoregressive (AR) models Autoregressive moving average (ARMA) models Application: specifying and estimating models for forecasting employment 9 / 323

10 Forecasting Cycles Optimal forecasts Forecasting moving average processes Forecasting infinite-ordered moving averages Making the forecasts operational The chain rule of forecasting Application: forecasting employment 10 / 323

11 Putting it all Together: A Forecasting Model with Trend, Seasonal and Cyclical Components Assembling what we ve learned Application: forecasting liquor sales Recursive estimation procedures for diagnosing and selecting forecasting models 11 / 323

12 Forecasting with Regression Models Conditional forecasting models and scenario analysis Accounting for parameter uncertainty in confidence intervals for conditional forecasts Unconditional forecasting models Distributed lags, polynomial distributed lags, and rational distributed lags Regressions with lagged dependent variables, regressions with ARMA disturbances, and transfer function models Vector autoregressions Predictive causality Impulse-response functions and variance decomposition Application: housing starts and completions 12 / 323

13 Evaluating and Combining Forecasts Evaluating a single forecast Evaluating two or more forecasts: comparing forecast accuracy Forecast encompassing and forecast combination Application: OverSea shipping volume on the Atlantic East trade lane 13 / 323

14 Unit Roots, Stochastic Trends, ARIMA Forecasting Models, and Smoothing Stochastic trends and forecasting Unit roots: estimation and testing Application: modeling and forecasting the yen/dollar exchange rate Smoothing Exchange rates, continued 14 / 323

15 Volatility Measurement, Modeling and Forecasting The basic ARCH process The GARCH process Extensions of ARCH and GARCH models Estimating, forecasting and diagnosing GARCH models Application: stock market volatility 15 / 323

16 Useful Books, Journals and Software Books Statistics review, etc.: Wonnacott, T.H. and Wonnacott, R.J. (1990), Introductory Statistics, Fifth Edition. New York: John Wiley and Sons. Pindyck, R.S. and Rubinfeld, D.L. (1997),Econometric Models and Economic Forecasts, Fourth Edition. New York: McGraw-Hill. Maddala, G.S. (2001), Introduction to Econometrics, Third Edition. New York: Macmillan. Kennedy, P. (1998), A Guide to Econometrics, Fourth Edition. Cambridge, Mass.: MIT Press. 16 / 323

17 Useful Books, Journals and Software cont. Time series analysis: Chatfield, C. (1996), The Analysis of Time Series: An Introduction, Fifth Edition. London: Chapman and Hall. Granger, C.W.J. and Newbold, P. (1986), Forecasting Economic Time Series, Second Edition. Orlando, Florida: Academic Press. Harvey, A.C. (1993), Time Series Models, Second Edition. Cambridge, Mass.: MIT Press. Hamilton, J.D. (1994), Time Series Analysis, Princeton: Princeton University Press. 17 / 323

18 Useful Books, Journals and Software cont. Special insights: Armstrong, J.S. (Ed.) (1999), The Principles of Forecasting. Norwell, Mass.: Kluwer Academic Forecasting. Makridakis, S. and Wheelwright S.C. (1997), Forecasting: Methods and Applications, Third Edition. New York: John Wiley. Bails, D.G. and Peppers, L.C. (1997), Business Fluctuations. Englewood Cliffs: Prentice Hall. Taylor, S. (1996), Modeling Financial Time Series, Second Edition. New York: Wiley. 18 / 323

19 Useful Books, Journals and Software cont. Journals Journal of Forecasting Journal of Business Forecasting Methods and Systems Journal of Business and Economic Statistics Review of Economics and Statistics Journal of Applied Econometrics 19 / 323

20 Useful Books, Journals and Software cont. Software General: Eviews S+ Minitab SAS R Python Many more... Cross-section: Stata Open-ended: Matlab 20 / 323

21 Useful Books, Journals and Software cont. Online Information Resources for Economists: 21 / 323

22 A Brief Review of Probability, Statistics, and Regression for Forecasting Topics Discrete Random Variable Discrete Probability Distribution Continuous Random Variable Probability Density Function Moment Mean, or Expected Value Location, or Central Tendency Variance Dispersion, or Scale Standard Deviation Skewness Asymmetry Kurtosis Leptokurtosis 22 / 323

23 A Brief Review of Probability, Statistics, and Regression for Forecasting Topics cont. Skewness Asymmetry Kurtosis Leptokurtosis Normal, or Gaussian, Distribution Marginal Distribution Joint Distribution Covariance Correlation Conditional Distribution Conditional Moment Conditional Mean Conditional Variance 23 / 323

24 A Brief Review of Probability, Statistics, and Regression for Forecasting cont. Topics cont. Population Distribution Sample Estimator Statistic, or Sample Statistic Sample Mean Sample Variance Sample Standard Deviation Sample Skewness Sample Kurtosis χ 2 Distribution t Distribution F Distribution Jarque-Bera Test 24 / 323

25 Regression as Curve Fitting Least-squares estimation: Fitted values: min β T t = 1 [ y t β 0 β 1 x t ] 2 Residuals: ŷ t = ˆβ 0 + ˆβ 1 x t e t = y t ŷ t 25 / 323

26 Regression as a probabilistic model Simple regression: y t = β 0 + β 1 x t + ε t ε t iid (0, σ 2 ) Multiple regression: y t = β 0 + β 1 x t + β 2 z t + ε t ε t iid (0, σ 2 ) 26 / 323

27 Regression as a probabilistic model cont. Mean dependent var S.D. dependent var 1.49 ȳ = 1 T T t=1 y t SD = Sum squared resid T t=1 (y t ȳ) 2 T 1 SSR = T t=1 e 2 t 27 / 323

28 Regression as a probabilistic model cont. F statistic S.E. of regression 0.99 F = (SSR res SSR) / (k 1) SSR / (T k) s 2 = T t=1 e2 t T k SER = s 2 = T t=1 e2 t T k 28 / 323

29 Regression as a probabilistic model cont. R squared 0.58 R 2 = 1 T t=1 e2 t T t=1 (y t ȳ t ) 2 or R 2 = 1 Adjusted R squared T 1 T T t=1 e2 t T t=1 (y t ȳ t ) 2 R 2 = 1 1 T k 1 T 1 T t=1 e2 t T t=1 (y t ȳ t ) 2 29 / 323

30 Regression as a probabilistic model cont. Akaike info criterion 0.03 Schwarz criterion 0.15 AIC = e ( 2k T ) SIC = T ( k T) T t=1 e2 t T T t=1 e2 t T 30 / 323

31 Regression as a probabilistic model cont. Durbin Watson stat 1.97 y t = β 0 + β 1 x t + ε t v t iid N(0, σ 2 ) ε t = φε t 1 + v t DW = T t=2 (e t e t 1 ) 2 T t=1 e2 t 31 / 323

32 Regression of y on x and z 32 / 323

33 Scatterplot of y versus x 33 / 323

34 Scatterplot of y versus x Regression Line Superimposed 34 / 323

35 Scatterplot of y versus z Regression Line Superimposed 35 / 323

36 Residual Plot Regression of y on x and z 36 / 323

37 Six Considerations Basic to Successful Forecasting 1. The Decision Environment and Loss Function L(e) = e 2 L(e) = e 2. The Forecast Object Event outcome, event timing, time series. 3. The Forecast Statement Point forecast, interval forecast, density forecast, probability forecast 37 / 323

38 Six Considerations Basic to Successful Forecasting cont. 4. The Forecast Horizon h-step ahead forecast h-step-ahead extrapolation forecast 5. The Information Set Ω univariate T = {y T, y T 1,..., y 1 } Ω multivariate T = {y T, x T, y T 1, x T 1,..., y 1, x 1 } 6. Methods and Complexity, the Parsimony Principle, and the Shrinkage Principle Signal vs. noise Smaller is often better Even incorrect restrictions can help 38 / 323

39 Six Considerations Basic to Successful Forecasting cont. Decision Making with Symmetric Loss Demand High Demand Low Build Inventory 0 $10,000 Reduce Inventory $10,000 0 Decision Making with Asymmetric Loss Demand High Demand Low Build Inventory 0 $10,000 Reduce Inventory $20, / 323

40 Six Considerations Basic to Successful Forecasting cont. Forecasting with Symmetric Loss High Actual Sales Low Actual Sales High Forecasted Sales 0 $10,000 Low Forecasted Sales $10,000 0 Forecasting with Asymmetric Loss High Actual Sales Low Actual Sales High Forecasted Sales 0 $10,000 Low Forecasted Sales $20, / 323

41 Quadratic Loss 41 / 323

42 Absolute Loss 42 / 323

43 Asymmetric Loss 43 / 323

44 Forecast Statement 44 / 323

45 Forecast Statement cont. 45 / 323

46 Extrapolation Forecast 46 / 323

47 Extrapolation Forecast cont. 47 / 323

48 Statistical Graphics For Forecasting 1. Why Graphical Analysis is Important Graphics helps us summarize and reveal patterns in data Graphics helps us identify anomalies in data Graphics facilitates and encourages comparison of different pieces of data Graphics enables us to present a huge amount of data in a small space, and it enables us to make huge data sets coherent 2. Simple Graphical Techniques Univariate, multivariate Time series vs. distributional shape Relational graphics 3. Elements of Graphical Style Know your audience, and know your goals. Show the data, and appeal to the viewer. Revise and edit, again and again. 4. Application: Graphing Four Components of Real GNP 48 / 323

49 Anscombe s Quartet (1) (2) (3) (4) x1 y1 x2 y2 x3 y3 x / 323

50 Anscombe s Quartet 50 / 323

51 Anscombe s Quartet Bivariate Scatterplot 51 / 323

52 1-Year Treasury Bond Rates 52 / 323

53 Change in 1-Year Treasury Bond Rates 53 / 323

54 Liquor Sales 54 / 323

55 Histogram and Descriptive Statistics Change in 1-Year Treasury Bond Rates 55 / 323

56 Scatterplot 1-Year vs. 10-year Treasury Bond Rates 56 / 323

57 Scatterplot Matrix 1-, 10-, 20-, and 30-Year Treasury Bond Rates 57 / 323

58 Time Series Plot Aspect Ratio 1: / 323

59 Time Series Plot Banked to 45 Degrees 59 / 323

60 Time Series Plot Aspect Ratio 1: / 323

61 Graph 61 / 323

62 Graph 62 / 323

63 Graph 63 / 323

64 Graph 64 / 323

65 Graph 65 / 323

66 Components of Real GDP (Millions of Current Dollars, Annual 66 / 323

67 Modeling and Forecasting Trend 1. Modeling Trend T t = β 0 + β 1 TIME t T t = β 0 + β 1 TIME t + β 2 TIME 2 t T t = β 0 e β 1TIME t ln(t t ) = ln(β 0 ) + β 1 TIME t 67 / 323

68 Modeling and Forecasting Trend 2. Estimating Trend Models ( ˆβ 0, ˆβ1 ) = arg min β 0,β 1 T t = 1 [ y t β 0 β 1 TIME t ] 2 ( ˆβ 0, ˆβ1, ˆβ2 ) = arg min β 0,β 1,β 2 ( ˆβ 0, ˆβ1 ) = arg min β 0,β 1 T t = 1 T t = 1 [ yt β 0 β 1 TIME t β 2 TIME 2 t [ y t β 0 e β 1TIME t ] 2 ( ˆβ 0, ˆβ1 ) = arg min β 0,β 1 T t = 1 [ ln y t ln β 0 β 1 TIME t ] 2 68 / 323

69 Modeling and Forecasting Trend 3. Forecasting Trend y t = β 0 + β 1 TIME t + ε t y T+h = β 0 + β 1 TIME T+h + ε T+h y T+h,T = β 0 + β 1 TIME T+h ŷ T+h,T = ˆβ 0 + ˆβ 1 TIME T+h 69 / 323

70 Modeling and Forecasting Trend 3. Forecasting Trend cont. y T+h,T ± 1.96σ ŷ T+h,T ± 1.96ˆσ N(y T+h,T, σ 2 ) N(ŷ T+h,T, ˆσ 2 ) 70 / 323

71 Modeling and Forecasting Trend 4. Selecting Forecasting Models MSE = T t=1 e2 t T R 2 = 1 T t=1 e2 t T t=1 (y t ȳ) 2 71 / 323

72 Modeling and Forecasting Trend 4. Selecting Forecasting Models cont. s 2 = T t=1 e2 t T k s 2 = ( T ) T t=1 e2 t T k T R 2 = 1 T t=1 e2 t / T k T t=1 (y t ȳ) 2 / T 1 = 1 s 2 Tt=1 (y t ȳ) 2 / T 72 / 323

73 Modeling and Forecasting Trend 4. Selecting Forecasting Models cont. AIC = e ( 2k T ) SIC = T ( k T) T t=1 e2 t T T t=1 e2 t T Consistency Efficiency 73 / 323

74 Labor Force Participation Rate 74 / 323

75 Increasing and Decreasing Labor Trends 75 / 323

76 Labor Force Participation Rate 76 / 323

77 Linear Trend Female Labor Force Participation Rate 77 / 323

78 Linear Trend Male Labor Force Participation Rate 78 / 323

79 Volume on the New York Stock Exchange 79 / 323

80 Various Shapes of Quadratic Trends 80 / 323

81 Quadratic Trend Volume on the New York Stock Exchange 81 / 323

82 Log Volume on the New York Stock Exchange 82 / 323

83 Various Shapes of Exponential Trends 83 / 323

84 Linear Trend Log Volume on the New York Stock Exchange 84 / 323

85 Exponential Trend Volume on the New York Stock Exchange 85 / 323

86 Degree-of-Freedom Penalties Various Model Selection Criteria 86 / 323

87 Retail Sales 87 / 323

88 Retail Sales Linear Trend Regression 88 / 323

89 Retail Sales Linear Trend Residual Plot 89 / 323

90 Retail Sales Quadratic Trend Regression 90 / 323

91 Retail Sales Quadratic Trend Residual Plot 91 / 323

92 Retail Sales Log Linear Trend Regression 92 / 323

93 Retail Sales Log Linear Trend Residual Plot 93 / 323

94 Retail Sales Exponential Trend Regression 94 / 323

95 Retail Sales Exponential Trend Residual Plot 95 / 323

96 Model Selection Criteria Linear, Quadratic and Exponential Trend Models Linear Trend Quadratic Trend Exponential Trend AIC SIC / 323

97 Retail Sales History January, 1990 December, / 323

98 Retail Sales History January, 1990 December, / 323

99 Retail Sales History January, 1990 December, / 323

100 Retail Sales History January, 1990 December, / 323

101 Modeling and Forecasting Seasonality 1. The Nature and Sources of Seasonality 2. Modeling Seasonality D 1 = (1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0,...) D 2 = (0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0,...) D 3 = (0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0,...) D 4 = (0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1,...) s y t = γ i D it + ε t i=1 s y t = β 1 TIME t + γ i D it i=1 + ε t y t = β 1 TIME t + s γ i D it + i=1 v 1 i=1 δ HD i HDV it + v 2 i=1 δi TD TDV i 101 / 323

102 Modeling and Forecasting Seasonality 3. Forecasting Seasonal Series y t = β 1 TIME t + s γ i D it + i=1 v 1 i=1 δ HD i HDV it + v 2 i=1 δi TD TDV i y T+h = β 1 TIME T+h + s γ i D i,t+h + i=1 v 1 i=1 δ HD i HDV i,t+h + y T+h,T = β 1 TIME T+h + s γ i D i,t+h + i=1 v 1 i=1 δ HD i HDV i,t+h + ŷ T+h,T = ˆβ 1 TIME T+h + sˆγ i D i,t+h + i=1 v 1 i=1 ˆδ HD i HDV i,t+h / 323

103 Modeling and Forecasting Seasonality 3. Forecasting Seasonal Series cont. y T+h,T ± 1.96σ ŷ T+h,T ± 1.96ˆσ N(y T+h,T, σ 2 ) N(ŷ T+h,T, ˆσ 2 ) 103 / 323

104 Gasoline Sales 104 / 323

105 Liquor Sales 105 / 323

106 Durable Goods Sales 106 / 323

107 Housing Starts, January, 1946 November, / 323

108 Housing Starts, January, 1990 November, / 323

109 Housing Starts Regression Results - Seasonal Dummy Variable Model 109 / 323

110 Residual Plot 110 / 323

111 Housing Starts Estimated Seasonal Factors 111 / 323

112 Housing Starts 112 / 323

113 Housing Starts 113 / 323

114 Characterizing Cycles 1. Covariance Stationary Time Series Realization Sample Path Covariance Stationary Ey t = µ t Ey t = µ γ(t, τ) = cov(y t, y t τ ) = E(y t µ)(y t τ µ) γ(t, τ) = γ(τ) ρ(τ) = cov(y t, y t τ ) var(yt ) var(y t τ ) = γ(τ) = γ(τ) γ(0) γ(0) γ(0) 114 / 323

115 1.Characterizing Cycles Cont. corr(x, y) = cov(x, y) σ x σ y ρ(τ) = γ(τ), τ = 0, 1, 2,... γ(0) ρ(τ) = cov(y t, y t τ ) var(yt ) var(y t τ ) = γ(τ) = γ(τ) γ(0) γ(0) γ(0) p(τ) regression of y t on y t 1,..., y t τ 115 / 323

116 2.White Noise y t WN(0, σ 2 ) y t iid (0, σ 2 ) y t iid N(0, σ 2 ) 116 / 323

117 2.White Noise Cont. E(y t ) = 0 var(y t ) = σ 2 E(y t Ω t 1 ) = 0 var(y t Ω t 1 ) = E[(y t E(y t Ω t 1 )) 2 Ω t 1 ] = σ / 323

118 3.The Lag Operator L y t = y t 1 L 2 y t = L(L(y t )) = L(y t 1 ) = y t 2 B(L) = b 0 + b 1 L + b 2 L b m L m L m y t = y t m y t = (1 L)y t = y t y t 1 (1 +.9L +.6L 2 )y t = y t +.9y t 1 +.6y t 2 B(L) = b 0 + b 1 L + b 2 L = b i L i B(L) ε t = b 0 ε t + b 1 ε t 1 + b 2 ε t = b i ε t i i=0 i=0 118 / 323

119 4.Wold s Theorem, the General Linear Process, and Rational Distributed Lags Wold s Theorem Let {y t } be any zero-mean covariance-stationary process. Then: y t = B(L)ε t = b i ε t i i=0 ε t WN(0, σ 2 ) where b 0 = 1 and i=0 b 2 i < 119 / 323

120 The General Linear Process y t = B(L)ε t = ε t b i ε t i i=0 iid WN(0, σ 2 ), where b 0 = 1 and b 2 i < i=0 120 / 323

121 The General Linear Process Cont. E(y t ) = E( b i ε t i ) = i=0 b i Eε t i = i=0 b i 0 = 0 i=0 var(y t ) = var( b i ε t i ) = i=0 b 2 i var(ε t i ) = i=0 b 2 i σ 2 i=0 = σ 2 b i=0 E(y t Ω t 1 ) = E(ε t Ω t 1 ) + b 1 E(ε t 1 Ω t 1 ) + b 2 E(ε t 2 Ω t 1 ) +... = var(y t Ω t 1 ) = E[(y t E(y t Ω t 1 )) 2 Ω t 1 ] = E(ε 2 t Ω t 1 ) = E(ε 2 t ) 121 / 323

122 Rational Distributed Lags B(L) = Θ (L) Φ (L) Θ(L) = Φ(L) = q θ i L i i=0 p φ i L i i=0 B(L) Θ (L) Φ (L) 122 / 323

123 5.Estimation and Inference for the Mean, Auto Correlation and Partial Autocorrelation Functions ˆρ(τ) = 1 T ȳ = 1 T T t=1 y t ρ(τ) = E [(y t µ) (y t τ µ)] E[(y t µ) 2 ] T t = τ+1 [(y t ȳ) (y t τ ȳ)] T t = τ+1 1 T T t=1 (y = [(y t ȳ) ( t ȳ) 2 T t=1 (y t ŷ t = ĉ + ˆβ 1 y t ˆβ τ y t τ ˆp(τ) ˆβ τ ( ˆρ(τ), ˆp(τ) N 0, ) 1 T 123 / 323

124 5.Estimation and Inference for the Mean, Auto Correlation and Partial Autocorrelation Functions Cont. ˆρ(τ) N(0, 1 T ) roottˆρ(τ) N(0, 1) T ˆρ 2 (τ) χ 2 1 Q BP = T Q LB = T (T + 2) m ˆρ 2 (τ) τ = 1 m τ = 1 ( 1 T τ ) ˆρ 2 (τ) 124 / 323

125 A Rigid Cycle Pattern 125 / 323

126 Autocorrelation Function, One-Sided Gradual Damping 126 / 323

127 Autocorrelation Function, Non-Damping 127 / 323

128 Autocorrelation Function, Gradual Damped Oscillation 128 / 323

129 Autocorrelation Function, Sharp Cutoff 129 / 323

130 Realization of White Noise Process 130 / 323

131 Population Autocorrelation Function of White Noise Process 131 / 323

132 Population Partialautocorrelation Function of White Noise Process 132 / 323

133 Canadian Employment Index 133 / 323

134 Canadian Employment Index Correlogram Sample: 1962:1 1993:4 Included observations: 128 Acorr. P. Acorr. Std. Error Ljung-Box p-v v / 323

135 Canadian Employment Index, Sample Autocorrelation and Partial Autocorrelation Functions 135 / 323

136 Modeling Cycles: MA,AR, and ARMA Models The MA(1) Process y t = ε t + θε t 1 = (1 + θl)ε t ε t WN(0, σ 2 ) If invertible: y t = ε t + θy t 1 θ 2 y t 2 + θ 3 y t / 323

137 Modeling Cycles: MA,AR, and ARMA Models Cont. Ey t = E(ε t ) + θe(ε t 1 ) = 0 var(y t ) = var(ε t ) + θ 2 var(ε t 1 ) = σ 2 + θ 2 σ 2 = σ 2 (1 + θ 2 ) E(y t Ω t 1 ) = E((ε t + θε t 1 ) Ω t 1 ) = E(ε t Ω t 1 ) + θe(ε t 1 Ω t 1 ) var(y t Ω t 1 ) = E[(y t E(y t Ω t 1 )) 2 Ω t 1 ] = E(ε 2 t Ω t 1 ) = E(ε 2 t ) 137 / 323

138 The MA(q) Process where y t = ε t + θ 1 ε t θ q ε t q = Θ(L)ε t ε t WN(0, σ 2 ) Θ(L) = 1 + θ 1 L θ q L q 138 / 323

139 The AR(1) Process y t = φy t 1 + ε t ε t WN(0, σ 2 ) If covariance stationary: y t = ε t + φε t 1 + φ 2 ε t / 323

140 Moment Structure E(y t ) = E(ε t + φε t 1 + φ 2 ε t ) = E(ε t ) + φe(ε t 1 ) + φ 2 E(ε t 2 ) +... = 0 var(y t ) = var(ε t + φε t 1 + φ 2 ε t ) = σ 2 + φ 2 σ 2 + φ 4 σ = σ 2 i=0 φ2i = σ 2 1 φ / 323

141 Moment Structure Cont. E(y t y t 1 ) = E((φy t 1 + ε t ) y t 1 ) = φe(y t 1 y t 1 ) + E(ε t y t 1 ) = φy t = φy t 1 var(y t y t 1 ) = var((φy t 1 + ε t ) y t 1 ) = φ 2 var(y t 1 y t 1 ) + var(ε t y t 1 ) = 0 + σ 2 = σ / 323

142 Moment Structure Cont. Autocovariances and autocorrelations: y t = φy t 1 + ε t y t y t τ = φy t 1 y t τ + ε t y t τ For τ 1, (Yule-Walker equation) But γ(τ) = φγ(τ 1). γ(0) = σ 2 1 φ / 323

143 Moment Structure Cont.. Thus and γ(τ) = φ τ σ 2 1 φ 2, τ = 0, 1, 2,... Partial autocorrelations: ρ(τ) = φ τ, τ = 0, 1, 2, / 323

144 The AR(p) Process y t = φ 1 y t 1 + φ 2 y t φ p y t p + ε t ε t WN(0, σ 2 ) 144 / 323

145 The ARMA(1,1) Process MA representation if invertible: y t = φy t 1 + ε t + θε t 1 ε t WN(0, σ 2 ) y t = (1 + θ L) (1 φ L) ε t AR representation of covariance stationary: (1 φ L) (1 + θ L) y t = ε t 145 / 323

146 The ARMA(p,q) Process y t = φ 1 y t φ p y t p + ε t + θ 1 ε t θ q ε t q ε t WN(0, σ 2 ) Φ(L)y t = Θ(L)ε t 146 / 323

147 Realization of Two MA(1) Processes 147 / 323

148 Population Autocorrelation Function MA(1) Process θ = / 323

149 Population Autocorrelation Function MA(1) Process θ = / 323

150 Population Partial Autocorrelation Function MA(1) Process θ = / 323

151 Population Partial Autocorrelation Function MA(1) Process θ = / 323

152 Realization of Two AR(1) Processes 152 / 323

153 Population Autocorrelation Function AR(1) Process φ = / 323

154 Population Autocorrelation Function AR(1) Process φ = / 323

155 Population Partial Autocorrelation Function AR(1) Process φ = / 323

156 Population Partial Autocorrelation Function AR(1) Process φ = / 323

157 Population Autocorrelation Function AR(2) Process with Complex Roots 157 / 323

158 Employment: MA(4) Model Sample: 1962:1 1993:4 Included observations: / 323

159 Employment MA(4) Residual Plot 159 / 323

160 Employment: MA(4) Model Residual Correlogram Sample: 1962:1 1993:4 Included observations: 128 Q statistic probabilities adjusted for 4 ARMA term(s) Acorr. P. Acorr. Std. Error Ljung-Box p-v / 323

161 Employment: MA(4) Model Residual Sample Autocorrelation and Partial Autocorrelation Functions, With Plus or Minus Two Standard Error Bands 161 / 323

162 Employment: AR(2) Model LS // Dependent Variable is CANEMP Sample: 1962:1 1993:4 Included observations: 128 Convergence achieved after 3 iterations Variable Coefficient Std. Error t Statistic Prob. 162 / 323

163 Employment AR(2) Model Residual Plot 163 / 323

164 Employment AIC Values of Various ARMA Models MA Order AR Order / 323

165 Employment SIC Values of Various ARMA Models MA Order AR Order / 323

166 Employment: ARMA(3,1) Model LS // Dependent Variable is CANEMP Sample: 1962:1 1993:4 Included observations: 128 Convergence achieved after 17 iterations Variable Coefficient Std. Error t Statistic Prob. 166 / 323

167 Employment ARMA(3) Model Residual Plot 167 / 323

168 Employment: ARMA(3,1) Model Residual Correlogram Sample: 1962:1 1993:4 Included observations: 128 Q statistic probabilities adjusted for 4 ARMA term(s) Acorr. P. Acorr. Std. Error Ljung-Box p-v / 323

169 Employment: ARMA(3) Model Residual Sample Autocorrelation and Partial Autocorrelation Functions, With Plus or Minus Two Standard Error Bands 169 / 323

170 Forecasting Cycles Ω T = {y T, y T 1, y T 2,...}, Ω T = {ε T, ε T 1, ε T 2,...}. Optimal Point Forecasts for Infinite-Order Moving Averages y t = b i ε t i, i=0 where ε t WN(0, σ 2 ), 170 / 323

171 Forecasting Cycles Cont. b 0 = 1, and. σ 2 i=0 b 2 i < y T+h = ε T+h + b 1 ε T+h b h ε T + b h+1 ε T y T+h,T = b h ε T + b h+1 ε T e T+h,T = (y T+h y T+h,T ) = σ 2 h = σ2 h 1 b 2 i. i=0 h 1 b i ε T+h i, i=0 171 / 323

172 Interval and Density Forecasts y T+h = y T+h,T + e T+h,T. 95% h-step-ahead interval forecast: y T+h,T ± 1.96σ h h-step-ahead density forecast: N(y T+h,T, σ 2 h ) Making the Forecasts Operational The Chain Rule of Forecasting 172 / 323

173 Employment History and Forecast MA(4) Model 173 / 323

174 Employment History and Long-Horizon Forecast MA(4) Model 174 / 323

175 Employment History, Forecast and Realization MA(4) Model 175 / 323

176 Employment History and Forecast AR(2) Model 176 / 323

177 Employment History and Long-Horizon Forecast AR(2) Model 177 / 323

178 Employment History and Very Long-Horizon Forecast AR(2) Model 178 / 323

179 Employment History, Forecast and Realization AR(2) Model 179 / 323

180 Putting it all Together A Forecast Model with Trend, Seasonal and Cyclical Components The full model: y t = T t (θ) + s γ i D it + i=1 v 1 i=1 δ HD i HDV it + v 2 i=1 δ TD i TDV it + ε t Φ(L)ε t = Θ(L)v t Φ(L) = 1 φ 1 L... φ p L p Θ(L) = 1 + θ 1 L θ q L q v t WN(0, σ 2 ). 180 / 323

181 Point Forecasting y T+h = T T+h (θ) + s γ i D i,t+h + i=1 v 1 i=1 δ HD i HDV i,t+h + v 2 i=1 δi TD T y T+h,T = T T+h (θ) + s γ i D i,t+h + i=1 v 1 i=1 δ HD i HDV i,t+h + v 2 δi TD i=1 ŷ T+h,T = T T+h (ˆθ) + sˆγ i D i,t+h + i=1 v 1 i=1 ˆδ HD i HDV i,t+h + v 2 i=1 ˆδ i TD T 181 / 323

182 Interval Forecasting and Density Forecasting Interval Forecasting: ŷ T+h,T ± z α/2ˆσ h e.g.: (95% interval) ŷ T+h,T ± 1.96ˆσ h Density Forecasting: N(ŷ T+h,T, ˆσ 2 h ) 182 / 323

183 Recursive Estimation y t = K β k x kt + ε t k=1 ε t iidn(0, σ 2 ), t = 1,..., T. OLS estimation uses the full sample, t = 1,..., T. Recursive least squares uses an expanding sample. Begin with the first K observations and estimate the model. Then estimate using the first K + 1 observations, and so on. At the end we have a set of recursive parameter estimates: ˆβ k,t, for k = 1,..., K and t = K,..., T. 183 / 323

184 Recursive Residuals At each t, t = K,..., T 1, compute a 1-step forecast, K ŷ t+1,t = ˆβ kt x k,t+1. k=1 The corresponding forecast errors, or recursive residuals, are ê t+1,t = y t+1 ŷ t+1,t. ê t+1,t N(0, σ 2 r t ) where r t > 1 for all t 184 / 323

185 Standardized Recursive Residuals and CUSUM w t+1,t êt+1,t σ r t, t = K,..., T 1. Under the maintained assumptions, CUSUM t w t+1,t iidn(0, 1). t t=k Then w t+1,t, t = K,..., T 1 is just a sum of iid N(0, 1) s. 185 / 323

186 Liquor Sales, / 323

187 Log Liquor Sales, / 323

188 Log Liquor Sales: Quadratic Trend Regression 188 / 323

189 Liquor Sales Quadratic Trend Regression Residual Plot 189 / 323

190 Liquor Sales Quadratic Trend Regression Residual Correlogram Acorr. P. Acorr. Std. Error Ljung-Box / 323

191 Liquor Sales Quadratic Trend Regression Residual Sample Autocorrelation Functions 191 / 323

192 Liquor Sales Quadratic Trend Regression Residual Partial Autocorrelation Functions 192 / 323

193 Log Liquor Sales: Quadratic Trend Regression With Seasonal Dummies and AR(3) Disturbances 193 / 323

194 Liquor Sales Quadratic Trend Regression with Seasonal Dummies Residual Plot 194 / 323

195 Liquor Sales Quadratic Trend Regression with Seasonal Dummies Residual Correlogram Acorr. P. Acorr.Std.Error Ljung-Box p-value / 323

196 Liquor Sales Quadratic Trend Regression with Seasonal Dummies Residual Sample Autocorrelation Functions 196 / 323

197 Liquor Sales Quadratic Trend Regression with Seasonal Dummies Residual Sample Partial Autocorrelation Functions 197 / 323

198 Liquor Sales Quadratic Trend Regression with Seasonal Dummies and AR(3) Disturbances Residual Plot 198 / 323

199 Liquor Sales Quadratic Trend Regression with Seasonal Dummies and AR(3) Disturbances Residual Correlogram Acorr. P. Acorr. Std. Error Ljung-Box p-v / 323

200 Liquor Sales Quadratic Trend Regression with Seasonal Dummies and AR(3) Disturbances Residual Sample Autocorrelation Functions 200 / 323

201 Liquor Sales Quadratic Trend Regression with Seasonal Dummies and AR(3) Disturbances Residual Sample Partial Autocorrelation Functions 201 / 323

202 Liquor Sales Quadratic Trend Regression with Seasonal Dummies and AR(3) Disturbances Residual Histogram and Normality Test 202 / 323

203 Log Liquor Sales History and 12-Month-Ahead Forecast 203 / 323

204 Log Liquor Sales History, 12-Month-Ahead Forecast, and Realization 204 / 323

205 Log Liquor Sales History and 60-Month-Ahead Forecast 205 / 323

206 Log Liquor Sales Long History and 60-Month-Ahead Forecast 206 / 323

207 Liquor Sales Long History and 60-Month-Ahead Forecast 207 / 323

208 Recursive Analysis Constant Parameter Model 208 / 323

209 Recursive Analysis Breaking Parameter Model 209 / 323

210 Log Liquor Sales: Quadratic Trend Regression with Seasonal Dummies and AR(3) Residuals and Two Standard Errors Bands 210 / 323

211 Log Liquor Sales: Quadratic Trend Regression with Seasonal Dummies and AR(3) Disturbances Recursive Parameter Estimates 211 / 323

212 Log Liquor Sales: Quadratic Trend Regression with Seasonal Dummies and AR(3) Disturbances CUMSUM Analysis 212 / 323

213 Forecasting with Regression Models Conditional Forecasting Models and Scenario Analysis Density forecast: y t = β 0 + β 1 x t + ε t ε t N(0, σ 2 ) y T+h,T x T+h = β 0 + β 1 x T+h N(y T+h,T x T+h, σ2 ) Scenario analysis, contingency analysis No forecasting the RHS variables problem 213 / 323

214 Unconditional Forecasting Models y T+h,T = β 0 + β 1 x T+h,T Forecasting the RHS variables problem Could fit a model to x (e.g., an autoregressive model) Preferably, regress y on x t h, x t h 1,... No problem in trend and seasonal models 214 / 323

215 Distributed Lags Start with unconditional forecast model: Generalize to y t = β 0 + δx t 1 + ε t y t = β 0 + distributed lag model lag weights lag distribution N x i=1 δ i x t i + ε t 215 / 323

216 Polynomial Distributed Lags min β 0, δ i [ T y t β 0 t = N x+1 N x i=1 δ i x t i ] 2 subject to δ i = P(i) = a + bi + ci 2, i = 1,..., N x Lag weights constrained to lie on low-order polynomial Additional constraints can be imposed, such as Smooth lag distribution Parsimonious P(N x ) = / 323

217 Rational Distributed Lags Equivalently, y t = A(L) B(L) x t + ε t B(L)y t = A(L)x t + B(L) ε t Lags of x and y included Important to allow for lags of y, one way or another 217 / 323

218 Another way: distributed lag regression with lagged dependent variables y t = β 0 + N y α i y t i + i=1 N x j=1 δ j x t j + ε t Another way: distributed lag regression with ARMA disturbances y t = β 0 + ε t N x i=1 δ i x t i = Θ(L) Φ(L) v t v t WN(0, σ 2 ) + ε t 218 / 323

219 Another Way: The Transfer function Model and Various Special Cases Univariate ARMA Distributed Lag with y t = C(L) D(L) ε t A(, or Lagged Dep. Variables B(L) y t = A(L) x t + ε t y t = A(L) B(L) x t + 1 B(L) ε t C( Distributed Lag with 219 / 323

220 Vector Autoregressions e.g., bivariate VAR(1) Estimation by OLS y 1,t = φ 11 y 1,t 1 + φ 12 y 2,t 1 + ε 1,t y 2,t = φ 21 y 1,t 1 + φ 22 y 2,t 1 + ε 2,t ε 1,t WN(0, σ 2 1) ε 2,t WN(0, σ 2 2) cov(ε 1,t, ε 2,t ) = σ 12 Order selection by information criteria Impulse-response functions, variance decompositions, predictive causality Forecasts via Wold s chain rule 220 / 323

221 Point and Interval Forecast 221 / 323

222 U.S. Housing Starts and Completions, / 323

223 Starts Correlogram Sample: 1968: :12 Included observations: 288 Acorr. P. Acorr. Std. Error Ljung-Box p-v / 323

224 Starts Sample Autocorrelations 224 / 323

225 Starts Sample Partial Autocorrelations 225 / 323

226 Completions Correlogram Completions Correlogram Sample: 1968: :12 Included observations: 288 Acorr. P. Acorr. Std. Error Ljung-Box p-v / 323

227 Completions Sample Autocorrelations 227 / 323

228 Completions Partial Autocorrelations 228 / 323

229 Starts and Completions: Sample Cross Correlations 229 / 323

230 VAR Order Selection with AIC and SIC 230 / 323

231 VAR Starts Equation 231 / 323

232 VAR Start Equation Residual Plot 232 / 323

233 VAR Starts Equation Residual Correlogram Sample: 1968: :12 Included observations: 284 Acorr. P. Acorr. Std. Error Ljung-Box p-v / 323

234 VAR Starts Equation Residual Sample Autocorrelations 234 / 323

235 Var Starts Equation Residual Sample Partial Autocorrelations 235 / 323

236 Evaluating and Combining Forecasts Evaluating a single forecast Process: y t = µ + ε t + b 1 ε t 1 + b 2 ε t ε t WN(0, σ 2 ), h-step-ahead linear least-squares forecast: y t+h,t = µ + b h ε t + b h+1 ε t Corresponding h-step-ahead forecast error: e t+h,t = y t+h y t+h,t = ε t+h + b 1 ε t+h b h 1 ε t+1 with variance σh 2 = h 1 σ2 (1 + b 2 i ) i=1 236 / 323

237 Evaluating and Combining Forecasts So, four key properties of optimal forecasts: a. Optimal forecasts are unbiased b. Optimal forecasts have 1-step-ahead errors that are white noise c. Optimal forecasts have h-step-ahead errors that are at most MA(h-1) d. Optimal forecasts have h-step-ahead errors with variances that are non-decreasing in h and that converge to the unconditional variance of the process 1. All are easily checked. How? 237 / 323

238 Assessing optimality with respect to an information set Unforecastability principle: The errors from good forecasts are not be forecastable! Regression: k 1 e t+h,t = α + α i x it + u t 1. Test whether α 0,..., α k 1 are 0 Important case: i=1 e t+h,t = α + α 1 y t+h,t + u t 1. Test whether (α 0, α 1 ) = (0, 0) Equivalently, = β + β 1 y t+h,t + u t y t+h,t 1. Test whether (β 0, β 1 ) = (0, 1) 238 / 323

239 Evaluating multiple forecasts: comparing forecast accuracy Forecast errors, e t+h,t p t+h,t = y t+h y t+h,t y t+h EV = = y t+h y t+h,t Forecast percent errors, ME = 1 T 1 T T t=1 e t+h,t T (e t+h,t ME) 2 1 MSE = 1 T T t=1 MSPE = 1 T T t=1 RMSE = 1 T T e 2 t+h,t t=1 p 2 t+h,t e 2 t+h,t 239 / 323

240 Forecast encompassing y t+h = β a y a t+h,t + β by b t+h,t + ε t+h,t 1. If (β a, β b ) = (1, 0), model a forecast-encompasses model b 2. If (β a, β b ) = (0, 1), model b forecast-encompasses model a 3. Otherwise, neither model encompasses the other Alternative approach: (y t+h y t = β(y a t+h,t y t) + β b (y b t+h,t y t) + ε t+h,t 1. Useful in I(1) situations 240 / 323

241 Variance-covaiance forecast combination Composite formed from two unbiased forecasts: y c t+h = ωya t+h,t + (1 ω)yb t+h,t e c t+h = ωea t+h,t + (1 ω)eb t+h,t σ 2 c = ω 2 σ 2 aa + (1 ω) 2 σ 2 bb + 2ω(1 ω)σ2 ab ω = σ 2 bb σ2 ab σ 2 bb + σ2 aa 2sigma 2 ab ˆω = ˆσ 2 bb ˆσ2 ab ˆσ 2 bb + ˆσ2 aa 2ˆσ 2 ab 241 / 323

242 Regression-based forecast combination y t+h = β 0 + β 1 y a t+h,t + β 2y b t+h,t + ε t+h,t 1. Equivalent to variance-covariance combination if weights sum to unity and intercept is excluded 2. Easy extension to include more than two forecasts 3. Time-varying combining weights 4. Dynamic combining regressions 5. Shrinkage of combining weights toward equality 6. Nonlinear combining regressions 242 / 323

243 Unit Roots, Stochastic Trends, ARIMA Forecasting Models, and Smoothing 1. Stochastic Trends and Forecasting Φ(L)y t = Θ(L)ε t Φ(L) = Φ (L)(1 L) Φ (L)(1 L)y t = Θ(L)ε t I(0) vs I(1) processes Φ (L) y t = Θ(L)ε t 243 / 323

244 Unit Roots, Stochastic Trends, ARIMA Forecasting Models, and Smoothing Cont. Random Walk y t = y t 1 + ε t ε t WN(0, σ 2 ), Random walk with drift y t = δ + y t 1 + ε t ε t WN(0, σ 2 ), Stochastic trend vs deterministic trend 244 / 323

245 Properties of random walks y t = y t 1 + ε t ε t WN(0, σ 2 ), With time 0 value y 0 : y t = y 0 + t i=1 E(y t ) = y 0 var(y t ) = tσ 2 ε i lim var(y t) = t 245 / 323

246 Random walk with drift Random walk with drift y t = δ + y t 1 + ε t ε t WN(0, σ 2 ), Assuming time 0 value y 0 : y t = tδ + y 0 + t i=1 E(y t ) = y 0 + tδ var(y t ) = tσ 2 lim var(y t) = t ε i 246 / 323

247 ARIMA(p,1,q) model or Φ(L)(1 L)y t = c + Θ(L)ε t (1 L)y t = cφ 1 (L) + Φ 1 (L)Θ(L)ε t where Φ(L) = 1 Φ 1 L... Φ p L p Θ(L) = 1 Θ 1 L... Θ q L q and all the roots of both lag operator polynomials are outside the unit circle. 247 / 323

248 ARIMA(p,d,q) model or Φ(L)(1 L) d y t = c + Θ(L)ε t (1 L) d y t = cφ 1 (L) + Φ 1 (L)Θ(L)ε t where Φ(L) = 1 Φ 1 L... Φ p L p Θ(L) = 1 Θ 1 L... Θ q L q and all the roots of both lag operator polynomials are outside the unit circle. 248 / 323

249 Properties of ARIMA(p,1,q) processes Appropriately made stationary by differencing Shocks have permanent effects Forecasts don t revert to a mean Variance grows without bound as time progresses Interval forecasts widen without bound as horizon grows 249 / 323

250 Random walk example Point forecast Recall that for the AR(1) process, the optimal forecast is y t = φy t 1 + ε t y t WN(0, σ 2 ) y T+h,T Thus in the random walk case, = φ h y T y T+h,T = y T, for all h 250 / 323

251 Random walk example Cont. Interval and density forecasts Recall error associated with optimal AR(1) forecast: e T+h,T = (y T+h y T+h,T ) = ε T+h + φε T+h φ h 1 ε T+1 with variance σ 2 h Thus in the random walk case, e T+h,T = h 1 = σ2 φ 2i i=0 h 1 ε T+h i i=0 σ 2 h = hσ2 h step ahead 95% interval : y T ± 1.96σ h h step ahead density forecast : N(y T, hσ 2 ) 251 / 323

252 Effects of Unit Roots Sample autocorrelation function fails to damp Sample partial autocorrelation function near 1 for τ = 1, and then damps quickly Properties of estimators change e.g., least-squares autoregression with unit roots True process: y t = y t 1 + ε t Estimated model: y t = φy t 1 + ε t Superconsistency:T ( ˆφ LS 1) stabilizes as sample size grows Bias: E(ˆφ LS ) < 1 Offsetting effects of bias and superconsistency 252 / 323

253 Unit Root Tests y t ε t = φy t 1 + ε t iid N(0, σ 2 ) ˆτ = ˆφ 1 s 1 T t=2 y2 t 1 Dickey-Fuller ˆτ distribution Trick regression: y t y t = (φ 1)y t 1 + ε t 253 / 323

254 Allowing for nonzero mean under the alternative Basic model: (y t µ) = φ(y t 1 µ) + ε t which we rewrite as y t = α + φy t 1 + ε t where α = µ(1 φ) α vanishes when φ = 1 (null) α is nevertheless present under the alternative, so we include an intercept in the regression Dickey-Fuller τ ˆ µ distribution 254 / 323

255 Allowing for deterministic linear trend under the alternative Basic model: (y t a btime t ) = φ(y t 1 a b TIME t 1 ) + ε t or y t = α + βtime t + φy t 1 + ε t where α = a(1 φ) + bφ and β = b(1 φ). Under the null hypothesis we have a random walk with drift, y t = b + y t 1 + ε t Under the deterministic-trend alternative hypothesis both the intercept and the trend enter and so are included in the regression. 255 / 323

256 Allowing for higher-order autoregressive dynamics AR(p) process: Rewrite: y t + y t = ρ 1 y t 1 + p φ j y t j j=1 = ε t p ρ j (y t j+1 y t j ) + ε t j=2 where p 2, ρ 1 = p j=1 φ j, and ρ i = p j=1 φ j, i = 2,..., p. Unit root: ρ 1 = 1 (AR(p 1) in first differences) ˆτ distribution holds asymptotically. 256 / 323

257 Allowing for a nonzero mean in the AR(p) case or (y t µ) + p φ j (y t j µ) = ε t j=1 y t = α + ρy t 1 + p ρ j (y t j+1 y t j ) + ε t, j=2 where α = µ(1 + p j=1 φ j), and the other parameters are as above. In the unit root case, the intercept vanishes, because p j=1 φ j = 1. ˆτ µ distribution holds asymptotically. 257 / 323

258 Allowing for trend under the alternative or where and (y t a btime t ) + p φ j (y t j a btime t j ) = ε t j=1 y t = k 1 + k 2 TIME t + ρ 1 y t 1 + k 1 = a(1 + p ρ j (y t j+1 y t j ) + ε t j=2 p φ i ) b i=1 k 2 = btime t (1 + p iφ i i=1 p φ i ) i=1 In the unit root case, k 1 = b p i=1 iφ i and k 2 = 0. ˆτ τ distribution holds asymptotically. 258 / 323

259 General ARMA representations: augmented Dickey-Fuller tests k 1 y t = ρ 1 y t 1 + ρ j (y t j+1 y t j ) + ε t j=2 k 1 y t = α + ρ 1 y t 1 + ρ j (y t j+1 y t j ) + ε t j=2 k 1 y t = k 1 + k 2 TIME t + ρ 1 y t 1 + ρ j (y t j+1 y t j ) + ε t j=2 k-1 augmentation lags have been included ˆτ, ˆτ µ, and ˆτ τ hold asymptotically under the null 259 / 323

260 Simple moving average smoothing 1. Original data: {y t } T t=1 2. Smoothed data: {ȳ t } 3. Two-sided moving average is ȳ t = (2m + 1) 1 m i= m y t i 4. One-sided moving average is ȳ t = (m + 1) 1 m i=0 y t i 5. One-sided weighted moving average is ȳ t = m i=0 w iy t i Must choose smoothing parameter, m 260 / 323

261 Exponential Smoothing Local level model: y t = c 0t + ε t c 0t = c 0,t 1 + η t η t WN(0, σ 2 η) Exponential smoothing can construct the optimal estimate of c 0 - and hence the optimal forecast of any future value of y - on the basis of current and past y What if the model is misspecified? 261 / 323

262 Exponential smoothing algorithm Observed series, {y t } T t=1 Smoothed series, {ȳ t } T t=1 (estimate of the local level) Forecasts, ŷ T +h,t 1. Initialize at t = 1: ȳ 1 = y 1 2. Update: ȳ t = αy t + (1 α)ȳ t 1, t = 2,...T 3. Forecast: ŷ T + h, T = ȳ T Smoothing parameter α [0, 1] 262 / 323

263 Demonstration that the weights are exponential Start: ȳ t = αy t + (1 α)ȳ t 1 Substitute backward for ȳ: ȳ t = t 1 w j y t j j=0 where w j = α(1 α) j Exponential weighting, as claimed Convenient recursive structure 263 / 323

264 Holt-Winters Smoothing y t = c 0t + c 1t TIME t + ε t c 0t = c 0,t 1 + η t c 1t = c 1,t 1 + ν t Local level and slope model Holt-Winters smoothing can construct optimal estimates of c 0 and c 1 - hence the optimal forecast of any future value of y by extrapolating the trend - on the basis of current and past y 264 / 323

265 Holt-Winters smoothing algorithm 1. Initialize at t = 2: ȳ 2 = y 2 F 2 = y 2 y 1 2. Update: ȳ t = αy t + (1 α)(ȳ t 1 + F t 1 ), 0 < α < 1 F t = β(ȳ t ȳ t 1 ) + (1 β)f t 1, 0 < β < 1 t = 3, 4,..., T. 3. Forecast: ŷ T +h,t = ȳ T + hf T ȳ t is the estimated level at time t F t is the estimated slope at time t 265 / 323

266 Random Walk Level and Change 266 / 323

267 Random Walk With Drift Level and Change 267 / 323

268 U.S. Per Capita GNP History and Two Forecasts 268 / 323

269 U.S. Per Capita GNP History, Two Forecasts, and Realization 269 / 323

270 Random Walk, Levels Sample Autocorrelation Function (Top Panel) and Sample Partial Autocorrelation Function (Bottom Panel) 270 / 323

271 Random Walk, First Differences Sample Autocorrelation Function (Top Panel) and Sample Partial Autocorrelation Function (Bottom Panel) 271 / 323

272 Log Yen / Dollar Exchange Rate (Top Panel) and Change in Log Yen / Dollar Exchange Rate (Bottom Panel) 272 / 323

273 Log Yen / Dollar Exchange Rate Sample Autocorrelations (Top Panel) and Sample Partial Autocorrelations (Bottom Panel) 273 / 323

274 Log Yen / Dollar Exchange Rate, First Differences Sample Autocorrelations (Top Panel) and Sample Partial Autocorrelations (Bottom Panel) 274 / 323

275 Log Yen / Dollar Rate, Levels AIC and SIC Values of Various ARMA Models 275 / 323

276 Log Yen / Dollar Exchange Rate Best-Fitting Deterministic-Trend Model 276 / 323

277 Log Yen / Dollar Exchange Rate Best-Fitting Deterministic-Trend Model : Residual Plot 277 / 323

278 Log Yen / Dollar Rate History and Forecast : AR(2) in Levels with Linear Trend 278 / 323

279 Log Yen / Dollar Rate History and Long-Horizon Forecast : AR(2) in Levels with Linear Trend 279 / 323

280 Log Yen / Dollar Rate History, Forecast and Realization : AR(2) in Levels with Linear Trend 280 / 323

281 Log Yen / Dollar Exchange Rate Augmented Dickey-Fuller Unit Root Test 281 / 323

282 Log Yen / Dollar Rate, Changes AIC and SIC Values of Various ARMA Models 282 / 323

283 Log Yen / Dollar Exchange Rate Best-Fitting Stochastic-Trend Model 283 / 323

284 Log Yen / Dollar Exchange Rate Best-Fitting Stochastic-Trend Model : Residual Plot 284 / 323

285 Log Yen / Dollar Rate History and Forecast : AR(1) in Differences with Intercept 285 / 323

286 Log Yen / Dollar Rate History and Long-Horizon Forecast : AR(1) in Differences with Intercept 286 / 323

287 Log Yen / Dollar Rate History, Forecast and Realization : AR(1) in Differences with Intercept 287 / 323

288 Log Yen / Dollar Exchange Rate Holt-Winters Smoothing 288 / 323

289 Log Yen / Dollar Rate History and Forecast : Holt-Winters Smoothing 289 / 323

290 Log Yen / Dollar Rate History and Long-Horizon Forecast : Holt-Winters Smoothing 290 / 323

291 Log Yen / Dollar Rate History, Forecast and Realization : Holt-Winters Smoothing 291 / 323

292 Volatility Measurement, Modeling and Forecasting The main idea: ε t Ω t 1 (0, σ 2 t ) Ω t 1 = {ε t 1, ε t 2,...} We ll look at: Basic Structure and properties Time variation in volatility and prediction-error variance ARMA representation in squares GARCH(1,1) and exponential smoothing Unconditional symmetry and leptokurtosis Convergence to normality under temporal aggregation Estimation and testing 292 / 323

293 Basic Structure and Properties Standard models (e.g., ARMA): Unconditional mean: constant Unconditional variance: constant Conditional mean: varies Conditional variance: constant (unfortunately) k-step-ahead forecast error variance: depends only on k, not on Ω t (again unfortunately) 293 / 323

294 The Basic ARCH Process B(L) = y t b i L i i = 0 = B(L)ε t i = 0 b 2 i < b 0 = 1 ε t Ω t 1 N(0, σ 2 t ) σ 2 t = ω + γ(l)ε 2 t ω > 0 γ(l) = p γ i L i i = 1 γ i 0 for all i γi < / 323

295 The Basic ARCH Process cont. ARCH(1) process: r t Ω t 1 (0, σ 2 t ) σ 2 t = ω + αr 2 t 1 Unconditional mean: E(r t ) = 0 Unconditional variance: E(r t E(r t )) 2 = ω 1 α Conditional mean: E(r t Ω t 1 ) = 0 Conditional variance: E([r t E(r t Ω t 1 )] 2 Ω t 1 ) = ω + αr 2 t / 323

296 The GARCH Process y t = ε t ε t Ω t 1 N(0, σ 2 t ) σ 2 t = ω + α(l)ε 2 t + β(l)σ 2 t α(l) = p α i L i, β(l) = i = 1 q β i L i i = 1 ω > 0, α i 0, β i 0, αi + β i < / 323

297 Time Variation in Volatility and Prediction Error Variance Prediction error variance depends on Ω t 1 e.g. 1-step-ahead prediction error variance is now σ 2 t = ω + αr 2 t 1 + βσ 2 t 1 Conditional variance is a serially correlated RV Again, follows immediately from σ 2 t = ω + αr 2 t 1 + βσ 2 t / 323

298 ARMA Representation in Squares r 2 t has the ARMA(1,1) representation: r 2 t = ω + (α + β)r 2 t 1 βν t 1 + ν t where ν t = r 2 t σ 2 t Important result: The above equation is simply r 2 t = (ω + (α + β)r 2 t 1 βν t 1 ) + ν t = σ 2 t + ν t Thus r 2 t is a noisy indicator of σ 2 t 298 / 323

299 GARCH(1,1) and Exponential Smoothing Exponential smoothing recursion: r 2 t = γr 2 t + (1 γ) r 2 t 1 Back substitution yields: where GARCH(1,1) σ 2 t Back substitution yields: σ 2 t = r 2 t = w j r 2 t j w j = γ(1 γ) j = ω + αr 2 t 1 + βσ 2 t 1 ω 1 β + α β j 1 r 2 t j 299 / 323

300 Unconditional Symmetry and Leptokurtosis Volatility clustering produces unconditional leptokurtosis Conditional symmetry translates into unconditional symmetry Unexpected agreement with the facts! Convergence to Normality under Temporal Aggregation Temporal aggregation of covariance stationary GARCH processes produces convergence to normality. Again, unexpected agreement with the facts! 300 / 323

301 Estimation and Testing Estimation: easy! Maximum Likelihood Estimation L(θ; r 1,..., r T ) = f(r T Ω T 1 ; θ) f(r T 1 Ω T 2 ; θ)... If the conditional densities are Gaussian, f(r t Ω t 1 ; θ) = 1 ( σt 2 (θ) 1/2 exp 1 r 2 ) t 2π 2 σt 2(θ). We can ignore the f (r p,..., r 1 ; θ) term, yielding the likelihood: T p 2 ln(2π) 1 2 T t=p+1 ln σ 2 t (θ) 1 2 T t=p+1 r 2 t σ 2 t (θ). Testing: likelihood ratio tests Graphical diagnostics: Correlogram of squares, correlogram of squared standardized residuals 301 / 323

302 Variations on Volatility Models We will look at: Asymmetric response and the leverage effect Exogenous variables GARCH-M and time-varying risk premia 302 / 323

303 Asymmetric Response and the Leverage Effect: TGARCH and EGARCH Asymmetric response I: TARCH Standard GARCH: TARCH: σ 2 t σ 2 t = ω + αr 2 t 1 + βσ 2 t 1 = ω + αr 2 t 1 + γr 2 t 1D t 1 + βσ 2 t 1 where positive return (good news): α effect on volatility negative return (bad news): α + γ effect on volatility γ 0 : Asymmetric news response γ > 0 : Leverage effect 303 / 323

304 Asymmetric Response and the Leverage Effect Cont. Asymmetric Response II: E-GARCH ln(σt 2 ) = ω + α r t 1 + γ r t 1 + β ln(σ σ t 1) 2 t 1 σ t 1 Log specification ensures that the conditional variance is positive. Volatility driven by both size and sign of shocks Leverage effect when γ < / 323

305 Introducing Exogenous Variables r t Ω t 1 N(0, σ 2 t ) σ 2 t = ω + αr 2 t 1 + βσ 2 t 1 + γx t where: γ is a parameter vector X is a set of positive exogenous variables. 305 / 323

306 Component GARCH Standard GARCH: (σ 2 t ω) = α(r 2 t 1 ω) + β(σ 2 t 1 ω), for constant long-run volatility ω. Component GARCH: (σ 2 t q t ) = α(r 2 t 1 q t 1 ) + β(σ 2 t 1 q t 1 ), for time-varying long-run volatility q t, where q t = ω + ρ(q t 1 ω) + φ(r 2 t 1 σ 2 t 1) 306 / 323

307 Component GARCH Cont. Transitory dynamics governed by α + β Persistent dynamics governed by ρ Equivalent to nonlinearly restricted GARCH(2,2) Exogenous variables and asymmetry can be allowed: (σt 2 q t ) = α(r 2 t 1 q t 1 ) + γ(r 2 t 1 q t 1 )D t 1 + β(σt 1 2 q t 1 ) + θ 307 / 323

308 Regression with GARCH Disturbances y t = x tβ + ε t ε t Ω t 1 N(0, σ 2 t ) 308 / 323

309 GARCH-M and Time-Varying Risk Premia Standard GARCH regression model: y t = x tβ + ε t ε t Ω t 1 N(0, σ 2 t ) GARCH-M model is a special case: y t = x tβ + γσ 2 t + ε t ε t Ω t 1 N(0, σ 2 t ) Time-varying risk premia in excess returns 309 / 323

310 Time Series Plot NYSE Returns 310 / 323

311 Histogram and Related Diagnostic Statistics NYSE Returns 311 / 323

312 Correlogram NYSE Returns 312 / 323

313 Time Series Plot Squared NYSE Returns 313 / 323

314 Correlogram Squared NYSE Returns 314 / 323

315 AR(5) Model Squared NYSE Returns 315 / 323

316 ARCH(5) Model NYSE Returns 316 / 323

317 Correlogram Standardized ARCH(5) Residuals : NYSE Returns 317 / 323

318 GARCH(1,1) Model NYSE Returns 318 / 323

319 Correlogram Standardized GARCH(1,1) Residuals : NYSE Returns 319 / 323

320 Estimated Conditional Standard Deviation GARCH(1,1) Model : NYSE Returns 320 / 323

321 Estimated Conditional Standard Deviation Exponential Smoothing : NYSE Returns 321 / 323

322 Conditional Standard Deviation History and Forecast : GARCH(1,1) Model 322 / 323

323 Conditional Standard Deviation Extended History and Extended Forecast : GARCH(1,1) Model 323 / 323