ST 810-006 Statistics and Financial Risk Section 1 Time Series Modeling 1 / 37 Time Series Modeling
ST 810-006 Statistics and Financial Risk Time Domain Approach The time domain approach to modeling a time series { Y t } focuses on the conditional distribution of Y t | Y t − 1 , Y t − 2 , . . . . One reason for this focus is that the joint distribution of Y 1 , Y 2 , . . . , Y n can be factorized as f 1: n ( y 1 , y 2 , . . . , y n ) = f 1 ( y 1 ) f 2 | 1 ( y 2 | y 1 ) . . . f n | n − 1:1 ( y n | y n − 1 , y n − 2 , . . . , y 1 ) . So the likelihood function is determined by these conditional distributions. 2 / 37 Time Series Modeling
ST 810-006 Statistics and Financial Risk The conditional distribution may be defined by: the conditional mean, µ t = E( Y t | Y t − 1 = y t − 1 , Y t − 2 = y t − 2 , . . . ) ; the conditional variance, h t = Var( Y t | Y t − 1 = y t − 1 , Y t − 2 = y t − 2 , . . . ) ; the shape of the conditional distribution. 3 / 37 Time Series Modeling
ST 810-006 Statistics and Financial Risk Forecasting The conditional distribution of Y t | Y t − 1 , Y t − 2 , . . . also gives the most complete solution to the forecasting problem: We observe Y t − 1 , Y t − 2 , . . . ; what statements can we make about Y t ? The conditional mean is our best forecast, and the conditional standard deviation measures how far we believe the actual value might differ from the forecast. The conditional shape, usually a fixed distribution such as the normal, allows us to make probability statements about the actual value. 4 / 37 Time Series Modeling
ST 810-006 Statistics and Financial Risk First Wave The first wave of time series methods focused on the conditional mean, µ t . The conditional variance was assumed to be constant. The conditional shape was either normal or unspecified. Need only to specify the form of µ t = µ t ( y t − 1 , y t − 2 , . . . ) . Time-homogeneous: µ t = µ ( y t − 1 , y t − 2 , . . . ) , depends on t only through y t − 1 , y t − 2 , . . . . 5 / 37 Time Series Modeling
ST 810-006 Statistics and Financial Risk Autoregression Simplest form: µ t = a linear function of a small number of values: µ t = φ 1 y t − 1 + · · · + φ p y t − p . Equivalently, and more familiarly, Y t = φ 1 Y t − 1 + · · · + φ p Y t − p + ǫ t , where ǫ t = Y t − µ t satisfies E( ǫ t | Y t − 1 , Y t − 2 , . . . ) = 0 , Var( ǫ t | Y t − 1 , Y t − 2 , . . . ) = h . 6 / 37 Time Series Modeling
ST 810-006 Statistics and Financial Risk Recursion Problem: some time series need large p . Solution: recursion; include also some past values of µ t : µ t = φ 1 y t − 1 + · · · + φ p y t − p + ψ 1 µ t − 1 + · · · + ψ q µ t − q . Equivalently, and more familiarly, Y t = φ 1 Y t − 1 + · · · + φ p Y t − p + ǫ t + θ 1 ǫ t − 1 + · · · + θ q ǫ t − q . This is the ARMA (AutoRegressive Moving Average) model of order ( p , q ). 7 / 37 Time Series Modeling
ST 810-006 Statistics and Financial Risk Two Years of S&P 500: Changing Variance 10 5 0 −5 −10 2008 2009 8 / 37 Time Series Modeling
ST 810-006 Statistics and Financial Risk Second Wave The second wave of time series methods added a focus on the conditional variance, h t . Now need to specify the form of h t = h t ( y t − 1 , y t − 2 , . . . ) . Time-homogeneous: h t = h ( y t − 1 , y t − 2 , . . . ) , depends on t only through y t − 1 , y t − 2 , . . . . 9 / 37 Time Series Modeling
ST 810-006 Statistics and Financial Risk ARCH Simplest form: h t a linear function of a small number of squared ǫ s: h t = ω + α 1 ǫ 2 t − 1 + · · · + α q ǫ 2 t − q . Engle, ARCH (AutoRegressive Conditional Heteroscedasticity): proposed in 1982; Nobel Prize in Economics, 2003 (shared with the late Sir Clive Granger). 10 / 37 Time Series Modeling
ST 810-006 Statistics and Financial Risk Recursion Problem: some time series need large q . Solution: recursion; include also some past values of h t : h t = ω + α 1 ǫ 2 t − 1 + · · · + α q ǫ 2 t − q + β 1 h t − 1 + · · · + β p h t − p . Bollerslev, 1987; GARCH (Generalized ARCH; no Nobel yet, nor yet a Knighthood). Warning! note the reversal of the roles of p and q from the convention of ARMA( p , q ). 11 / 37 Time Series Modeling
ST 810-006 Statistics and Financial Risk GARCH(1,1) The simplest GARCH model has p = 1 , q = 1: h t = ω + αǫ 2 t − 1 + β h t − 1 If α + β < 1, there exists a stationary process with this structure. If α + β = 1, the model is called integrated : IGARCH(1 , 1). 12 / 37 Time Series Modeling
ST 810-006 Statistics and Financial Risk Section 2 Stochastic Volatility 13 / 37 Stochastic Volatility
ST 810-006 Statistics and Financial Risk Stochastic Volatility In a stochastic volatility model, an unobserved ( latent ) process { X t } affects the distribution of the observed process { Y t } , specifically the variance of Y t . Introducing a “second source of variability” is appealing from a modeling perspective. 14 / 37 Stochastic Volatility
ST 810-006 Statistics and Financial Risk Simple Example For instance: { X t } satisfies X t − µ = φ ( X t − 1 − µ ) + ξ t , � � 0 , σ 2 where { ξ t } are i.i.d. N . ξ If | φ | < 1, this is a (stationary) autoregression, but if φ = 1 it is a (non-stationary) random walk. Y t = σ t η t , where σ 2 t = σ 2 ( X t ) is a non-negative function such as σ 2 ( X t ) = exp( X t ) and { η t } are i.i.d. (0 , 1)–typically Gaussian, but also t . 15 / 37 Stochastic Volatility
ST 810-006 Statistics and Financial Risk Conditional Distributions So the conditional distribution of Y t given Y t − 1 , Y t − 2 , . . . and X t , X t − 1 , . . . is simple: 0 , σ 2 ( X t ) � � Y t | Y t − 1 , Y t − 2 , . . . , X t , X t − 1 , · · · ∼ N . But the conditional distribution of Y t given only Y t − 1 , Y t − 2 , . . . is not analytically tractable. In particular, h t ( y t − 1 , y t − 2 , . . . ) = Var( Y t | Y t − 1 = y t − 1 , Y t − 2 = y t − 2 , . . . ) is not a simple function. 16 / 37 Stochastic Volatility
ST 810-006 Statistics and Financial Risk Difficulties Analytic difficulties cause problems in: estimation; forecasting. Computationally intensive methods, e.g.: Particle filtering; Numerical quadrature. 17 / 37 Stochastic Volatility
ST 810-006 Statistics and Financial Risk Section 3 Stochastic Volatility and GARCH 18 / 37 Stochastic Volatility and GARCH
ST 810-006 Statistics and Financial Risk Stochastic Volatility and GARCH Stochastic volatility models have the attraction of an explicit model for the volatility, or variance. Is analytic difficulty the unavoidable cost of that advantage? 19 / 37 Stochastic Volatility and GARCH
ST 810-006 Statistics and Financial Risk A Simple Tractable Model: The Latent Process We construct a latent process by: 2 , τ 2 � ν � X 0 ∼ Γ , 2 and for t > 0 X t = B t X t − 1 , where � ν 2 , 1 � θ B t ∼ β 2 and { B t } are i.i.d. and independent of X 0 . 20 / 37 Stochastic Volatility and GARCH
ST 810-006 Statistics and Financial Risk The Observed Process The observed process is defined for t ≥ 0 by Y t = σ t η t where 1 σ t = √ X t , and { η t } are i.i.d. N (0 , 1) and independent of { X t } . Equivalently: given X u = x u , 0 ≤ u , and Y u = y u , 0 ≤ u < t , Y t ∼ N (0 , σ 2 t ) with the same definition of σ t . 21 / 37 Stochastic Volatility and GARCH
ST 810-006 Statistics and Financial Risk Constraints Since X − 1 � � Var( Y 0 ) = E , 0 we constrain ν > 2 to ensure that X − 1 � � E < ∞ . 0 Requiring X − 1 X − 1 � � � � E = E 0 t for all t > 0 is also convenient, and is met if θ = ν − 2 ν − 1 . 22 / 37 Stochastic Volatility and GARCH
ST 810-006 Statistics and Financial Risk Comparison with Earlier Example This is quite similar to the earlier example, with φ = 1: Write X ∗ t = − log ( X t ). Then X ∗ t = X ∗ t − 1 + ξ ∗ t , where ξ ∗ t = − log ( B t ) . In terms of X ∗ t , σ 2 t = exp( X ∗ t ) . 23 / 37 Stochastic Volatility and GARCH
ST 810-006 Statistics and Financial Risk Differences A key constraint is that now φ = 1, so { X ∗ t } is a (non-stationary) random walk, instead of a (stationary) auto-regression. Also { X ∗ t } is non-Gaussian, where in the earlier example, the latent process was Gaussian. Also { X ∗ t } has a drift, because E( ξ ∗ t ) � = 0 . Of course, we could include a drift in the earlier example. 24 / 37 Stochastic Volatility and GARCH
ST 810-006 Statistics and Financial Risk Matched Simulated Random Walks 0.0 −1.0 −2.0 nu = 10 Gaussian 0 20 40 60 80 100 25 / 37 Stochastic Volatility and GARCH
ST 810-006 Statistics and Financial Risk Matched Simulated Random Walks 0 −2 −4 −6 nu = 5 Gaussian −8 0 20 40 60 80 100 26 / 37 Stochastic Volatility and GARCH
ST 810-006 Statistics and Financial Risk So What? So our model is not very different from (a carefully chosen instance of) the earlier example. So does it have any advantage? Note: the inverse Gamma distribution is the conjugate prior for the variance of the Gaussian distribution. 27 / 37 Stochastic Volatility and GARCH
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