Economics 672 Fall 2017 Tauchen. Jump Regression

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1 Economics 672 Fall 2017 Tauchen 1 Main Model In the jump regression setting we have Jump Regression X = ( Z Y where Z is the log of the market index and Y is the log of an asset price. The dynamics are dz t = σ zt dw 1t + Z t dy t = β c t dz c 1t + σ yt dw 2t + β J t Z t + Ỹt where the W s are independent Brownian motions and rest of the notation is as follows: σ zt : Market diffusive volatility Z t : jump part of the market return β c t : beta on diffusive (continuous) moves in Z dz c 1t : the diffusive (continuous) move in Z, dz c 1t = σ zt dw 1t σ yt : idiosyncratic diffusive volatility β J t : beta on jump moves in Z Ỹt : idiosyncratic jumps in the asset return The key aspect to understanding jump regression is that, by definition, the idiosyncratic jumps in Y are independent of the market jumps, and two independent processes never jump at the same time. Thus, Z t Ỹt = 0. t [0, T ]. Much of the consequences of the theory follow from this understanding. This setup fits into the general framework, where ) (1) (2) (3) dx t = c t dw t + X t (4) ( X t = ) Z t βt J Z t + Ỹt Determination of the c t process in terms of the processes of (2) is left as an exercise. We assume X is finitely active, which means that it only jumps a finite number of times on any time interval. It proves useful to separate X into its continuous and discontinuous parts, (5) X t = X c t + X d t. (6)

2 Economics 672, Fall Within this setup we have then that Y tp = β J t p Z tp (7) at the jump times t p of the Z process, t p, p = 1, 2,..., P. Of course at the jump times t r of Ỹ we have Y t = r Ỹt, and Y r t = 0 at all other times t t p t r. The above just defines βt J p, without empirical content. However, when we are willing to assume that there is a constant linear relationship so βt J p = β, constant, then we get Y tp = β Z tp (8) The relationship (8) is powerful, in that it means the relationship between the Z jumps and the Y jumps at the jump times is exact, without error. The strength of (8) will become apparent as move on through jump regression. There are two ways in which it can fail. First, the relationship is always linear but the slope varies over time, where we write Or the relationship is non-linear and we write We will explore these possibilities more later. 2 Detecting the Market Jumps Y tp = β tp Z tp. (9) Y tp = β( Z tp ) Z tp. (10) Under the usual sampling scheme we observe the 2 1 vector of returns n i X, i = 1, 2,..., nt. For the jump regression task, we only retain the data points where a jump in the market index is detected. To this end, let u ni denote the jump threshold for the market return as in our earlier work on separating returns into jump and diffusive parts. In other words, in the data the event n i Z > u ni marks a detected jump in the market return. Since we want to be sure to restrict the jump regression only to points where we are quite confident the market jumped, we set the the threshold quite high, i.e., use a high value of α. We need to be a little bit careful with the sets of jump indices and jump times. The true (unknown) jump times are T = {t [0, nt ] : Z t 0}, (11) The set T is a finite number of times with P elements. For large enough n, the jump times will lie in distinct discrete intervals, i.e., no discrete interval contains more than one actual jump; thus, to each t p T there is a unique index i p such that t p ((i p 1) n, i p n ). Let denote those P indices. Now let I = {i p } p=1,2,...,p, (12) I n = {i p I : Z > u n,ip }, (13)

3 Economics 672, Fall The set I n is the set of indexes of actual jumps that were detected by the jump detection scheme. Necessarily, I n I because some jumps might be missed by the detector. Lastly let I n = {1 i nt : n i Z > u n,i }, (14) In general, I n I because the detector might err in one of two ways: missing an actual jump or by incorrectly declaring an interval to contain a jump. Li et al. (2017) show that for large n both I n I and I n I. That is, the jump detector is correct in the limit. Using advanced methods, one can also show that the jump detector narrows in so fast onto the set of actual jumps that in much of what follows we can act as if I n coincides with I. 3 Inference for the Jump Beta 3.1 When c t is continuous and β c t = β The theory becomes simpler and easier to understand if we assume that the 2 2 variance matrix c t is continuous and that the diffusive beta equals the jump beta across the jump interval. The theory can handle deviations from these presumptions but at the expense of extra theoretical derivations that take us too far afield for now. Recall the classical linear regression model of basic econometrics, y = Xβ + u, Var(u) = Ω where Ω σ 2 I and we have non-spherical disturbances. The OLS estimator is b OLS = (X X) 1 X y and so b OLS β = (X X) 1 X u (15) Var(b OLS ) = (X X) 1 X ΩX (X X) 1, an expression in the Stock-Watson text and many others as well. Of course, we could do GLS, but let s set that aside for now because modern econometrics tends to use b OLS and estimate the covariance matrix above. Very similar expressions arise in jump regression, but the core theory is entirely different. The OLS jump beta estimator is b OLS = p=1 n i p Z Y. Keep in mind that the setup with constant jump beta implies the model Y t = β Z t + Ỹt, Z t Ỹt = 0, t [0, T ].

4 Economics 672, Fall After much tedious (but useful) calculations we can get and expression that is the analogue of (15) for the jump regression setting: p=1 b OLS β = n i p Z E, where E t = Y t βz t is the error process with time-varying error process c e,t. Note that E = Y β Z, and separating E into jump and continuous parts we have Since Y tp β Z tp = 0, then E = Y tp β Z tp + Y c β Z c. E = Y c β Z c. Jacod and Protter (2012) show that for continuous moves like E ip E n 1/2 σ e,p Z p where Z p N(0, 1) and σ e,p is the standard deviation: σ e,p = c e,tp. Using the above, the sampling error in the jump beta estimate is b OLS β p=1 n i p Z n 1/2 σ e,p Z p, we have that The (theoretical form) of asymptotic distribution of the jump regression OLS estimator is thus 1 1/2 n (b OLS β) N(0, V b ), V b = The applied form is σ 2 e,p = 1 (2k n + 1) n p=1 b OLS N β, ( n i p σ e,p 2 n ( Pn ) 2 k n j= k n P σe,p 2 ( P ) 2 ( +jy b OLS +j 1{ n ip+j Z u n,ip+j}. In practice we might set k n = 7 or so depending upon how smooth we think the diffusion variance is across the jump. 1 Although the asymptotic theory is quite different the result is to a large extent parallel to that of the usual textbook regression theory where the theoretical form of the asymptotic variance matrix is Mxx 1 M xx Mxx 1 where M xx = lim n X X/n and M xx = lim n X ΩX/n

5 Economics 672, Fall When c t is discontinuous at the jump times t p, p = 1, 2,..., P This topic will be considered separately in a subsequent lecture. References Jacod, J. and P. Protter (2012). Discretization of Processes. Springer-Verlag. Li, J., V. Todorov, and G. Tauchen (2017). Jump regressions. Econometrica 85,