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5.2 Stochastic Chaos Model 119 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y 0 = .001 y 0 = .5 y 0 = .99 FIGURE 5.2. Stochastic chaos process for different initial conditions TABLE 5.1. In-Sample Diagnostics: Stochastic Chaos Model (Structure: 4 Lags, 3 Neurons) Diagnostic Linear Model (Network Model) Estimate R 2 .29 (.53) HQIF 1534 (1349) L-B ∗ .251 M-L ∗ .0001 E-N ∗ .0000 J-B ∗ .55 L-W-G 1000 B-D-S∗ .0000 ∗ marginal significance levels network model, appearing in parentheses, explains 53%. The Hannan- Quinn information criterion favors, not surprisingly, the network model. The significance test of the Q statistic shows that we cannot reject serial independence of the regression residuals. By all other criteria, the linear 120 5. Estimating and Forecasting with Artificial Data 0 50 100 150 200 250 300 350 400 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 Linear Model Network Model FIGURE 5.3. In-sample errors: stochastic chaos model specification suffers from serious specification error. There is evidence of serial correlation in squared errors, as well as non-normality, asymmetry, and neglected nonlinearity in the residuals. Such indicators would suggest the use of nonlinear models as alternatives to the linear autoregressive structure. Figure 5.3 pictures the error paths predicted by the linear and network models. The linear model errors are given by the solid curve and the net- work errors by dotted paths. As expected, we see that the dotted curves generally are closer to zero. 5.2.2 Out-of-Sample Performance The path of the out-of-sample prediction errors appears in Figure 5.4. The solid path represents the forecast error of the linear model while the dotted curves are for the network forecast errors. This shows the improved per- formance of the network relative to the linear model, in the sense that its errors are usually closer to zero. Table 5.2 summarizes the out-of-sample statistics. These are the root mean squared error statistics (RMSQ), the Diebold-Mariano statistics for lags zero through four (DM-0 to DM-4), the success ratio for percentage 5.2 Stochastic Chaos Model 121 0 10 20 30 40 50 60 70 80 90 100 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 Linear Model Network Model FIGURE 5.4. Out-of-sample prediction errors: stochastic chaos model TABLE 5.2. Forecast Tests: Stochastic Chaos Model (Structure: 5 Lags, 4 Neurons) Diagnostic Linear Neural Net RMSQ .147 .117 DM-0 ∗ — .000 DM-1 ∗ — .004e-5 DM-2 ∗ — .032e-5 DM-3 ∗ — .115e-5 DM-4 ∗ — .209e-5 SR 1 1 B-Ratio — .872 ∗ marginal significance levels of correct sign predictions (SR), and the bootstrap ratio (B-Ratio), which is the ratio of the network bootstrap error statistic to the linear boot- strap error measure. A value less than one, of course, represents a gain for network estimation. 122 5. Estimating and Forecasting with Artificial Data The results show that the root mean squared error statistic of the network model is almost 20% lower than that of the linear model. Not surprisingly, the Diebold-Mariano tests with lags zero through four are all significant. The success ratio for both models is perfect, since all of the returns in the stochastic chaos model are positive. The final statistic is the boot- strap ratio, the ratio of the network bootstrap error relative to the linear bootstrap error. We see that the network reduces the bootstrap error by almost 13%. Clearly, if underlying data were generated by a stochastic process, networks are to be preferred over linear models. 5.3 Stochastic Volatility/Jump Diffusion Model The SVJD model is widely used for representing highly volatile asset returns in emerging markets such as Russia or Brazil during periods of extreme macroeconomic instability. The model combines a stochastic volatility component, which is a time-varying variance of the error term, as well as a jump diffusion component, which is a Poisson jump process. Both the stochastic volatility component and the Poisson jump components directly affect the mean of the asset return process. They are realistic para- metric representations of the way many asset returns behave, particularly in volatile emerging-market economies. Following Bates (1996) and Craine, Lochester, and Syrtveit (1999), we present this process in continuous time by the following equations: dS S =(µ −λ k) · dt + √ V · dZ + k ·dq (5.2) dV =(α −βV ) · dt + σ v √ V · dZ v (5.3) Corr(dZ, dZ v )=ρ (5.4) prob(dq =1)=λ ·dt (5.5) ln(1 + k) ∼ φ(ln[1 + k] − .5κ, κ 2 ) (5.6) where dS/S is the rate of return on an asset, µ is the expected rate of appreciation, λ the annual frequency of jumps, and k is the random per- centage jump conditional on the jump occurring. The variable ln(1 + k)is distributed normally with mean ln[1+ k]−.5κ and variance κ 2 . The symbol φ represents the normal distribution. The advantage of the continuous time representation is that the time interval can become arbitrarily smaller and approximate real time changes. 5.3 Stochastic Volatility/Jump Diffusion Model 123 TABLE 5.3. Parameters for SVJD Process Mean return µ .21 Mean volatility α .0003 Mean reversion of volatility β .7024 Time interval (daily) dt 1/250 Expected jump k .3 Standard deviation of percentage jump κ .0281 Annual frequency of jumps λ 2 Correlation of Weiner processes ρ .6 The instantaneous conditional variance V follows a mean-reverting square root process. The parameter α is the mean of the conditional vari- ance, while β is the mean-reversion coefficient. The coefficient σ v is the variance of the volatility process, while the noise terms dZ and dZ v are the standard continuous-time white noise Weiner processes, with correlation coefficient ρ. Bates (1996) points out that this process has two major advantages. First, it allows systematic volatility risk, and second, it generates an “ana- lytically tractable method” for pricing options without sacrificing accuracy or unnecessary restrictions. This model is especially useful for option pricing in emerging markets. The parameters used to generate the SVJD process appear in Table 5.3. In this model, S t+1 is equal to S t +[S t ·(µ−λk)] ·dt, and for a small value of dt will be unit-root nonstationary. After first-differencing, the model will be driven by the components of dV and k·dq, which are random terms. We should not expect the linear or neural network model to do particularly well. Put another way, we should be suspicious if the network model significantly outperforms a rather poor linear model. One realization of the SVJD process, after first-differencing, appears in Figure 5.5. As in the case of the stochastic chaos model, there are periods of high volatility followed by more tranquil periods. Unlike the stochastic chaos model, however, the periods of tranquility are not perfectly flat. We also notice that the returns in the SVJD model are both positive and negative. 5.3.1 In-Sample Performance Table 5.4 gives the in-sample regression diagnostics of the linear model. Clearly, the linear approach suffers serious specification error in the error structure. Although the network multiple correlation coefficient is higher than that of the linear model, the Hannan-Quinn information criterion only slightly favors the network model. The slight improvement of the R 2 statistic does not outweigh by too much the increase in complexity due to 124 5. Estimating and Forecasting with Artificial Data 0 50 100 150 200 250 300 350 400 450 500 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 FIGURE 5.5. Stochastic volatility/jump diffusion process TABLE 5.4. In-Sample Diagnostics: First-Differenced SVJD Model (Structure: 4 Lags, 3 Neurons) Diagnostic Linear Model (Network Model) Estimate R 2 .42 (.45) HQIF 935 (920) L-B ∗ .783 M-L ∗ .025 E-N ∗ .0008 J-B ∗ 0 L-W-G 11 B-D-S∗ .0000 ∗ marginal significance levels the larger number of parameters to be estimated. While the Lee-White- Granger test does not turn up evidence of neglected nonlinearity, the BDS test does. Figure 5.6 gives in-sample errors for the SVJD realizations. We do not see much difference. 5.4 The Markov Regime Switching Model 125 0 50 100 150 200 250 300 350 400 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 Linear Network FIGURE 5.6. In-sample errors: SVJD model 5.3.2 Out-of-Sample Performance Figure 5.7 pictures the out-of-sample errors of the two models. As expected, we do not see much difference in the two paths. The out-of-sample statistics appearing in Table 5.5 indicate that the network model does slightly worse, but not significantly worse, than the lin- ear model, based on the Diebold-Mariano statistic. Both models do equally well in terms of the success ratio for correct sign predictions, with slightly better performance by the network model. The bootstrap ratio favors the network model, reducing the error percentage of the linear model by slightly more than 3%. 5.4 The Markov Regime Switching Model The Markov regime switching model is widely used in time-series analysis of aggregate macro data such as GDP growth rates. The basic idea of the 126 5. Estimating and Forecasting with Artificial Data 0 10 20 30 40 50 60 70 80 90 100 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 Linear Model Network Model FIGURE 5.7. Out-of-sample prediction errors: SVJD model TABLE 5.5. Forecast Tests: SVJD Model (Structure: 4 Lags, 3 Neurons) Diagnostic Linear Neural Net RMSQ .157 .167 DM-0 ∗ — .81 DM-1 ∗ — .74 DM-2 ∗ — .73 DM-3 ∗ — .71 DM-4 ∗ — .71 SR .646 .656 B-Ratio —– .968 ∗ marginal significance levels regime switching model is that the underlying process is linear. However, the process follows different regimes when the economy is growing and when the economy is shrinking. Originally due to Hamilton (1990), it was applied to GDP growth rates in the United States. 5.4 The Markov Regime Switching Model 127 Following Tsay (2002, p. 135–137), we simulate the following model rep- resenting the rate of growth of GDP for the U.S. economy for two states in the economy, S 1 and S 2 : x t = c c + p  i−1 φ 1,i x t−i + ε 1,i ,ε 1 ˜φ(0,σ 2 1 ), if S = S 1 = c 2 + p  i−1 φ 2,i x t−i + ε 2,i ε 2 ˜φ(0,σ 2 2 )ifS = S 2 (5.7) where φ represents the Gaussian density function. These states have the following transition matrix, P, describing the probability of moving from one state to the next, from time (t −1) to time t: P =  (S 1 t, |S 1 t−1, )(S 1 t, |S 2 t−1, ) (S 2 t, |S 1 t−1, )(S 2 t, |S 2 t−1, )  =  (1 −w 2 ) w 2 w 1 (1 −w 1 )  (5.8) The MRS model is essentially a combination of two linear models with different coefficients, with a jump or switch pushing the data-generating mechanism from one model to the other. So there is only a small degree of nonlinearity in this system. The parameters used for generating 500 realizations of the MRS model appear in Table 5.6. Notice that in the specification of the transition probabilities, as Tsay (2002) points out, “it is more likely for the U.S. GDP to get out of a contraction period than to jump into one” [Tsay (2002), p. 137]. In our simulation of the model, the transition probability matrix is called from a uniform random number generator. If, for example, in state S = S 1 , a random value of .1 is drawn, the regime will switch to the second state, S = S 2 . If a value greater than .118 is drawn, then the regime will remain in the first state, S = S 1 . TABLE 5.6. Parameters for MRS Process Parameter State 1 State 2 c i .909 −.420 φ i,1 .265 .216 φ i,2 .029 .628 φ i,3 −.126 −.073 φ i,4 −.110 −.097 σ i .816 1.01 w i .118 .286 128 5. Estimating and Forecasting with Artificial Data 0 50 100 150 200 250 300 350 400 450 500 −6 −5 −4 −3 −2 −1 0 1 2 3 4 FIGURE 5.8. Markov switching process The process {x t } exhibits periodic regime changes, with different dynam- ics in each regime or state. Since the representative forecasting agent does not know that the true data-generating mechanism for {x t } is a Markov regime switching model, a unit root test for this variable cannot reject an I(1) or nonstationary process. However, work by Lumsdaine and Papell (1997) and Cook (2001) has drawn attention to the bias of unit root tests when structural breaks take place. We thus approximate the process {x t } as a stationary process. The underlying data-generating mechanism is, of course, near linear, so we should not expect great improvement from neural network approxi- mation. One realization, for 500 observations, appears in Figure 5.8. 5.4.1 In-Sample Performance Table 5.7 gives the in-sample regression diagnostics of the linear model. The linear regression model does not do a bad job, up to a point: there is no significant evidence of serial correlation in the residuals, and we cannot [...]... out-of-sample forecasting Since the MRS is basically a linear model with a small probability of a switch in the coefficients of the linear data-generating process, the network simply does about as well as the linear model What will be more interesting is the forecasting of the switches in volatility, rather than the return itself, in this series We return to this subject in the following section 5.5 Volatility... not perform worse in any noticeable way than the linear model While these results do not show overwhelming support for the superiority of network forecasting for the volatility regime switching model, they do 134 5 Estimating and Forecasting with Artificial Data 5 Linear 4 3 Network 2 1 0 −1 −2 −3 0 10 20 30 40 50 60 70 80 90 100 FIGURE 5.13 Out-of-sample prediction errors: VRS model TABLE 5.10 Forecast... 400 FIGURE 5.15 Actual and in- sample predictions: DLM model 5.6.1 In- Sample Performance The in- sample statistics and time paths appear in Table 5.11 and Figure 5.15, respectively We see that the in- sample power of the linear 5.7 Black-Sholes Option Pricing Model: Implied Volatility Forecasting 137 TABLE 5.12 Forecast Tests: DLM Model (Structure: 4 Lags, 3 Neurons) Diagnostic Linear Neural Net RMSQ DM-0∗... X, r, τ Hutchinson, Lo, and Poggio (1994) have extensively explored how well neural network methods (including both radial basis and feedforward networks) approximate call option prices.6 As these authors point out, were we working with time-series observations, it would be necessary to transform the independent variables S, X ,and C into ratios, St /Xt and Ct /Xt 5.7.1 In- Sample Performance Table... used for estimation and forecasting performance against the linear model I invite the reader to continue these experiments with artificial data 6 Times Series: Examples from Industry and Finance This chapter moves the analysis away from artificially generated data to real-world data, to see how well the neural network model performs relative to the linear model We focus on three examples: one from industry,... Model: Implied Volatility Forecasting The Black-Sholes (1973) option pricing model is a well-known method for calculating arbitrage-free prices for options As Peter Bernstein (1998) points out, this formula was widely in use by practitioners before it was recognized through publication in academic journals 138 5 Estimating and Forecasting with Artificial Data 20 15 10 5 0 −5 −10 −15 −20 0 10 20 30 40 50... much better than the linear model, since the underlying data generating processes were almost linear The results of our investigation of these diverse stochastic experiments suggest that the real payoff from neural networks will come from volatility forecasting rather than pure return forecasting in financial markets, as we see in the high payoff from the implied volatility forecasting exercise with the... but 4 See Neft¸i (2000) for a concise treatment of the theory and derivation of optionc pricing models 5 The firm may also enter into a forward contract on foreign exchange markets While preventing loss due to a collapse of sterling, the forward contract also prevents any gain due to an appreciation of sterling 140 5 Estimating and Forecasting with Artificial Data 60 120 CALL 110 40 100 20 90 MARKET PRICE... While the tests for serial dependence in the residuals and squared residuals, as well as for symmetry and normality in the residuals, are not significant, the BDS test for neglected nonlinearity is significant Figure 5.12 pictures the in- sample error paths of the two models 5.5.2 Out-of-Sample Performance Figure 5.13 and Table 5.10 show the out-of-sample performance of the two models Again, there is not... model), forecast onevar svjdmodel new1 .m (for the stochastic volatility jump diffusion model), forecast onevar markovmodel new1 .m (for the Markov regime switching model), and forecast onevar dlm new1 .m (for the distorted long-memory model) 5.8.2 Suggested Exercises The programs in the previous section can be modified to generate alternative series of artificial data, extend the length of the sample, and modify . the returns r t . 132 5. Estimating and Forecasting with Artificial Data 0 50 100 150 200 250 300 350 400 450 50 0 −6 −4 −2 0 2 4 6 8 0 50 100 150 200 250 300 350 400 450 50 0 0 1 2 3 4 5 First-Differenced. much the increase in complexity due to 124 5. Estimating and Forecasting with Artificial Data 0 50 100 150 200 250 300 350 400 450 50 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 FIGURE 5. 5. Stochastic. 400 −30 −20 −10 0 10 20 30 Linear Network FIGURE 5. 15. Actual and in- sample predictions: DLM model 5. 6.1 In- Sample Performance The in- sample statistics and time paths appear in Table 5. 11 and Figure 5. 15, respectively.