New Frontiers in Banking Services Emerging Needs and Tailored Products for Untapped Markets_5 doc

In-Sample Diagnostics: Stochastic Chaos Model Structure: 4 Lags, 3 Neurons Diagnostic Linear Model Network Model ∗marginal significance levels network model, appearing in parentheses, exp

0 5 10 15 20 25 30 0

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)

∗marginal significance levels

network model, appearing in parentheses, explains 53% The Quinn information criterion favors, not surprisingly, the network model.The significance test of the Q statistic shows that we cannot reject serialindependence of the regression residuals By all other criteria, the linear

FIGURE 5.3 In-sample errors: stochastic chaos model

specification suffers from serious specification error There is evidence ofserial correlation in squared errors, as well as non-normality, asymmetry,and neglected nonlinearity in the residuals Such indicators would suggestthe use of nonlinear models as alternatives to the linear autoregressivestructure

Figure 5.3 pictures the error paths predicted by the linear and networkmodels 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 curvesgenerally 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 Thesolid path represents the forecast error of the linear model while the dottedcurves are for the network forecast errors This shows the improved per-formance of the network relative to the linear model, in the sense that itserrors are usually closer to zero

Table 5.2 summarizes the out-of-sample statistics These are the rootmean squared error statistics (RMSQ), the Diebold-Mariano statistics forlags zero through four (DM-0 to DM-4), the success ratio for percentage

FIGURE 5.4 Out-of-sample prediction errors: stochastic chaos model

TABLE 5.2 Forecast Tests: Stochastic Chaos Model

(Structure: 5 Lags, 4 Neurons)

∗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 strap error measure A value less than one, of course, represents a gain fornetwork estimation

boot-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 inthe stochastic chaos model are positive The final statistic is the boot-strap ratio, the ratio of the network bootstrap error relative to the linearbootstrap error We see that the network reduces the bootstrap error byalmost 13%.

Clearly, if underlying data were generated by a stochastic process,networks are to be preferred over linear models

The SVJD model is widely used for representing highly volatile assetreturns in emerging markets such as Russia or Brazil during periods

of extreme macroeconomic instability The model combines a stochasticvolatility 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), wepresent this process in continuous time by the following equations:

φ represents the normal distribution The advantage of the continuous time

representation is that the time interval can become arbitrarily smaller andapproximate real time changes

TABLE 5.3 Parameters for SVJD Process

Mean reversion of volatility β 7024

Standard deviation of percentage jump κ 0281

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 thestandard 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 optionpricing 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 significantlyoutperforms a rather poor linear model

One realization of the SVJD process, after first-differencing, appears inFigure 5.5 As in the case of the stochastic chaos model, there are periods

of high volatility followed by more tranquil periods Unlike the stochasticchaos model, however, the periods of tranquility are not perfectly flat

We also notice that the returns in the SVJD model are both positive andnegative

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 errorstructure Although the network multiple correlation coefficient is higherthan that of the linear model, the Hannan-Quinn information criterion

only slightly favors the network model The slight improvement of the R2

statistic does not outweigh by too much the increase in complexity due to

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)

∗marginal significance levels

the larger number of parameters to be estimated While the Granger test does not turn up evidence of neglected nonlinearity, the BDStest does Figure 5.6 gives in-sample errors for the SVJD realizations We

Lee-White-do not see much difference

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 thenetwork model does slightly worse, but not significantly worse, than the lin-ear model, based on the Diebold-Mariano statistic Both models do equallywell in terms of the success ratio for correct sign predictions, with slightlybetter performance by the network model The bootstrap ratio favors thenetwork model, reducing the error percentage of the linear model by slightlymore than 3%

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

FIGURE 5.7 Out-of-sample prediction errors: SVJD model

TABLE 5.5 Forecast Tests: SVJD Model (Structure:

∗marginal significance levels

regime switching model is that the underlying process is linear However,the process follows different regimes when the economy is growing andwhen the economy is shrinking Originally due to Hamilton (1990), it wasapplied to GDP growth rates in the United States

Following Tsay (2002, p 135–137), we simulate the following model resenting the rate of growth of GDP for the U.S economy for two states in

rep-the economy, S1and S2:

φ 2,i x t −i + ε 2,i ε2˜φ(0, σ22) if S = S2 (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:

of nonlinearity in this system The parameters used for generating 500realizations 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 acontraction period than to jump into one” [Tsay (2002), p 137] In oursimulation of the model, the transition probability matrix is called from

a uniform random number generator If, for example, in state S = S1, a

random value of 1 is drawn, the regime will switch to the second state,

S = S2 If a value greater than 118 is drawn, then the regime will remain

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 doesnot 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 anI(1) or nonstationary process However, work by Lumsdaine and Papell(1997) and Cook (2001) has drawn attention to the bias of unit root testswhen 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 mation One realization, for 500 observations, appears in Figure 5.8

TABLE 5.7 In-Sample Diagnostics: MRSModel (Structure: 4 Lags, 3 Neurons)

Diagnostic Linear Model (Network Model)

Estimate

R2 35 (.38)HQIF 3291 (3268)

∗marginal significance levels

reject normality in the distribution of the residuals The BDS test showssome evidence of neglected nonlinearity, but the LWG test does not.Figure 5.9 pictures the error paths generated by the linear and neural net

models While the overall explanatory power or R2 statistic of the neural

Diagnostic Linear Neural Net

∗marginal significance levels

net is slightly higher and the Hannan-Quinn information criterion indicatesthat the network model should be selected, there is not much noticeabledifference in the two paths relative to the actual series

5.4.2 Out-of-Sample Performance

The forecast statistics appear in Table 5.8 We see that the root meansquared error is slightly higher for the network, but the Diebold-Marianostatistics indicate that the difference in the prediction errors is not statis-tically significant The bootstrap error ratio shows that the network modelgives a marginal improvement relative to the linear benchmark

The paths of the linear and network out-of-sample errors appear inFigure 5.10

We see, not surprisingly, that both the linear and network models deliverabout the same accuracy in out-of-sample forecasting Since the MRS isbasically a linear model with a small probability of a switch in the coeffi-cients 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 ity, rather than the return itself, in this series We return to this subject inthe following section

Building on the stochastic volatility and Markov regime switching modelsand following Tsay [(2002), p 133], we use a simple autoregressive modelwith a regime switching mechanism for its volatility, rather than the return

FIGURE 5.10 Out-of-sample prediction errors: MRS model

process itself Specifically, we simulate the following model, similar to theone Tsay estimated as a process representing the daily log returns, includingdividend payments, of IBM stock:2

r t = 043 − 022r t −1 + σ t + u t (5.9)

u t = σ t ε t , ε t ˜φ(0, 1) (5.10)

σ2t = 098u2t −1 + 954σ t2−1 if u t −1 ≤ 0

= 060 + 046u2t −1 + 8854σ t2−1 if u t −1 > 0 (5.11)

where φ(0, 1) is the standard normal or Gaussian density Notice that this

VRS model will have drift in its volatility when the shocks are positive,but not when the shocks are negative However, as Tsay points out, the

2Tsay (2002) omits the GARCH-in-Mean term 5σ t in his specification of the

returns r t.

FIGURE 5.11 First-differenced returns and volatility of the VRS model

model essentially follows an IGARCH (integrated GARCH) when shocksare negative, since the coefficients sum to a value greater than unity.Figure 5.11 pictures the first-differenced series of {r t }, since we could

not reject a unit-root process, as well as the volatility process{σ2

t }.

5.5.1 In-Sample Performance

Table 5.9 gives the linear regression results for the returns We see thatthe in-sample explanatory power of both models is about the same Whilethe tests for serial dependence in the residuals and squared residuals, aswell as for symmetry and normality in the residuals, are not significant,the BDS test for neglected nonlinearity is significant Figure 5.12 picturesthe 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 thetwo models Again, there is not much to recommend the network model

TABLE 5.9 In-Sample Diagnostics: VRS

Model (Structure: 4 Lags, 3 Neurons)

Diagnostic Linear Model (Network Model)

FIGURE 5.12 In-sample errors: VRS model

for return forecasting, but in its favor, it does not perform worse in anynoticeable 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

FIGURE 5.13 Out-of-sample prediction errors: VRS model

TABLE 5.10 Forecast Tests: VRS Model(Structure: 4 Lags, 3 Neurons)

Diagnostic Linear Neural Net

∗marginal significance levels

show improved out-of-sample performance both by the root mean squarederror and the bootstrap criteria It should be noted once more that thereturn process is highly linear by design While the network does not dosignificantly better by the Diebold-Mariano test, it does buy a forecastingimprovement at little cost

5.6 Distorted Long-Memory Model

Originally put forward by Kantz and Schreiber (1997), the distorted memory (DLM) model was recently analyzed for stochastic neural networkapproximation by Lai and Wong (2001) The model has the following form:

long-y t = x2t −1 x t (5.12)

x t = 99x t −1 +  t (5.13)

Following Lai and Wong, we specify σ = 5 and x0 = 5 One realization

appears in Figure 5.14 It pictures a market or economy subject to bubbles.Since we can reject a unit root in this series, we analyze it in levels ratherthan in first differences.3

FIGURE 5.14 Returns of DLM model

3 We note, however, the unit root tests are designed for variables emanating from a linear data-generating process.

Diagnostic Linear Model

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 andFigure 5.15, respectively We see that the in-sample power of the linear

TABLE 5.12 Forecast Tests: DLM Model(Structure: 4 Lags, 3 Neurons)

Diagnostic Linear Neural Net

∗marginal significance levels

model is quite high The network model is slightly higher, and it is favored

by the Hannan-Quinn criterion Except for insignificant tests for serial pendence, however, the diagnostics all indicate lack of serial independence,

inde-in terms of serial correlation of the squared errors, as well as non-normality,asymmetry, and neglected nonlinearity (given by the BDS test result) Sincethe in-sample predictions of the linear and neural network models so closelytrack the actual path of the dependent variable, we cannot differentiate themovements of these variables in Figure 5.15

5.6.2 Out-of-Sample Performance

The relevant out-of-sample statistics appear in Table 5.12 and the tion error paths are in Figure 5.16 We see that the root mean squared errorsare significantly lower, while the success ratio for the sign predictions areperfect for both models The network bootstrap error is also practicallyidentical Thus, the network gives a significantly improved performanceover the linear alternative, on the basis of the Diebold-Mariano statistics,even when the linear alternative gives a very high in-sample fit

Volatility Forecasting

The Black-Sholes (1973) option pricing model is a well-known methodfor calculating arbitrage-free prices for options As Peter Bernstein (1998)points out, this formula was widely in use by practitioners before it wasrecognized through publication in academic journals