ing factor sensitivities. Factor models enableinvestment managers, quantitative traders and risk managers to model co-movementsamong assets in an efﬁcient way by concentrating the correlation structure into a smallnumber of factors. Unfortunately, the correlation structure is not constant but evolvesin time and so do the factor sensitivities. As a result, the sensitivity estimates have tobe constantly updated in order to keep up with the changes. The ﬁrst method, basedon rolling regressions, is the most popular but also the least accurate. The secondmethod is based on a weighted regression approach which overcomes some of thelimitations of the ﬁrst method by giving more importance to recent observations.Finally, a Kalman ﬁlter-based stochastic parameter regression model is shown tooptimally estimate nonstationary factor exposures.8. “Stochastic Volatility Models: A Survey with Applications to Option Pricing andValue at Risk” by M. Billio and D. Sartore: this chapter analyses the impact on Valueat Risk and option pricing of the presence of stochastic volatility, using data for theFTSE100 stock index. Given the time-varying volatility exhibited by most ﬁnancialdata, there has been a growing interest in time series models of changing variance inrecent years and the literature on stochastic volatility models has expanded greatly:for these models, volatility depends on some unobserved components or a latentstructure. This chapter discusses some of the most important ideas, focusing on thesimplest forms of the techniques and models available. It considers some motivationsfor stochastic volatility models: empirical stylised facts, pricing of contingent assets
Preface
xix
and risk evaluation, and distinguishes between models with continuous and discretevolatility, the latter depending on a hidden Markov chain. A stochastic volatilityestimation program is presented and several applications to option pricing and riskevaluation are discussed.9. “Portfolio Analysis Using Excel” by J. Laws analyses the familiar Markovitz modelusing Excel. This topic is taught on Finance degrees and Master’s programmes allover the world, increasingly through the use of Excel. The author takes time out toexplain how the spreadsheet is set up and how simple short-cuts can make analysis ofthis type of problem quick and straightforward. In the ﬁrst section of the chapter theauthor uses a two-variable example to show how portfolio risk and return vary withthe input weights, then he goes on to show how to determine the optimal weights,in a risk minimisation sense, using both linear expressions and matrix algebra. Inthe second part of the chapter the author extends the number of assets to sevenand illustrates that using matrix algebra within Excel, the Markovitz analysis of ann-asset portfolio is as straightforward as the analysis of a two-asset portfolio. Theauthor takes special care in showing how the correlation matrix can be generatedmost efﬁciently and how within the same framework the optimisation objective canbe modiﬁed without fuss.10. “Applied Volatility and Correlation Modelling Using Excel” by F. Bourgoin. Theoriginality of this chapter lies in the fact that the author manages to implementa range of univariate and multivariate models within the software package, Excel.This is extremely useful as a large proportion of ﬁnance practitioners, students andresearchers are familiar with this package. Using S&P500 return data the authorgenerates one-step-ahead forecasts of volatility using the J.P. Morgan RiskMetricsmodel, the J.P. Morgan RiskMetrics model with optimal decay, a GARCH(1,1) modelwith and without a variance reduction technique and ﬁnally using the GJR model toaccount for asymmetric reaction to news. A comparison of forecasts is made andsome useful insights into the efﬁcacy of the models highlighted. In the second partof the chapter the author uses return data on the DAX30 and CAC40 to model thecorrelation structure using a number of models. As with the univariate approachthis includes the J.P. Morgan RiskMetrics model with and without optimal decay, aGARCH model with and without variance reduction and ﬁnally the so-called “FastGARCH” model of which the author has previously made signiﬁcant contributionsto the literature.11. “Optimal Allocation of Trend-Following Rules: An Application Case of TheoreticalResults” by P. Lequeux uses sophisticated Excel modelling tools to determine whatshould be the optimal weighting of trading rules to maximise the information ratio.The trading rules utilised in the chapter are moving average trading rules ranging inorder from 2 to 117 days and they are applied to a sample of ﬁve currency pairs(USD–JPY, EUR–USD, GBP–USD, USD–CAD and AUD–USD) over the period15/02/1996 to 12/03/2002. The analysis could however be applied to any ﬁnancialasset and any linear trading rule. In the applied example the author attempts todetermine ex-ante what would be the optimal weighting between moving averagesof order 2, 3, 5, 9, 32, 61 and 117 to maximise the delivered information ratio.To assist in understanding, the model has been programmed into a spreadsheet togive the reader the possibility to experiment. The results show that in four currency
xx
Preface
pairs out of ﬁve the optimal weighting procedure is superior, when measured by theinformation ratio, to an equally weighted basket of trading rules.12. “Portfolio Management and Information from Over-the-Counter Currency Options”by J. B. Luı́s: this chapter looks at the informational content of risk-reversals andstrangles derived from OTC at-the-money forward volatilities. Three empirical applications of the literature are presented: one on the EUR/USD, followed by the analysisof implied correlations and the credibility of the Portuguese exchange rate policy during the transition to the EMU, and of the Danish exchange rate policy around theeuro referendum in September 2000. This chapter is supported by the necessary Excelﬁles to allow the reader to validate the author’s results and/or apply the analysis to adifferent dataset.13. “Filling Analysis for Missing Data: An Application to Weather Risk Management” byC. L. Dunis and V. Karalis: this chapter analyses the use of alternative methods whenconfronted with missing data, a common problem when not enough historical data orclean historical data exist, which will typically be the case when trying to develop adecision tool either for a new asset in a given asset class (say a recently issued stockin a given company sector) or for a new asset class as such (for instance weatherderivatives). The application to weather data derives from the fact that most weatherderivatives pricing methodologies rely heavily on clean data. The statistical imputationaccuracy of different ﬁlling methods for missing historical records of temperature datais compared: the Expectation Maximisation (EM) algorithm, the Data Augmentation(DA) algorithm, the Kalman Filter (KF), Neural Networks Regression (NNR) modelsand, ﬁnally, Principal Component Analysis (PCA). Overall, it is found that, for theperiods and the data series concerned, the results of PCA outperformed the othermethodologies in all cases of missing observations analysed.Overall, the objective of Applied Quantitative Methods for Trading and Investment is notto make new contributions to ﬁnance theory and/or ﬁnancial econometrics: more simply,but also more practically, it is to enable its readers to make competent use of advancedmethods for modelling ﬁnancial markets.We hope that, with the numerous ﬁles and software programs made available on theaccompanying CD-Rom, it will constitute a valuable reference textbook for quantitativemarket professionals, academics and ﬁnance graduate students.Many of the authors of chapters contained in this book have an afﬁliation to the Forecasting Financial Markets (FFM) conference which has been held each May since 1993.The editors of the text and several of the authors are members or associates of the Centrefor International Banking, Economics and Finance (CIBEF) at Liverpool John MooresUniversity. Details of both the conference and CIBEF may be found at www.cibef.com.February 2003
1
Applications of Advanced RegressionAnalysis for Trading and Investment∗CHRISTIAN L. DUNIS AND MARK WILLIAMS
ABSTRACTThis chapter examines and analyses the use of regression models in trading and investmentwith an application to foreign exchange (FX) forecasting and trading models. It is notintended as a general survey of all potential applications of regression methods to theﬁeld of quantitative trading and investment, as this would be well beyond the scope ofa single chapter. For instance, time-varying parameter models are not covered here asthey are the focus of another chapter in this book and Neural Network Regression (NNR)models are also covered in yet another chapter.In this chapter, NNR models are benchmarked against some other traditional regressionbased and alternative forecasting techniques to ascertain their potential added value as aforecasting and quantitative trading tool.In addition to evaluating the various models using traditional forecasting accuracymeasures, such as root-mean-squared errors, they are also assessed using ﬁnancial criteria,such as risk-adjusted measures of return.Having constructed a synthetic EUR/USD series for the period up to 4 January 1999, themodels were developed using the same in-sample data, leaving the remainder for out-ofsample forecasting, October 1994 to May 2000, and May 2000 to July 2001, respectively.The out-of-sample period results were tested in terms of forecasting accuracy, and interms of trading performance via a simulated trading strategy. Transaction costs are alsotaken into account.It is concluded that regression models, and in particular NNR models do have the abilityto forecast EUR/USD returns for the period investigated, and add value as a forecastingand quantitative trading tool.
1.1 INTRODUCTIONSince the breakdown of the Bretton Woods system of ﬁxed exchange rates in 1971–1973and the implementation of the ﬂoating exchange rate system, researchers have been motivated to explain the movements of exchange rates. The global FX market is massive with∗
The views expressed herein are those of the authors, and not necessarily those of Girobank.
Applied Quantitative Methods for Trading and Investment. 2003 John Wiley & Sons, Ltd ISBN: 0-470-84885-5
Edited by C.L. Dunis, J. Laws and P. Naı̈m
2
Applied Quantitative Methods for Trading and Investment
an estimated current daily trading volume of USD 1.5 trillion, the largest part concerningspot deals, and is considered deep and very liquid. By currency pairs, the EUR/USD isthe most actively traded.The primary factors affecting exchange rates include economic indicators, such asgrowth, interest rates and inﬂation, and political factors. Psychological factors also play apart given the large amount of speculative dealing in the market. In addition, the movementof several large FX dealers in the same direction can move the market. The interactionof these factors is complex, making FX prediction generally difﬁcult.There is justiﬁable scepticism in the ability to make money by predicting price changesin any given market. This scepticism reﬂects the efﬁcient market hypothesis accordingto which markets fully integrate all of the available information, and prices fully adjustimmediately once new information becomes available. In essence, the markets are fullyefﬁcient, making prediction useless. However, in actual markets the reaction to new information is not necessarily so immediate. It is the existence of market inefﬁciencies thatallows forecasting. However, the FX spot market is generally considered the most efﬁcient,again making prediction difﬁcult.Forecasting exchange rates is vital for fund managers, borrowers, corporate treasurers,and specialised traders. However, the difﬁculties involved are demonstrated by the factthat only three out of every 10 spot foreign exchange dealers make a proﬁt in any givenyear (Carney and Cunningham, 1996).It is often difﬁcult to identify a forecasting model because the underlying laws maynot be clearly understood. In addition, FX time series may display signs of nonlinearitywhich traditional linear forecasting techniques are ill equipped to handle, often producingunsatisfactory results. Researchers confronted with problems of this nature increasinglyresort to techniques that are heuristic and nonlinear. Such techniques include the use ofNNR models.The prediction of FX time series is one of the most challenging problems in forecasting.Our main motivation in this chapter is to determine whether regression models and, amongthese, NNR models can extract any more from the data than traditional techniques. Overthe past few years, NNR models have provided an attractive alternative tool for researchersand analysts, claiming improved performance over traditional techniques. However, theyhave received less attention within ﬁnancial areas than in other ﬁelds.Typically, NNR models are optimised using a mathematical criterion, and subsequentlyanalysed using similar measures. However, statistical measures are often inappropriatefor ﬁnancial applications. Evaluation using ﬁnancial measures may be more appropriate,such as risk-adjusted measures of return. In essence, trading driven by a model with asmall forecast error may not be as proﬁtable as a model selected using ﬁnancial criteria.The motivation for this chapter is to determine the added value, or otherwise, of NNRmodels by benchmarking their results against traditional regression-based and other forecasting techniques. Accordingly, ﬁnancial trading models are developed for the EUR/USDexchange rate, using daily data from 17 October 1994 to 18 May 2000 for in-sampleestimation, leaving the period from 19 May 2000 to 3 July 2001 for out-of-sample forecasting.1 The trading models are evaluated in terms of forecasting accuracy and in termsof trading performance via a simulated trading strategy.1The EUR/USD exchange rate only exists from 4 January 1999: it was retropolated from 17 October 1994 to31 December 1998 and a synthetic EUR/USD series was created for that period using the ﬁxed EUR/DEMconversion rate agreed in 1998, combined with the USD/DEM daily market rate.
Applications of Advanced Regression Analysis
3
Our results clearly show that NNR models do indeed add value to the forecasting process.The chapter is organised as follows. Section 1.2 presents a brief review of some of theresearch in FX markets. Section 1.3 describes the data used, addressing issues such asstationarity. Section 1.4 presents the benchmark models selected and our methodology.Section 1.5 brieﬂy discusses NNR model theory and methodology, raising some issuessurrounding the technique. Section 1.6 describes the out-of-sample forecasting accuracyand trading simulation results. Finally, Section 1.7 provides some concluding remarks.
1.2 LITERATURE REVIEWIt is outside the scope of this chapter to provide an exhaustive survey of all FX applications. However, we present a brief review of some of the material concerning ﬁnancialapplications of NNR models that began to emerge in the late 1980s.Bellgard and Goldschmidt (1999) examined the forecasting accuracy and trading performance of several traditional techniques, including random walk, exponential smoothing,and ARMA models with Recurrent Neural Network (RNN) models.2 The research wasbased on the Australian dollar to US dollar (AUD/USD) exchange rate using half hourlydata during 1996. They conclude that statistical forecasting accuracy measures do nothave a direct bearing on proﬁtability, and FX time series exhibit nonlinear patterns thatare better exploited by neural network models.Tyree and Long (1995) disagree, ﬁnding the random walk model more effective than theNNR models examined. They argue that although price changes are not strictly random,in their case the US dollar to Deutsche Mark (USD/DEM) daily price changes from 1990to 1994, from a forecasting perspective what little structure is actually present may wellbe too negligible to be of any use. They acknowledge that the random walk is unlikelyto be the optimal forecasting technique. However, they do not assess the performance ofthe models ﬁnancially.The USD/DEM daily price changes were also the focus for Refenes and Zaidi (1993).However they use the period 1984 to 1992, and take a different approach. They developeda hybrid system for managing exchange rate strategies. The idea was to use a neuralnetwork model to predict which of a portfolio of strategies is likely to perform bestin the current context. The evaluation was based upon returns, and concludes that thehybrid system is superior to the traditional techniques of moving averages and meanreverting processes.El-Shazly and El-Shazly (1997) examined the one-month forecasting performance ofan NNR model compared with the forward rate of the British pound (GBP), GermanMark (DEM), and Japanese yen (JPY) against a common currency, although they do notstate which, using weekly data from 1988 to 1994. Evaluation was based on forecastingaccuracy and in terms of correctly forecasting the direction of the exchange rate. Essentially, they conclude that neural networks outperformed the forward rate both in terms ofaccuracy and correctness.Similar FX rates are the focus for Gençay (1999). He examined the predictability ofdaily spot exchange rates using four models applied to ﬁve currencies, namely the Frenchfranc (FRF), DEM, JPY, Swiss franc (CHF), and GBP against a common currency from2
A brief discussion of RNN models is presented in Section 1.5.
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Applied Quantitative Methods for Trading and Investment
1973 to 1992. The models include random walk, GARCH(1,1), NNR models and nearestneighbours. The models are evaluated in terms of forecasting accuracy and correctness ofsign. Essentially, he concludes that non-parametric models dominate parametric ones. Ofthe non-parametric models, nearest neighbours dominate NNR models.Yao et al. (1996) also analysed the predictability of the GBP, DEM, JPY, CHF, andAUD against the USD, from 1984 to 1995, but using weekly data. However, they take anARMA model as a benchmark. Correctness of sign and trading performance were usedto evaluate the models. They conclude that NNR models produce a higher correctnessof sign, and consequently produce higher returns, than ARMA models. In addition, theystate that without the use of extensive market data or knowledge, useful predictions canbe made and signiﬁcant paper proﬁt can be achieved.Yao et al. (1997) examine the ability to forecast the daily USD/CHF exchange rateusing data from 1983 to 1995. To evaluate the performance of the NNR model, “buy andhold” and “trend following” strategies were used as benchmarks. Again, the performancewas evaluated through correctness of sign and via a trading simulation. Essentially, compared with the two benchmarks, the NNR model performed better and produced greaterpaper proﬁt.Carney and Cunningham (1996) used four data sets over the period 1979 to 1995to examine the single-step and multi-step prediction of the weekly GBP/USD, dailyGBP/USD, weekly DEM/SEK (Swedish krona) and daily GBP/DEM exchange rates.The neural network models were benchmarked by a naı̈ve forecast and the evaluationwas based on forecasting accuracy. The results were mixed, but concluded that neuralnetwork models are useful techniques that can make sense of complex data that deﬁestraditional analysis.A number of the successful forecasting claims using NNR models have been published. Unfortunately, some of the work suffers from inadequate documentation regardingmethodology, for example El-Shazly and El-Shazly (1997), and Gençay (1999). Thismakes it difﬁcult to both replicate previous work and obtain an accurate assessment ofjust how well NNR modelling techniques perform in comparison to other forecastingtechniques, whether regression-based or not.Notwithstanding, it seems pertinent to evaluate the use of NNR models as an alternativeto traditional forecasting techniques, with the intention to ascertain their potential addedvalue to this speciﬁc application, namely forecasting the EUR/USD exchange rate.
1.3 THE EXCHANGE RATE AND RELATED FINANCIAL DATAThe FX market is perhaps the only market that is open 24 hours a day, seven days aweek. The market opens in Australasia, followed by the Far East, the Middle East andEurope, and ﬁnally America. Upon the close of America, Australasia returns to the marketand begins the next 24-hour cycle. The implication for forecasting applications is that incertain circumstances, because of time-zone differences, researchers should be mindfulwhen considering which data and which subsequent time lags to include.In any time series analysis it is critical that the data used is clean and error free sincethe learning of patterns is totally data-dependent. Also signiﬁcant in the study of FX timeseries forecasting is the rate at which data from the market is sampled. The samplingfrequency depends on the objectives of the researcher and the availability of data. Forexample, intraday time series can be extremely noisy and “a typical off-ﬂoor trader. . .
Applications of Advanced Regression Analysis
5
would most likely use daily data if designing a neural network as a component of anoverall trading system” (Kaastra and Boyd, 1996: 220). For these reasons the time seriesused in this chapter are all daily closing data obtained from a historical database providedby Datastream.The investigation is based on the London daily closing prices for the EUR/USDexchange rate.3 In the absence of an indisputable theory of exchange rate determination, we assumed that the EUR/USD exchange rate could be explained by that rate’srecent evolution, volatility spillovers from other ﬁnancial markets, and macro-economicand monetary policy expectations. With this in mind it seemed reasonable to include,as potential inputs, other leading traded exchange rates, the evolution of important stockand commodity prices, and, as a measure of macro-economic and monetary policy expectations, the evolution of the yield curve. The data retained is presented in Table 1.1along with the relevant Datastream mnemonics, and can be reviewed in Sheet 1 of theDataAppendix.xls Excel spreadsheet.Table 1.1 Data and Datastream mnemonicsNumber
Variable
Mnemonics
123456789101112131415161718192021222324252627
FTSE 100 – PRICE INDEXDAX 30 PERFORMANCE – PRICE INDEXS&P 500 COMPOSITE – PRICE INDEXNIKKEI 225 STOCK AVERAGE – PRICE INDEXFRANCE CAC 40 – PRICE INDEXMILAN MIB 30 – PRICE INDEXDJ EURO STOXX 50 – PRICE INDEXUS EURO-$ 3 MONTH (LDN:FT) – MIDDLE RATEJAPAN EURO-$ 3 MONTH (LDN:FT) – MIDDLE RATEEURO EURO-CURRENCY 3 MTH (LDN:FT) – MIDDLE RATEGERMANY EURO-MARK 3 MTH (LDN:FT) – MIDDLE RATEFRANCE EURO-FRANC 3 MTH (LDN:FT) – MIDDLE RATEUK EURO-£ 3 MONTH (LDN:FT) – MIDDLE RATEITALY EURO-LIRE 3 MTH (LDN:FT) – MIDDLE RATEJAPAN BENCHMARK BOND-RYLD.10 YR (DS) – RED. YIELDECU BENCHMARK BOND 10 YR (DS) ‘DEAD’ – RED. YIELDGERMANY BENCHMARK BOND 10 YR (DS) – RED. YIELDFRANCE BENCHMARK BOND 10 YR (DS) – RED. YIELDUK BENCHMARK BOND 10 YR (DS) – RED. YIELDUS TREAS. BENCHMARK BOND 10 YR (DS) – RED. YIELDITALY BENCHMARK BOND 10 YR (DS) – RED. YIELDJAPANESE YEN TO US $ (WMR) – EXCHANGE RATEUS $ TO UK £ (WMR) – EXCHANGE RATEUS $ TO EURO (WMR) – EXCHANGE RATEBrent Crude-Current Month, fob US $/BBLGOLD BULLION $/TROY OUNCEBridge/CRB Commodity Futures Index – PRICE INDEX
FTSE100DAXINDXS&PCOMPJAPDOWAFRCAC40ITMIB30DJES50IECUS$3MECJAP3MECEUR3MECWGM3MECFFR3MECUK£3MECITL3MJPBRYLDECBRYLDBDBRYLDFRBRYLDUKMBRYDUSBD10YITBRYLDJAPAYE$USDOLLRUSEURSPOILBRENGOLDBLNNYFECRB
3EUR/USD is quoted as the number of USD per euro: for example, a value of 1.2657 is USD1.2657 per euro.The EUR/USD series for the period 1994–1998 was constructed as indicated in footnote 1.
6
Applied Quantitative Methods for Trading and Investment
All the series span the period from 17 October 1994 to 3 July 2001, totalling 1749trading days. The data is divided into two periods: the ﬁrst period runs from 17 October1994 to 18 May 2000 (1459 observations) used for model estimation and is classiﬁedin-sample, while the second period from 19 May 2000 to 3 July 2001 (290 observations) is reserved for out-of-sample forecasting and evaluation. The division amounts toapproximately 17% being retained for out-of-sample purposes.Over the review period there has been an overall appreciation of the USD againstthe euro, as presented in Figure 1.1. The summary statistics of the EUR/USD for theexamined period are presented in Figure 1.2, highlighting a slight skewness and lowkurtosis. The Jarque–Bera statistic conﬁrms that the EUR/USD series is non-normal at the99% conﬁdence interval. Therefore, the indication is that the series requires some type oftransformation. The use of data in levels in the FX market has many problems, “FX pricemovements are generally non-stationary and quite random in nature, and therefore not verysuitable for learning purposes. . . Therefore for most neural network studies and analysisconcerned with the FX market, price inputs are not a desirable set” (Mehta, 1995: 191).To overcome these problems, the EUR/USD series is transformed into rates of return.Given the price level P1 , P2 , . . . , Pt , the rate of return at time t is formed by:PtRt =−1(1.1)Pt−1
EUR/USD
An example of this transformation can be reviewed in Sheet 1 column C of theoos Naı̈ve.xls Excel spreadsheet, and is also presented in Figure 1.5. See also the commentin cell C4 for an explanation of the calculations within this column.An advantage of using a returns series is that it helps in making the time series stationary, a useful statistical property.Formal conﬁrmation that the EUR/USD returns series is stationary is conﬁrmed at the1% signiﬁcance level by both the Augmented Dickey–Fuller (ADF) and Phillips–Perron(PP) test statistics, the results of which are presented in Tables 1.2 and 1.3.The EUR/USD returns series is presented in Figure 1.3. Transformation into returnsoften creates a noisy time series. Formal conﬁrmation through testing the signiﬁcance of1.601.501.401.301.201.101.000.900.800.700.6095
Figure 1.14
96
9798990017 October 1994 to 3 July 2001
01
EUR/USD London daily closing prices (17 October 1994 to 3 July 2001)4
Retropolated series for 17 October 1994 to 31 December 1998.
Applications of Advanced Regression Analysis
7
200Series:USEURSPSample 1 1749Observations 1749150
100
50
MeanMedianMaximumMinimumStd. Dev.SkewnessKurtosis
1.1176971.1174001.3470000.8287000.136898−0.3297112.080124
Jarque–BeraProbability
93.353500.000000
00.9
1.0
1.1
1.2
1.3
Figure 1.2 EUR/USD summary statistics (17 October 1994 to 3 July 2001)Table 1.2ADF test statistic
a
EUR/USD returns ADF test
−18.37959
1%5%10%
critical valueacritical valuecritical value
−3.4371−2.8637−2.5679
MacKinnon critical values for rejection of hypothesis of a unit root.
Augmented Dickey–Fuller Test EquationDependent Variable: D(DR− USEURSP)Method: Least SquaresSample(adjusted): 7 1749Included observations: 1743 after adjusting endpointsVariable
Coefﬁcient
Std. error
t-Statistic
Prob.
DR− USEURSP(−1)D(DR− USEURSP(−1))D(DR− USEURSP(−2))D(DR− USEURSP(−3))D(DR− USEURSP(−4))C
−0.979008−0.002841−0.015731−0.011964−0.014248−0.000212
0.0532660.0476410.0412880.0336840.0240220.000138
−18.37959−0.059636−0.381009−0.355179−0.593095−1.536692
0.00000.95250.70320.72250.55320.1246
R-squaredAdjusted R-squaredS.E. of regressionSum squared resid.Log likelihoodDurbin–Watson stat.
0.4912770.4898120.0057480.0573946521.6971.999488
Mean dependent var.S.D. dependent var.Akaike info. criterionSchwarz criterionF -statisticProb(F -statistic)
1.04E-060.008048−7.476417−7.457610335.48580.000000
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Applied Quantitative Methods for Trading and