Chapter 1:Applications of Advanced Regression Analysis for Trading and Investment
The prediction of Forex time series is one of the most challenging problems in forecasting.
a parametric model in statistics is a family of distributions that can be described using a finite number of parameters.
a model is considered non-parametric if all the parameters are in infinite dimensional parameter space.
Essentially, he concludes that non-parametric models dominate parametric ones. Of the non-parametric models, nearest neighbours dominate NNR models.
the data is obtained from Datastream for the historical forex.
FX price movements are generally non-stationary and quite random in nature, not suitable for learning purposes. To overcome this problem the series is transformed into rates of return given a formula:
Rt=( Pt/pt-1) -1
The advantage of using a return series is that it helps making the time series stationary (use statistical property)
To confirm that a series is stationary we perform the Augmented Dickey-fuller and Phillips-Perron test statistics to show 1% significance level.
ADF and PP from the above mentioned is a test for a unit root in a time series sample.
a unit root is a feature of processes that evolve through time that can cause problems in statistical inference if its not adequately dealt with.
Augmented Dickey-Fuller
Uses a negative number, the more negative it is, the stronger the rejection of the hypothesis that there is a unit root at some level of confidence
Phillips-perron
it is used in time series analysis to test the null hypothesis that a time series is integrated of order 1.
the Phillips–Perron test makes a non-parametric correction to the t-test statistic
The benchmark models used:
Naive strategy:
Assumes that the most recent period change is the best predictor of the future.
MACD Strategy:
a moving average is obtained by finding the mean for a specified set of values and then using it to forecast the next period
ARMA Methodology:
useful to a single stationary series or when economic theory is not useful. a highly refined curve fitting device that uses current and past valuesof the dependent variables to produce accurate short term forecast.
Does not assume any particular pattern in a time series,but uses an iterative approach to identify a possible model from a general class of models.
Tests of adequacy determines a satisfactory model. The general class of ARMA models is for stationary time series, if the series is not stationary an appropriate transformation is required.
Likelihood ratio(LR) is used for redundant or omitted variables
*used to compare the fit of two models, one the null model is a special case of the other alternative model
Ramseys RESET test was used for model miss-specification.
*a general specification test for the linear regression model.it tests whether nonlinear combinations of the fitted values help explain the response variable.
significance of the model is tested via F-Test
*a statistical test in which the test statistics has an F-distribution under the null hypothesis. its used to compare statistical models that have been for to a data set, in order to identity the model that best fits the population from which the data were sampled.
serial correlation LM test(Breusch–Godfrey test) shows further confirmation of the model at 99% confidence interval.
*used to assess the validity of some of the modelling assumptions inherent in applying regression-like models to observed data series
Logit estimation:
logit model belongs to a group of models termed "Classification models"
a multivariate statistical technique used to estimate the probability of an upward or downward movement in a variable.
Neural Network are universal appropriators capable of approximating any continuous function.
The advantage of NNR models over traditional forecasting methods is that the model best adapted to a particular problem cannot be identified. therefore its better to resort to a method that is a generalization of the many models that rely on an a priori model.
The problem of NNR models is because of their Black-box nature. excessive training times, overfitting, large number of parameters required for training are some of the problems.
Therefore deciding on the appropriate network involves much TRIAL AND ERROR.
financial applications time series may well be quasi-random or at least contain noise.
quasi-random: n-tuple to fill n-space uniformly.
Occam's Razor: selecting among competing hypothesis which makes the fewest assumptions.
unnecessary complex models should not be preferred to simpler ones.
the objective is to find a model with the smallest possible complexity and yet still describe the dataset without overfitting
a reasonable strategy in desigining NNR models is to start with one later containing a few hidden nodes and increase the complexity while monitoring the generalisation ability. a crucial factor is determining the number of layers and hidden nodes.
Backpropagation networks are the most common multilayer network and are the most used type in financial time series forecasting (Kaastra and Boyd, 1996).
Because of the pattern matching of NNR models the representation of data is critical for a successful network design.raw data is rarely fed into the network. they are scaled between the upper and lower bounds of the activation function.
another crucial parameter of the network is the learning rate. smaller learning rate slows the learning process. while larger rates cause the rror function to change wildly without continuously improving.
linear cross-correlation analysis: give some indication of which variables to include in a model, or atleast a starting point to the analysis
How to perform Linear-Cross-Correlation?
explained variance:how much variation in that model given a dataset
post training weight analysis helps establish the importance of the explanatory variables because of no standard statistical tests for NNR models. the idea is to find a measure of contribution a given weight has to the overall output of the network. Such analysis includes examination of a Hinton Graph.
Hinton Graph represents graphically the weight matrix within a network
The MAE and RMSE statistics are scale-dependent measures but allow a comparison between the actual and forecast values, the lower the values the better the forecasting accuracy.
When it is more important to evaluate the forecast errors independently of the scale of the variables, the MAPE and Theil-U are used. They are constructed to lie within [0,1], zero indicating a perfect fit.
The study used rates of return. Mehta 1995 suggests the use of first difference as a way to generate data sets for neural networks.
CONCLUSION
in order to use a NNR model we need to process the time series to a stable non moving series and arrange the input of the network based on the activation function available to the network. Everything else about the NNR remains the same, the regular network parameters should be tested via trial and error to find the best number of hidden nodes or hidden layers.
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