Abstract

The debate covering stock return predictability is now shifted towards the investigation of changing patterns of return predictability as suggested by the adaptive market hypothesis (AMH). The present article inspects the varying return predictability pertaining to the equity market in Pakistan under AMH framework.  A nonlinear autoregressive neural network (NARNN) model is employed to investigate the nonlinear dependency of returns over a period of eighteen years. NARNN is a robust and flexible technique that is free from any restrictive assumptions. Under a rolling window framework, the repeating patterns of predictability and unpredictability are observed. This finding confirms the idea of AMH.

Key Words

Adaptive Market Hypothesis, Efficient Market Hypothesis, Artificial Neural Network,

                          Rolling Window Analysis

Introduction

The neoclassical thoughts of finance ignore the behavioral aspects of decision making. It demands homogeneity in participant’s beliefs while in practice the investor’s decisions have psychological effects. The behavioral school criticizes the traditional finance on the basis of psychological factors which they ignored and provided a new interaction between financial activity and psychology Fromlet (2001).  The emerging field of behavioral finance which brings the most contradictory research findings makes the validity of efficient market hypothesis (EMH) uncertain (Kahneman & Tversky (1979), Grossman & Stiglitz (1980), De Bondt & Thaler (1985).  Shlifer (2000) critically reviews the three main assumptions of EMH and argues that in real-world situations the decisions of irrational investors are based on the perception framed by them, the price fluctuations are attributed to underreaction and overreaction of the market participants and arbitrage opportunity is risky and limited. 

Traditionally the level of market efficiency was examined for some predetermined sample period (Campbell et al., 1997). The static approach of efficiency was easy to judge and this approach tells that the weather market is efficient or market is not efficient over the entire sample period. The viewpoint on the traditional focus of absolute, static and short-horizon predictability of EMH is now relocating and the time-varying  degree of efficiency is taking place over them  Campbell et al., (1997), Lo & MacKinlay (2002) and Lo (2008).  

The reconciliation approach of Lo (2004, 2005) brings the unique hypothesis introduced with the adaptive market hypothesis (AMH). The theories of traditional finance and behavioral finance are connected together through this hypothesis. It assumes that the efficiency level in a market remains to change and does not remain static. Arrival of new players, change in level of competition, different economic and political situations  and the  ability of investors  to adapt the new market conditions causes the change in level of efficiency in the market Kim et al., (2011).  Under an AMH framework the profit opportunities do exist on varying time period and market may be efficient on some times and inefficient on some other times Lim et al., (2013) and Urquhart and Mc Groarty (2014). 

Aforementioned theory of AMH is in its formative stage and only limited number of research studies available which empirically investigated its implications.  Underneath the AMH point of view, the nonlinear dependency of return over changing time period is desirable to detect. Only employing linear test to detect dependency is not sufficient as the nonlinear dependencies are still possible to exist if no dependency is suggested by linear techniques   Hiremath & Kamaiah (2010), Alagidede (2011), Caraiani (2012) Urquhart & Hudson (2013). Literature characterized the emerging economies as non-linear in terms of their stock prices movement Todea (2009).  Hinich & Patterson (2005) investigated the nonlinear dependency by using non-overlapped moving time windows on equally divided length of sample periods.

 These studies modeled the pattern of return predictability through traditional statistical techniques. Traditional statistical techniques are criticizes due to the imposed restriction on the data while using it for forecasting and for not being able to correctly model some complex nonlinear relationship. The Artificial neural networks (ANN) having the ability to learn complex relationships between variables, provide an alternative to overcome the shortcomings of conventional techniques to model the return predictability (Oliveira et al 2011, Maqaleh et al 2016 ).  ANNs gained wide attention in the domain of financial time series forecasting due to their nonlinear, nonparametric, self-adaptive and noise-tolerant properties Hussain et al., (2008), Khashei & Bijari (2010).  As compared to other linear and nonlinear statistical techniques, the ANN technique is consider more accurate to model a time series having  more nonlinearities  Lee & Chen (2005). ANNs with their inherent capabilities to identify any nonlinear complex relationship present in time series are being widely used in stock return prediction Dase & Pawar (2010). 

For an informationally inefficient stock market of Pakistan Haque et al (2011), Nisar, S., & Hanif, M. (2012) and with highly volatile economic condition, it is highly relevant to investigate the implications of AMH in Pakistan financial market. Urquhart & McGoarty (2016) in accordance to the proposed framework of AMH discover the direct relationship between fluctuating market situation and the ability to forecast the market. Possibility to predict the expected returns over different time periods relates to the different market conditions Noda (2016). And each and every market is characterized by dynamic market conditions so there is a need to evaluate each market individually based on unique circumstances. For evaluation such dynamic financial market one need to follow a dynamic procedure. There is a need to explore the complex relation of return predictability over varying time periods and different market conditions in Pakistan financial market. Therefore to explore the validity of AMH, in explaining the stock behavior over a long interval, the present study is conducted. By having the ability to model complex nonlinear relationship between variables applying ANN can we able to assess the implications of AMH in Pakistan financial market.  




Data 

The monthly return data of bench market index of Pakistan Stock Exchange (PSX) that is KSE-100 is taken as data sample for this study.  The sample period is comprises on data from January 2000 to December 2018. The sample period is taken due to the reasons because to make the study up to date by including current data in the article and this large data set will enable us to produce workable results by ensuring adequate data for rolling window analysis. It will also cover the most of the major events of Pakistan stock exchange and other economic and political events relevant to stock market functioning.  

Methodology

The methodology of the study is comprises on two stages. The first stage involves the steps to choose an optimal neural network model. At this stage the parameters for the optimal model are selected. Selection of parameters is done by considering performance measure errors. All the parameters which are helpful in minimising the error term is consider for the optimal model.

 At second stage this optimal neural network model will be used for modelling return predictability pattern.  A nonlinear auto regressive neural network model is used under rolling window frame work. The rolling window analysis enables us to examine the pattern of predictability over the changing time period without any delay in time (Charles et al., 2012). By plotting the performance measure errors through rolling window analysis on time will give insights to the changing degree of predictability. These stages are discussed in detail in the next section.




First stage: steps for optimal selection of ANN Model 

For designing an optimal ANN model we follow the eight step process presented by Kaastra and Boyd (1996).   




Step I: Variable Selection 

This step involves the selection of dependent and independent variables. In this study dependent variable is the natural logarithm return of KSE-100 index. Return is calculated according to the annual continuous compounded rate for a mentioned time period Nisar & Hanif (2012). Simply, the index value at time ‘t’ is divided by index value at time ‘t-1’ and then the log of this value represents the return. This may be represented in the form of equation as follows:

Rt = ln( Pt / Pt-1 ) ……………….. (1)

Where, 

Rt = Return for time‘t’

Ln = Natural log

Pt = Current price for time‘t’

Pt-1 = previous price of time‘t-1’

A lagged series is an independent variable under univariate time series analysis and this independent variable is an input variable for ANN Model. The selection of best lagged series is based on the low performance measure produced by using that lagged series.  Frequency of data is monthly Index values which are being used in the study. 

Step 2: Data Collection

The required data is taken from the web site of Pakistan Stock Exchange; www.psx.com.pk. And the information regarding relevant events is collected from 

 

Step 3: Data Pre-Processing

Variable are not fed to ANN in raw form, but they need to be analysed and transformed before using. Logarithmic transformation of the variables is done to minimise the noise and flatten the distribution of the data to assist ANN in learning the relevant patterns.

 

Step 4: Training, Testing, and Validation Sets

Data set is needed to divide into three distinctive sets. The data split ratio followed by the study is presented in table

Table 1.

> > > > > > >

Layers width="138"> Nodes width="124"> Training set width="116"> Testing set width="94"> Validation set

1,2 width="138"> 2, 3,4,…., 50 width="124"> 60% width="116"> 20% width="94"> 20%

1,2 width="138"> 2, 3,4,…., 50 width="124"> 65% width="116"> 15% width="94"> 20%

1,2 width="138"> 2, 3,4,…., 50 width="124"> 70% width="116"> 15% width="94"> 15%

1,2 width="138"> 2,3,4,…., 50 width="124"> 75% width="116"> 10% width="94"> 15%

1,2 width="138"> 2, 3,4,…., 50 width="124"> 80% width="116"> 10% width="94"> 10%

1,2 width="138"> 2, 3,4,…., 50 width="124"> 85% width="116"> 05% width="94"> 10%

1,2 width="138"> 2, 3,4,…., 50 width="124"> 90% width="116"> 05% width="94"> 05%

From the above possible combination of data split ratio, the best combination of data split ratio is selected after trial.

 

Step 5: Neural Network Paradigms

This step involves the selection of parameters involved in making neural network architecture. Nonlinear autoregressive feed-forward artificial neural network having multi-layer perceptions in Matlab R2018a, which this study is going to employ as (Adebiyi et al., 2014; Zhang 2003) with little modifications.  Equation 1 explains the relationship among the output and the inputs using linear and nonlinear activation functions, of a multilayer fed-forward neural network.

where j (j = 0; 1; 2;:::;h) and ij (i = 0; 1; 2;:::;p; j = 1; 2;:::;h) are the model parameters often called the connection weights; p is the number of input nodes and h is the number of hidden nodes. F and G are hidden and output layer activation functions, respectively. For hidden and out-put layers the sigmoid and linear functions are given in Eqs. 2  and 3, respectively Zhang (2003),  Khashei & Bijari (2011).

F(x) = …………………… (2)

G(x) = x ………………………. (3)  

In financial time series forecasting the minimum number of hidden layer is recommended for optimal working, because by increasing the hidden layers it increases its complexity. In time series forecasting mostly a feed forward network with single hidden layer is used for modelling the data set Zangh (2003). From one and two layers the best performing layer will be selected.

Under trial and error method backward approach is followed to select hidden nodes Panchal and Panchal (2014). Starting from 50 nodes and moving backward for selecting best performing nod. From 50 nodes the best performing node is selected for optimal ANN model.

 

Step 6: Evaluation criteria

Three statistical performance evaluation measures Mean Absolute Error (MAE), Mean squared error (MSE) and Root mean square error (RMSE) are being used by this study  (Shamisi, et al., 2013).. These performance error statistics can be defined as follows:

MAE=  ………………………............ (4)           

MSE=  ……………………………..  (5)                  

RMSE =  …………………. (6)

Where R_t and are, respectively, the actual returns and forecasted returns, and N is the size of the testing dataset.  Forecasting errors needs to be less for forecasting accuracies for financial time series. The point at which the minimum MSE is reported will be selected as best lag point for each country.

 

Step 7: Neural Network Training

In this study feed-forward neural networks is trained using Levenberg-Marquardt   learning algorithm as used by Shayea (2017). Matlab software is used for training the network. Matlab use train lm function to updates weight and bias values according to Levenberg-Marquardt optimization. This function uses 100 maximum numbers of epochs to train the network. The output is determined through a non-linear activation function. The activation function is usually a logistic function that transforms the output to a number that is between 0 &1.

 

Step 8: Implementation

Nonlinear autoregressive neural network is implemented under the above selected parameters. This experiment helps us to select optimal model for further analyses. 

Second stage: Using Optimal Architecture under Rolling Window Analysis

After construction of best combination of hidden layers (x), nodes (y) and lags (z) the point which reports minimum errors the next step is to constitute rolling window. Rolling window will be using the estimation window of 36 months with one month rolling. The equation for feed forward artificial neural network for rolling window is:

 ……………......... (7)

The training data window should be optimal because, “If the minimum training window is too long the model will be slow to respond to state changes., If the training window is too short, the model may overreact to noise” (Arlot & Celisse 2010). 

Figure 1

Schematic Diagram of Rolling Window Analysis



The optimal length of 36 months moving window is created and tested. The use of moving window approach enables us to capture the certain trends over moving time periods. 

The movement of error term in response to market conditions is analysed according to the following rule. If the error term are high in some market the prediction level is low and market is little efficient in that position as compare to the points where the error term are low. These levels of predictability are due to the prevailing market condition. So different scenarios are given in the table which will help to explain the KSE 100 movement towards efficiency or inefficiency or it is providing a better explanation of adaptive market hypothesis by showing cyclical efficiency.  



Figure 2

Movement of error term and levels of predictability






Empirical Results and Discussion

Descriptive statistics of the monthly return of the KSE-100 index show that the average monthly return at KSE-100 is 0.013879 with a standard deviation of 0.078258. The monthly return series is negatively skewed with a skewness value of -1.656 and a high value of kurtosis 13.747 which shows that distribution is not normal. The Jarque-Bera statistics for monthly return is 1175.235 and P- value is 0.0000 which shows that the daily return series does not follows normal distribution.

Table 3. Descriptive Statistics of KSE-100 Index Monthly Return

> > > > >

Statistics width="269" valign="top"> KSE-100 Monthly return

Mean width="269" valign="top"> 0.013879

Standard deviation width="269" valign="top"> 0.078258

Skewness width="269" valign="top"> -1.656

Kurtosis width="269" valign="top"> 13.747

Jarque–Bera width="269" valign="top"> 1175.235*

*denote significant at 1 % level

Results

Results of the study are divided in two stages as according to the described methodology. At the first stage the results for the selection of the optimal model for artificial neural network are presented. Nonlinear autoregressive neural network is run at different lags using fifty hidden neurons and at different combinations of data split ratio as of training, testing and validation set.  

Figures 3-6 reports the results of root mean square error at The arrival50 nodes for first four lags while using different combinations of data split ratios for training; testing and validation. A best combination of data split ratio which reports the lowest root mean square error, for each lag is selected from this exercise. .08,.15,.05, .70, .15, .15, .90,.05,.05 and .70, 10, 20 are the data split ratios which are showing lowest performance measure for the first, second, third and fourth lag respectively.    

Figure 3-6

RMSE-1, RMSE-2, RMSE-3 and RMSE-4 at different data split ratio



Above figures also provides valuable information about the number of nodes in the optimal model that by increasing the number of nodes in all combinations of data split ratios the value of performance error increases. The selection of nodes for the best combination of data split ratios is determined through the table 5-8. Table 5-8 reports the results of performance measure error at fifty nodes. The nodes at which the root mean square error is lowest are node 3, node 1, node 2 and node 4 for the first, second, third and fourth lag respectively.  So by keeping only lower number of hidden neuron we can have an optimal model for rolling window nonlinear auto regressive neural network in next stage.

Table 4. R MSE-1at node 1 to node 50

> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > >

Comb/H.N width="59" nowrap=""> N1 width="59" nowrap=""> N2 width="59" nowrap=""> N3 width="59" nowrap=""> N4 width="61" nowrap=""> N5 width="59" nowrap=""> N6 width="61" nowrap=""> N7 width="54" nowrap=""> N8

.6,.2,.2 width="59" nowrap=""> 0.0871429 width="59" nowrap=""> 0.0777065 width="59" nowrap=""> 0.0681314 width="59" nowrap=""> 0.0948506 width="61" nowrap=""> 0.08744 width="59" nowrap=""> 0.0723339 width="61" nowrap=""> 0.0804678 width="54" nowrap=""> 0.158522

.65, .15, .2 width="59" nowrap=""> 0.0875724 width="59" nowrap=""> 0.0777065 width="59" nowrap=""> 0.0662375 width="59" nowrap=""> 0.0962138 width="61" nowrap=""> 0.090858 width="59" nowrap=""> 0.0723339 width="61" nowrap=""> 0.0794958 width="54" nowrap=""> 0.158522

.65,.2,.15 width="59" nowrap=""> 0.0871429 width="59" nowrap=""> 0.0777065 width="59" nowrap=""> 0.0674325 width="59" nowrap=""> 0.0962138 width="61" nowrap=""> 0.08744 width="59" nowrap=""> 0.0723339 width="61" nowrap=""> 0.080145 width="54" nowrap=""> 0.158522

.7,.2,.1 width="59" nowrap=""> 0.0871429 width="59" nowrap=""> 0.0777065 width="59" nowrap=""> 0.0674325 width="59" nowrap=""> 0.0962138 width="61" nowrap=""> 0.090858 width="59" nowrap=""> 0.0723339 width="61" nowrap=""> 0.0794958 width="54" nowrap=""> 0.158522

.7,.15,.15 width="59" nowrap=""> 0.0878294 width="59" nowrap=""> 0.0777065 width="59" nowrap=""> 0.0662375 width="59" nowrap=""> 0.0962138 width="61" nowrap=""> 0.090858 width="59" nowrap=""> 0.0723339 width="61" nowrap=""> 0.0794958 width="54" nowrap=""> 0.158522

.7,.1,.2 width="59" nowrap=""> 0.0884779 width="59" nowrap=""> 0.0777322 width="59" nowrap=""> 0.0649009 width="59" nowrap=""> 0.0962138 width="61" nowrap=""> 0.0952367 width="59" nowrap=""> 0.0723339 width="61" nowrap=""> 0.0794958 width="54" nowrap=""> 0.158522

.75,.2,.05 width="59" nowrap=""> 0.0875724 width="59" nowrap=""> 0.0777065 width="59" nowrap=""> 0.0668866 width="59" nowrap=""> 0.0962138 width="61" nowrap=""> 0.090858 width="59" nowrap=""> 0.0723339 width="61" nowrap=""> 0.0794958 width="54" nowrap=""> 0.158522

.75,.15,.1 width="59" nowrap=""> 0.0882553 width="59" nowrap=""> 0.0777065 width="59" nowrap=""> 0.0662375 width="59" nowrap=""> 0.0962138 width="61" nowrap=""> 0.090858 width="59" nowrap=""> 0.0723339 width="61" nowrap=""> 0.0794958 width="54" nowrap=""> 0.158522

.75,.1,.15 width="59" nowrap=""> 0.0889562 width="59" nowrap=""> 0.0777322 width="59" nowrap=""> 0.0641353 width="59" nowrap=""> 0.0962138 width="61" nowrap=""> 0.0952367 width="59" nowrap=""> 0.0722395 width="61" nowrap=""> 0.0794958 width="54" nowrap=""> 0.158522

.75,.05,.2 width="59" nowrap=""> 0.0900455 width="59" nowrap=""> 0.077754 width="59" nowrap=""> 0.0633395 width="59" nowrap=""> 0.1086797 width="61" nowrap=""> 0.1079713 width="59" nowrap=""> 0.0709004 width="61" nowrap=""> 0.0775824 width="54" nowrap=""> 0.158522

.8,.05,.15 width="59" nowrap=""> 0.0904711 width="59" nowrap=""> 0.077754 width="59" nowrap=""> 0.0622665 width="59" nowrap=""> 0.1174934 width="61" nowrap=""> 0.1079713 width="59" nowrap=""> 0.069538 width="61" nowrap=""> 0.0775824 width="54" nowrap=""> 0.1845742

.8,.1,.1 width="59" nowrap=""> 0.0894551 width="59" nowrap=""> 0.0777322 width="59" nowrap=""> 0.0641353 width="59" nowrap=""> 0.1020115 width="61" nowrap=""> 0.1000195 width="59" nowrap=""> 0.0713379 width="61" nowrap=""> 0.0794958 width="54" nowrap=""> 0.158522

.8,.15,.05 width="59" nowrap=""> 0.0882553 width="59" nowrap=""> 0.0777065 width="59" nowrap=""> 0.0655928 width="59" nowrap=""> 0.0962138 width="61" nowrap=""> 0.0952367 width="59" nowrap=""> 0.0723339 width="61" nowrap=""> 0.0794958 width="54" nowrap=""> 0.158522

.85,.05,.1 width="59" nowrap=""> 0.0913187 width="59" nowrap=""> 0.077754 width="59" nowrap=""> 0.0623123 width="59" nowrap=""> 0.1300981 width="61" nowrap=""> 0.1176686 width="59" nowrap=""> 0.069538 width="61" nowrap=""> 0.0784604 width="54" nowrap=""> 0.1906585

.85,.1,.05 width="59" nowrap=""> 0.0894551 width="59" nowrap=""> 0.0777359 width="59" nowrap=""> 0.0641353 width="59" nowrap=""> 0.1020115 width="61" nowrap=""> 0.1000195 width="59" nowrap=""> 0.0709004 width="61" nowrap=""> 0.0788605 width="54" nowrap=""> 0.158522

.9,.05,.05 width="59" nowrap=""> 0.0925586 width="59" nowrap=""> 0.0777839 width="59" nowrap=""> 0.063988 width="59" nowrap=""> 0.1543938 width="61" nowrap=""> 0.1176686 width="59" nowrap=""> 0.0679255 width="61" nowrap=""> 0.0784604 width="54" nowrap=""> 0.2442362

Comb/H.N width="59" nowrap=""> N15 width="59" nowrap=""> N20 width="59" nowrap=""> N25 width="59" nowrap=""> N30 width="61" nowrap=""> N35 width="59" nowrap=""> N40 width="61" nowrap=""> N45 width="54" nowrap=""> N50

.6,.2,.2 width="59" nowrap=""> 0.8540058 width="59" nowrap=""> 0.6168502 width="59" nowrap=""> 1.1183273 width="59" nowrap=""> 0.4948634 width="61" nowrap=""> 1.7217955 width="59" nowrap=""> 0.9181224 width="61" nowrap=""> 0.8662499 width="54" nowrap=""> 1.6956986

.65, .15, .2 width="59" nowrap=""> 0.8540058 width="59" nowrap=""> 0.6168502 width="59" nowrap=""> 1.1183273 width="59" nowrap=""> 0.4948634 width="61" nowrap=""> 1.7217955 width="59" nowrap=""> 0.9181224 width="61" nowrap=""> 0.8662499 width="54" nowrap=""> 1.6956986

.65,.2,.15 width="59" nowrap=""> 0.8540058 width="59" nowrap=""> 0.6168502 width="59" nowrap=""> 1.1183273 width="59" nowrap=""> 0.4948634 width="61" nowrap=""> 1.7217955 width="59" nowrap=""> 0.9181224 width="61" nowrap=""> 0.8662499 width="54" nowrap=""> 1.6956986

.7,.2,.1 width="59" nowrap=""> 0.8540058 width="59" nowrap=""> 0.6168502 width="59" nowrap=""> 1.1183273 width="59" nowrap=""> 0.4948634 width="61" nowrap=""> 1.7217955 width="59" nowrap=""> 0.9181224 width="61" nowrap=""> 0.8662499 width="54" nowrap=""> 1.6956986

.7,.15,.15 width="59" nowrap=""> 0.8540058 width="59" nowrap=""> 0.6168502 width="59" nowrap=""> 1.1183273 width="59" nowrap=""> 0.4948634 width="61" nowrap=""> 1.7217955 width="59" nowrap=""> 0.9181224 width="61" nowrap=""> 0.8662499 width="54" nowrap=""> 1.6956986

.7,.1,.2 width="59" nowrap=""> 0.8540058 width="59" nowrap=""> 0.6168502 width="59" nowrap=""> 1.1183273 width="59" nowrap=""> 0.4948634 width="61" nowrap=""> 1.7217955 width="59" nowrap=""> 0.9181224 width="61" nowrap=""> 0.8662499 width="54" nowrap=""> 1.6956986

.75,.2,.05 width="59" nowrap=""> 0.8540058 width="59" nowrap=""> 0.6168502 width="59" nowrap=""> 1.1183273 width="59" nowrap=""> 0.4948634 width="61" nowrap=""> 1.7217955 width="59" nowrap=""> 0.9181224 width="61" nowrap=""> 0.8662499 width="54" nowrap=""> 1.6956986

.75,.15,.1 width="59" nowrap=""> 0.8540058 width="59" nowrap=""> 0.6168502 width="59" nowrap=""> 1.1183273 width="59" nowrap=""> 0.4948634 width="61" nowrap=""> 1.7217955 width="59" nowrap=""> 0.9181224 width="61" nowrap=""> 0.8662499 width="54" nowrap=""> 1.6956986

.75,.1,.15 width="59" nowrap=""> 0.8540058 width="59" nowrap=""> 0.6168502 width="59" nowrap=""> 1.1183273 width="59" nowrap=""> 0.4948634 width="61" nowrap=""> 1.7217955 width="59" nowrap=""> 0.9181224 width="61" nowrap=""> 0.8662499 width="54" nowrap=""> 1.6956986

.75,.05,.2 width="59" nowrap=""> 0.8540058 width="59" nowrap=""> 0.6168502 width="59" nowrap=""> 1.1651408 width="59" nowrap=""> 0.4948634 width="61" nowrap=""> 1.7217955 width="59" nowrap=""> 0.9181224 width="61" nowrap=""> 0.8662499 width="54" nowrap=""> 1.6956986

.8,.05,.15 width="59" nowrap=""> 0.8540058 width="59" nowrap=""> 0.6168502 width="59" nowrap=""> 1.1651408 width="59" nowrap=""> 0.4948634 width="61" nowrap=""> 1.7217955 width="59" nowrap=""> 0.9181224 width="61" nowrap=""> 0.8662499 width="54" nowrap=""> 1.6956986

.8,.1,.1 width="59" nowrap=""> 0.8540058 width="59" nowrap=""> 0.6168502 width="59" nowrap=""> 1.1651408 width="59" nowrap=""> 0.4948634 width="61" nowrap=""> 1.7217955 width="59" nowrap=""> 0.9181224 width="61" nowrap=""> 0.8662499 width="54" nowrap=""> 1.6956986

.8,.15,.05 width="59" nowrap=""> 0.8540058 width="59" nowrap=""> 0.6168502 width="59" nowrap=""> 1.1183273 width="59" nowrap=""> 0.4948634 width="61" nowrap=""> 1.7217955 width="59" nowrap=""> 0.9181224 width="61" nowrap=""> 0.8662499 width="54" nowrap=""> 1.6956986

.85,.05,.1 width="59" nowrap=""> 0.8540058 width="59" nowrap=""> 0.6168502 width="59" nowrap=""> 1.1651408 width="59" nowrap=""> 0.4948634 width="61" nowrap=""> 1.7217955 width="59" nowrap=""> 0.9181224 width="61" nowrap=""> 1.2830719 width="54" nowrap=""> 1.6956986

.85,.1,.05 width="59" nowrap=""> 0.8540058 width="59" nowrap=""> 0.6168502 width="59" nowrap=""> 1.1651408 width="59" nowrap=""> 0.4948634 width="61" nowrap=""> 1.7217955 width="59" nowrap=""> 0.9181224 width="61" nowrap=""> 0.8662499 width="54" nowrap=""> 1.6956986

.9,.05,.05 width="59" nowrap=""> 0.8540058 width="59" nowrap=""> 0.6168502 width="59" nowrap=""> 1.1651408 width="59" nowrap=""> 0.4948634 width="61" nowrap=""> 1.7217955 width="59" nowrap=""> 0.9181224 width="61" nowrap=""> 1.2830719 width="54" nowrap=""> 1.6956986

Table 5. R MSE-2at node 1 to node 5

> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > >

Comb/H.N width="58" nowrap=""> N1 width="57" nowrap=""> N2 width="56" nowrap=""> N3 width="61" nowrap=""> N4 width="56" nowrap=""> N5 width="66" nowrap=""> N6 width="60" nowrap=""> N7 width="56" nowrap=""> N8

.6,.2,.2 width="58" nowrap=""> 0.076488 width="57" nowrap=""> 0.08083 width="56" nowrap=""> 0.080475 width="61" nowrap=""> 0.080454 width="56" nowrap=""> 0.077759 width="66" nowrap=""> 0.082688 width="60" nowrap=""> 0.088633 width="56" nowrap=""> 0.077956

.65,.15,.2 width="58" nowrap=""> 0.076459 width="57" nowrap=""> 0.081464 width="56" nowrap=""> 0.081179 width="61" nowrap=""> 0.080789 width="56" nowrap=""> 0.078423 width="66" nowrap=""> 0.082599 width="60" nowrap=""> 0.090661 width="56" nowrap=""> 0.078178

.65,.2,.15 width="58" nowrap=""> 0.076623 width="57" nowrap=""> 0.081223 width="56" nowrap=""> 0.080475 width="61" nowrap=""> 0.080454 width="56" nowrap=""> 0.077759 width="66" nowrap=""> 0.082599 width="60" nowrap=""> 0.090661 width="56" nowrap=""> 0.077956

.7,.2,.1 width="58" nowrap=""> 0.076459 width="57" nowrap=""> 0.08133 width="56" nowrap=""> 0.080475 width="61" nowrap=""> 0.080454 width="56" nowrap=""> 0.078038 width="66" nowrap=""> 0.082599 width="60" nowrap=""> 0.090661 width="56" nowrap=""> 0.077956

.7,.15,.15 width="58" nowrap=""> 0.076459 width="57" nowrap=""> 0.081464 width="56" nowrap=""> 0.081882 width="61" nowrap=""> 0.080832 width="56" nowrap=""> 0.078423 width="66" nowrap=""> 0.082692 width="60" nowrap=""> 0.090661 width="56" nowrap=""> 0.078558

.7,.1,.2 width="58" nowrap=""> 0.076506 width="57" nowrap=""> 0.081892 width="56" nowrap=""> 0.082999 width="61" nowrap=""> 0.08141 width="56" nowrap=""> 0.078423 width="66" nowrap=""> 0.082857 width="60" nowrap=""> 0.09079 width="56" nowrap=""> 0.078278

.75,.2,.05 width="58" nowrap=""> 0.076459 width="57" nowrap=""> 0.081464 width="56" nowrap=""> 0.081087 width="61" nowrap=""> 0.080561 width="56" nowrap=""> 0.078423 width="66" nowrap=""> 0.082599 width="60" nowrap=""> 0.090661 width="56" nowrap=""> 0.078178

.75,.15,.1 width="58" nowrap=""> 0.076462 width="57" nowrap=""> 0.081892 width="56" nowrap=""> 0.081882 width="61" nowrap=""> 0.080975 width="56" nowrap=""> 0.078423 width="66" nowrap=""> 0.082748 width="60" nowrap=""> 0.090661 width="56" nowrap=""> 0.078278

.75,.1,.15 width="58" nowrap=""> 0.076467 width="57" nowrap=""> 0.08255 width="56" nowrap=""> 0.082999 width="61" nowrap=""> 0.082025 width="56" nowrap=""> 0.07906 width="66" nowrap=""> 0.082857 width="60" nowrap=""> 0.09079 width="56" nowrap=""> 0.078278

.75,.05,.2 width="58" nowrap=""> 0.076467 width="57" nowrap=""> 0.083644 width="56" nowrap=""> 0.08384 width="61" nowrap=""> 0.08268 width="56" nowrap=""> 0.083312 width="66" nowrap=""> 0.083106 width="60" nowrap=""> 0.099939 width="56" nowrap=""> 0.078579

.8,.05,.15 width="58" nowrap=""> 0.076552 width="57" nowrap=""> 0.084454 width="56" nowrap=""> 0.085112 width="61" nowrap=""> 0.084488 width="56" nowrap=""> 0.086542 width="66" nowrap=""> 0.083577 width="60" nowrap=""> 0.099939 width="56" nowrap=""> 0.079257

.8,.1,.1 width="58" nowrap=""> 0.076467 width="57" nowrap=""> 0.08255 width="56" nowrap=""> 0.08384 width="61" nowrap=""> 0.08268 width="56" nowrap=""> 0.079995 width="66" nowrap=""> 0.083106 width="60" nowrap=""> 0.094729 width="56" nowrap=""> 0.078404

.8,.15,.05 width="58" nowrap=""> 0.076462 width="57" nowrap=""> 0.081892 width="56" nowrap=""> 0.082999 width="61" nowrap=""> 0.080975 width="56" nowrap=""> 0.078423 width="66" nowrap=""> 0.082857 width="60" nowrap=""> 0.090661 width="56" nowrap=""> 0.078278

.85,.05,.1 width="58" nowrap=""> 0.076552 width="57" nowrap=""> 0.085593 width="56" nowrap=""> 0.085112 width="61" nowrap=""> 0.084488 width="56" nowrap=""> 0.091624 width="66" nowrap=""> 0.084625 width="60" nowrap=""> 0.103356 width="56" nowrap=""> 0.079257

.85,.1,.05 width="58" nowrap=""> 0.076467 width="57" nowrap=""> 0.082927 width="56" nowrap=""> 0.08384 width="61" nowrap=""> 0.08268 width="56" nowrap=""> 0.081217 width="66" nowrap=""> 0.083106 width="60" nowrap=""> 0.094729 width="56" nowrap=""> 0.078579

.9,.05,.05 width="58" nowrap=""> 0.076987 width="57" nowrap=""> 0.087791 width="56" nowrap=""> 0.08744 width="61" nowrap=""> 0.088327 width="56" nowrap=""> 0.101376 width="66" nowrap=""> 0.100363 width="60" nowrap=""> 0.103356 width="56" nowrap=""> 0.086524

Comb/H.N width="58" nowrap="" valign="bottom"> N15 width="57" nowrap="" valign="bottom"> N20 width="56" nowrap="" valign="bottom"> N25 width="61" nowrap="" valign="bottom"> N30 width="56" nowrap="" valign="bottom"> N35 width="66" nowrap="" valign="bottom"> N40 width="60" nowrap="" valign="bottom"> N45 width="56" nowrap="" valign="bottom"> N50

.6,.2,.2 width="58" nowrap="" valign="bottom"> 0.329589 width="57" nowrap="" valign="bottom"> 0.809611 width="56" nowrap="" valign="bottom"> 1.010588 width="61" nowrap="" valign="bottom"> 0.365689 width="56" nowrap="" valign="bottom"> 0.666941 width="66" nowrap="" valign="bottom"> 0.675425 width="60" nowrap="" valign="bottom"> 0.990145 width="56" nowrap="" valign="bottom"> 1.742632

.65, .15, .2 width="58" nowrap="" valign="bottom"> 0.329589 width="57" nowrap="" valign="bottom"> 0.809611 width="56" nowrap="" valign="bottom"> 1.010588 width="61" nowrap="" valign="bottom"> 0.365689 width="56" nowrap="" valign="bottom"> 0.666941 width="66" nowrap="" valign="bottom"> 0.675425 width="60" nowrap="" valign="bottom"> 0.990145 width="56" nowrap="" valign="bottom"> 1.742632

.65,.2,.15 width="58" nowrap="" valign="bottom"> 0.329589 width="57" nowrap="" valign="bottom"> 0.809611 width="56" nowrap="" valign="bottom"> 1.010588 width="61" nowrap="" valign="bottom"> 0.365689 width="56" nowrap="" valign="bottom"> 0.666941 width="66" nowrap="" valign="bottom"> 0.675425 width="60" nowrap="" valign="bottom"> 0.990145 width="56" nowrap="" valign="bottom"> 1.742632

.7,.2,.1 width="58" nowrap="" valign="bottom"> 0.329589 width="57" nowrap="" valign="bottom"> 0.809611 width="56" nowrap="" valign="bottom"> 1.010588 width="61" nowrap="" valign="bottom"> 0.365689 width="56" nowrap="" valign="bottom"> 0.666941 width="66" nowrap="" valign="bottom"> 0.675425 width="60" nowrap="" valign="bottom"> 0.990145 width="56" nowrap="" valign="bottom"> 1.742632

.7,.15,.15 width="58" nowrap="" valign="bottom"> 0.329589 width="57" nowrap="" valign="bottom"> 0.809611 width="56" nowrap="" valign="bottom"> 1.010588 width="61" nowrap="" valign="bottom"> 0.365689 width="56" nowrap="" valign="bottom"> 0.666941 width="66" nowrap="" valign="bottom"> 0.675425 width="60" nowrap="" valign="bottom"> 0.990145 width="56" nowrap="" valign="bottom"> 1.742632

.7,.1,.2 width="58" nowrap="" valign="bottom"> 0.329589 width="57" nowrap="" valign="bottom"> 0.809611 width="56" nowrap="" valign="bottom"> 1.010588 width="61" nowrap="" valign="bottom"> 0.306386 width="56" nowrap="" valign="bottom"> 0.666941 width="66" nowrap="" valign="bottom"> 0.675425 width="60" nowrap="" valign="bottom"> 0.990145 width="56" nowrap="" valign="bottom"> 1.742632

.75,.2,.05 width="58" nowrap="" valign="bottom"> 0.329589 width="57" nowrap="" valign="bottom"> 0.809611 width="56" nowrap="" valign="bottom"> 1.010588 width="61" nowrap="" valign="bottom"> 0.365689 width="56" nowrap="" valign="bottom"> 0.666941 width="66" nowrap="" valign="bottom"> 0.675425 width="60" nowrap="" valign="bottom"> 0.990145 width="56" nowrap="" valign="bottom"> 1.742632

.75,.15,.1 width="58" nowrap="" valign="bottom"> 0.329589 width="57" nowrap="" valign="bottom"> 0.809611 width="56" nowrap="" valign="bottom"> 1.010588 width="61" nowrap="" valign="bottom"> 0.365689 width="56" nowrap="" valign="bottom"> 0.666941 width="66" nowrap="" valign="bottom"> 0.675425 width="60" nowrap="" valign="bottom"> 0.990145 width="56" nowrap="" valign="bottom"> 1.742632

.75,.1,.15 width="58" nowrap="" valign="bottom"> 0.318707 width="57" nowrap="" valign="bottom"> 0.809611 width="56" nowrap="" valign="bottom"> 1.010588 width="61" nowrap="" valign="bottom"> 0.306988 width="56" nowrap="" valign="bottom"> 0.666941 width="66" nowrap="" valign="bottom"> 0.675425 width="60" nowrap="" valign="bottom"> 0.990145 width="56" nowrap="" valign="bottom"> 1.742632

.75,.05,.2 width="58" nowrap="" valign="bottom"> 0.330477 width="57" nowrap="" valign="bottom"> 0.809611 width="56" nowrap="" valign="bottom"> 1.010588 width="61" nowrap="" valign="bottom"> 0.365818 width="56" nowrap="" valign="bottom"> 0.666941 width="66" nowrap="" valign="bottom"> 0.675425 width="60" nowrap="" valign="bottom"> 0.990145 width="56" nowrap="" valign="bottom"> 1.742632

.8,.05,.15 width="58" nowrap="" valign="bottom"> 0.34309 width="57" nowrap="" valign="bottom"> 0.809611 width="56" nowrap="" valign="bottom"> 1.010588 width="61" nowrap="" valign="bottom"> 0.365818 width="56" nowrap="" valign="bottom"> 0.666941 width="66" nowrap="" valign="bottom"> 0.675425 width="60" nowrap="" valign="bottom"> 0.990145 width="56" nowrap="" valign="bottom"> 1.742632

.8,.1,.1 width="58" nowrap="" valign="bottom"> 0.318707 width="57" nowrap="" valign="bottom"> 0.809611 width="56" nowrap="" valign="bottom"> 1.010588 width="61" nowrap="" valign="bottom"> 0.306988 width="56" nowrap="" valign="bottom"> 0.666941 width="66" nowrap="" valign="bottom"> 0.675425 width="60" nowrap="" valign="bottom"> 0.990145 width="56" nowrap="" valign="bottom"> 1.742632

.8,.15,.05 width="58" nowrap="" valign="bottom"> 0.329589 width="57" nowrap="" valign="bottom"> 0.809611 width="56" nowrap="" valign="bottom"> 1.010588 width="61" nowrap="" valign="bottom"> 0.306386 width="56" nowrap="" valign="bottom"> 0.666941 width="66" nowrap="" valign="bottom"> 0.675425 width="60" nowrap="" valign="bottom"> 0.990145 width="56" nowrap="" valign="bottom"> 1.742632

.85,.05,.1 width="58" nowrap="" valign="bottom"> 0.34911 width="57" nowrap="" valign="bottom"> 0.809611 width="56" nowrap="" valign="bottom"> 1.010588 width="61" nowrap="" valign="bottom"> 0.365818 width="56" nowrap="" valign="bottom"> 0.666941 width="66" nowrap="" valign="bottom"> 0.675425 width="60" nowrap="" valign="bottom"> 0.990145 width="56" nowrap="" valign="bottom"> 1.742632

.85,.1,.05 width="58" nowrap="" valign="bottom"> 0.321971 width="57" nowrap="" valign="bottom"> 0.809611 width="56" nowrap="" valign="bottom"> 1.010588 width="61" nowrap="" valign="bottom"> 0.365818 width="56" nowrap="" valign="bottom"> 0.666941 width="66" nowrap="" valign="bottom"> 0.675425 width="60" nowrap="" valign="bottom"> 0.990145 width="56" nowrap="" valign="bottom"> 1.742632

.9,.05,.05 width="58" nowrap="" valign="bottom"> 0.34911 width="57" nowrap="" valign="bottom"> 0.809611 width="56" nowrap="" valign="bottom"> 1.010588 width="61" nowrap="" valign="bottom"> 0.365818 width="56" nowrap="" valign="bottom"> 0.666941 width="66" nowrap="" valign="bottom"> 0.675425 width="60" nowrap="" valign="bottom"> 0.990145 width="56" nowrap="" valign="bottom"> 1.742632

 

Table 6. R MSE-3 at node 1 to node 50

> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > >

Comb/H.N width="63" nowrap="" valign="bottom"> N1 width="57" nowrap="" valign="bottom"> N2 width="57" nowrap="" valign="bottom"> N3 width="57" nowrap="" valign="bottom"> N4 width="57" nowrap="" valign="bottom"> N5 width="57" nowrap="" valign="bottom"> N6 width="57" nowrap="" valign="bottom"> N7 width="62" nowrap="" valign="bottom"> N8

.6,.2,.2 width="63" nowrap="" valign="bottom"> 0.078692 width="57" nowrap="" valign="bottom"> 0.077616 width="57" nowrap="" valign="bottom"> 0.07873 width="57" nowrap="" valign="bottom"> 0.079196 width="57" nowrap="" valign="bottom"> 0.081837 width="57" nowrap="" valign="bottom"> 0.082009 width="57" nowrap="" valign="bottom"> 0.081951 width="62" nowrap="" valign="bottom"> 0.084031

.65, .15, .2 width="63" nowrap="" valign="bottom"> 0.078763 width="57" nowrap="" valign="bottom"> 0.077618 width="57" nowrap="" valign="bottom"> 0.078708 width="57" nowrap="" valign="bottom"> 0.079542 width="57" nowrap="" valign="bottom"> 0.082142 width="57" nowrap="" valign="bottom"> 0.083106 width="57" nowrap="" valign="bottom"> 0.082582 width="62" nowrap="" valign="bottom"> 0.084031

.65,.2,.15 width="63" nowrap="" valign="bottom"> 0.078692 width="57" nowrap="" valign="bottom"> 0.077629 width="57" nowrap="" valign="bottom"> 0.078708 width="57" nowrap="" valign="bottom"> 0.079196 width="57" nowrap="" valign="bottom"> 0.081917 width="57" nowrap="" valign="bottom"> 0.082009 width="57" nowrap="" valign="bottom"> 0.08219 width="62" nowrap="" valign="bottom"> 0.084031

.7,.2,.1 width="63" nowrap="" valign="bottom"> 0.078715 width="57" nowrap="" valign="bottom"> 0.077629 width="57" nowrap="" valign="bottom"> 0.078708 width="57" nowrap="" valign="bottom"> 0.079196 width="57" nowrap="" valign="bottom"> 0.082142 width="57" nowrap="" valign="bottom"> 0.082537 width="57" nowrap="" valign="bottom"> 0.08219 width="62" nowrap="" valign="bottom"> 0.084031

.7,.15,.15 width="63" nowrap="" valign="bottom"> 0.078777 width="57" nowrap="" valign="bottom"> 0.077618 width="57" nowrap="" valign="bottom"> 0.078783 width="57" nowrap="" valign="bottom"> 0.079542 width="57" nowrap="" valign="bottom"> 0.082684 width="57" nowrap="" valign="bottom"> 0.083106 width="57" nowrap="" valign="bottom"> 0.083788 width="62" nowrap="" valign="bottom"> 0.084896

.7,.1,.2 width="63" nowrap="" valign="bottom"> 0.078777 width="57" nowrap="" valign="bottom"> 0.077671 width="57" nowrap="" valign="bottom"> 0.078958 width="57" nowrap="" valign="bottom"> 0.080269 width="57" nowrap="" valign="bottom"> 0.084078 width="57" nowrap="" valign="bottom"> 0.083757 width="57" nowrap="" valign="bottom"> 0.087859 width="62" nowrap="" valign="bottom"> 0.087568

.75,.2,.05 width="63" nowrap="" valign="bottom"> 0.078715 width="57" nowrap="" valign="bottom"> 0.077629 width="57" nowrap="" valign="bottom"> 0.078708 width="57" nowrap="" valign="bottom"> 0.079196 width="57" nowrap="" valign="bottom"> 0.082142 width="57" nowrap="" valign="bottom"> 0.082537 width="57" nowrap="" valign="bottom"> 0.08219 width="62" nowrap="" valign="bottom"> 0.084031

.75,.15,.1 width="63" nowrap="" valign="bottom"> 0.078777 width="57" nowrap="" valign="bottom"> 0.077618 width="57" nowrap="" valign="bottom"> 0.078947 width="57" nowrap="" valign="bottom"> 0.079542 width="57" nowrap="" valign="bottom"> 0.082917 width="57" nowrap="" valign="bottom"> 0.083106 width="57" nowrap="" valign="bottom"> 0.084939 width="62" nowrap="" valign="bottom"> 0.086199

.75,.1,.15 width="63" nowrap="" valign="bottom"> 0.07879 width="57" nowrap="" valign="bottom"> 0.077587 width="57" nowrap="" valign="bottom"> 0.079033 width="57" nowrap="" valign="bottom"> 0.081136 width="57" nowrap="" valign="bottom"> 0.084078 width="57" nowrap="" valign="bottom"> 0.085188 width="57" nowrap="" valign="bottom"> 0.087859 width="62" nowrap="" valign="bottom"> 0.087568

.75,.05,.2 width="63" nowrap="" valign="bottom"> 0.07879 width="57" nowrap="" valign="bottom"> 0.077603 width="57" nowrap="" valign="bottom"> 0.079033 width="57" nowrap="" valign="bottom"> 0.081899 width="57" nowrap="" valign="bottom"> 0.176237 width="57" nowrap="" valign="bottom"> 0.087636 width="57" nowrap="" valign="bottom"> 0.087859 width="62" nowrap="" valign="bottom"> 0.090254

.8,.05,.15 width="63" nowrap="" valign="bottom"> 0.07879 width="57" nowrap="" valign="bottom"> 0.077564 width="57" nowrap="" valign="bottom"> 0.079033 width="57" nowrap="" valign="bottom"> 0.082995 width="57" nowrap="" valign="bottom"> 0.222392 width="57" nowrap="" valign="bottom"> 0.09185 width="57" nowrap="" valign="bottom"> 0.090443 width="62" nowrap="" valign="bottom"> 0.094965

.8,.1,.1 width="63" nowrap="" valign="bottom"> 0.07879 width="57" nowrap="" valign="bottom"> 0.077587 width="57" nowrap="" valign="bottom"> 0.079033 width="57" nowrap="" valign="bottom"> 0.081832 width="57" nowrap="" valign="bottom"> 0.087906 width="57" nowrap="" valign="bottom"> 0.085188 width="57" nowrap="" valign="bottom"> 0.087859 width="62" nowrap="" valign="bottom"> 0.087568

.8,.15,.05 width="63" nowrap="" valign="bottom"> 0.078777 width="57" nowrap="" valign="bottom"> 0.077556 width="57" nowrap="" valign="bottom"> 0.078947 width="57" nowrap="" valign="bottom"> 0.080269 width="57" nowrap="" valign="bottom"> 0.082917 width="57" nowrap="" valign="bottom"> 0.083106 width="57" nowrap="" valign="bottom"> 0.086181 width="62" nowrap="" valign="bottom"> 0.086199

.85,.05,.1 width="63" nowrap="" valign="bottom"> 0.078846 width="57" nowrap="" valign="bottom"> 0.077564 width="57" nowrap="" valign="bottom"> 0.079622 width="57" nowrap="" valign="bottom"> 0.084757 width="57" nowrap="" valign="bottom"> 0.218313 width="57" nowrap="" valign="bottom"> 0.101089 width="57" nowrap="" valign="bottom"> 0.090443 width="62" nowrap="" valign="bottom"> 0.102392

.85,.1,.05 width="63" nowrap="" valign="bottom"> 0.07879 width="57" nowrap="" valign="bottom"> 0.077604 width="57" nowrap="" valign="bottom"> 0.079033 width="57" nowrap="" valign="bottom"> 0.081832 width="57" nowrap="" valign="bottom"> 0.087906 width="57" nowrap="" valign="bottom"> 0.085188 width="57" nowrap="" valign="bottom"> 0.087859 width="62" nowrap="" valign="bottom"> 0.090254

.9,.05,.05 width="63" nowrap="" valign="bottom"> 0.079304 width="57" nowrap="" valign="bottom"> 0.077547 width="57" nowrap="" valign="bottom"> 0.079964 width="57" nowrap="" valign="bottom"> 0.084757 width="57" nowrap="" valign="bottom"> 0.21555 width="57" nowrap="" valign="bottom"> 0.101089 width="57" nowrap="" valign="bottom"> 0.092693 width="62" nowrap="" valign="bottom"> 0.119395

Comb/H.N width="63" nowrap="" valign="bottom"> N15 width="57" nowrap="" valign="bottom"> N20 width="57" nowrap="" valign="bottom"> N25 width="57" nowrap="" valign="bottom"> N30 width="57" nowrap="" valign="bottom"> N35 width="57" nowrap="" valign="bottom"> N40 width="57" nowrap="" valign="bottom"> N45 width="62" nowrap="" valign="bottom"> N50

.6,.2,.2 width="63" nowrap="" valign="bottom"> 0.224166 width="57" nowrap="" valign="bottom"> 0.581126 width="57" nowrap="" valign="bottom"> 0.649279 width="57" nowrap="" valign="bottom"> 0.780372 width="57" nowrap="" valign="bottom"> 1.076284 width="57" nowrap="" valign="bottom"> 0.658626 width="57" nowrap="" valign="bottom"> 1.105486 width="62" nowrap="" valign="bottom"> 1.583643

.65, .15, .2 width="63" nowrap="" valign="bottom"> 0.236535 width="57" nowrap="" valign="bottom"> 0.581126 width="57" nowrap="" valign="bottom"> 0.649279 width="57" nowrap="" valign="bottom"> 0.780372 width="57" nowrap="" valign="bottom"> 1.076284 width="57" nowrap="" valign="bottom"> 0.652432 width="57" nowrap="" valign="bottom"> 1.105486 width="62" nowrap="" valign="bottom"> 1.583643

.65,.2,.15 width="63" nowrap="" valign="bottom"> 0.224166 width="57" nowrap="" valign="bottom"> 0.581126 width="57" nowrap="" valign="bottom"> 0.649279 width="57" nowrap="" valign="bottom"> 0.780372 width="57" nowrap="" valign="bottom"> 1.076284 width="57" nowrap="" valign="bottom"> 0.658626 width="57" nowrap="" valign="bottom"> 1.105486 width="62" nowrap="" valign="bottom"> 1.583643

.7,.2,.1 width="63" nowrap="" valign="bottom"> 0.224166 width="57" nowrap="" valign="bottom"> 0.581126 width="57" nowrap="" valign="bottom"> 0.649279 width="57" nowrap="" valign="bottom"> 0.780372 width="57" nowrap="" valign="bottom"> 1.076284 width="57" nowrap="" valign="bottom"> 0.658626 width="57" nowrap="" valign="bottom"> 1.105486 width="62" nowrap="" valign="bottom"> 1.583643

.7,.15,.15 width="63" nowrap="" valign="bottom"> 0.22791 width="57" nowrap="" valign="bottom"> 0.581126 width="57" nowrap="" valign="bottom"> 0.649279 width="57" nowrap="" valign="bottom"> 0.780372 width="57" nowrap="" valign="bottom"> 1.076284 width="57" nowrap="" valign="bottom"> 0.652432 width="57" nowrap="" valign="bottom"> 1.105486 width="62" nowrap="" valign="bottom"> 1.583643

.7,.1,.2 width="63" nowrap="" valign="bottom"> 0.226902 width="57" nowrap="" valign="bottom"> 0.581126 width="57" nowrap="" valign="bottom"> 0.628242 width="57" nowrap="" valign="bottom"> 0.780372 width="57" nowrap="" valign="bottom"> 1.076284 width="57" nowrap="" valign="bottom"> 0.694743 width="57" nowrap="" valign="bottom"> 1.105486 width="62" nowrap="" valign="bottom"> 1.583643

.75,.2,.05 width="63" nowrap="" valign="bottom"> 0.224166 width="57" nowrap="" valign="bottom"> 0.581126 width="57" nowrap="" valign="bottom"> 0.649279 width="57" nowrap="" valign="bottom"> 0.780372 width="57" nowrap="" valign="bottom"> 1.076284 width="57" nowrap="" valign="bottom"> 0.658626 width="57" nowrap="" valign="bottom"> 1.105486 width="62" nowrap="" valign="bottom"> 1.583643

.75,.15,.1 width="63" nowrap="" valign="bottom"> 0.229767 width="57" nowrap="" valign="bottom"> 0.581126 width="57" nowrap="" valign="bottom"> 0.649279 width="57" nowrap="" valign="bottom"> 0.780372 width="57" nowrap="" valign="bottom"> 1.076284 width="57" nowrap="" valign="bottom"> 0.652432 width="57" nowrap="" valign="bottom"> 1.105486 width="62" nowrap="" valign="bottom"> 1.583643

.75,.1,.15 width="63" nowrap="" valign="bottom"> 0.22998 width="57" nowrap="" valign="bottom"> 0.581126 width="57" nowrap="" valign="bottom"> 0.628242 width="57" nowrap="" valign="bottom"> 0.780372 width="57" nowrap="" valign="bottom"> 1.076284 width="57" nowrap="" valign="bottom"> 0.694743 width="57" nowrap="" valign="bottom"> 1.105486 width="62" nowrap="" valign="bottom"> 1.583643

.75,.05,.2 width="63" nowrap="" valign="bottom"> 0.238733 width="57" nowrap="" valign="bottom"> 0.581126 width="57" nowrap="" valign="bottom"> 0.668724 width="57" nowrap="" valign="bottom"> 0.780372 width="57" nowrap="" valign="bottom"> 1.076284 width="57" nowrap="" valign="bottom"> 0.727426 width="57" nowrap="" valign="bottom"> 1.105486 width="62" nowrap="" valign="bottom"> 1.583643

.8,.05,.15 width="63" nowrap="" valign="bottom"> 0.238733 width="57" nowrap="" valign="bottom"> 0.63271 width="57" nowrap="" valign="bottom"> 0.668724 width="57" nowrap="" valign="bottom"> 0.780372 width="57" nowrap="" valign="bottom"> 1.076284 width="57" nowrap="" valign="bottom"> 0.727426 width="57" nowrap="" valign="bottom"> 1.105486 width="62" nowrap="" valign="bottom"> 1.583643

.8,.1,.1 width="63" nowrap="" valign="bottom"> 0.230094 width="57" nowrap="" valign="bottom"> 0.581126 width="57" nowrap="" valign="bottom"> 0.594439 width="57" nowrap="" valign="bottom"> 0.780372 width="57" nowrap="" valign="bottom"> 1.076284 width="57" nowrap="" valign="bottom"> 0.694743 width="57" nowrap="" valign="bottom"> 1.105486 width="62" nowrap="" valign="bottom"> 1.583643

.8,.15,.05 width="63" nowrap="" valign="bottom"> 0.226902 width="57" nowrap="" valign="bottom"> 0.581126 width="57" nowrap="" valign="bottom"> 0.628242 width="57" nowrap="" valign="bottom"> 0.780372 width="57" nowrap="" valign="bottom"> 1.076284 width="57" nowrap="" valign="bottom"> 0.672239 width="57" nowrap="" valign="bottom"> 1.105486 width="62" nowrap="" valign="bottom"> 1.583643

.85,.05,.1 width="63" nowrap="" valign="bottom"> 0.240567 width="57" nowrap="" valign="bottom"> 0.63271 width="57" nowrap="" valign="bottom"> 0.702359 width="57" nowrap="" valign="bottom"> 0.853942 width="57" nowrap="" valign="bottom"> 1.076284 width="57" nowrap="" valign="bottom"> 0.741446 width="57" nowrap="" valign="bottom"> 1.105486 width="62" nowrap="" valign="bottom"> 1.583643

.85,.1,.05 width="63" nowrap="" valign="bottom"> 0.233305 width="57" nowrap="" valign="bottom"> 0.581126 width="57" nowrap="" valign="bottom"> 0.594439 width="57" nowrap="" valign="bottom"> 0.780372 width="57" nowrap="" valign="bottom"> 1.076284 width="57" nowrap="" valign="bottom"> 0.727426 width="57" nowrap="" valign="bottom"> 1.105486 width="62" nowrap="" valign="bottom"> 1.583643

.9,.05,.05 width="63" nowrap="" valign="bottom"> 0.240675 width="57" nowrap="" valign="bottom"> 0.63271 width="57" nowrap="" valign="bottom"> 0.702359 width="57" nowrap="" valign="bottom"> 0.853942 width="57" nowrap="" valign="bottom"> 1.141387 width="57" nowrap="" valign="bottom"> 0.743971 width="57" nowrap="" valign="bottom"> 1.105486 width="62" nowrap="" valign="bottom"> 1.583643

Table 7. R MSE-3 at node 1 to node 50

> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > >

Comb/H.N width="57" nowrap="" valign="bottom"> N1 width="57" nowrap="" valign="bottom"> N2 width="60" nowrap="" valign="bottom"> N3 width="57" nowrap="" valign="bottom"> N4 width="57" nowrap="" valign="bottom"> N5 width="57" nowrap="" valign="bottom"> N6 width="57" nowrap="" valign="bottom"> N7 width="65" nowrap="" valign="bottom"> N8

.6,.2,.2 width="57" nowrap="" valign="bottom"> 0.078881 width="57" nowrap="" valign="bottom"> 0.077755 width="60" nowrap="" valign="bottom"> 0.080605 width="57" nowrap="" valign="bottom"> 0.077593 width="57" nowrap="" valign="bottom"> 0.081087 width="57" nowrap="" valign="bottom"> 0.132182 width="57" nowrap="" valign="bottom"> 0.083044 width="65" nowrap="" valign="bottom"> 0.082383

.65, .15, .2 width="57" nowrap="" valign="bottom"> 0.080061 width="57" nowrap="" valign="bottom"> 0.07777 width="60" nowrap="" valign="bottom"> 0.080993 width="57" nowrap="" valign="bottom"> 0.077945 width="57" nowrap="" valign="bottom"> 0.085759 width="57" nowrap="" valign="bottom"> 0.132182 width="57" nowrap="" valign="bottom"> 0.083439 width="65" nowrap="" valign="bottom"> 0.082383

.65,.2,.15 width="57" nowrap="" valign="bottom"> 0.078881 width="57" nowrap="" valign="bottom"> 0.077755 width="60" nowrap="" valign="bottom"> 0.080699 width="57" nowrap="" valign="bottom"> 0.077593 width="57" nowrap="" valign="bottom"> 0.081796 width="57" nowrap="" valign="bottom"> 0.132182 width="57" nowrap="" valign="bottom"> 0.083439 width="65" nowrap="" valign="bottom"> 0.082383

.7,.2,.1 width="57" nowrap="" valign="bottom"> 0.079361 width="57" nowrap="" valign="bottom"> 0.07777 width="60" nowrap="" valign="bottom"> 0.080699 width="57" nowrap="" valign="bottom"> 0.077593 width="57" nowrap="" valign="bottom"> 0.081796 width="57" nowrap="" valign="bottom"> 0.132182 width="57" nowrap="" valign="bottom"> 0.083439 width="65" nowrap="" valign="bottom"> 0.082383

.7,.15,.15 width="57" nowrap="" valign="bottom"> 0.080061 width="57" nowrap="" valign="bottom"> 0.07777 width="60" nowrap="" valign="bottom"> 0.080993 width="57" nowrap="" valign="bottom"> 0.077945 width="57" nowrap="" valign="bottom"> 0.085759 width="57" nowrap="" valign="bottom"> 0.132182 width="57" nowrap="" valign="bottom"> 0.084693 width="65" nowrap="" valign="bottom"> 0.082383

.7,.1,.2 width="57" nowrap="" valign="bottom"> 0.081031 width="57" nowrap="" valign="bottom"> 0.077755 width="60" nowrap="" valign="bottom"> 0.08124 width="57" nowrap="" valign="bottom"> 0.078455 width="57" nowrap="" valign="bottom"> 0.085759 width="57" nowrap="" valign="bottom"> 0.132182 width="57" nowrap="" valign="bottom"> 0.085809 width="65" nowrap="" valign="bottom"> 0.083516

.75,.2,.05 width="57" nowrap="" valign="bottom"> 0.079361 width="57" nowrap="" valign="bottom"> 0.07777 width="60" nowrap="" valign="bottom"> 0.080882 width="57" nowrap="" valign="bottom"> 0.077945 width="57" nowrap="" valign="bottom"> 0.083213 width="57" nowrap="" valign="bottom"> 0.132182 width="57" nowrap="" valign="bottom"> 0.083439 width="65" nowrap="" valign="bottom"> 0.082383

.75,.15,.1 width="57" nowrap="" valign="bottom"> 0.080061 width="57" nowrap="" valign="bottom"> 0.077755 width="60" nowrap="" valign="bottom"> 0.08124 width="57" nowrap="" valign="bottom"> 0.078031 width="57" nowrap="" valign="bottom"> 0.085759 width="57" nowrap="" valign="bottom"> 0.132182 width="57" nowrap="" valign="bottom"> 0.084693 width="65" nowrap="" valign="bottom"> 0.082383

.75,.1,.15 width="57" nowrap="" valign="bottom"> 0.082619 width="57" nowrap="" valign="bottom"> 0.077755 width="60" nowrap="" valign="bottom"> 0.081532 width="57" nowrap="" valign="bottom"> 0.078835 width="57" nowrap="" valign="bottom"> 0.085759 width="57" nowrap="" valign="bottom"> 0.132182 width="57" nowrap="" valign="bottom"> 0.085809 width="65" nowrap="" valign="bottom"> 0.083516

.75,.05,.2 width="57" nowrap="" valign="bottom"> 0.084571 width="57" nowrap="" valign="bottom"> 0.077755 width="60" nowrap="" valign="bottom"> 0.082224 width="57" nowrap="" valign="bottom"> 0.079838 width="57" nowrap="" valign="bottom"> 0.09076 width="57" nowrap="" valign="bottom"> 0.132182 width="57" nowrap="" valign="bottom"> 0.090362 width="65" nowrap="" valign="bottom"> 0.092941

.8,.05,.15 width="57" nowrap="" valign="bottom"> 0.088195 width="57" nowrap="" valign="bottom"> 0.077755 width="60" nowrap="" valign="bottom"> 0.08237 width="57" nowrap="" valign="bottom"> 0.081885 width="57" nowrap="" valign="bottom"> 0.09076 width="57" nowrap="" valign="bottom"> 0.132182 width="57" nowrap="" valign="bottom"> 0.095644 width="65" nowrap="" valign="bottom"> 0.092941

.8,.1,.1 width="57" nowrap="" valign="bottom"> 0.082619 width="57" nowrap="" valign="bottom"> 0.077755 width="60" nowrap="" valign="bottom"> 0.081765 width="57" nowrap="" valign="bottom"> 0.079116 width="57" nowrap="" valign="bottom"> 0.089927 width="57" nowrap="" valign="bottom"> 0.132182 width="57" nowrap="" valign="bottom"> 0.087268 width="65" nowrap="" valign="bottom"> 0.085545

.8,.15,.05 width="57" nowrap="" valign="bottom"> 0.082619 width="57" nowrap="" valign="bottom"> 0.077755 width="60" nowrap="" valign="bottom"> 0.081765 width="57" nowrap="" valign="bottom"> 0.079116 width="57" nowrap="" valign="bottom"> 0.089927 width="57" nowrap="" valign="bottom"> 0.132182 width="57" nowrap="" valign="bottom"> 0.087268 width="65" nowrap="" valign="bottom"> 0.085545

.85,.05,.1 width="57" nowrap="" valign="bottom"> 0.088195 width="57" nowrap="" valign="bottom"> 0.077765 width="60" nowrap="" valign="bottom"> 0.082595 width="57" nowrap="" valign="bottom"> 0.085051 width="57" nowrap="" valign="bottom"> 0.096161 width="57" nowrap="" valign="bottom"> 0.132182 width="57" nowrap="" valign="bottom"> 0.103502 width="65" nowrap="" valign="bottom"> 0.092941

.85,.1,.05 width="57" nowrap="" valign="bottom"> 0.088195 width="57" nowrap="" valign="bottom"> 0.077765 width="60" nowrap="" valign="bottom"> 0.082595 width="57" nowrap="" valign="bottom"> 0.085051 width="57" nowrap="" valign="bottom"> 0.096161 width="57" nowrap="" valign="bottom"> 0.132182 width="57" nowrap="" valign="bottom"> 0.103502 width="65" nowrap="" valign="bottom"> 0.092941

.9,.05,.05 width="57" nowrap="" valign="bottom"> 0.10324 width="57" nowrap="" valign="bottom"> 0.077763 width="60" nowrap="" valign="bottom"> 0.082669 width="57" nowrap="" valign="bottom"> 0.092433 width="57" nowrap="" valign="bottom"> 0.097842 width="57" nowrap="" valign="bottom"> 0.132182 width="57" nowrap="" valign="bottom"> 0.118835 width="65" nowrap="" valign="bottom"> 0.102265

Comb/H.N width="57" nowrap="" valign="bottom"> N15 width="57" nowrap="" valign="bottom"> N20 width="60" nowrap="" valign="bottom"> N25 width="57" nowrap="" valign="bottom"> N30 width="57" nowrap="" valign="bottom"> N35 width="57" nowrap="" valign="bottom"> N40 width="57" nowrap="" valign="bottom"> N45 width="65" nowrap="" valign="bottom"> N50

.6,.2,.2 width="57" nowrap="" valign="bottom"> 0.132588 width="57" nowrap="" valign="bottom"> 1.367142 width="60" nowrap="" valign="bottom"> 1.341733 width="57" nowrap="" valign="bottom"> 0.476854 width="57" nowrap="" valign="bottom"> 1.738251 width="57" nowrap="" valign="bottom"> 0.9577 width="57" nowrap="" valign="bottom"> 0.742921 width="65" nowrap="" valign="bottom"> 0.967878

.65, .15, .2 width="57" nowrap="" valign="bottom"> 0.132588 width="57" nowrap="" valign="bottom"> 1.367142 width="60" nowrap="" valign="bottom"> 1.341733 width="57" nowrap="" valign="bottom"> 0.476854 width="57" nowrap="" valign="bottom"> 1.738251 width="57" nowrap="" valign="bottom"> 0.9577 width="57" nowrap="" valign="bottom"> 0.742921 width="65" nowrap="" valign="bottom"> 0.967878

.65,.2,.15 width="57" nowrap="" valign="bottom"> 0.132588 width="57" nowrap="" valign="bottom"> 1.367142 width="60" nowrap="" valign="bottom"> 1.341733 width="57" nowrap="" valign="bottom"> 0.476854 width="57" nowrap="" valign="bottom"> 1.738251 width="57" nowrap="" valign="bottom"> 0.9577 width="57" nowrap="" valign="bottom"> 0.742921 width="65" nowrap="" valign="bottom"> 0.967878

.7,.2,.1 width="57" nowrap="" valign="bottom"> 0.132588 width="57" nowrap="" valign="bottom"> 1.367142 width="60" nowrap="" valign="bottom"> 1.341733 width="57" nowrap="" valign="bottom"> 0.476854 width="57" nowrap="" valign="bottom"> 1.738251 width="57" nowrap="" valign="bottom"> 0.9577 width="57" nowrap="" valign="bottom"> 0.742921 width="65" nowrap="" valign="bottom"> 0.967878

.7,.15,.15 width="57" nowrap="" valign="bottom"> 0.130603 width="57" nowrap="" valign="bottom"> 1.367142 width="60" nowrap="" valign="bottom"> 1.341733 width="57" nowrap="" valign="bottom"> 0.476854 width="57" nowrap="" valign="bottom"> 1.738251 width="57" nowrap="" valign="bottom"> 0.9577 width="57" nowrap="" valign="bottom"> 0.742921 width="65" nowrap="" valign="bottom"> 0.967878

.7,.1,.2 width="57" nowrap="" valign="bottom"> 0.192835 width="57" nowrap="" valign="bottom"> 1.367142 width="60" nowrap="" valign="bottom"> 1.341733 width="57" nowrap="" valign="bottom"> 0.476854 width="57" nowrap="" valign="bottom"> 1.738251 width="57" nowrap="" valign="bottom"> 0.9577 width="57" nowrap="" valign="bottom"> 0.742921 width="65" nowrap="" valign="bottom"> 0.967878

.75,.2,.05 width="57" nowrap="" valign="bottom"> 0.132588 width="57" nowrap="" valign="bottom"> 1.367142 width="60" nowrap="" valign="bottom"> 1.341733 width="57" nowrap="" valign="bottom"> 0.476854 width="57" nowrap="" valign="bottom"> 1.738251 width="57" nowrap="" valign="bottom"> 0.9577 width="57" nowrap="" valign="bottom"> 0.742921 width="65" nowrap="" valign="bottom"> 0.967878

.75,.15,.1 width="57" nowrap="" valign="bottom"> 0.178954 width="57" nowrap="" valign="bottom"> 1.367142 width="60" nowrap="" valign="bottom"> 1.341733 width="57" nowrap="" valign="bottom"> 0.476854 width="57" nowrap="" valign="bottom"> 1.738251 width="57" nowrap="" valign="bottom"> 0.9577 width="57" nowrap="" valign="bottom"> 0.742921 width="65" nowrap="" valign="bottom"> 0.967878

.75,.1,.15 width="57" nowrap="" valign="bottom"> 0.331235 width="57" nowrap="" valign="bottom"> 1.367142 width="60" nowrap="" valign="bottom"> 1.341733 width="57" nowrap="" valign="bottom"> 0.476854 width="57" nowrap="" valign="bottom"> 1.738251 width="57" nowrap="" valign="bottom"> 0.9577 width="57" nowrap="" valign="bottom"> 0.742921 width="65" nowrap="" valign="bottom"> 0.967878

.75,.05,.2 width="57" nowrap="" valign="bottom"> 0.719118 width="57" nowrap="" valign="bottom"> 1.367142 width="60" nowrap="" valign="bottom"> 1.341733 width="57" nowrap="" valign="bottom"> 0.476854 width="57" nowrap="" valign="bottom"> 1.738251 width="57" nowrap="" valign="bottom"> 0.9577 width="57" nowrap="" valign="bottom"> 0.91157 width="65" nowrap="" valign="bottom"> 0.967878

.8,.05,.15 width="57" nowrap="" valign="bottom"> 0.576865 width="57" nowrap="" valign="bottom"> 1.367142 width="60" nowrap="" valign="bottom"> 1.341733 width="57" nowrap="" valign="bottom"> 0.476854 width="57" nowrap="" valign="bottom"> 1.738251 width="57" nowrap="" valign="bottom"> 0.9577 width="57" nowrap="" valign="bottom"> 0.91157 width="65" nowrap="" valign="bottom"> 0.967878

.8,.1,.1 width="57" nowrap="" valign="bottom"> 0.474361 width="57" nowrap="" valign="bottom"> 1.367142 width="60" nowrap="" valign="bottom"> 1.341733 width="57" nowrap="" valign="bottom"> 0.476854 width="57" nowrap="" valign="bottom"> 1.738251 width="57" nowrap="" valign="bottom"> 0.9577 width="57" nowrap="" valign="bottom"> 0.742921 width="65" nowrap="" valign="bottom"> 0.967878

.8,.15,.05 width="57" nowrap="" valign="bottom"> 0.474361 width="57" nowrap="" valign="bottom"> 1.367142 width="60" nowrap="" valign="bottom"> 1.341733 width="57" nowrap="" valign="bottom"> 0.476854 width="57" nowrap="" valign="bottom"> 1.738251 width="57" nowrap="" valign="bottom"> 0.9577 width="57" nowrap="" valign="bottom"> 0.742921 width="65" nowrap="" valign="bottom"> 0.967878

.85,.05,.1 width="57" nowrap="" valign="bottom"> 0.636563 width="57" nowrap="" valign="bottom"> 1.367142 width="60" nowrap="" valign="bottom"> 1.341733 width="57" nowrap="" valign="bottom"> 0.476854 width="57" nowrap="" valign="bottom"> 1.738251 width="57" nowrap="" valign="bottom"> 0.9577 width="57" nowrap="" valign="bottom"> 0.91157 width="65" nowrap="" valign="bottom"> 0.967878

.85,.1,.05 width="57" nowrap="" valign="bottom"> 0.636563 width="57" nowrap="" valign="bottom"> 1.367142 width="60" nowrap="" valign="bottom"> 1.341733 width="57" nowrap="" valign="bottom"> 0.476854 width="57" nowrap="" valign="bottom"> 1.738251 width="57" nowrap="" valign="bottom"> 0.9577 width="57" nowrap="" valign="bottom"> 0.91157 width="65" nowrap="" valign="bottom"> 0.967878

.9,.05,.05 width="57" nowrap="" valign="bottom"> 0.656375 width="57" nowrap="" valign="bottom"> 1.367142 width="60" nowrap="" valign="bottom"> 1.341733 width="57" nowrap="" valign="bottom"> 0.476854 width="57" nowrap="" valign="bottom"> 1.738251 width="57" nowrap="" valign="bottom"> 1.355673 width="57" nowrap="" valign="bottom"> 0.91157 width="65" nowrap="" valign="bottom"> 0.967878

Table 4-7.  RMSE-1, RMSE-2, RMSE-3 and RMSE-4 at fifty nodes

From the above experiment we come to know about the optimal parameters which are reporting lowest root mean square error for all four lags.  The next step reports the non linear autoregressive rolling window analysis by using these optimal parameters for model construction for all four lags. The selected optimal parameters are used to run rolling window non-liner autoregressive neural network.  From this rolling analysis the lag series  which  report the lowest root mean square error will be selected as best lag for further analysis.

Non linear auto regressive neural network rolling window analysis

This second stage reports the results of performance measure errors through NAR neural network under rolling window analysis. A thirty six month rolling window is taken to run NAR neural network on all the four lags by using optimal parameters selected in the previous stage.  The optimal models to run rolling window analysis for the four lags are reported in table 8.  

Table 8. Optimal parameters

> > > >

Lag width="96"> Training ratio width="90"> Testing ratio width="108"> Validation ratio width="41"> Nodes width="51"> Layers width="89"> Window size

First lag width="96" valign="top"> 80 width="90" valign="top"> 15 width="108" valign="top"> 05 width="41" valign="top"> 03 width="51" valign="top"> 1 width="89" valign="top"> 36

Second lag width="96" valign="top"> 70 width="90" valign="top"> 15 width="108" valign="top"> 15 width="41" valign="top"> 1 width="51" valign="top"> 1 width="89" valign="top"> 36

Third lag width="96" valign="top"> 90 width="90" valign="top"> 05 width="108" valign="top"> 05 width="41" valign="top"> 02 width="51" valign="top"> 1 width="89" valign="top"> 36

Fourth lag width="96" valign="top"> 70 width="90" valign="top"> 10 width="108" valign="top"> 20 width="41" valign="top"> 04 width="51" valign="top"> 1 width="89" valign="top"> 36

The rolling window analysis enables us to capture the variations in stock return predictability. Rolling window analysis also proves that the chances of predictability is high at lag one, as the lags values are increased the forecasting ability of the market start vanishing. As shown in figure 6 the rolling root mean square error is moving upward as with the increase of lag value.  This observation might mean that all the information is absorbed by the market after one month of trading. From this analysis we can chose lag one as the best lag for further analysis. 

It is apparent in the figure 6 that the rolling root mean square error of all four lags is moving in cyclical fashion.   It elucidate that a cyclical fashion changing degree of efficiency is observed and there are periods in where the market is more efficient than others (Urquhart & McGroarty 2016).  The low level of performance measuring errors shows that the forecasting error is least and it’s relatively easy to forecast the return opportunities in these time periods. And the time periods, in which performance measuring errors are higher, it’s difficult to forecast the market. However EMH theory does not explain such cycles (Lo, 2005).

Figure 7 and 8

RMSE all lags and threshold point



Lag one is showing the lowest root mean square error so by introducing a threshold point on the performance measure errors of lag one captured thorough 36 months rolling window the picture becomes clearer. When the performance measure error is less than 0.05 it shows that the index was predictable at that time and can be concluded as week form inefficient. Figure 7 exhibits the movement of performance error against the .05 threshold point.

Table 9. Degree of predictability

> > > > >

Time period width="202" valign="top"> Performance measure error width="159" valign="top"> Rejection /acceptance

Feb03-Dec04 width="202" valign="top"> Large RMSE width="159" valign="top"> rejection

Jan05-Feb09 width="202" valign="top"> Low RMSE width="159" valign="top"> acceptance

Jan09-Nov11 width="202" valign="top"> High RMSE width="159" valign="top"> rejection

Dec11-April16 width="202" valign="top"> Low RMSE width="159" valign="top"> acceptance

May16-Dec18 width="202" valign="top"> Mix outcome width="159" valign="top"> Mix outcome

Table 9 reports the periods of predictability and no predictability. The period which is showing the high-performance measure error is rejecting the possibility for prediction. And the periods with low-performance measure error representing the time period in which there is no possibility to predict the future return. These patterns of predictability and no predictability are consistent with the findings of Kim et al. (2011), Alvarez-Ramirez et al. (2012), Urquhart and Hudson (2013), Noda (2016). It indicates that the markets follow the learning process, they learn from the environment and adapt accordingly. Markets are dependent on changing market conditions. 

Conclusion

This paper examines the degree of return predictability in the Pakistan stock market to test the new proposed framework of the adaptive market hypothesis. The monthly returns of the KSE-100 index from January 2000 to December 2018 are selected to investigate the changing patterns of return predictability. The degree of return predictability is measured by the nonlinear autoregressive neural network under the rolling window framework. In the two-stage methodology first, the parameters are selected for the optimal model on the bases of the lowest performance measure error. Selected parameters are used to model the optimal structure of the nonlinear autoregressive neural network model. 

In the second stage, the rolling nonlinear autoregressive neural network analysis is conducted. This analysis is used to track the time-varying efficiency and inefficiency consistent with AMH. The rolling window analysis is helpful to fully cover the dynamics of time series.  The results indicate that there are some periods in which the Pakistan stock market is showing a low level of predictability than others where the index is not predictable. This observation is consistent with AMH's evolving view of market efficiency.

The changing nature of predictability can further be elaborated by considering the market dynamics. Every market has its own characteristics on which its level of predictability is dependent. 

Figure 2-a



Figure 2-b



Figure 3-a



Figure 3-b



Figure 3-c



Figure 3-d



Figure 7



Figure 8

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