SAC 306PROJECT REPORT.

A Time Series Analysis of the Apple Inc. Stock Price.

PRESENTED BY

Malit Keldine Owande……………..………………………I07/2930/2009 June Agnes Njeri Mwangi…………………………………I07/28782/2009 Omondi Joseph Ochieng……..………………….………I07/2931/2009 Kyalo Lydia Mukonyo………………………………….……

I07/29037/2009 Kihuni Margaret Wanjiku …………………………..……I07/28988/2009    

PRESENTED TO; Dr. Ivivi Mwaniki  

DATE:11th May, 2012.

TABLE OF CONTENTSCHAPTER ONE: INTRODUCTION

1.1Background of the study 1.2 Research Objectives 1.3 Significance of the study 1.4 Scope of the study

CHAPTER TWO: LITERATURE REVIEW

2.1 Introduction 2.2 Methodology 2.2.1 ARIMA Process 2.2.2 Stochastic Process 2.2.3 Spectral Density 2.2.4 Measures of central tendency 2.2.5 Autocorrelation function 2.2.6 Partial Autocorrelation Function

CHAPTER THREE: DATA ANALYSIS AND INTERPRATATION 3.1Introduction 3.2 Basic Statistics 3.3 Data Cleaning 3.4 Time Series Analysis 3.5 The Normal Q-Q Plot 3.6 Kernel Density Estimation 3.6 Residuals

CHAPTER 4: CONCLUSION REFERENCES

CHAPTER 1: INTRODUCTION1.1Background of the study In the financial field, and any field for that matter, Information is key in the decision

making process. Financial analysts, fund managers and individual investors aim to make numerous decisions ranging from what best to invest in, when best to invest and countless others. This decision making process is made complex by the fact that numerous factors, including social, political and economic factors, influence the investment market. It is therefore near impossible to predict market movements with a 100% certainty. A detailed scientific study helps to reduce the uncertainty and therefore investors are able to make well informed decisions. In our case of studying Apple Inc.’s stock options and stock price movements, a regression and time series analysis, will help especially with the identification of the main components; trend, seasonality, cyclical nature and distribution of random or irregular nature of the stock. From that we`ll be able to get the put-call parity and also predict future stock prices given past data.

Apple Inc. is an American multinational corporation that designs and sells consumer electronics, computer software, and personal computers. The company's best-known hardware products are the Macintosh line of computers, the iPod, the iPhone and the iPad.

It is the largest publicly traded company in the world by market capitalization, as well as the largest technology company in the world by revenue and profit, as of May 2012 with worldwide annual revenue of $108 billion in 2011.

Apple Inc. is headquartered in California. Its stock is listed on the New York Stock Exchange and is part of S&P 500 Index, the Russell 1000 Index and the Russell 1000 Growth Stock Index (NYSE, 2011).

1.2 Research Objectives The main objectives of the study were as follows; Carry out a time series analysis on a historical set of data. Figure out the trend,

cyclicality and come up with summary statistics on the data like the mean, median, mode, variance and the correlations between the values in the data set.

Investigate the residuals and find out whether they have a normal or non-normal distribution.

Fit a time series model and come up with a graphical representation of the data being analyzed and forecast future behaviour, using information and properties obtained from analysis of the residuals.

1.3 Significance of the study This study on the historical share prices will help in determining the trend of the

stock and forecast share prices. This information coupled with some knowledge of the stock market will give an investor more confidence on whether to buy or sell the stock.

1.4 Scope of the study The study has taken a sample of 3110 observations of the stock price of Apple

Inc. trading in the New York Stock Exchange. The data was observed from 1nd January 2000 to 10th May 2012. The stock price is the adjusted closing price of the stock.

2.1 IntroductionWe referred to a number of books and websites in the study of

Apple Inc. stock prices and options. These include:

Time Series with Applications in R by Jonathan D. Cryer and Kung-Sik Chan.

This provided relevant R codes for the analysis of the process.

CT6 Statistical Methods ActEd Notes, for an in-depth understanding of the various Time Series models.

Practical Regression and ANOVA using R by Julian Faraway, to help us with various codes used in analyzing and fitting the data.

http://www.marketwatch.com/optionscenter, to help us with understanding the operations of stock options and for provision of data

http://en.wikipedia.org/wiki/Apple_Inc. and www.apple.com for info on Apple in. and its stock options

www.finance.yahoo.com or the historical Apple inc.’s share price data.

CHAPTER TWO: LITERATURE REVIEW

2.2 Methodology

Studying of the autocorrelation function (ACF) and partial autocorrelation function (PACF) in order to fit an ARMA model of the data. Autocorrelation describes the correlation between different values of the process at different points in time, as a function of the two times or as a function of the time difference, if process is stationary.

Differencing of the data to eliminate the trend which the data takes, whether linear, quadratic or nonlinear.

Studying of the residual properties with the help of a QQ plot and histogram.

Smoothing of the data by an analysis of the trend, seasonality, cyclicality, and noise through the use of the spragga command.

Use of visual aids like histograms, boxplots and line graphs to help in our analysis.

Studying of frequency of data by making use of the spectral density frequency.

2.2.1 ARIMA Process ARIMA processes are mathematical models used for forecasting. The acronym

ARIMA stands for "Auto-Regressive Integrated Moving Average." The ARIMA approach to forecasting is based on the following ideas:

1. The forecasts are based on linear functions of the sample observations; 2. The aim is to find the simplest models that provide an adequate description

of the observed data. This is sometimes known as the principle of parsimony.

Each ARIMA process has three parts: the autoregressive (or AR) part; the integrated (or I) part; and the moving average (or MA) part. The models are often written in shorthand as ARIMA (p, d, q) where p describes the AR part, d describes the integrated part and q describes the MA part. (Pankratz, 1983)

AR: This part of the model describes how each observation is a function of the previous p observations. For example, if p = 1, then each observation is a function of only one previous observation. That is,

Y (t) = c + φ (1) Y (t−1) + e (t) Where Y (t) represents the observed value at time t, Y (t−1) represents the

previous observed value at time t − 1, e (t) represents some random error and c and φ (1) are both constants. Other observed values of the series can be included in the right-hand side of the equation if p > 1:

Y (t) = c + φ (1) Y (t−1) + φ (2) Y (t−2) + · · · + φ (p) Y (t−p) + e (t)

I: This part of the model determines whether the observed values are modeled directly, or whether the differences between consecutive observations are modeled instead. If d = 0, the observations are modeled directly. If d = 1, the differences between consecutive observations are modeled. If d = 2, the differences of the differences are modeled. In practice, d is rarely more than 2.

MA: This part of the model describes how each observation is a function of the previous q errors. For example, if q = 1, then each observation is a function of only one previous error. That is,

Y (t) = c + θ (1) e (t−1) + e (t). Here e (t) represents the random error at time t and e (t−1) represents the

previous random error at time t − 1. Other errors can be included in the right-hand side of the equation if q > 1.

Combining these three parts gives the diverse range of ARIMA models.

2.2.2 Stochastic Process The sequence of random variables {Yt : t = 0, ±1, ±2, ±3,…} is called a stochastic

process and serves as a model for an observed time series. It is known that the complete probabilistic structure of such a process is determined by the set of distributions of all finite collections of the Y’s.

2.2.3 Spectral Density

For estimating spectral densities of stationary seasonal time series processes, a new kernel is proposed. The proposed kernel is of the shape which is in harmony with oscillating patterns of the autocorrelation functions of typical seasonal time series process. Basic properties such as consistency and non-negativity of the spectral density estimator are vital. (Box, 1970)

2.2.4 Measures of central tendency

This includes mean, mode and median. Other basic statistics operations are skewness, kurtosis and variance.

The mode is the most frequently appearing value in the population or sample.

To find the median, we arrange the observations in order from smallest to largest value. If there are an odd number of observations, the median is the middle value. If there is an even number of observations, the median is the average of the two middle values.

The mean of a sample or a population is computed by adding all of the observations and dividing by the number of observations.

2.2.5 Autocorrelation function

In statistics, the autocorrelation of a random process describes the correlation between values of the process at different points in time, as a function of the two times or of the time difference. Let X be some repeatable process, and i be some point in time after the start of that process. (i may be an integer for a discrete-time process or a real number for a continuous-time process.) Then Xi is the value (or realization) produced by a given run of the process at time i. Suppose that the process is further known to have defined values for mean μi and variance σi2 for all times i. Then the definition of the autocorrelation between times s and t is

2.2.6 Partial Autocorrelation Function

Given a time series zt, the partial autocorrelation of lag k is the autocorrelation between zt and zt + k with the linear dependence of zt + 1 through to zt + k − 1 removed; equivalently, it is the autocorrelation between zt and zt − k that is not accounted for by lags 1 to k-1, inclusive.

Partial autocorrelation plots are a commonly used tool for model identification in Box-Jenkins models. Specifically, partial autocorrelations are useful in identifying the order of an autoregressive model. The partial autocorrelation of an AR(p) process is zero at lag p+1 and greater.

An approximate test that a given partial correlation is zero (at a 5% significance level) is given by comparing the sample partial autocorrelations against the critical region with upper and lower limits given by , where n is the record length (number of points) of the time-series being analyzed. This approximation relies on the assumption that the record length is moderately large (say n>30) and that the underlying process has a multivariate normal distribution.

CHAPTER THREE: DATA ANALYSIS AND INTERPRATATION

3.1Introduction This section describes how the study was conducted. It explains the

sample used and the time series analysis methods applied. This section also explains how the data was analyzed to produce the required information required for the study and graphical representation of analysis

3.2 Apple Inc.’s Stock Options Put-call parity analysis

1. Results of the estimates using R for stock with a maturity days of less than 100

 

1.1 LINEAR MODEL (c=Bo+B1p+B2k)

Estimation or r

Given that the estimate of Beta2 is -0.9983, the estimated r is 0.03653106

Estimate t-value Std.Error

B0 559.4097 101.79 6.945e+00

B1 0.9977 -71.57 2.621e-02

B2 -0.9983 54.44 2.240e-05

1.2 QUADRATIC MODEL (c=Bo+B1k+B2k2)

Residuals:

Coefficients:

Min 1Q Median 3Q Max

-38.813 -8.375 1.480 8.787 14.747

Estimate t-value Std.Error

B0 7.070e+02 101.79 6.945e+00

B1 -1.876e+00

-71.57 2.621e-02

B2 1.219e-03 54.44 2.240e-05

Residual standard error: 10.49 on 130 degrees of freedom Multiple R-squared: 0.9913, Adjusted R-squared: 0.9912 F-statistic: 7406 on 2 and 130 DF p-value: < 2.2e-16

The resulting quadratic fit is

c=7.070e+02+(-1.876e+00k)+ 1.219e-03k2

Figure 1.2 Plot of the quadratic model of Call price against strike price

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1.2.1 HYPOTHESIS TEST To test the hypothesis Ho: B2=0 Vs H1: B2≠0

We consider the p-value of the test statistic of B2. This p-value is very small (< 2.2e-16) as expected from its corresponding t-value

which is large (54.44) implying that the test is significant. i.e. Reject H0. Rejection of H0 implies that B2≠0 which implies that B2 is actually significant i.e. it is better to use the quadratic fit instead of the linear fit.

1.2.2 COEFFICIENT OF DETERMINATION () The adjusted R-squared is 0.9912, implying that 99.12% of the total variability is

explained by the model, which means that the very little variability is not explained (error) i.e 0.88%.

1.3 THE (Y=B0+B1X1+B2X2+e) MODEL

Coefficients:

The Summary is as follows:

Residual standard error: 0.01843 on 130 degrees of freedom Multiple R-squared: 0.9913, Adjusted R-squared: 0.9912

1.3.1 COEFFICIENT OF DETERMINATION () The adjusted R-squared is 0.9912, implying that 99.12% of the total variability

is explained by the model, which means that only 0.88% is not explained.

An therefore our (Y=B0+B1X1+B2X2+e) model is

y2=1.2421119+(-1.875733*x)+( 0.0012193*x1)

Intercept X x1

1.2421119 -1.8757353 0.0012193

Figure 1.3 Superimposed Plot of call price against strike price comparing the quadratic and the Y=B0+B1X+B2X2

The above graph (call value against strike price) compares the graphs of the model (Y=B0+B1X1+B2X2+e)(red) against the graph of the QUADRATIC(blue) model.

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1.4 BLACK SCHOLES MODELFigure 1.4.1 Plot of the black scholes model of call price against strike price

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Figure 1.4.2 Superimposed plot of the black scholes, quadratic, and Y=B0+B1X+B2X2+e model of call price against strike price

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2. Results of the estimates using R for stock with a maturity days of more than 100

1.1 LINEAR MODEL (c=Bo+B1p+B2k)

Estimation or r

Given that the estimate of Beta2 is -1.004 , the estimated r is -0.007065742

Estimate t-value Std.Error

B0 558.200 1294.7 0.431159

B1 1.008 419.6 0.002402

B2 -1.004 -824.3 0.001218

1.2 QUADRATIC MODEL (c=Bo+B1k+B2k2)

Residuals:

Coefficients:

Min 1Q Median 3Q Max

-11.9776 -3.5845 0.0835 4.2055 6.0975

Estimate t-value Std.Error

B0 617.2 333.8 1.849

B1 -1.47 -191.3 7.686e-03

B2 8.762e-04 124.9 7.017e-06

Residual standard error: 4.313 on 148 degrees of freedom Multiple R-squared: 0.9989, Adjusted R-squared: 0.9989 F-statistic: 7.004e+04 on 2 and 148 DF p-value: < 2.2e-16

The resulting quadratic fit is

c=617.2+(-1.47k)+ 8.762e-04k2

Figure 1.2 Plot of the quadratic model of Call price against strike price

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1.2.1 HYPOTHESIS TEST To test the hypothesis Ho: B2=0 Vs H1: B2≠0

We consider the p-value of the test statistic of B2. This p-value is very small (< 2.2e-16) as expected from its corresponding t-value which is large

(124.9) implying that the test is significant. i.e. Reject H0. Rejection of H0 implies that B2≠0 which implies that B2 is actually significant i.e. it is better to use the quadratic fit instead of the linear fit.

1.2.2 COEFFICIENT OF DETERMINATION () The adjusted R-squared is 0.9989, implying that 99.89% of the total variability is explained by

the model, which means that the very little variability is not explained (error) i.e 0.11%.  

1.3 THE (Y=B0+B1X1+B2X2+e) MODELCoefficients:

The Summary is as follows:

Residual standard error: 0.01965 on 148 degrees of freedom Multiple R-squared: 0.9932, Adjusted R-squared: 0.9931

1.3.1 COEFFICIENT OF DETERMINATION () The adjusted R-squared is 0.9931, implying that 99.31% of the total variability

is explained by the model, which means that only 0.69% is not explained.

An therefore our (Y=B0+B1X1+B2X2+e) model is

y2=0.9496+(-0.9011*x)+(4.449e-10*x1)

Intercept X x1

0.9496 -0.9011 4.449e-10

Figure 1.3 Superimposed Plot of call price against strike price comparing the quadratic and the Y=B0+B1X+B2X2

The above graph (call value against strike price) compares the graphs of the model (Y=B0+B1X1+B2X2+e)(red) against the graph of the QUADRATIC(blue) model.

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1.4 BLACK SCHOLES MODELFigure 1.4.1 Plot of the black scholes model of call price against strike price

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Strike Price

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Figure 1.4.2 Superimposed plot of the black scholes, quadratic, and Y=B0+B1X+B2X2+e model of call price against strike price

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Strike Price

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3.4 Basic Statistics The stock prices fluctuated from a minimum price of $6.56 and maximum price of $ 636.23. There are

3109 observations. Using R, we obtained the following information

 

The standard deviation for the data is 8.391015. The mode of the data is 40.98. Kurtosis is a measure of whether the data are peaked or flat relative to a normal

distribution. The kurtosis of the data is -0.23948. Skewness is a measure of symmetry of the data. The skewness of the data is 0.728988.

Thus positive skewness shows that the tail on the right side is longer than that on the left and that most of the values lie on the left as indicated on the histogram.

3.3 Data Cleaning

Data Cleaning refers to the process used to determine inaccurate, incomplete, or unreasonable data and then improving the quality through correction of detected errors and omissions. The process may include format checks, completeness checks, reasonableness checks, limit checks, review of the data to identify outliers or other errors, and assessment of data. These processes usually result in flagging, documenting and subsequent checking and correction of suspect records.

Data cleaning was done by the use of histograms and box plots as shown below:

mean(y) 114.922

median(y) 64.98

Figure3.1 Histogram of the movement of prices for the past 13 years

Histogram of Apple inc. stock prices

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As shown in the histogram, figure 3.1, most values lie on the left and the data has a positive skewness.

The spread of the data can be better observed in the box plot.

A box plot or a box-and-whisker diagram is ideal for identifying any outliers. Since our data set is obtained from real observations of the stock market and all the recordings are true we would consider outliers. This is because there are many factors that affect the stock market and a sudden dip or peak in the market is a real factor to consider in our analysis.

 

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Boxplot of Apple inc. stock prices

Figure 3.2 Boxplot of the price movements

3.4 Time Series Analysis Our data is obtained through real observations of the New York Stock

Exchange that runs on Monday through Friday between 9.30am and 4.00pm with holiday exceptions. The purpose of time series analysis is generally twofold: to understand or model the stochastic mechanism that gives rise to an observed series and to predict or forecast the future values of a series based on the history of that series and, possibly, other related series or factors.

Using R we were able to construct a time plot of our findings as in figure 3.3

Apple Inc. share Close prices Jan 2000 to May 2012

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Figure 3.3. Graph of movement of adjusted closing price for 12 year period

The red line indicates the mean of the distribution of the prices. From the time series plot we can see the price fluctuation of the stock which has a general rise in prices.

The ACF and PACF of our time series is shown below

Figure 3.4 The ACF of Apple inc. stock price

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ACF of Apple inc. stock prices

The ACF gradually declines with increase in lag period, indicating presence of trend. Thus we can infer that the data is not independent, hence it is not stationary.

The PACF, as shown below, further proves the existence of trend.

Figure 3.5 PACF of Apple inc. Stock Price

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PACF of Apple inc. stock prices

So as to perform any operations on our data we need to make sure it is stationary (A time series with mean and variance which are constant over time)

A stationarized series is relatively easy to predict: we simply predict that its statistical properties will be the same in the future as they have been in the past. The predictions for the stationarized series can then be "untransformed," by reversing the mathematical transformations were previously used, to obtain predictions for the original series. Thus, finding the sequence of transformations needed to stationarize a time series often provides important clues in the search for an appropriate forecasting model.

We begin by logging our data to remove any component on none linearity and obtaining the first difference of the logged data.

Figure 3.6 Graphical representations of differenced data.

Fisrt Differenced Data

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Figure 3.7 ACF of first differenced logged data

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ACF of First Difference

Figure 3.8 PACF of first differenced logged data

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PACF of Fisrt Difference

From our ACF and PACF it is evidently clear that our data is not yet stationary as the pattern in our ACF does not show gradual exponential decay. We will then need to get the second difference of our logged data

The second difference will give us stationary data

 

Figure 3.9 Distribution of second differenced logged data

Second Differenced Data

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Figure 3.10 ACF of second differenced logged data (Stationary data)

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ACF of Second Difference

Figure 3.11 PACF of second differenced logged data (Stationary data)

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PACF of Second Difference

From our PACF it is evidently clear that our data is now stationary as evidenced by the exponential decay in the PACF correlogram and the spike t lag 1 in the ACF

correlogram.Figure 3.12. Histogram of stationary data

Histogram of Stationary data

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We can observe the box plot of our new detrended data. This will give us the spread of our stationary data.

Figure 3.13 Box plot of detrended data

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Boxplot of detrended data

Fitting the ARIMA(p,d,q) for Apple inc. Stock Price

We then estimate the order of our ARIMA model by trying different estimates of terms (orders of the process) and trying to determine what works best. One way of doing this is by choosing the number of terms that produces the least AIC (Akaike Information Criterion) value.

ARIMA(p,d,q) Variance Log likelihood AIC

arima(0,2,0) 0.02047 1631.67 -3261.34

arima(1,2,0) 0.008622 2973.46 -5942.91

arima(2,2,0) 0.005715 3611.48 -7216.95

arima(3,2,0) 0.0042 4089.11 -8170.21

arima(0,2,1) 0.006106 3505.53 -7007.06

arima(0,2,2) 0.002015 5220.01 -10434.02

arima(0,2,3) 0.001119 6133.22 -12258.44

arima(1,2,1) 0.0033 4460.29 -8914.59

arima(1,2,2) 0.001536 5645.14 -11282.27

arima(1,2,3) 0.00172 5466.44 -10922.88

arima(2,2,2) 0.001314 5882.15 -11754.30

arima(3,2,2) 0.00129 5912.4 -11812.80

arima(2,2,3) 0.001562 5616.94 -11221.89

arima(3,2,3) 0.001318 5878.33 -11742.67

As in the table above, the best fit model that gives the lowest variance is the ARIMA (0, 2, 3) model.

The results are as follows

Coefficients Standard error Variance Log-likelihood AICma1 ma2 ma3-2.7530 2.5100 -0.7568 0.0051 0.0126 0.0077 0.001119 6133.22 -12258.44

Forecasting using the ARIMA(0,2,3) model

The next step is to forecast data. Having fitted ARIMA(0,2,3), we use R to predict the share prices for the next 10 days with an error measurement. Our results are as follows;

Time Predicted value Standard Error

11/05/2012 571.1766 3.405162

12/05/2012 571.8284 4.836982

13/05/2012 572.4819 5.933826

14/05/2012 573.1353 6.862424

15/05/2012 573.7887 7.684031

16/05/2012 574.4422 8.430000

17/05/2012 575.0956 9.118922

18/05/2012 575.7491 9.762893

19/05/2012 576.4025 10.370299

3.5 Analysis of The residuals

From the fitted model, we analyze the residuals for any signs of non-randomness. We outline the results of our diagnostic check below. The diagram shows a plot of the residuals, the autocorrelation of the residuals and the p-values of the Ljung-box statistic for the first 10 lags.

Standardized Residuals

Time

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p values for Ljung-Box statistic

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As seen above, the ACF of the residuals shows that the residuals do not have any serial dependency (are random) and hence we can conclude that our ARIMA model is adequate. We can therefore assume that the residuals are not correlated and that they are normally distributed

3.5 The Normal Q-Q Plot A Q-Q plot is a probability plot, which is a graphical method for

comparing two probability distributions by plotting their quantiles against each other. First, the set of intervals for the quantiles are chosen. A point (x,y) on the plot corresponds to one of the quantiles of the second distribution (y-coordinate) plotted against the same quantile of the first distribution (x-coordinate). Thus the line is a parametric curve with the parameter which is the (number of the) interval for the quantile

The points plotted in a Q-Q plot are always non-decreasing when viewed from left to right. If the two distributions being compared are identical, the Q-Q plot follows the 45° line y = x. If the two distributions agree after linearly transforming the values in one of the distributions, then the Q-Q plot follows some line, but not necessarily the line y = x. If the general trend of the Q-Q plot is flatter than the line y = x, the distribution plotted on the horizontal axis is more dispersed than the distribution plotted on the vertical axis.

We compared our sample quartiles with a theoretical quartile.

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Normal Q-Q Plot

Theoretical Quantiles

Sa

mp

le Q

ua

ntile

sFigure 3.16 Normal Q-Q plot of detrended data

We tested our data for normality and we concluded that our data follows a fairly normal distribution as evidenced by the diagram above where most of the plotted points lie on the line.

3.6 Kernel Density Estimation

Kernel density estimation is a non-parametric way of estimating the probability density function of a random variable. Kernel density estimation is a fundamental data smoothing problem where inferences about the population are made, based on a finite data sample.Let (x1, x2, …, xn) be an identically and independent sample drawn from some distribution with an unknown density ƒ. We are interested in estimating the shape of this function ƒ. Its kernel density estimator is

where K(·) is the kernel — a symmetric but not necessarily positive function that integrates to one — and h > 0 is a smoothing parameter called the bandwidth.

Figure 3.17 Kernel density estimation of differenced data

-0.5 0.0 0.5

02

46

81

01

2

Kernel density estimation

Differenced Data

De

nsity

The diagram shows that our residuals follow a normal distribution which is in tandem with our qqplot analysis

CHAPTER 4: CONCLUSION From the project study carried out, the forecasted information as given above can

be said to be credible by looking at the past trends.

The time series was made stationary after getting the second difference of the log of our data. We fitted an ARIMA (0, 2, 3) model that helped us forecast accurate future price of the Apple Inc. stock.

We can confidently state that the price of the Apple Inc. share will increase for the next 10 days and vary around 574.4422, with the highest price in the 10 days expected to be 577.0560 and the lowest expected to be 571.1766

REFERENCES

Box, G.E.P. and G.M. Jenkins (1970) Time series analysis: Forecasting and Control, San Francisco: Holden-Day.

Makridakis, S., S.C. Wheelwright, and R.J. Hyndman (1998) Forecasting:

Methods and Applications, New York: John Wiley & Sons.

Yahoo finance. The apple Inc. [online] (Updated 14th May 2012) Available at http://finance.yahoo.com/q/hp?s=AAPL&a=00&b=1&c=2007&d=04&e=10&f=2012&g=d

Pankratz, A. (1983) Forecasting with Univariate Box–Jenkins models: concepts and cases, New York: John Wiley & Sons.