Finished Assignment 1

8/2/2019 Finished Assignment 1

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Assignment #1 1

Economic Forecasting and Analysis

Assignment #1

Song Li

Schulich School of Business, York University


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Assignment #1 2

Reference: http://finance.yahoo.com/q/hp?s=AAPL

Apple Inc. ( AAPL ) Historical prices from Dec 2004 to June 2010 ( see attached appendix)

1. To interpret the AAPL historical price from Dec 2004 to June 2010, I have retrieved its

monthly historical price records from Yahoo Finance. This sample with 67 selectedmonthly prices represents the general performance of the AAPL during this period. 118.55

is the mean value of 67 sample prices which is obtained by divide the total amount of the

sample price by number 67. Standard Deviation of 61.40 indicates the unit of measurement

for measuring distance and variation from the mean. Standard Deviation on the rate of

return is the measure of the volatility of an investment. AAPL Standard Deviation on the

rate of return is 0.117 from the mean of 0.038 which indicates a high volatility. The lowest

stock price during this period is 32.20 and the highest price is 261.09. The lowest rate of

return is minus 33% and highest rate of return is 24%. The mean return is 3.8%.

Both sets of data do not follow perfect normal distribution. The stock price has positiveskew which means that the mass of the distribution is concentrated on the left of the figure

and the curved has been pulled to the right by several extremely large numbers in the right.

The mean is bigger than the median. It means that the stock price is significantly higher in

some years. The stock return has negative skew which means that the mass of distribution

is concentrated on the right of the figure and the curved has been pulled to the left by

several extremely low numbers in the left. The mean is smaller than the median. The stock

had several really bad returns during these years.

95% Confidence interval for mean lie between 103.57 and 133.52 which means that with

95% confidence that the mean of the population ( population here refers to the stock priceat any time during the indicated period) is between 103.57 and 133.52. Same concept can

be applied to the confidence interval to the mean rate of return. There is 95% confidence

level that the mean return of the stock at any point in time is between 0.99% and 6.75%.

The box plot is used to display the distributional characteristics of the data. The stock price

does not have any outliers. However the stock return has a few outliers which indicate that

there are several significantly larger negative returns than the rest. Quartiles divide a set of

data into four equal parts. Therefore Q1=67.96 means that 25% of the stock price will be

less than 67.96 and Q3=168.21 means that 75% of the stock price will be less than 168.21.

P value < 0.005 for stock price summary which shows that P is significantly less than 0.05,

Ho needs to be rejected. It once again confirms that the data is not normally distributed.
http://finance.yahoo.com/q/hp?s=AAPLhttp://finance.yahoo.com/q/hp?s=AAPL


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Assignment #1 3

250200150100500

14

12

10

8

6

4

2

0

Pr i ce

Fequency

Mean 118.5

StDev 61.40

N 67

H i s t o g r am o f P r i c eNormal

25020015010050

Median

Mean

1401301201101009080

1st Quartile 67.96

M edian 105.12

3rd Quarti le 168.21

Maximum 261.09

103.57 133.52

85.34 135.37

52.48 74.01

A -S quared 1.16

P -V alue < 0.005

Mean 118.55

StDev 61.40

Variance 3770.33

Skewness 0.540878

Kurtosis -0.650723

N 67

M inimum 32.20

A nderson-Darling Normality T est

95% C onfidence Interval for Mean

95% C onfidence Interval for Median

95% C onfidence Interv al for StDev9 5 % C o n f id e nc e I n t e r v a ls

S u m m a r y f o r P r i c e


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Assignment #1 4

Descriptive Statistics: Price

Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3

Price 67 0 118.55 7.50 61.40 32.20 67.96 105.12 168.21

Variable Maximum

Price 261.09

0.240.12-0.00-0.12-0.24

Median

Mean

0.080.060.040.020.00

1st Q uartile -0.044046Median 0.058008

3rd Q uartile 0.124339

Maximum 0.237701

0.009922 0.067512

0.014398 0.081817

0.100001 0.141400

A -S quared 0.66

P-V alue 0.081

M ean 0.038717

S tD ev 0.117132

Variance 0.013720

Skewness -0.83611

Kurtosis 1. 08264

N 66

Minimum -0.329558

A nderson-Darling Normality T est

95% C onfidence Interval for Mean

95% C onfidence Interval for Median

95% C onfidence Interv al for StDev9 5 % C o n f id e nc e I n t e r v a ls

S u m m a r y f o r R e t u r n

Descriptive Statistics: return

Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3

return 66 0 0.0387 0.0144 0.1171 -0.3296 -0.0440 0.0580 0.1244

Variable Maximum

return 0.2377


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Assignment #1 5

635649423528211471

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Index

Prce

T ime S e r i e s P l o t o f P r i c e

The time series plot of the price shows some smooth upward trend since end of year 2004

to approximately end of year 2006. Since 2007 to end of year 2008 it has some significant

peaks and valleys. Starting from 2009 onwards, it resumes a smooth upward trend again.

However the plot in general shows an upward trend.


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Assignment #1 6

2. This selected set of data has a very strong linear trend. The trend equation Yt = 26.56 +

2.71*t. The slope of this trend equation indicates that the stock prices are estimated to

increase an average of 2.71 each month. As the following figure shows the straight-line

trend fitted to the actual data with a reasonable MSD of 976.

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Index

Prce

MAPE 19.320

MAD 22.285

MSD 976.538

Accuracy Measures

Actual

Fits

Variable

T rend Ana l y s i s P l o t f o r P r i c eLinear Trend Model

Yt = 26.56 + 2.71*t

3. By analyzing the autocorrelation of stock price, I have noticed that the autocorrelations for

the first four time lags are significantly different from zero and that the values then

gradually drop to zero. The Q statistic for 10 time lag is 253.12, which is greater than the

chi-square value 18.3 (the upper 0.05 point of a chi-square distribution with 10 degrees of

freedom.) This result indicates the autocorrelation of the first 10 lag as a group are

significantly different from zero. Based on this I can conclude that the data are highly

autocorrelated and exhibit a fairly strong trend like behaviour. Since the trend is so

overwhelming, it is very difficult to detect the seasonality.

In order to test its seasonality, the series are differenced and the percentage return of each

month stock price is calculated to remove the trend and to create a stationary series. As

shown in the following, the autocorrelations at each lag are almost close to zero. The LBQ

statistic for 10 lags is also relatively small, none of the autocorrelations falls out-side the

corresponding confidence limits, so there is little evidence to suggest the differenced data

are autocorrelated. Therefore the conclusion is, the stock return series shows no evidence

of a trend. The data are random with no seasonality.


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Assignment #1 7

10987654321

1.0

0.80.6

0.4

0.2

0.0

-0.2

-0.4

-0.6

-0.8

-1.0

La g

Autocorelaton

Au toco r re l a t i on Func t i on fo r p r i c e(with 5% significance limits for the autocorrelations)

Autocorrelation Function: price

Lag ACF T LBQ

1 0.914876 7.49 58.63

2 0.818941 4.10 106.33

3 0.717545 2.93 143.52

4 0.634952 2.31 173.10

5 0.570605 1.93 197.38

6 0.505797 1.62 216.77

7 0.432729 1.34 231.20

8 0.365107 1.10 241.64

9 0.295491 0.87 248.60

10 0.235950 0.69 253.12


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Assignment #1 8

10987654321

1.0

0.80.6

0.4

0.2

0.0

-0.2

-0.4

-0.6

-0.8

-1.0

La g

Autocorelaton

Au toco r re l a t i on Func t i on fo r r e tu rn(with 5% significance limits for the autocorrelations)

Autocorrelation Function: return

Lag ACF T LBQ

1 0.130771 1.06 1.18

2 0.068775 0.55 1.51

3 -0.080664 -0.64 1.98

4 -0.122039 -0.96 3.05

5 -0.044806 -0.35 3.20

6 -0.117848 -0.92 4.24

7 -0.068394 -0.53 4.60

8 0.013994 0.11 4.61

9 -0.101782 -0.78 5.43

10 0.050144 0.38 5.63


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Assignment #1 9

4. By comparing the following options (3 months MA, 10 months MA, and single exponential

smoothing), I concluded that the single exponential smoothing model fits my data set the

best with the lowest MSD. For the simple moving average, the smaller the number, the

larger the weight given to recent periods. The larger the number, the smaller the weight

given the more recent periods. Therefore the moving average of longer period (10MA)

tends to smooth out the peaks and valleys of the curve and therefore is the one which

deviates from my data set the most. The drawback of the moving average method is that it

assigns an equal weight to each past value involved in the average. It is usually used when

a series has stabilized and generally unchanged environment. It is not suitable for dealing

with trend and seasonality.

Exponential smoothing is often a good forecasting procedure when a non-random time

series exhibits trending behaviour. It generates forecast by using all the observations and

assigning weights that decline exponentially as the observations get older. In my analysis,

the system chooses the optimized Alpha, the smoothing constant, at 1.12. Alpha is a

weighting factor that determines the extent to which the current observation influences the

forecast of the next observation. Alpha 1.12 means that 120% of weighting is assigned

towards the most current observation. That explains why the exponential smoothing

follows the data sets more closely because todays stock price is more influenced by the

most recent stock price than the price in older time period.


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Assignment #1 10

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Index

Prce

Length 3

Moving Average

MAPE 14.083

MAD 17.078

MSD 499.905

Accuracy Measures

Actual

Fits

Variable

M o v i n g A v e r a g e P l o t f o r P r i c e

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Prce

Length 10

Moving Av erage

MAPE 24.00

MA D 31.62

MSD 1458.71

Accuracy Measures

Actual

Fits

Variable

M o v i n g A v e r a g e P l o t f o r P r i c e


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Index

Prce

Alpha 1.12563

Smoothing Constant

MAPE 9.571

MAD 10.920

MSD 252.328

Accuracy Measures

Actual

Fits

Variable

S mo o th i ng P l o t f o r P r i c eSingle Exponential Method

Single Exponential Smoothing for PriceData Price

Length 67

Smoothing Constant

Alpha 1.12563

Accuracy Measures

MAPE 9.571

MAD 10.920

MSD 252.328

------------------------------------------------------------------------------------

( Notes: the following drawing and descriptive data are optional and for additional

reference only.)


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Assignment #1 12

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Index

Prce

Alpha (level) 1.12033

Gamma (trend) 0.03385

Smoothing C onstants

MAPE 9.688

MAD 10.555

MSD 254.398

Accuracy Measures

Actual

Fits

Variable

S mo o th i ng P l o t f o r P r i c eDouble Exponential Method

Double Exponential Smoothing for PriceData Price

Length 67

Smoothing Constants

Alpha (level) 1.12033

Gamma (trend) 0.03385

Accuracy Measures

MAPE 9.688

MAD 10.555

MSD 254.398

Finished Assignment 1

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