Econ2209 Week 3

Business Forecasting ECON2209

Slides 03

Lecturer: Minxian Yang

BF-03 1 my, School of Economics, UNSW

Ch.4 Statistical Graphics

• Lecture Plan – Graphs of data: merits and limitations – Examples: use graphs to show data features – Time series differ from random sample – Components in time series – Classical decomposition

BF-03 my, School of Economics, UNSW 2


Statistical Graphics • Example: Anscombe’s quartet

– Why identical regression line?


obs X1 Y1 X2 Y2 X3 Y3 X4 Y4 1 10 8.04 10 9.14 10 7.46 8 6.58 2 8 6.95 8 8.14 8 6.77 8 5.76 3 13 7.58 13 8.74 13 12.74 8 7.71 4 9 8.81 9 8.77 9 7.11 8 8.84 5 11 8.33 11 9.26 11 7.81 8 8.47 6 14 9.96 14 8.10 14 8.84 8 7.04 7 6 7.24 6 6.13 6 6.08 8 5.25 8 4 4.26 4 3.10 4 5.39 19 12.5 9 12 10.84 12 9.13 12 8.15 8 5.56 10 7 4.82 7 7.26 7 6.42 8 7.91 11 5 5.68 5 4.74 5 5.73 8 6.89

(0.12) (1.12) 67.0 ,50.0 00.3ˆ 2 =⋅+= Rxy


• Example – Scatter plots explain it vividly.


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X1

Y1

Y1 vs. X1

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Y2

Y2 vs. X2

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Y3 vs. X3

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Y4 vs. X4


• Advantages of graphs – Graphs represent data visually and help us to see

data features/patterns. • “A graph is worth a thousand of words.”

– Graphs make anomalies/outliers apparent. – Graphs are effective in comparing data sets.

– But it is hard to visualise high dimensional data.



• Scatter plots Useful to reveal the relationship between two

variables. eg. “xyz.dat”


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X

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Y vs. X

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Z

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Y vs. Z

ls y c z ====================================================== Variable Coefficient Std. Error t-Statistic Prob. C 10.04891 0.272825 36.83280 0.0000 Z -0.364783 0.242923 -1.501641 0.1400

Weak relation? It could be

y = b0+b1z+b2x + u


• Scatter plots eg. “xyz.dat”. Partial relationship of y and z (after controlling for x):

regress y on c x to get resid01; regress z on c x to get resid02; scatter plot resid01 against resid02.


EViews: Type in top panel ls y c x On the result window, click Proc, Make Residual Series, resid01 (in Name for resid series), OK Type in top panel ls z c x On the result window, click Proc, Make Residual Series, resid02 (in Name for resid series), OK Type in top panel group grp resid02 resid01 grp.linefit Also compare the result of ls resid01 resid02 against the result of ls y c x z

Dependent Variable: Y, Sample: 1 48 Variable Coefficient Std. Error t-Statistic Prob. C 9.884732 0.190297 51.94359 0.0000 X 1.073140 0.150341 7.138031 0.0000 Z -0.638011 0.172499 -3.698642 0.0006

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RESID02

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RESID01 vs. RESID02


• Time series plot eg. “liquor.dat”, monthly sales, 1967.01 – 1994.12 trend (increase over time on average) seasonality (reoccurring pattern: Dec high, Feb low)


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68 70 72 74 76 78 80 82 84 86 88 90 92

Liquor Sales


• Time series plot eg. US 10-year treasury bond yield (monthly, %), 530obs persistent (gentle moves with few large jumps) random trend (ups & downs without a clear pattern)


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10-year T-bond Yield


• Time series plot eg. Change of US 10-year treasury bond yield (monthly) fluctuate about 0 (never move away for long) volatile (change direction frequently with large spikes)


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Change of 10-year T-bond Yield

1−−=∆ ttt yyy

EViews: Read in bond10y.csv File, Open, Foreign Data as Workfile, bond10y.csv (in File name), Open, Finish Type in top panel plot close genr dy=close-close(-1) plot dy hist dy


• Time series plot – Time series plots reveal

• trends • seasonalities • volatilities (mount of variation) • breaks (pattern changes) • outliers (unusual observations)

in data. – First thing in time series analysis: plotting data



• Histogram Describes how data are distributed. (frequency distribution) eg. Change of US 10-year treasury bond yield (monthly) Thick-tailed (caused by a small number of large jumps)


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-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Sample 1962M01 2006M02Observations 529

Mean 0.000926Median 0.010000Maximum 1.590000Minimum -1.880000Std. Dev. 0.348751Skewness -0.280792Kurtosis 6.565498

Jarque-Bera 287.1622Probability 0.000000

Change of 10-year T-bond Yield

Normal distribution: skewness = 0, kurtosis = 3. At the 5% level, reject normality if Jarque-Bera > 5.99.


• Empirical cumulative distribution (cdf) Another way to look at how data are distributed. eg. Change of US 10-year treasury bond yield (monthly)


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-1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2

Change

Prob

abilit

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Empirical CDF

80% of observations are

below 0.23


• QQ-plot Check how a theoretical distribution fits data. For a perfect fit, the QQ-plot is a straight line. eg. Change of US 10-year treasury bond yield: It appears non-normal.

data with 529 observations: (1/529)th quantile = -1.88

std normal distribution: (1/529)th quantile = -2.90


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Change

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Theoretical Quantile-Quantile

(-1.88, -2.90)


• EViews Commands for the examples


EViews: Read in bond10y.csv File, Open, Foreign Data as Workfile, bond10y.csv (in File name), Open, Finish Type in top panel genr dy=close-close(-1) plot close dy dy.line dy.hist dy.distplot cdf dy.qqplot scalar q80=@quantile(dy, .8) scalar qn529=@qnorm(1/529)


• Style of graphs – Easy to understand eg. indicate the meaning of axes Highlight the point you are trying to make – Informative eg. make self-explained graphs – Attractive eg. use of proper colours and symbols – Avoid “chart junk” eg. abuse of colours, shadings, grids,…



Decomposition of a Time Series • Time series versus random sample

– A random sample is a set of independent observations on a variable,

often collected at one point in time. eg. Income of a randomly-selected household Med-insurance status of a randomly-selected household

– A time series is a set of observations on a variable observed

over consecutive time intervals. eg. T-bond rate, gold price, retail sales, … (daily, annual,...)



• Features of economic time series data – Trend, seasonality, fluctuation/cycle – Autocorrelation (future is influenced by present) eg. Department stores turnover: 1982.04 – 1999.10 Trend (sales growing), Seasonality (peak & trough repeats) Cycle (random fluctuations)


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Retail Turnover ($M)

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log Retail Turnover


• Features of economic time series data eg. Gold price ($US per fine ounce, London 3pm, 1/80-11/99) - sub-samples very different; varying trends (randomly); - persistent with few large jumps


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Gold Price (USD/fine ounce, London 3pm)


• Trend, seasonality and cycle Let yt be a time series (observable). – Trend, mt, is the smoothly evolving part of yt. It represents the long-run movement of yt.

eg. Trend in log retail turnover


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log Retail Turnover: Trend


• Trend, seasonality and cycle – Seasonality, st, is the repetitive part of yt. It repeats over a fixed number of periods. eg. quartly seasonality repeats over 4 quarters. eg. Log retail turnover: raw – trend = detreded = seasonality + cycle


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log Retail Turnover: Trend & Seasonality

-.12-.08-.04.00.04.08.12

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Cycle Raw-Trend Seasonality


• Trend, seasonality and cycle – Cycle, xt, is the random fluctuation in yt, aka

irregular component. It is a RV for each t. eg. Interesting to know how xt and xt -1 are associated. eg. Log retail turnover: cycle = raw – trend – seasonality


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ntile

Theoretical Quantile-Quantile

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82 84 86 88 90 92 94 96 98

Cycle


• Classical decomposition of yt – CD is a model that splits the observable time

series, yt, into three unobserved components: trend (mt), seasonality(st), cycle(xt) – Additive decomposition

where p is the number of periods in a season. eg. p = 12 for monthly series – Multiplicative decomposition: Yt = Mt St Xt But log(Yt) has an additive decomposition.


,0)(E ,0 , ,1

===++= ∑=

++ t

p

iittpttttt xsssxsmy

Normalized to zero: so that each component is identified.


• Classical decomposition of yt e.g. Department stores turnover: 1982.04 – 1999.10


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82 84 86 88 90 92 94 96 98

log Retail Turnover

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82 84 86 88 90 92 94 96 98

Cycle

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82 84 86 88 90 92 94 96 98

Seasonality

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82 84 86 88 90 92 94 96 98

Trend

EViews: Read in data by clicking File, New, Workfile, Dated (in Workfile structure), Monthly (in Frequency), 1982:01 (in Start date), 1999:10 (in End date), OK; Proc (on Workfile window), Import, Import from file…, departstoresTurnover05.xls (in File name), Open, Finish, OK Time series plot of log sales by typing in top panel genr y=log(sales) y.line Generate MA trend genr ma12b=(y(-6)+y(-5)+y(-4)+y(-3)+y(-2)+y(-1)+y+y(1)+y(2)+y(3)+y(4)+y(5))/12 genr ma12f=(y(-5)+y(-4)+y(-3)+y(-2)+y(-1)+y+y(1)+y(2)+y(3)+y(4)+y(5)+y(6))/12 genr ma12=0.5*(ma12b+ma12f) De-trended series genr ydt=y-m12 plot y ma12 plot y ma12 ydt


• Data and EViews



• Summary – List the types of graphs we used today. – Why plotting data is important? – What are the major components in a time series? – What is the additive decomposition? – Why must the seasonal component sum to zero

over a season? – Why is the mean of cycle component normalised

to zero? BF-03 my, School of Economics, UNSW 26

Econ2209 Week 3

Documents

school of economics

panel ls y c x

panel ls z c x

zls y c z

resid01 regress z

result of ls resid01

regress y

data features time series