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TS_Fall_2011_Lecture_1

TS_Fall_2011_Lecture_1 - FIN 271 FINANCIAL MODELING AND...

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FIN 271 FINANCIAL MODELING AND ECONOMETRICS TIME SERIES MODELING LECTURE SET 1 REFIK SOYER THE GEORGE WASHINGTON UNIVERSITY

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2 INTRODUCTION • WHAT IS A TIME SERIES (TS)? It is a sequence of observations which are ordered in time. ]] á ] 12 n , , , ; 1, 2, , n. ÊÖ]×>œ á > Examples : GNP, Dow-Jones Index, interest rates, stock returns, exchange rates, mortgage rates, temperature, brain waves (electroencephographic-EEG), etc. • TS plots: Monthly housing starts in the US 30 40 50 60 70 80 90 100 110 120 130 140 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 1
3 • Seasonal patterns: seasonal box plots HSTARTS By MONTH 30 40 50 60 70 80 90 100 110 120 130 140 1 2 3 4 5 6 7 8 9 10 11 12 MONTH • Random sample versus a TS sample What happens when you reorder the observations in a random sample ? What happens when you reorder the observations in a TS sample ? 2

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4 IMPORTANT FEATURES OF TS a) observations arrive according to some order. b) in practice measurements are made in discrete (equispaced) time intervals, in principle some quantities such as voltage or temperature can be measured continuously. c) Measurements can be discrete or continuous; but for many applications they are assumed to consist of continuous values. d) observations are necessarily independent. They may be correlated and the not degree of correlation may depend on their positions in the sequence. closer ones might be more related to each other. Ê e) only series can be analyzed stable f) financial TS may have additional uncertainty due to volatility. - also skewness and heavy tails of asset returns TS. 3
5 HIGHER ORDER MOMENTS The first moment, mean, is a measure of location and the standard deviation is a scale parameter. For distributions other than normal, higher order moments are of interest. : The third central moment measures the asymmetry of with respect to its Skewness ] mean WÐ]Ñ œ I Ð]  Ñ Ó ’“ . 5 ] \$ \$ ] . Symmetry implies zero skewness; positive skewness indicates a long right tail and negative skewness indicates the opposite. Normal distribution has zero skewness. : The fourth central moment measures the tail thickness of O?<>9=3= ] OÐ]Ñ œ I Ð]  Ñ Ó . 5 ] % % ] . Kurtosis indicates the extent to which probability is concentrated in the tails. 4

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6 Normal distribution has ; the quantity is called the OÐ]Ñ œ \$ OÐ]Ñ  \$ excess kurtosis . A distribution with positive excess has heavier tails than the normal distribution. • Under normality, for large sample size, sample skewness, will have normal distribution with mean and variance , that is ! 'Î8 µRÐ! ß"Ñ WÐ]Ñ s 'Î8 È where . 1 1 _ WÐ]Ñœ Ð] ]Ñ s Ð8  Ñ= ] \$ 8 >œ" \$ " t • Under normality, for large sample size, sample kurtosis, will have normal distribution with mean and variance , that is \$ #%Î8 s #%Î8 È where . 1 _ OÐ]Ñœ s Ð8  Ñ= ] % 8 >œ" % " t 5
7 ESTIMATION USING SAS __________________________ DATA GE; INFILE "c:\TSDATA\ge.txt"; INPUT PRICE; LOGRETURN=DIF(LOG(PRICE)); PROC UNIVARIATE; VAR LOGRETURN; RUN; _____________________________ Moments N 252 Sum Weights 252 Mean -0.0000115 Sum Observations -0.0028973 Std Deviation 0.0175854 Variance 0.00030925 Skewness -0.1500695 Kurtosis 0.14401017 Uncorrected SS 0.0776214 Corrected SS 0.07762133 Coeff Variation -152954.42 Std Error Mean 0.00110778 Are the values of the sample skewness and excess kurtosis different than zero ?

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TS_Fall_2011_Lecture_1 - FIN 271 FINANCIAL MODELING AND...

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