CS代写 ECON3206/5206 Financial Econometrics School of Economics, UNSW 1

Microsoft Word – TOPIC1notes.doc

Course materials subject to Copyright

UNSW Sydney owns copyright in these materials (unless stated otherwise). The material is subject to copy-
right under Australian law and overseas under international treaties. The materials are provided for use by
enrolled UNSW students. The materials, or any part, may not be copied, shared or distributed, in print or
digitally, outside the course without permission. Students may only copy a reasonable portion of the material
for personal research or study or for criticism or review. Under no circumstances may these materials be
copied or reproduced for sale or commercial purposes without prior written permission of UNSW Sydney.

Statement on class recording

To ensure the free and open discussion of ideas, students may not record, by any means, classroom lectures,
discussion and/or activities without the advance written permission of the instructor, and any such recording
properly approved in advance can be used solely for the students own private use.

WARNING: Your failure to comply with these conditions may lead to disciplinary action, and may give rise
to a civil action or a criminal offence under the law.

THE ABOVE INFORMATION MUST NOT BE REMOVED FROM THIS MATERIAL.

ECON3206/5206 Financial Econometrics School of Economics, UNSW 1

UNDERSTANDING FINANCIAL DATA

1. Introduction

Before building financial models, it is important to understand the empirical
characteristics of financial data. Some key empirical properties investigated here are:

(i) The shape of the empirical distribution of returns
(ii) Autocorrelation structure in the mean of returns
(iii) Autocorrelation structure in the variance of returns

The return on a stock with price tP and dividend tD is computed as

1log( ) log( )t t t tR P D P−= + −

In the case where dividends are incorporated in prices 1log( ) log( )t t tR P P−= − .

2. Descriptive Statistics

There exists a number of descriptive measures which can be used to summarize the
distribution of a financial time series 1 2{ , , , }TR R RK where tR is the return on an asset
(stock) at time t. We assume that this distribution is stationary, so that the distribution of
tR at each point in time is the same.

The descriptive statistics considered in this course are as follows:

2.1 Measure of Location (Mean)

The mean is computed as

If tR is the return on a stock, µ is the average return on the stock

2.2 Measures of Variation

Standard Deviation

The standard deviation of the return on a stock is computed as:

ECON3206/5206 Financial Econometrics School of Economics, UNSW 2

It can be interpreted as a measure of the stock’s risk.

The variance of the stocks return is computed as

and is also a measure of the stock’s risk.

Covariance

Let tX be the return on stock X and tY the return on stock Y. The covariance between the
returns on the two stocks is:

XY t x t y

The covariance measures the degree of association between tX and tY .

Correlation

The correlation between tX and tY is

where 1 1X Yρ− ≤ ≤ . The correlation coefficient is a dimensionless quantity and it also
measures the degree of association between tX and tY .

2.3 is computed as

For symmetric distributions, such as the normal distribution, there is no skewness. For
some distributions, however, high (low) values can be more common than low (high)
values. In this case the distribution is skewed to the left (right).

ECON3206/5206 Financial Econometrics School of Economics, UNSW 3

2.4 is computed as

An important stylised fact concerning financial data is that there are frequent extreme
observations in both tails of the empirical distribution of many financial series, which is
not consistent with the assumption of normality. For the normal distribution, K = 3. For
financial data, we frequently observe K > 3. This “excess kurtosis” is caused by the
“fatness” in the tails of the data distribution.

3. Predictability

3.1 Autocorrelation of Returns

Definition
Let tX be the return of an asset at time t. The autocorrelation between tX and

t jX − is estimated as

Distribution

Under the hypothesis of no autocorrelation, that is, 0 : 0H ρ = ,

)/1,0(~ TNrj ,

where T is the sample size.

A joint test of autocorrelation up to lag m, can be undertaken by using the Ljung-

Box statistic, ∑

)2()( , which is approximately )(2 mχ under H0.

Observation
Most empirical studies show that there is very little evidence of autocorrelation in
returns data so that there is very little evidence of dependence in the mean.

ECON3206/5206 Financial Econometrics School of Economics, UNSW 4

3.2 Autocorrelation of Squared Returns

This can be tested by computing both jr and the Ljung-Box statistic but with tX
replaced by 2tX , that is, using squared returns. It is common to denote the Ljung-Box
statistic when based on squared returns as )(mQxx .

Interpretation
Significant autocorrelation in squared returns reflects the volatility clustering
characteristically observed in returns; namely, large (small) changes in returns tend to be
followed by large (small) changes. As will be discussed later, significant autocorrelation
in squared returns is evidence of ARCH (Autoregressive Conditional Heteroscedasticity)
effects, that is, of a time-varying conditional variance in returns.

Observation
In contrast to the autocorrelation structure of returns, there is evidence of
significant autocorrelation in squared returns. This implies that returns are not
independent.

3.3 Application: Testing for Efficiency in Stock Returns

An important model used in finance to explain financial prices is based on the
efficient markets hypothesis. A market is said to be weakly efficient if the most recent
price reflects the available information. This implies that the price tP , of a financial asset
follows a random walk:

1t t tP P ε−= +

where tε is a disturbance term. Alternatively, the logarithmic form is

1log( ) log( )t t tP P µ−= +

where tµ is a disturbance term.

Implication
If the market is weakly efficient there should be no information contained in the
disturbance term tµ that is useful for predicting 1+tµ .

This suggests that a simple test of weak efficiency is to compute the returns

ECON3206/5206 Financial Econometrics School of Economics, UNSW 5

1log( ) log( )t t tP Pµ −= −

and test for autocorrelation. If there is no significant autocorrelation, this provides
support for the efficient markets hypotheis.

Application
The data are the log returns on the NYSE Composite Index, expressed as a
percentage (by multiplying the daily log return by 100). The sample period is from
3/1/1995 to 30/8/2002, a total of 1931 observations. A graph of the data is shown below.
The graph shows volatility clustering in the sense that large movements in returns are
accompanied by further large movements in returns resulting in periods of high volatility.
Similarly, tranquil periods are also evident in the graph.

250 500 750 1000 1250 1500 1750

Percentage daily (log) return on the NYSE Composite Index
3/1/1995 to 30/8/2002

Some descriptive statistics are shown in the table below. Some key points are:

1. Returns show significant autocorrelation of various orders as based on the
autocorrelation coefficient ( jr ) and the Ljung-Box statistic ( ( )xQ j ). This
suggests that there is some dependency in the mean and hence the hypothesis
that the stock market is efficient is rejected for this data set.

2. The autocorrelations of squared returns are much larger than those of returns.

Moreover, the Ljung-Box test statistic applied to squared returns is also much
larger and very significant. Both results suggest that there is considerable

ECON3206/5206 Financial Econometrics School of Economics, UNSW 6

dependency in the variance.

ECON3206/5206 Financial Econometrics School of Economics, UNSW 7

Table 1. Descriptive Statistics of daily (log) percentage return on the NYSE Composite
Index: 3/1/95-30/8/02

Statistic Return ( NYSER ) p-value

2r -0.046

3r -0.031

4r -0.001

5r -0.052

(1)xQ 9.0448 0.003
(5)xQ 20.226 0.001
(10)xQ 26.723 0.003
(15)xQ 36.726 0.001
(20)xQ 39.710 0.005

Statistic Squared Return 2( )NYSER p-value

(1)xxQ 62.044 0.000
(5)xxQ 264.80 0.000
(10)xxQ 389.80 0.000
(15)xxQ 449.58 0.000
(20)xxQ 491.70 0.000

jr is the autocorrelation coefficient at lag j. ( )xQ j and ( )xxQ j is the Ljung-Box Q
statistic for the first j lags of the autocorrelation function of returns and squared returns,
respectively

ECON3206/5206 Financial Econometrics School of Economics, UNSW 8

4. Distribution of Returns

The above example demonstrates the autocorrelations in returns and squared returns. We
now look at the shape of the empirical distribution of asset returns through the mean,
variance, skewness and kurtosis. We discuss the shape of the empirical distribution of
returns in relation to the normal distribution.

4.1 The Normal Distribution

Definition
The normal distribution of a random variable X with mean µ and variance 2σ , is
denoted as 2( , )N µ σ and is given by

xf x e xµ σ

− −= − ∞ < < ∞ If a normal random variable has been standardized to have zero mean and unit variance, then the standard normal distribution is denoted as N(0,1) and is given by 2 /21( ) , −= − ∞ < < ∞ Properties (1) The normal distribution is bell-shaped and symmetric around the origin. Thus the normal distribution exhibits no skewness. (2) The skewness and kurtosis coefficients for the normal distribution are respectively 4.2 Testing for Normality Single Tests A simple way to test for normality is to compare the computed skewness and kurtosis coefficients with the theoretical values under the assumption of normality; namely 0 and 3 respectively. Thus, the tests are Skewness'Test:' 3Kurtosis'Test:' © Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this ECON3206/5206 Financial Econometrics School of Economics, UNSW 9 where S and K are the estimated statistics for skewness and kurtosis, respectively. Both test statistics are distributed under the null hypothesis of normality as N(0,1). Thus “large” values of the test statistics, say in excess of two standard deviations (that is, greater than 2 or less than -2) constitute rejection of the hull hypothesis of normality. Joint Test A joint test can also be constructed as 2 2Sk K tJ B Z Z= + which is distributed as a chi-square with two degrees of freedom (i.e. 2(2)χ ). The null hypothesis of normality is rejected at the 5% level when the p-value is less than α =0.05. This is commonly referred to as the Jarque-Bera test for normality. 4.3 Leptokurtosis The distribution of many asset returns series have empirical distributions which differ from normality in two respects: (1) Fatness in the tails, which corresponds to points in time where large movements in returns have been excessive relative to the normal distribution. (2) Sharp peaks, which corresponds to periods when there is very little movement in the return series. Distributions which have these two properties are known as leptokurtic. 4.4 Application: The Distribution of Stock Returns The empirical distribution for (log) daily percentage return on the NYSE Composite Index is shown in the figure below. As before the data covers the period 3/1/95 to © Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this ECON3206/5206 Financial Econometrics School of Economics, UNSW 10 -6 -4 -2 0 2 4 Series: SR Sample 1 1931 Observations 1930 Mean 0.035300 Median 0.052285 Maximum 5.178704 Minimum -6.791142 Std. Dev. 1.006207 Skewness -0.315728 Kurtosis 7.224376 Jarque-Bera 1467.129 Probability 0.000000 Some key features are: 1. The mean represents the average (percentage) daily return on the NYSE Composite Index. The annual average return is 0.035×250=8.75% assuming that there are 250 trading days in a year. 2. The skewness appears to be small. The test statistic 66407.5 indicates that distribution of returns is negatively skewed. 3. The distribution is fat-tailed as evidenced by the high coefficient of kurtosis with the test statistic 37.89205 4. This is further supported by the Jarque-Bera statistic, JB=1467.13 which is significant at the 1% level (p-value<0.01). 5. Conclude that returns on the NYSE Composite Index are not normally distributed. 程序代写 CS代考加微信: powcoder QQ: 1823890830 Email: powcoder@163.com

Related Posts