KYIV SCHOOL OF ECONOMICS

Financial Econometrics (2nd part):

Introduction to Financial Time Series

May 2011

Instructor: Maksym Obrizan

Lecture notes I

# 2. Purposes of the short course:

(i) Review some theoretical models of financial time series

(ii) Develop practical skills of applied financial time series

Main Text: Analysis of Financial Time Series, by Ruey Tsay (selected chapters)

Data: mostly US time series (the most liquid market) but methods are applicable to transition countries as well

# 3. This course assumes some basic knowledge of time series econometrics but the most important concepts will be (quickly) reviewed

This lecture: very brief review of AR(p), and applications of linear time series models

What is Financial Time Series?

# 4. Financial Time Series

Of course, financial time series analysis has to incorporate uncertainty about asset returns

The most recent global crisis indicates that pricing bubbles and inadequate risk management are still present even in the most (financially) developed markets.

# 5. Although, stock returns are often the focus of financial theory other important financial time series include:

In addition, some of these methods can be used to study macroeconomic time series such as GDP or its components

# 6. Most financial studies use returns, instead of prices of assets

Campbell, Lo and MacKinlay (1997) give two reasons for this:

(i) return is a scale-free summary of investment opportunity;

(ii) returns have more attractive statistical properties than prices

# 7. Basic concepts

Let

The sample mean is

# 8. The sample variance is

# 9. The third central moment measures the symmetry of X with respect to its mean (skewness)

For figure on slide 10: Skewness is 0.007 for standard normal and -0.839 for skewed to the left

# 10.

# 11. The 4th central moment measures the tail behavior of X

Excess kurtosis K(x)-3 :

Kurtosis of Student t distrbution with 1 df is 3.43 on slide 12

# 12.

-25 -20 -15 -10 -5 0 50

0.05

0.1

0.15

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Skewed to the left

Standard normal

-5 -4 -3 -2 -1 0 1 2 3 4 50

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t with 1 df

Standard normal

# 13. Skewness and Kurtosis in returns data

Skewness

High excess kurtosis

In practice, high excess kurtosis means that the distribution of returns tends to contain more extreme values than the standard normal

# 14. Stationarity –

A time series {rt} is strictly stationary if

A time series {rt} is weakly stationary if

In this course: weakly stationary time series

# 15. Linear time series

The mean

and the variance

# 16. Quick review of autoregressive models

AR(p) model is

Meaning: the past p values of rt-I (i=1,…,p) jointly determine the conditional expectation of rt given the past data

# 17. To identify the order p of AR(p) model in practice one can use:

(i) PACF

(ii) information criteria (AIC)

Partial Autocorrelation Function (PACF):

# 18. The estimate of the second equation is called the lag-2 sample PACF of rt.

Intuitively, for an AR(p) model the lag-p sample PACF should not be zero but lag-j PACF should be close to zero for all j>p.

# 19. Indeed, under certain regularity conditions the sample PACF of an AR(p) process has the following properties

# 20. Alternatively, we can use Akaike Information Criterion for a Gaussian AR(k) model

The second term is called the penalty function for adding additional parameters

# 21. Monthly Value-Weighted Index Returns # 22. NOTES

# 23. Parameter Estimation # 24. Model checking: ACF

The sample autocorrelation of series {rt}

Notice: is biased (but consistent) estimate of . However, if sample is large then bias is not serious

# 25. After you fit the model obtain residual series to check for remaining autocorrelation

Suggestion:

Also plot 95% confidence intervals

Graph to the left?

Graph below?

# 26.

# 28. Ljung-Box (1978) statistics –

In practice, the choice of m may affect the performance of Q(m) statistics

Simulations suggest to set m to approx. ln(T)

# 27.

0 2 4 6 8 10 12 14 16 18 20-0.2

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Lag

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Sample Autocorrelation Function (ACF)

0 2 4 6 8 10 12 14 16 18 20-0.2

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Sample Autocorrelation Function (ACF)

# 29. For an AR(p) model, the Ljung-Box statistics Q(m) follows asymptotically a chi-squared distribution with m-p degrees of freedom

# 30. Implications

# 31. Forecasting: we are at time h and are interested in forecasting {rh+b} where b>0

Forecast often employs the minimum squared error term loss function

# 32. Multistep Ahead Forecast

This forecast can be obtained recursively

Important: for a stationary AR(p) model the long term forecast converges to unconditional mean (mean reversion) and the variance of forecast error approaches the unconditional variance

# 33. NOTES # 34. NOTES

# 35. Application: AR(2) model and business cycles

Consider an AR(2) model

It can be shown that the ACF of a stationary AR(2) model satisfies

# 36. This equation can be re-written as the second-order difference equation

where B is called back-shift operator such that

Sometimes lag operator L is used instead of B

# 37. Corresponding to the difference equation there is quadratic equation

which can be solved for characteristic roots

Interesting case when

# 38. If characteristic roots are complex numbers (complex conjugate pair) then the ACF of this series shows damping sine and cosine waves

For example, AR(2) model

The graph is in the bottom left corner

# 39. # 40. In business and economic applications complex characteristic roots give rise to business cycles

For an AR(2) model on slide # 35 with a pair of complex characteristic roots the average length of the stochastic cycles is

where the cosine inverse is stated in degrees

0 5 10 15 20 25 30 35 40-0.6

-0.4

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1

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Sample Autocorrelation Function (ACF)

# 41. Illustration: US GNP seasonally adjusted from QII.1947 to Q1.1991

Fit AR(3) model

Obtain a corresponding third-order difference equation

# 42. Factor out as

For the second-order factor is then

1-0.87B-(-0.27)B2 =0 we have 0.872+4(-0.27)<0

The average length of the stochastic cycles is

# 43. Application: Seasonal Models

Quarterly earnings per share of a company may exhibit cyclical or periodic behavior – seasonal time series

# 44.

# 45. Seasonal differencing

In general, for a time series with periodicity s:

# 46.

# 47. Multiplicative Seasonal Models: The airline model

# 48. Application to log series of Johnson and Johnson

# 49. # 50. Regression models with time series errors

Suppose, we are interested in term structure of interest rates

# 51. If the error term is a white noise then the LS method results in consistent estimates

# 52. Application to the US weekly interest rate series:

r1t - 1-year Treasury constant maturity rate

r2t - 3-year Treasury constant maturity rate

Simple but inadequate model:

# 53. Developing a more adequate model # 54. Cont’d

# 55. Fitting a linear regression model with time series errors

# 56. NOTES